text:
# Planet R

## March 30, 2017

### CRANberries

#### New package mudfold with initial version 1.0

Package: mudfold
Type: Package
Title: A Nonparametric Model for Unfolding Scale Analysis
Version: 1.0
Date: 2017-3-30
Author: Spyros Balafas
Maintainer: Spyros Balafas <s.balafas@rug.nl>
Depends: R (>= 3.3.3), gtools, zoo, ggplot2, reshape2, stats, utils
Description: Nonparametric item response theory model fruitful for the analysis of proximity data.
URL: http://www.math.rug.nl/stat/Main/HomePage
NeedsCompilation: yes
Repository: CRAN
Packaged: 2017-03-30 14:26:20 UTC; balaf
Date/Publication: 2017-03-30 16:36:48 UTC

#### New package spatstat.local with initial version 3.5-6

Package: spatstat.local
Type: Package
Title: Extension to 'spatstat' for Local Composite Likelihood
Version: 3.5-6
Date: 2017-03-30
Depends: R (>= 3.3.0), spatstat (>= 1.50-0), stats, graphics
Imports: tensor, spatstat.utils
Description: Extension to the 'spatstat' package, enabling the user to fit point process models to point pattern data by local composite likelihood ('geographically weighted regression').
ByteCompile: true
NeedsCompilation: no
Packaged: 2017-03-30 09:56:51 UTC; 214132e
Repository: CRAN
Date/Publication: 2017-03-30 11:28:32 UTC

#### New package optim.functions with initial version 0.1

Package: optim.functions
Title: Standard Benchmark Optimization Functions
Version: 0.1
Authors@R: person("Thomas", "Torsney-Weir", email="torsneyt@gmail.com", role=c("aut", "cre"))
Description: A set of standard benchmark optimization functions for R and a common interface to sample them.
Depends: R (>= 3.0)
LazyData: true
Suggests: testthat, covr
Imports: lhs, randtoolbox, stats, stringr
Collate: 'optim.functions.R' 'vectorize.R' 'sampling.R' 'single.objective.R' 'functions.R'
RoxygenNote: 5.0.1
NeedsCompilation: no
Packaged: 2017-03-30 06:13:39 UTC; tom
Author: Thomas Torsney-Weir [aut, cre]
Maintainer: Thomas Torsney-Weir <torsneyt@gmail.com>
Repository: CRAN
Date/Publication: 2017-03-30 11:28:25 UTC

#### New package mstR with initial version 1.0

Package: mstR
Type: Package
Title: Procedures to Generate Patterns under Multistage Testing
Version: 1.0
Date: 2017-03-30
Author: David Magis (U Liege, Belgium), Duanli Yan (ETS, USA), Alina von Davier (ACTNext, USA)
Maintainer: David Magis <david.magis@ulg.ac.be>
Depends: R (>= 2.8.0)
Description: Generation of response patterns under dichotomous and polytomous computerized multistage testing (MST) framework. It holds various IRT- and score-based methods to select the next module and estimate ability levels.
NeedsCompilation: no
Packaged: 2017-03-30 07:48:04 UTC; David
Repository: CRAN
Date/Publication: 2017-03-30 11:28:38 UTC

## March 29, 2017

### Dirk Eddelbuettel

#### #1: Easy Package Registration

Welcome to the first actual post in the R4 series, following the short announcement earlier this week.

### Context

Last month, Brian Ripley announced on r-devel that registration of routines would now be tested for by R CMD check in r-devel (which by next month will become R 3.4.0). A NOTE will be issued now, this will presumably turn into a WARNING at some point. Writing R Extensions has an updated introduction) of the topic.

Package registration has long been available, and applies to all native (i.e. "compiled") function via the .C(), .Call(), .Fortran() or .External() interfaces. If you use any of those -- and .Call() may be the only truly relevant one here -- then is of interest to you.

Brian Ripley and Kurt Hornik also added a new helper function: tools::package_native_routine_registration_skeleton(). It parses the R code of your package and collects all native function entrypoints in order to autogenerate the registration. It is available in R-devel now, will be in R 3.4.0 and makes adding such registration truly trivial.

But as of today, it requires that you have R-devel. Once R 3.4.0 is out, you can call the helper directly.

As for R-devel, there have always been at least two ways to use it: build it locally (which we may cover in another R4 installment), or using Docker. Here will focus on the latter by relying on the Rocker project by Carl and myself.

### Use the Rocker drd Image

We assume you can run Docker on your system. How to add it on Windows, macOS or Linux is beyond our scope here today, but also covered extensively elsewhere. So we assume you can execute docker and e.g. bring in the 'daily r-devel' image drd from our Rocker project via

~$docker pull rocker/drd With that, we can use R-devel to create the registration file very easily in a single call (which is a long command-line we have broken up with one line-break for the display below). The following is real-life example when I needed to add registration to the RcppTOML package for this week's update: ~/git/rcpptoml(master)$ docker run --rm -ti -v $(pwd):/mnt rocker/drd \ ## line break RD --slave -e 'tools::package_native_routine_registration_skeleton("/mnt")' #include <R.h> #include <Rinternals.h> #include <stdlib.h> // for NULL #include <R_ext/Rdynload.h> /* FIXME: Check these declarations against the C/Fortran source code. */ /* .Call calls */ extern SEXP RcppTOML_tomlparseImpl(SEXP, SEXP, SEXP); static const R_CallMethodDef CallEntries[] = { {"RcppTOML_tomlparseImpl", (DL_FUNC) &RcppTOML_tomlparseImpl, 3}, {NULL, NULL, 0} }; void R_init_RcppTOML(DllInfo *dll) { R_registerRoutines(dll, NULL, CallEntries, NULL, NULL); R_useDynamicSymbols(dll, FALSE); } edd@max:~/git/rcpptoml(master)$

### Decompose the Command

We can understand the docker command invocation above through its components:

• docker run is the basic call to a container
• --rm -ti does subsequent cleanup (--rm) and gives a terminal (-t) that is interactive (-i)
• -v $(pwd):/mnt uses the -v a:b options to make local directory a available as b in the container; here $(pwd) calls print working directory to get the local directory which is now mapped to /mnt in the container
• rocker/drd invokes the 'drd' container of the Rocker project
• RD is a shorthand to the R-devel binary inside the container, and the main reason we use this container
• -e 'tools::package_native_routine_registration_skeleton("/mnt") calls the helper function of R (currently in R-devel only, so we use RD) to compute a possible init.c file based on the current directory -- which is /mnt inside the container

That it. We get a call to the R function executed inside the Docker container, examining the package in the working directory and creating a registration file for it which is display to the console.

### Copy the Output to src/init.c

We simply copy the output to a file src/init.c; I often fold one opening brace one line up.

### Update NAMESPACE

We also change one line in NAMESPACE from (for this package) useDynLib("RcppTOML") to useDynLib("RcppTOML", .registration=TRUE). Adjust accordingly for other package names.

### That's it!

And with that we a have a package which no longer provokes the NOTE as seen by the checks page. Calls to native routines are now safer (less of a chance for name clashing), get called quicker as we skip the symbol search (see the WRE discussion) and best of all this applies to all native routines whether written by hand or written via a generator such as Rcpp Attributes.

So give this a try to get your package up-to-date.

This post by Dirk Eddelbuettel originated on his Thinking inside the box blog. Please report excessive re-aggregation in third-party for-profit settings.

## March 28, 2017

### Dirk Eddelbuettel

#### nanotime 0.1.2

A new minor version of the nanotime package for working with nanosecond timestamps arrived yesterday on CRAN.

nanotime uses the RcppCCTZ package for (efficient) high(er) resolution time parsing and formatting up to nanosecond resolution, and the bit64 package for the actual integer64 arithmetic.

This release just arranges things neatly before Leonardo Silvestri and I may shake things up with a possible shift to doing it all in S4 as we may need the added rigour for nanotime object operations for use in his ztsdb project.

#### Changes in version 0.1.2 (2017-03-27)

• The as.integer64 function is now exported as well.

We also have a diff to the previous version thanks to CRANberries. More details and examples are at the nanotime page; code, issue tickets etc at the GitHub repository.

This post by Dirk Eddelbuettel originated on his Thinking inside the box blog. Please report excessive re-aggregation in third-party for-profit settings.

## March 27, 2017

### Dirk Eddelbuettel

#### #0: Introducing R^4

So I had been toying with the idea of getting back to the blog and more regularly writing / posting little tips and tricks. I even started taking some notes but because perfect is always the enemy of the good it never quite materialized.

But the relatively broad discussion spawned by last week's short rant on Suggests != Depends made a few things clear. There appears to be an audience. It doesn't have to be long. And it doesn't have to be too polished.

So with that, let's get the blogging back from micro-blogging.

This note forms post zero of what will a new segment I call R4 which is shorthand for relatively random R rambling.

Stay tuned.

This post by Dirk Eddelbuettel originated on his Thinking inside the box blog. Please report excessive re-aggregation in third-party for-profit settings.

## March 26, 2017

### Dirk Eddelbuettel

#### RcppTOML 0.1.2

A new release of RcppTOML is now on CRAN. This release fixes a few parsing issues for less frequently-used inputs: vectors of boolean or date(time) types, as well as table array input.

RcppTOML brings TOML to R. TOML is a file format that is most suitable for configurations, as it is meant to be edited by humans but read by computers. It emphasizes strong readability for humans while at the same time supporting strong typing as well as immediate and clear error reports. On small typos you get parse errors, rather than silently corrupted garbage. Much preferable to any and all of XML, JSON or YAML -- though sadly these may be too ubiquitous now. TOML is making good inroads with newer and more flexible projects such as the Hugo static blog compiler, or the Cargo system of Crates (aka "packages") for the Rust language.

#### Changes in version 0.1.2 (2017-03-26)

• Dates and Datetimes in arrays in the input now preserve their types instead of converting to numeric vectors (#13)

• Boolean vectors are also correctly handled (#14)

• TableArray types are now stored as lists in a single named list (#15)

• The README.md file was expanded with an example and screenshot.

• Added file init.c with calls to R_registerRoutines() and R_useDynamicSymbols(); also use .registration=TRUE in useDynLib in NAMESPACE

• Two example files were updated.

Courtesy of CRANberries, there is a diffstat report for this release.

More information is on the RcppTOML page page. Issues and bugreports should go to the GitHub issue tracker.

This post by Dirk Eddelbuettel originated on his Thinking inside the box blog. Please report excessive re-aggregation in third-party for-profit settings.

## February 27, 2017

### Journal of Statistical Software

#### Introduction to stream: An Extensible Framework for Data Stream Clustering Research with R

In recent years, data streams have become an increasingly important area of research for the computer science, database and statistics communities. Data streams are ordered and potentially unbounded sequences of data points created by a typically non-stationary data generating process. Common data mining tasks associated with data streams include clustering, classification and frequent pattern mining. New algorithms for these types of data are proposed regularly and it is important to evaluate them thoroughly under standardized conditions. In this paper we introduce stream, a research tool that includes modeling and simulating data streams as well as an extensible framework for implementing, interfacing and experimenting with algorithms for various data stream mining tasks. The main advantage of stream is that it seamlessly integrates with the large existing infrastructure provided by R. In addition to data handling, plotting and easy scripting capabilities, R also provides many existing algorithms and enables users to interface code written in many programming languages popular among data mining researchers (e.g., C/C++, Java and Python). In this paper we describe the architecture of stream and focus on its use for data stream clustering research. stream was implemented with extensibility in mind and will be extended in the future to cover additional data stream mining tasks like classification and frequent pattern mining.

#### R Package gdistance: Distances and Routes on Geographical Grids

The R package gdistance provides classes and functions to calculate various distance measures and routes in heterogeneous geographic spaces represented as grids. Least-cost distances as well as more complex distances based on (constrained) random walks can be calculated. Also the corresponding routes or probabilities of passing each cell can be determined. The package implements classes to store the data about the probability or cost of transitioning from one cell to another on a grid in a memory-efficient sparse format. These classes make it possible to manipulate the values of cell-to-cell movement directly, which offers flexibility and the possibility to use asymmetric values. The novel distances implemented in the package are used in geographical genetics (applying circuit theory), but also have applications in other fields of geospatial analysis.

#### Identifying Causal Effects with the R Package causaleffect

Do-calculus is concerned with estimating the interventional distribution of an action from the observed joint probability distribution of the variables in a given causal structure. All identifiable causal effects can be derived using the rules of do-calculus, but the rules themselves do not give any direct indication whether the effect in question is identifiable or not. Shpitser and Pearl (2006b) constructed an algorithm for identifying joint interventional distributions in causal models, which contain unobserved variables and induce directed acyclic graphs. This algorithm can be seen as a repeated application of the rules of do-calculus and known properties of probabilities, and it ultimately either derives an expression for the causal distribution, or fails to identify the effect, in which case the effect is non-identifiable. In this paper, the R package causaleffect is presented, which provides an implementation of this algorithm. Functionality of causaleffect is also demonstrated through examples.

#### medflex: An R Package for Flexible Mediation Analysis using Natural Effect Models

Mediation analysis is routinely adopted by researchers from a wide range of applied disciplines as a statistical tool to disentangle the causal pathways by which an exposure or treatment affects an outcome. The counterfactual framework provides a language for clearly defining path-specific effects of interest and has fostered a principled extension of mediation analysis beyond the context of linear models. This paper describes medflex, an R package that implements some recent developments in mediation analysis embedded within the counterfactual framework. The medflex package offers a set of ready-made functions for fitting natural effect models, a novel class of causal models which directly parameterize the path-specific effects of interest, thereby adding flexibility to existing software packages for mediation analysis, in particular with respect to hypothesis testing and parsimony. In this paper, we give a comprehensive overview of the functionalities of the medflex package.

## February 22, 2017

### Bioconductor Project Working Papers

#### Estimating the Probability of Clonal Relatedness of Pairs of Tumors in Cancer Patients

Next generation sequencing panels are being used increasingly in cancer research to study tumor evolution. A specific statistical challenge is to compare the mutational profiles in different tumors from a patient to determine the strength of evidence that the tumors are clonally related, i.e. derived from a single, founder clonal cell. The presence of identical mutations in each tumor provides evidence of clonal relatedness, although the strength of evidence from a match is related to how commonly the mutation is seen in the tumor type under investigation. This evidence must be weighed against the evidence in favor of independent tumors from non-matching mutations. In this article we frame this challenge in the context of diagnosis using a novel random effects model. In this way, by analyzing a set of tumor pairs, we can estimate the proportion of cases that are clonally related in the sample as well as the individual diagnostic probabilities for each case. The method is illustrated using data from a study to determine the clonal relationship of lobular carcinoma in situ with subsequent invasive breast cancers where each tumor in the pair was subjected to whole exome sequencing. The statistical properties of the method are evaluated using simulations, demonstrating that the key model parameters are estimated with only modest bias in small samples.

## February 21, 2017

### Bioconductor Project Working Papers

#### Evaluation of Progress Towards the UNAIDS 90-90-90 HIV Care Cascade: A Description of Statistical Methods Used in an Interim Analysis of the Intervention Communities in the SEARCH Study

WHO guidelines call for universal antiretroviral treatment, and UNAIDS has set a global target to virally suppress most HIV-positive individuals. Accurate estimates of population-level coverage at each step of the HIV care cascade (testing, treatment, and viral suppression) are needed to assess the effectiveness of "test and treat" strategies implemented to achieve this goal. The data available to inform such estimates, however, are susceptible to informative missingness: the number of HIV-positive individuals in a population is unknown; individuals tested for HIV may not be representative of those whom a testing intervention fails to reach, and HIV-positive individuals with a viral load measured may not be representative of those for whom no viral load is obtained. We provide an in-depth description of the statistical methods (target parameters, assumptions, statistical estimands, and algorithms) used in an interim analysis of the intervention arm of the SEARCH Study (NCT01864603) to analyze progress towards the UNAIDS 90-90-90 target at study baseline and after one and two years. We describe the methods used to account for informative measurement in all analyses as well as for informative censoring in longitudinal analyses. We use targeted maximum likelihood estimation (TMLE) with Super Learning to generate semi-parametric efficient and double robust estimates of the care cascade among a open cohort of prevalent HIV-positive adults and among a closed cohort of baseline HIV-positive adults. TMLE is also used to evaluate predictors of poor outcomes.

## February 20, 2017

### mlpack

mlpack is, to quote, a scalable machine learning library, written in C++, that aims to provide fast, extensible implementations of cutting-edge machine learning algorithms. It has been written by Ryan Curtin and others, and is described in two papers in BigLearning (2011) and JMLR (2013). mlpack uses Armadillo as the underlying linear algebra library, which, thanks to RcppArmadillo, is already a rather well-known library in the R ecosystem.

### RcppMLPACK1

Qiang Kou has created the RcppMLPACK package on CRAN for easy-to-use integration of mlpack with R. It integrates the mlpack sources, and is, as a CRAN package, widely available on all platforms.

However, this RcppMLPACK package is also based on a by-now dated version of mlpack. Quoting again: mlpack provides these algorithms as simple command-line programs and C++ classes which can then be integrated into larger-scale machine learning solutions. Version 2 of the mlpack sources switched to a slightly more encompassing build also requiring the Boost libraries ‘program_options’, ‘unit_test_framework’ and ‘serialization’. Within the context of an R package, we could condition out the first two as R provides both the direct interface (hence no need to parse command-line options) and also the testing framework. However, it would be both difficult and potentially undesirable to condition out the serialization which allows mlpack to store and resume machine learning tasks.

We refer to this version now as RcppMLPACK1.

### RcppMLPACK2

As of February 2017, the current version of mlpack is 2.1.1. As it requires external linking with (some) Boost libraries as well as with Armadillo, we have created a new package RcppMLPACK2 inside a new GitHub organization RcppMLPACK.

#### Linux

This package works fine on Linux provided mlpack, Armadillo and Boost are installed.

#### OS X / macOS

For maxOS / OS X, James Balamuta has tried to set up a homebrew recipe but there are some tricky interaction with the compiler suites used by both brew and R on macOS.

#### Windows

For Windows, one could do what Jeroen Ooms has done and build (external) libraries. Volunteers are encouraged to get in touch via the issue tickets at GitHub.

#### Installation from source

Release are available from a drat repository hosted in the GitHub orgranization RcppMLPACK. So

will use this. If you prefer to rather pick a random commit state,

will work as well.

### Example: Logistic Regression

To illustrate mlpack we show a first simple example also included in the package. As the rest of the Rcpp Gallery, these are “live” code examples.

We can then call this function with the same (trivial) data set as used in the first unit test for it:

$parameters [1] 67.9550 -13.6328 -13.6328  ### Example: Naive Bayes Classifier A second examples shows the NaiveBayesClassifier class. We can use the sample data included in recent-enough version of the RcppMLPACK package: $means
[1] 2.75000 4.00000 3.68750 2.37500 8.33333 4.66667 3.66667 2.40000

$variances [1] 0.333333 0.800000 0.629167 0.383333 0.809524 3.380952 0.666667 0.400000$probabilities
[1] 0.516129 0.483871

$means [1] 2.75000 4.00000 3.68750 2.37500 8.33333 4.66667 3.66667 2.40000$variances
[1] 0.333333 0.800000 0.629167 0.383333 0.809524 3.380952 0.666667 0.400000

$probabilities [1] 0.516129 0.483871$classification
[1] 0 0 0 1 1 1 1

[1] TRUE

As we can see, the computed classification on the test set corresponds to the expected classification in testlabels.

## February 19, 2017

### Background

When we started the Rcpp Gallery in late 2012, a few of us spent the next four weeks diligently writing articles ensuring that at least one new article would be posted per day. Two early articles covered the then-budding support for C++11. Both First steps in using C++11 with Rcpp and A first lambda function with C++11 and Rcpp started by actually showing how to set the compiler directive via the PKG_CXXFLAGS environment variable.

Both posts were updated a few months later to demonstrate the C++11 plugin that was added in Rcpp release 0.10.3 in March of 2013, or almost four years ago. Many posts here, on StackOverflow and of course on the mailing list make use of the plugin to tell the compiler to turn on C++11 support. In the early days, this often meant partial support via (in the case of g++) the -std=c++0x switch. This is still supported by Rcpp as in particular the Windows side of things was relying on older compilers like g++ version 4.6.* until more recently (see below).

But some things got a lot better with R 3.1.0. As the NEWS announcement dryly noted: There is support for compiling C++11 code in packages on suitable platforms: see ‘Writing R Extensions’. which was coupled with support for selecting the C++11 language standard via either CXX_STD in src/Makevars, or the SystemRequirements line in DESCRIPTION. In late 2011, Martyn Plummer also wrote a helpful R Journal article on best practices for portable C++ packages.

And as we alluded to above, Rcpp has supported C++11 since the dawn of time (because, as we detail below, it ultimately comes down what your compiler supports, and what R facilitates). Many CRAN packages have by now taken advantage of this increased support for C++11 in particular. As of today, we see 88 CRAN packages declaring this via DESCRIPTION and 127 CRAN package via src/Makevars. And of course, almost all use Rcpp along with C++ to take advantage of the R and C++ integration it offers.

### Rcpp: Sitting between your Compiler and R

So what are the defining parameters for support by Rcpp? In essence, Rcpp is guided by just two (external) factors:

• the support in the provided compiler, and

• the support offered by R for package building,

and both are worth detailing as we do in the next two sections..

### Compiler Support

First, the choice of compiler matters. Rcpp operates on top of R, and (like any R package) it is driven entirely by the build instructions from R. It can therefore be dependent on the compiler choices offered by R. This meant g++ version 4.6.3 for Windows for years. And this got a lot better when Rtools switched to g++ version 4.9.3 with R 3.3.0 not quite a year ago. It now means that support for C++11 is almost universal. (Some small things are still missing in g++-4.9.3; notably complete support for the C++ Standard Library leading us recently to backport get_time from LLVM into RcppCCTZ. Also note that macOS / OS X has its own dependency chain due to compiler, and release, choices made by the R package build system for that platform.)

Now, that is just the minimum available compiler for a particular platform, albeit a very important one as it defines what binary CRAN packages on Windows can support. Other platforms, however, have a faster release cadence. g++-5 and clang++-3.3 are now fairly common and have (near-)complete C++11 support. The most recent Ubuntu release 16.10 even defaults to g++ version 6.2.0, and Debian already has version 6.3.0 (and binaries of version 7.* in its experimental branch). Similarly, recent version of clang are available both directly in the distributoin and via nightly builds from dedicated PPAs.

And for these ‘6.*’ version of g++, the default C++ standard is already C++14, the follow-up standard release to C++11. For example, C++14 extends support for auto to return values so that we can compile a simple program such as

withour any additional switches or flags if the compiler is as recent as g++ version 6. A plain g++ -o demoprog demoprog.cpp will do (provided the snippet was saved as file demoprogr.cpp) as C++14 is the default for this compiler. Otherwise, adding -std=c++14 explicitly instruct the compiler to compile as C++14.

Moreover, it also works for R:

[1] 2

[1] 2

(on a machine such as the one used to write this piece) without requiring any additional plugins. As a reminder, cppFunction() supports these via the plugin= argument. On another machine, we might use Rcpp::cppFunction("auto doubleMe(const int &x) { return x + x; }", plugin="cpp14"). Similarly, in code sourced via sourceCpp() we would use an attribute; see the Rcpp Attributes vignette for details.

### Support by R for Packages: C++14 coming soon via R-devel

As noted above, the R 3.1.0 release started to add support for modern C++ by enabling packages to select C++11.

The next release will be R 3.4.0. While as of now without an announced release date, it is likely to be shipped this April. It will bring support for C++14. The current draft of its version of Writing R Extenions shows that CXX_STD = CXX14 can be used to select this language standard in a package. Several build variables extend support to CXX1Y beyond the existing CXX1X, see the Writing R Extenions manual for full details.

To illustrate, on a machine with g++ version 6.*, nothing has to be turned on for C++14 in what will be R 3.4.0.

On a machine where g++-5 is the default, the CXX1XSTD value may be empty as this compiler defaults to C++11; we would expect CXX1YSTD = -std=c++14 there (or the ‘gnu’ variant).

As we noted above, well over one-hundred packages on CRAN already use C++11. We expect to see C++14 being used once R 3.4.0 is released later this spring

### Extensions on the Rcpp side: C++17

As we wrote in the opening paragraph, a plugin for C++11 has been part of Rcpp for several years. And a plugin for C++14 was added by Dan in Rcpp 0.12.4 about a year ago.

The current development sources of Rcpp, corresponding to interim GitHub release 0.12.9.3, added a plugin for C++17 as experimental support exists in g++ and clang++ right now. With that, a suitably recent compiler, and a version of Rcpp that is at least release 0.12.9.3, the following example (motivated by this example section of the C++ Standard post) also builds:

### Summary

This note discussed where Rcpp stands with respect to “modern C++”. As a brief summary:

• Rcpp supports any C++ language standard the underlying compiler supports: C++98, C++11, C++14, C++17;

• Packages using Rcpp can deploy every language standard suppported by R: currently C++, C++11 and very soon C++14;

• Package distribution may need to reflect the build infracture; on Windows this means g++-4.9.3 with near-complete C++11 support;

• Local developement can be more experimental and even C++17 is now supported by Rcpp as well;

• Portable packages should specify the C++ language standard they expect (unless it is C++98).

### Background

When we started the Rcpp Gallery in late 2012, a few of us spent the next four weeks diligently writing articles ensuring that at least one new article would be posted per day. Two early articles covered the then-budding support for C++11. Both First steps in using C++11 with Rcpp and A first lambda function with C++11 and Rcpp started by actually showing how to set the compiler directive via the PKG_CXXFLAGS environment variable.

Both posts were updated a few months later to demonstrate the C++11 plugin that was added in Rcpp release 0.10.3 in March of 2013, or almost four years ago. Many posts here, on StackOverflow and of course on the mailing list make use of the plugin to tell the compiler to turn on C++11 support. In the early days, this often meant partial support via (in the case of g++) the -std=c++0x switch. This is still supported by Rcpp as in particular the Windows side of things was relying on older compilers like g++ version 4.6.* until more recently (see below).

But some things got a lot better with R 3.1.0. As the NEWS announcement dryly noted: There is support for compiling C++11 code in packages on suitable platforms: see ‘Writing R Extensions’. which was coupled with support for selecting the C++11 language standard via either CXX_STD in src/Makevars, or the SystemRequirements line in DESCRIPTION. In late 2011, Martyn Plummer also wrote a helpful R Journal article on best practices for portable C++ packages.

And as we alluded to above, Rcpp has supported C++11 since the dawn of time (because, as we detail below, it ultimately comes down what your compiler supports, and what R facilitates). Many CRAN packages have by now taken advantage of this increased support for C++11 in particular. As of today, we see 88 CRAN packages declaring this via DESCRIPTION and 127 CRAN package via src/Makevars. And of course, almost all use Rcpp along with C++ to take advantage of the R and C++ integration it offers.

### Rcpp: Sitting between your Compiler and R

So what are the defining parameters for support by Rcpp? In essence, Rcpp is guided by just two (external) factors:

• the support in the provided compiler, and

• the support offered by R for package building,

and both are worth detailing as we do in the next two sections..

### Compiler Support

First, the choice of compiler matters. Rcpp operates on top of R, and (like any R package) it is driven entirely by the build instructions from R. It can therefore be dependent on the compiler choices offered by R. This meant g++ version 4.6.3 for Windows for years. And this got a lot better when Rtools switched to g++ version 4.9.3 with R 3.3.0 not quite a year ago. It now means that support for C++11 is almost universal. (Some small things are still missing in g++-4.9.3; notably complete support for the C++ Standard Library leading us recently to backport get_time from LLVM into RcppCCTZ. Also note that macOS / OS X has its own dependency chain due to compiler, and release, choices made by the R package build system for that platform.)

Now, that is just the minimum available compiler for a particular platform, albeit a very important one as it defines what binary CRAN packages on Windows can support. Other platforms, however, have a faster release cadence. g++-5 and clang++-3.3 are now fairly common and have (near-)complete C++11 support. The most recent Ubuntu release 16.10 even defaults to g++ version 6.2.0, and Debian already has version 6.3.0 (and binaries of version 7.* in its experimental branch). Similarly, recent version of clang are available both directly in the distributoin and via nightly builds from dedicated PPAs.

And for these ‘6.*’ version of g++, the default C++ standard is already C++14, the follow-up standard release to C++11. For example, C++14 extends support for auto to return values so that we can compile a simple program such as

withour any additional switches or flags if the compiler is as recent as g++ version 6. A plain g++ -o demoprog demoprog.cpp will do (provided the snippet was saved as file demoprogr.cpp) as C++14 is the default for this compiler. Otherwise, adding -std=c++14 explicitly instruct the compiler to compile as C++14.

Moreover, it also works for R:

[1] 2

[1] 2

(on a machine such as the one used to write this piece) without requiring any additional plugins. As a reminder, cppFunction() supports these via the plugin= argument. On another machine, we might use Rcpp::cppFunction("auto doubleMe(const int &x) { return x + x; }", plugin="cpp14"). Similarly, in code sourced via sourceCpp() we would use an attribute; see the Rcpp Attributes vignette for details.

### Support by R for Packages: C++14 coming soon via R-devel

As noted above, the R 3.1.0 release started to add support for modern C++ by enabling packages to select C++11.

The next release will be R 3.4.0. While as of now without an announced release date, it is likely to be shipped this April. It will bring support for C++14. The current draft of its version of Writing R Extenions shows that CXX_STD = CXX14 can be used to select this language standard in a package. Several build variables extend support to CXX1Y beyond the existing CXX1X, see the Writing R Extenions manual for full details.

To illustrate, on a machine with g++ version 6.*, nothing has to be turned on for C++14 in what will be R 3.4.0.

On a machine where g++-5 is the default, the CXX1XSTD value may be empty as this compiler defaults to C++11; we would expect CXX1YSTD = -std=c++14 there (or the ‘gnu’ variant).

As we noted above, well over one-hundred packages on CRAN already use C++11. We expect to see C++14 being used once R 3.4.0 is released later this spring

### Extensions on the Rcpp side: C++17

As we wrote in the opening paragraph, a plugin for C++11 has been part of Rcpp for several years. And a plugin for C++14 was added by Dan in Rcpp 0.12.4 about a year ago.

The current development sources of Rcpp, corresponding to interim GitHub release 0.12.9.3, added a plugin for C++17 as experimental support exists in g++ and clang++ right now. With that, a suitably recent compiler, and a version of Rcpp that is at least release 0.12.9.3, the following example (motivated by this example section of the C++ Standard post) also builds:

### Summary

This note discussed where Rcpp stands with respect to “modern C++”. As a brief summary:

• Rcpp supports any C++ language standard the underlying compiler supports: C++98, C++11, C++14, C++17;

• Packages using Rcpp can deploy every language standard suppported by R: currently C++, C++11 and very soon C++14;

• Package distribution may need to reflect the build infracture; on Windows this means g++-4.9.3 with near-complete C++11 support;

• Local developement can be more experimental and even C++17 is now supported by Rcpp as well;

• Portable packages should specify the C++ language standard they expect (unless it is C++98).

## February 18, 2017

### RCpp Gallery

Armadillo is a very versatile C++ library for linear algebra, brought to R via the RcppArmadillo package. It has proven to be very useful and popular, and is (as of February 2017) used by well over 300 CRAN packages as indicated by the reverse depends / linking-to on its CRAN page. Well over a dozen earlier posts on this Rcpp Gallery site also demonstrate the popularity of the package as well as different usage patterns.

Armadillo has a core focus on dense matrices, but continues to expand its capabilities for sparse matrices. Basic operation are supported directly via the templated header files, along with calls back into the default (dense) LAPACK and BLAS libraries we can access easily as R uses them.

Armadillo also supports dedicated sparse solvers via the SuperLU package. However, this requires access to the SuperLU library. Many Linux distributions ship it, see e.g. the Debian package page and the Ubuntu package page; there is also a homebrew recipe for OS X / macOS (or other systems using brew). As of this writing, the version in the current Ubuntu release is behind the version Debian. But it is the 5.2.* version that is in Debian that is also required with current Armadillo versions 7.700.* so we prepared ‘backports’ via Dirk’s PPA repo.)

Recently, a GitHub issue ticket asked how to use SuperLU along with RcppArmadillo. This post essentially repeats the main answer, which was spread out over multiple posts, in a single article.

In a nutshell, two things need to happen:

1. One needs to define the required variable ARMA_USE_SUPERLU which has to be done before the Armadillo headers are included. One possibility (shown below) is a #define statement right in the code file.

2. One needs to tell the linker to use the SuperLU library. This step is of course not perfectly portable, and does require that the library be installed. (A genuinely portable solution would either need to test for presence of SuperLU, or include its sources. Both aspects are beyond the scope of this post._

This code snippet defines a function superLuDemo() which we can call from R:

Done.

As the data generated here is random, we did not bother printing the (dense) result vector of length 1000.

## February 10, 2017

### Bioconductor Project Working Papers

#### IMPROVING POWER IN GROUP SEQUENTIAL, RANDOMIZED TRIALS BY ADJUSTING FOR PROGNOSTIC BASELINE VARIABLES AND SHORT-TERM OUTCOMES

In group sequential designs, adjusting for baseline variables and short-term outcomes can lead to increased power and reduced sample size. We derive formulas for the precision gain from such variable adjustment using semiparametric estimators for the average treatment effect, and give new results on what conditions lead to substantial power gains and sample size reductions. The formulas reveal how the impact of prognostic variables on the precision gain is modified by the number of pipeline participants, analysis timing, enrollment rate, and treatment effect heterogeneity, when the semiparametric estimator uses correctly specified models. Given set prognostic value of baseline variables and short-term outcomes within each arm, the precision gain is maximal when there is no treatment effect heterogeneity. In contrast, a purely predictive baseline variable, which only explains treatment effect heterogeneity but is marginally uncorrelated with the outcome, can lead to no precision gain. The theory is supported by simulation studies based on data from a trial of a new surgical intervention for treating stroke.

## February 08, 2017

### Bioconductor Project Working Papers

#### IT'S ALL ABOUT BALANCE: PROPENSITY SCORE MATCHING IN THE CONTEXT OF COMPLEX SURVEY DATA

Many research studies aim to draw causal inferences using data from large, nationally representative survey samples, and many of these studies use propensity score matching to make those causal inferences as rigorous as possible given the non-experimental nature of the data. However, very few applied studies are careful about incorporating the survey design with the propensity score analysis, which may mean that the results don’t generate population inferences. This may be because few methodological studies examine how to best combine these methods. Furthermore, even fewer of the methodological studies incorporate different non-response mechanisms in their analysis. This study examines methods for how to handle survey weights in propensity score matching analyses of survey data, under diferent non-response mechanisms. Based on the results from Monte Carlo simulations implemented on synthetic data as well as a data based application we developed suggestions regarding the implementation of propensity score methods to make causal inferences relevant to the target population of a sample survey. Our main conclusions are: (1) whether the survey weights are incorporated in the estimation of the propensity score does not impact estimation of the population treatment effect, as long as good population balance is achieved across confounders, (2) survey weights must be taken into account in the outcome analysis and (3) transfer of survey weights (i.e., matched comparison units are assigned the sampling weight of the treated unit they have been matched to) can be benefcial under certain non-response mechanisms.

## September 29, 2016

### Statistical Modelling

#### Time-dependent ROC methodology to evaluate the predictive accuracy of semiparametric multi-state models in the presence of competing risks: An application to peritoneal dialysis programme

Abstract:

The evaluation of peritoneal dialysis (PD) programmes requires the use of statistical methods that suit the complexity of such programmes. Multi-state regression models taking competing risks into account are a good example of suitable approaches. In this work, multi-state structured additive regression (STAR) models combined with penalized splines (P-splines) are proposed to evaluate peritoneal dialysis programmes. These models are very flexible since they may consider smooth estimates of baseline transition intensities and the inclusion of time-varying and smooth covariate effects at each transition. A key issue in survival analysis is the quantification of the time-dependent predictive accuracy of a given regression model, which is typically assessed using receiver operating characteristic (ROC)’based methodologies. The main objective of the present study is to adapt the concept of time-dependent ROC curve, and their corresponding area under the curve (AUC), to a multi-state competing risks framework. All statistical methodologies discussed in this work were applied to PD survival data. Using a multi-state competing risks framework, this study explored the effects of major clinical covariates on survival such as age, sex, diabetes and previous renal replacement therapy. Such multi-state model was composed of one transient state (peritonitis) and several absorbing states (death, transfer to haemodialysis and renal transplantation). The application of STAR models combined with time-dependent ROC curves revealed important conclusions not previously reported in the nephrology literature when using standard statistical methodologies. For practical application, all the statistical methods proposed in this article were implemented in R and we wrote and made available a script named as NestedCompRisks.

#### A multivariate single-index model for longitudinal data

Abstract:

Index measures are commonly used in medical research and clinical practice, primarily for quantification of health risks in individual subjects or patients. The utility of an index measure is ultimately contingent on its ability to predict health outcomes. Construction of medical indices has largely been based on heuristic arguments, although the acceptance of a new index typically requires objective validation, preferably with multiple outcomes. In this article, we propose an analytical tool for index development and validation. We use a multivariate single-index model to ascertain the best functional form for risk index construction. Methodologically, the proposed model represents a multivariate extension of the traditional single-index models. Such an extension is important because it assures that the resultant index simultaneously works for multiple outcomes. The model is developed in the general framework of longitudinal data analysis. We use penalized cubic splines to characterize the index components while leaving the other subject characteristics as additive components. The splines are estimated directly by penalizing nonlinear least squares, and we show that the model can be implemented using existing software. To illustrate, we examine the formation of an adiposity index for prediction of systolic and diastolic blood pressure in children. We assess the performance of the method through a simulation study.

#### Semi-parametric frailty model for clustered interval-censored data

Abstract:

The shared frailty model is a popular tool to analyze correlated right-censored time-to-event data. In the shared frailty model, the latent frailty is assumed to be shared by the members of a cluster and is assigned a parametric distribution, typically a gamma distribution due to its conjugacy. In the case of interval-censored time-to-event data, the inclusion of frailties results in complicated intractable likelihoods. Here, we propose a flexible frailty model for analyzing such data by assuming a smooth semi-parametric form for the conditional time-to-event distribution and a parametric or a flexible form for the frailty distribution. The results of a simulation study suggest that the estimation of regression parameters is robust to misspecification of the frailty distribution (even when the frailty distribution is multimodal or skewed). Given sufficiently large sample sizes and number of clusters, the flexible approach produces smooth and accurate posterior estimates for the baseline survival function and for the frailty density, and it can correctly detect and identify unusual frailty density forms. The methodology is illustrated using dental data from the Signal Tandmobiel® study.

#### Bayesian dynamic modelling to assess differential treatment effects on panic attack frequencies

Abstract:

To represent the complex structure of intensive longitudinal data of multiple individuals, we propose a hierarchical Bayesian Dynamic Model (BDM). This BDM is a generalized linear hierarchical model where the individual parameters do not necessarily follow a normal distribution. The model parameters can be estimated on the basis of relatively small sample sizes and in the presence of missing time points. We present the BDM and discuss the model identification, convergence and selection. The use of the BDM is illustrated using data from a randomized clinical trial to study the differential effects of three treatments for panic disorder. The data involves the number of panic attacks experienced weekly (73 individuals, 10–52 time points) during treatment. Presuming that the counts are Poisson distributed, the BDM considered involves a linear trend model with an exponential link function. The final model included a moving average parameter and an external variable (duration of symptoms pre-treatment). Our results show that cognitive behavioural therapy is less effective in reducing panic attacks than serotonin selective re-uptake inhibitors or a combination of both. Post hoc analyses revealed that males show a slightly higher number of panic attacks at the onset of treatment than females.

## July 18, 2016

### Data frames are lists

Most R users will know that data frames are lists. You can easily verify that a data frame is a list by typing

d <- data.frame(id=1:2, name=c("Jon", "Mark"))
d

 id name
1 1 Jon
2 2 Mark

is.list(d)

[1] TRUE

However, data frames are lists with some special properties. For example, all entries in the list must have the same length (here 2), etc. You can find a nice description of the differences between lists and data frames here. To access the first column of d, we find that it contains a vector (and a factor in case of column name). Note, that [[ ]] is an operator to select a list element. As data frames are lists, they will work here as well.

is.vector(d[[1]])

[1] TRUE

### Data frame columns can contain lists

A long time, I was unaware of the fact, that data frames may also contain lists as columns instead of vectors. For example, let’s assume Jon’s children are Mary and James, and Mark’s children are called Greta and Sally. Their names are stored in a list with two elements. We can add them to the data frame like this:

d$children <- list(c("Mary", "James"), c("Greta", "Sally")) d   id name children 1 1 Jon Mary, James 2 2 Mark Greta, Sally  A single data frame entry in column children now contains more than one value. Given that the column is a list, not a vector, we cannot go as usual when modifying an entry of the column. For example, to change Jon’s children, we cannot do > d[1 , "children"] <- c("Mary", "James", "Thomas") Error in [<-.data.frame(*tmp*, 1, "children", value = c("Mary", "James", : replacement has 3 rows, data has 1  Taking into account the list structure of the column, we can type the following to change the values in a single cell. d[1 , "children"][[1]] <- list(c("Mary", "James", "Thomas")) # or also d$children[1] <- list(c("Mary", "James", "Thomas"))
d

 id name children
1 1 Jon Mary, James, Thomas
2 2 Mark Greta, Sally

You can also create a data frame having a list as a column using the <tt>data.frame</tt> function, but with a little tweak. The list column has to be wrapped inside the function <tt>I</tt>. This will protect it from several conversions taking place in <tt>data.frame</tt> (see <tt>?I</tt> documentation).

d <- data.frame(id = 1:2,
name = c("Jon", "Mark"),
children = I(list(c("Mary", "James"),
c("Greta", "Sally")))
)

This is an interesting feature, which gives me a deeper understanding of what a data frame is. But when exactly would I want to use it? I have not encountered the need to use it very often yet (though of course there may be plenty of situations where it makes sense). But today I had a case where this feature seemed particularly useful.

### Converting lists and data frames to JSON

I had two separate types of information. One stored in a data frame and the other one in a list Referring to the example above, I had

d <- data.frame(id=1:2, name=c("Jon", "Mark"))
d

 id name
1 1 Jon
2 2 Mark

and

ch <- list(c("Mary", "James"), c("Greta", "Sally"))
ch

[[1]]
[1] "Mary" "James"

[[2]]
[1] "Greta" "Sally"

I needed to return an array of JSON objects which look like this.

[
{
"id": 1,
"name": "Jon",
"children": ["Mary", "James"]
},
{
"id": 2,
"name": "Mark",
"children": ["Greta", "Sally"]
}
]

Working with the superb jsonlite package to convert R to JSON, I could do the following to get the result above.

library(jsonlite)

l <- split(d, seq(nrow(d))) # convert data frame rows to list
l <- unname(l)              # remove list names
for (i in seq_along(l))     # add element from ch to list
l[[i]] <- c(l[[i]], children=ch[i])

toJSON(l, pretty=T, auto_unbox = T) # convert to JSON

The results are correct, but getting there involved quite a number of tedious steps. These can be avoided by directly placing the list into a column of the data frame. Then jsonlite::toJSON takes care of the rest.

d$children <- list(c("Mary", "James"), c("Greta", "Sally")) toJSON(d, pretty=T, auto_unbox = T)  [ { "id": 1, "name": "Jon", "children": ["Mary", "James"] }, { "id": 2, "name": "Mark", "children": ["Greta", "Sally"] } ]  Nice :) What we do here, is basically creating the same nested list structure as above, only now it is disguised as a data frame. However, this approach is much more convenient. ## December 27, 2015 ### Alstatr #### R and Python: Gradient Descent One of the problems often dealt in Statistics is minimization of the objective function. And contrary to the linear models, there is no analytical solution for models that are nonlinear on the parameters such as logistic regression, neural networks, and nonlinear regression models (like Michaelis-Menten model). In this situation, we have to use mathematical programming or optimization. And one popular optimization algorithm is the gradient descent, which we're going to illustrate here. To start with, let's consider a simple function with closed-form solution given by $$f(\beta) \triangleq \beta^4 - 3\beta^3 + 2.$$ We want to minimize this function with respect to$\beta. The quick solution to this, as what calculus taught us, is to compute for the first derivative of the function, that is $$\frac{\text{d}f(\beta)}{\text{d}\beta}=4\beta^3-9\beta^2.$$ Setting this to 0 to obtain the stationary point gives us \begin{align} \frac{\text{d}f(\beta)}{\text{d}\beta}&\overset{\text{set}}{=}0\nonumber\\ 4\hat{\beta}^3-9\hat{\beta}^2&=0\nonumber\\ 4\hat{\beta}^3&=9\hat{\beta}^2\nonumber\\ 4\hat{\beta}&=9\nonumber\\ \hat{\beta}&=\frac{9}{4}. \end{align} The following plot shows the minimum of the function at\hat{\beta}=\frac{9}{4}$(red line in the plot below). R ScriptNow let's consider minimizing this problem using gradient descent with the following algorithm: 1. Initialize$\mathbf{x}_{r},r=0$2. while$\lVert \mathbf{x}_{r}-\mathbf{x}_{r+1}\rVert > \nu$3.$\mathbf{x}_{r+1}\leftarrow \mathbf{x}_{r} - \gamma\nabla f(\mathbf{x}_r)$4.$r\leftarrow r + 1$5. end while 6. return$\mathbf{x}_{r}$and$r$where$\nabla f(\mathbf{x}_r)$is the gradient of the cost function,$\gamma$is the learning-rate parameter of the algorithm, and$\nu$is the precision parameter. For the function above, let the initial guess be$\hat{\beta}_0=4$and$\gamma=.001$with$\nu=.00001$. Then$\nabla f(\hat{\beta}_0)=112$, so that $\hat{\beta}_1=\hat{\beta}_0-.001(112)=3.888.$ And$|\hat{\beta}_1 - \hat{\beta}_0| = 0.112> \nu$. Repeat the process until at some$r$,$|\hat{\beta}_{r}-\hat{\beta}_{r+1}| \ngtr \nu$. It will turn out that 350 iterations are needed to satisfy the desired inequality, the plot of which is in the following figure with estimated minimum$\hat{\beta}_{350}=2.250483\approx\frac{9}{4}$. R Script with PlotPython ScriptObviously the convergence is slow, and we can adjust this by tuning the learning-rate parameter, for example if we try to increase it into$\gamma=.01$(change gamma to .01 in the codes above) the algorithm will converge at 42nd iteration. To support that claim, see the steps of its gradient in the plot below. If we try to change the starting value from 4 to .1 (change beta_new to .1) with$\gamma=.01$, the algorithm converges at 173rd iteration with estimate$\hat{\beta}_{173}=2.249962\approx\frac{9}{4}(see the plot below). Now let's consider another function known as Rosenbrock defined as $$f(\mathbf{w})\triangleq(1 - w_1) ^ 2 + 100 (w_2 - w_1^2)^2.$$ The gradient is \begin{align} \nabla f(\mathbf{w})&=[-2(1 - w_1) - 400(w_2 - w_1^2) w_1]\mathbf{i}+200(w_2-w_1^2)\mathbf{j}\nonumber\\ &=\left[\begin{array}{c} -2(1 - w_1) - 400(w_2 - w_1^2) w_1\\ 200(w_2-w_1^2) \end{array}\right]. \end{align} Let the initial guess be\hat{\mathbf{w}}_0=\left[\begin{array}{c}-1.8\\-.8\end{array}\right]$,$\gamma=.0002$, and$\nu=.00001$. Then$\nabla f(\hat{\mathbf{w}}_0)=\left[\begin{array}{c} -2914.4\\-808.0\end{array}\right]$. So that $$\nonumber \hat{\mathbf{w}}_1=\hat{\mathbf{w}}_0-\gamma\nabla f(\hat{\mathbf{w}}_0)=\left[\begin{array}{c} -1.21712 \\-0.63840\end{array}\right].$$ And$\lVert\hat{\mathbf{w}}_0-\hat{\mathbf{w}}_1\rVert=0.6048666>\nu$. Repeat the process until at some$r$,$\lVert\hat{\mathbf{w}}_r-\hat{\mathbf{w}}_{r+1}\rVert\ngtr \nu$. It will turn out that 23,374 iterations are needed for the desired inequality with estimate$\hat{\mathbf{w}}_{23375}=\left[\begin{array}{c} 0.9464841 \\0.8956111\end{array}\right]$, the contour plot is depicted in the figure below. R Script with Contour PlotPython ScriptNotice that I did not use ggplot for the contour plot, this is because the plot needs to be updated 23,374 times just to accommodate for the arrows for the trajectory of the gradient vectors, and ggplot is just slow. Finally, we can also visualize the gradient points on the surface as shown in the following figure. R ScriptIn my future blog post, I hope to apply this algorithm on statistical models like linear/nonlinear regression models for simple illustration. #### R: Principal Component Analysis on Imaging Ever wonder what's the mathematics behind face recognition on most gadgets like digital camera and smartphones? Well for most part it has something to do with statistics. One statistical tool that is capable of doing such feature is the Principal Component Analysis (PCA). In this post, however, we will not do (sorry to disappoint you) face recognition as we reserve this for future post while I'm still doing research on it. Instead, we go through its basic concept and use it for data reduction on spectral bands of the image using R. ### Let's view it mathematically Consider a line$L$in a parametric form described as a set of all vectors$k\cdot\mathbf{u}+\mathbf{v}$parameterized by$k\in \mathbb{R}$, where$\mathbf{v}$is a vector orthogonal to a normalized vector$\mathbf{u}$. Below is the graphical equivalent of the statement: So if given a point$\mathbf{x}=[x_1,x_2]^T$, the orthogonal projection of this point on the line$L$is given by$(\mathbf{u}^T\mathbf{x})\mathbf{u}+\mathbf{v}$. Graphically, we mean$Proj$is the projection of the point$\mathbf{x}$on the line, where the position of it is defined by the scalar$\mathbf{u}^{T}\mathbf{x}$. Therefore, if we consider$\mathbf{X}=[X_1, X_2]^T$be a random vector, then the random variable$Y=\mathbf{u}^T\mathbf{X}$describes the variability of the data on the direction of the normalized vector$\mathbf{u}$. So that$Y$is a linear combination of$X_i, i=1,2$. The principal component analysis identifies a linear combinations of the original variables$\mathbf{X}$that contain most of the information, in the sense of variability, contained in the data. The general assumption is that useful information is proportional to the variability. PCA is used for data dimensionality reduction and for interpretation of data. (Ref 1. Bajorski, 2012) To better understand this, consider two dimensional data set, below is the plot of it along with two lines ($L_1$and$L_2$) that are orthogonal to each other: If we project the points orthogonally to both lines we have, So that if normalized vector$\mathbf{u}_1$defines the direction of$L_1$, then the variability of the points on$L_1$is described by the random variable$Y_1=\mathbf{u}_1^T\mathbf{X}$. Also if$\mathbf{u}_2$is a normalized vector that defines the direction of$L_2$, then the variability of the points on this line is described by the random variable$Y_2=\mathbf{u}_2^T\mathbf{X}$. The first principal component is one with maximum variability. So in this case, we can see that$Y_2$is more variable than$Y_1$, since the points projected on$L_2$are more dispersed than in$L_1$. In practice, however, the linear combinations$Y_i = \mathbf{u}_i^T\mathbf{X}, i=1,2,\cdots,p$is maximized sequentially so that$Y_1$is the linear combination of the first principal component,$Y_2$is the linear combination of the second principal component, and so on. Further, the estimate of the direction vector$\mathbf{u}$is simply the normalized eigenvector$\mathbf{e}$of the variance-covariance matrix$\mathbf{\Sigma}$of the original variable$\mathbf{X}$. And the variability explained by the principal component is the corresponding eigenvalue$\lambda$. For more details on theory of PCA refer to (Bajorski, 2012) at Reference 1 below. As promised we will do dimensionality reduction using PCA. We will use the Airborne Visible/Infrared Imaging Spectrometer (AVIRIS) data from (Barjorski, 2012), you can use other locations of AVIRIS data that can be downloaded here. However, since for most cases the AVIRIS data contains thousands of bands so for simplicity we will stick with the data given in (Bajorski, 2012) as it was cleaned reducing to 152 bands only. ### What is spectral bands? In imaging, spectral bands refer to the third dimension of the image usually denoted as$\lambda$. For example, RGB image contains red, green and blue bands as shown below along with the first two dimensions$x$and$y$that define the resolution of the image. These are few of the bands that are visible to our eyes, there are other bands that are not visible to us like infrared, and many other in electromagnetic spectrum. That is why in most cases AVIRIS data contains huge number of bands each captures different characteristics of the image. Below is the proper description of the data. ### Data The Airborne Visible/Infrared Imaging Spectrometer (AVIRIS), is a sensor collecting spectral radiance in the range of wavelengths from 400 to 2500 nm. It has been flown on various aircraft platforms, and many images of the Earth’s surface are available. A 100 by 100 pixel AVIRIS image of an urban area in Rochester, NY, near the Lake Ontario shoreline is shown below. The scene has a wide range of natural and man-made material including a mixture of commercial/warehouse and residential neighborhoods, which adds a wide range of spectral diversity. Prior to processing, invalid bands (due to atmospheric water absorption) were removed, reducing the overall dimensionality to 152 bands. This image has been used in Bajorski et al. (2004) and Bajorski (2011a, 2011b). The first 152 values in the AVIRIS Data represent the spectral radiance values (a spectral curve) for the top left pixel. This is followed by spectral curves of the pixels in the first row, followed by the next row, and so on. (Ref. 1 Bajorski, 2012) To load the data, run the following code: Above code uses EBImage package, and can be installed from my previous post. ### Why do we need to reduce the dimension of the data? Before we jump in to our analysis, in case you may ask why? Well sometimes it's just difficult to do analysis on high dimensional data, especially on interpreting it. This is because there are dimensions that aren't significant (like redundancy) which adds to our problem on the analysis. So in order to deal with this, we remove those nuisance dimension and deal with the significant one. To perform PCA in R, we use the function princomp as seen below: The structure of princomp consist of a list shown above, we will give description to selected outputs. Others can be found in the documentation of the function by executing ?princomp. • sdev - standard deviation, the square root of the eigenvalues$\lambda$of the variance-covariance matrix$\mathbf{\Sigma}$of the data, dat.mat; • loadings - eigenvectors$\mathbf{e}$of the variance-covariance matrix$\mathbf{\Sigma}$of the data, dat.mat; • scores - the principal component scores. Recall that the objective of PCA is to find for a linear combination$Y=\mathbf{u}^T\mathbf{X}$that will maximize the variance$Var(Y)$. So that from the output, the estimate of the components of$\mathbf{u}$is the entries of the loadings which is a matrix of eigenvectors, where the columns corresponds to the eigenvectors of the sequence of principal components, that is if the first principal component is given by$Y_1=\mathbf{u}_1^T\mathbf{X}$, then the estimate of$\mathbf{u}_1$which is$\mathbf{e}_1$(eigenvector) is the set of coefficients obtained from the first column of the loadings. The explained variability of the first principal component is the square of the first standard deviation sdev, the explained variability of the second principal component is the square of the second standard deviation sdev, and so on. Now let's interpret the loadings (coefficients) of the first three principal components. Below is the plot of this, Base above, the coefficients of the first principal component (PC1) are almost all negative. A closer look, the variability in this principal component is mainly explained by the weighted average of radiance of the spectral bands 35 to 100. Analogously, PC2 mainly represents the variability of the weighted average of radiance of spectral bands 1 to 34. And further, the fluctuation of the coefficients of PC3 makes it difficult to tell on which bands greatly contribute on its variability. Aside from examining the loadings, another way to see the impact of the PCs is through the impact plot where the impact curve$\sqrt{\lambda_j}\mathbf{e}_j$are plotted, I want you to explore that. Moving on, let's investigate the percent of variability in$X_i$explained by the$j$th principal component, below is the formula of this, $$\nonumber \frac{\lambda_j\cdot e_{ij}^2}{s_{ii}},$$ where$s_{ii}$is the estimated variance of$X_i$. So that below is the percent of explained variability in$X_i$of the first three principal components including the cumulative percent variability (sum of PC1, PC2, and PC3), For the variability of the first 33 bands, PC2 takes on about 90 percent of the explained variability as seen in the above plot. And still have great contribution further to 102 to 152 bands. On the other hand, from bands 37 to 100, PC1 explains almost all the variability with PC2 and PC3 explain 0 to 1 percent only. The sum of the percentage of explained variability of these principal components is indicated as orange line in the above plot, which is the cumulative percent variability. To wrap up this section, here is the percentage of the explained variability of the first 10 PCs. PC1PC2PC3PC4PC5PC6PC7PC8PC9PC10 Table 1: Variability Explained by the First Ten Principal Components for the AVIRIS data. 82.05717.1760.3200.1820.0940.0650.0370.0290.0140.005 Above variability were obtained by noting that the variability explained by the principal component is simply the eigenvalue (square of the sdev) of the variance-covariance matrix$\mathbf{\Sigma}$of the original variable$\mathbf{X}$, hence the percentage of variability explained by the$j$th PC is equal to its corresponding eigenvalue$\lambda_j$divided by the overall variability which is the sum of the eigenvalues,$\sum_{j=1}^{p}\lambda_j$, as we see in the following code, ### Stopping Rules Given the list of percentage of variability explained by the PCs in Table 1, how many principal components should we take into account that would best represent the variability of the original data? To answer that, we introduce the following stopping rules that will guide us on deciding the number of PCs: 1. Scree plot; 2. Simple fare-share; 3. Broken-stick; and, 4. Relative broken-stick. The scree plot is the plot of the variability of the PCs, that is the plot of the eigenvalues. Where we look for an elbow or sudden drop of the eigenvalues on the plot, hence for our example we have Therefore, we need return the first two principal components based on the elbow shape. However, if the eigenvalues differ by order of magnitude, it is recommended to use the logarithmic scale which is illustrated below, Unfortunately, sometimes it won't work as we can see here, it's just difficult to determine where the elbow is. The succeeding discussions on the last three stopping rules are based on (Bajorski, 2012). The simple fair-share stopping rule identifies the largest$k$such that$\lambda_k$is larger than its fair share, that is larger than$(\lambda_1+\lambda_2+\cdots+\lambda_p)/p$. To illustrate this, consider the following: Thus, we need to stop at second principal component. If one was concerned that the above method produces too many principal components, a broken-stick rule could be used. The rule is that it identifies the principal components with largest$k$such that$\lambda_j/(\lambda_1+\lambda_2+\cdots +\lambda_p)>a_j$, for all$j\leq k$, where $$\nonumber a_j = \frac{1}{p}\sum_{i=j}^{p}\frac{1}{i},\quad j =1,\cdots, p.$$ Let's try it, Above result coincides with the first two stopping rule. The draw back of simple fair-share and broken-stick rules is that it do not work well when the eigenvalues differ by orders of magnitude. In such case, we then use the relative broken-stick rule, where we analyze$\lambda_j$as the first eigenvalue in the set$\lambda_j\geq \lambda_{j+1}\geq\cdots\geq\lambda_{p}$, where$j < p$. The dimensionality$k$is chosen as the largest value such that$\lambda_j/(\lambda_j+\cdots +\lambda_p)>b_j$, for all$j\leq k$, where $$\nonumber b_j = \frac{1}{p-j+1}\sum_{i=1}^{p-j+1}\frac{1}{i}.$$ Applying this to the data we have, According to the numerical output, the first 34 principal components are enough to represent the variability of the original data. ### Principal Component Scores The principal component scores is the resulting new data set obtained from the linear combinations$Y_j=\mathbf{e}_j(\mathbf{x}-\bar{\mathbf{x}}), j = 1,\cdots, p$. So that if we use the first three stopping rules, then below is the scores (in image) of PC1 and PC2, If we base on the relative broken-stick rule then we return the first 34 PCs, and below is the corresponding scores (in image).  Click on the image to zoom in. ### Residual Analysis Of course when doing PCA there are errors to be considered unless one would return all the PCs, but that would not make any sense because why would someone apply PCA when you still take into account all the dimensions? An overview of the errors in PCA without going through the theory is that, the overall error is simply the excluded variability explained by the$k$th to$p$th principal components,$k>j$. ### Reference #### R: k-Means Clustering on Imaging Enough with the theory we recently published, let's take a break and have fun on the application of Statistics used in Data Mining and Machine Learning, the k-Means Clustering. k-means clustering is a method of vector quantization, originally from signal processing, that is popular for cluster analysis in data mining. k-means clustering aims to partition n observations into k clusters in which each observation belongs to the cluster with the nearest mean, serving as a prototype of the cluster. (Wikipedia, Ref 1.) We will apply this method to an image, wherein we group the pixels into k different clusters. Below is the image that we are going to use,  Colorful Bird From Wall321 We will utilize the following packages for input and output: 1. jpeg - Read and write JPEG images; and, 2. ggplot2 - An implementation of the Grammar of Graphics. ### Download and Read the Image Let's get started by downloading the image to our workspace, and tell R that our data is a JPEG file. ### Cleaning the Data Extract the necessary information from the image and organize this for our computation: The image is represented by large array of pixels with dimension rows by columns by channels -- red, green, and blue or RGB. ### Plotting Plot the original image using the following codes: ### Clustering Apply k-Means clustering on the image: Plot the clustered colours: Possible clusters of pixels on different k-Means: Originalk = 6 Table 1: Different k-Means Clustering. k = 5k = 4 k = 3k = 2 I suggest you try it! ### Reference 1. K-means clustering. Wikipedia. Retrieved September 11, 2014. ## December 16, 2015 ### Alstatr #### R and Python: Theory of Linear Least Squares In my previous article, we talked about implementations of linear regression models in R, Python and SAS. On the theoretical sides, however, I briefly mentioned the estimation procedure for the parameter$\boldsymbol{\beta}$. So to help us understand how software does the estimation procedure, we'll look at the mathematics behind it. We will also perform the estimation manually in R and in Python, that means we're not going to use any special packages, this will help us appreciate the theory. ### Linear Least Squares Consider the linear regression model, $y_i=f_i(\mathbf{x}|\boldsymbol{\beta})+\varepsilon_i,\quad\mathbf{x}_i=\left[ \begin{array}{cccc} 1&x_{11}&\cdots&x_{1p} \end{array}\right],\quad\boldsymbol{\beta}=\left[\begin{array}{c}\beta_0\\\beta_1\\\vdots\\\beta_p\end{array}\right],$ where$y_i$is the response or the dependent variable at the$i$th case,$i=1,\cdots, N$. The$f_i(\mathbf{x}|\boldsymbol{\beta})$is the deterministic part of the model that depends on both the parameters$\boldsymbol{\beta}\in\mathbb{R}^{p+1}$and the predictor variable$\mathbf{x}_i$, which in matrix form, say$\mathbf{X}$, is represented as follows $\mathbf{X}=\left[ \begin{array}{cccccc} 1&x_{11}&\cdots&x_{1p}\\ 1&x_{21}&\cdots&x_{2p}\\ \vdots&\vdots&\ddots&\vdots\\ 1&x_{N1}&\cdots&x_{Np}\\ \end{array} \right].$$\varepsilon_i$is the error term at the$i$th case which we assumed to be Gaussian distributed with mean 0 and variance$\sigma^2$. So that $\mathbb{E}y_i=f_i(\mathbf{x}|\boldsymbol{\beta}),$ i.e.$f_i(\mathbf{x}|\boldsymbol{\beta})$is the expectation function. The uncertainty around the response variable is also modelled by Gaussian distribution. Specifically, if$Y=f(\mathbf{x}|\boldsymbol{\beta})+\varepsilon$and$y\in Y$such that$y>0, then \begin{align*} \mathbb{P}[Y\leq y]&=\mathbb{P}[f(x|\beta)+\varepsilon\leq y]\\ &=\mathbb{P}[\varepsilon\leq y-f(\mathbf{x}|\boldsymbol{\beta})]=\mathbb{P}\left[\frac{\varepsilon}{\sigma}\leq \frac{y-f(\mathbf{x}|\boldsymbol{\beta})}{\sigma}\right]\\ &=\Phi\left[\frac{y-f(\mathbf{x}|\boldsymbol{\beta})}{\sigma}\right], \end{align*} where\Phi$denotes the Gaussian distribution with density denoted by$\phi$below. Hence$Y\sim\mathcal{N}(f(\mathbf{x}|\boldsymbol{\beta}),\sigma^2). That is, \begin{align*} \frac{\operatorname{d}}{\operatorname{d}y}\Phi\left[\frac{y-f(\mathbf{x}|\boldsymbol{\beta})}{\sigma}\right]&=\phi\left[\frac{y-f(\mathbf{x}|\boldsymbol{\beta})}{\sigma}\right]\frac{1}{\sigma}=\mathbb{P}[y|f(\mathbf{x}|\boldsymbol{\beta}),\sigma^2]\\ &=\frac{1}{\sqrt{2\pi}\sigma}\exp\left\{-\frac{1}{2}\left[\frac{y-f(\mathbf{x}|\boldsymbol{\beta})}{\sigma}\right]^2\right\}. \end{align*} If the data are independent and identically distributed, then the log-likelihood function ofyis, \begin{align*} \mathcal{L}[\boldsymbol{\beta}|\mathbf{y},\mathbf{X},\sigma]&=\mathbb{P}[\mathbf{y}|\mathbf{X},\boldsymbol{\beta},\sigma]=\prod_{i=1}^N\frac{1}{\sqrt{2\pi}\sigma}\exp\left\{-\frac{1}{2}\left[\frac{y_i-f_i(\mathbf{x}|\boldsymbol{\beta})}{\sigma}\right]^2\right\}\\ &=\frac{1}{(2\pi)^{\frac{n}{2}}\sigma^n}\exp\left\{-\frac{1}{2}\sum_{i=1}^N\left[\frac{y_i-f_i(\mathbf{x}|\boldsymbol{\beta})}{\sigma}\right]^2\right\}\\ \log\mathcal{L}[\boldsymbol{\beta}|\mathbf{y},\mathbf{X},\sigma]&=-\frac{n}{2}\log2\pi-n\log\sigma-\frac{1}{2\sigma^2}\sum_{i=1}^N\left[y_i-f_i(\mathbf{x}|\boldsymbol{\beta})\right]^2. \end{align*} And because the likelihood function tells us about the plausibility of the parameter\boldsymbol{\beta}$in explaining the sample data. We therefore want to find the best estimate of$\boldsymbol{\beta}$that likely generated the sample. Thus our goal is to maximize the likelihood function which is equivalent to maximizing the log-likelihood with respect to$\boldsymbol{\beta}$. And that's simply done by taking the partial derivative with respect to the parameter$\boldsymbol{\beta}$. Therefore, the first two terms in the right hand side of the equation above can be disregarded since it does not depend on$\boldsymbol{\beta}$. Also, the location of the maximum log-likelihood with respect to$\boldsymbol{\beta}$is not affected by arbitrary positive scalar multiplication, so the factor$\frac{1}{2\sigma^2}$can be omitted. And we are left with the following equation, $$\label{eq:1} -\sum_{i=1}^N\left[y_i-f_i(\mathbf{x}|\boldsymbol{\beta})\right]^2.$$ One last thing is that, instead of maximizing the log-likelihood function we can do minimization on the negative log-likelihood. Hence we are interested on minimizing the negative of Equation (\ref{eq:1}) which is $$\label{eq:2} \sum_{i=1}^N\left[y_i-f_i(\mathbf{x}|\boldsymbol{\beta})\right]^2,$$ popularly known as the residual sum of squares (RSS). So RSS is a consequence of maximum log-likelihood under the Gaussian assumption of the uncertainty around the response variable$y$. For models with two parameters, say$\beta_0$and$\beta_1$the RSS can be visualized like the one in my previous article, that is Performing differentiation under$(p+1)$-dimensional parameter$\boldsymbol{\beta}is manageable in the context of linear algebra, so Equation (\ref{eq:2}) is equivalent to \begin{align*} \lVert\mathbf{y}-\mathbf{X}\boldsymbol{\beta}\rVert^2&=\langle\mathbf{y}-\mathbf{X}\boldsymbol{\beta},\mathbf{y}-\mathbf{X}\boldsymbol{\beta}\rangle=\mathbf{y}^{\text{T}}\mathbf{y}-\mathbf{y}^{\text{T}}\mathbf{X}\boldsymbol{\beta}-(\mathbf{X}\boldsymbol{\beta})^{\text{T}}\mathbf{y}+(\mathbf{X}\boldsymbol{\beta})^{\text{T}}\mathbf{X}\boldsymbol{\beta}\\ &=\mathbf{y}^{\text{T}}\mathbf{y}-\mathbf{y}^{\text{T}}\mathbf{X}\boldsymbol{\beta}-\boldsymbol{\beta}^{\text{T}}\mathbf{X}^{\text{T}}\mathbf{y}+\boldsymbol{\beta}^{\text{T}}\mathbf{X}^{\text{T}}\mathbf{X}\boldsymbol{\beta} \end{align*} And the derivative with respect to the parameter is \begin{align*} \frac{\operatorname{\partial}}{\operatorname{\partial}\boldsymbol{\beta}}\lVert\mathbf{y}-\mathbf{X}\boldsymbol{\beta}\rVert^2&=-2\mathbf{X}^{\text{T}}\mathbf{y}+2\mathbf{X}^{\text{T}}\mathbf{X}\boldsymbol{\beta} \end{align*} Taking the critical point by setting the above equation to zero vector, we have \begin{align} \frac{\operatorname{\partial}}{\operatorname{\partial}\boldsymbol{\beta}}\lVert\mathbf{y}-\mathbf{X}\hat{\boldsymbol{\beta}}\rVert^2&\overset{\text{set}}{=}\mathbf{0}\nonumber\\ -\mathbf{X}^{\text{T}}\mathbf{y}+\mathbf{X}^{\text{T}}\mathbf{X}\hat{\boldsymbol{\beta}}&=\mathbf{0}\nonumber\\ \mathbf{X}^{\text{T}}\mathbf{X}\hat{\boldsymbol{\beta}}&=\mathbf{X}^{\text{T}}\mathbf{y}\label{eq:norm} \end{align} Equation (\ref{eq:norm}) is called the normal equation. If\mathbf{X}$is full rank, then we can compute the inverse of$\mathbf{X}^{\text{T}}\mathbf{X}, \begin{align} \mathbf{X}^{\text{T}}\mathbf{X}\hat{\boldsymbol{\beta}}&=\mathbf{X}^{\text{T}}\mathbf{y}\nonumber\\ (\mathbf{X}^{\text{T}}\mathbf{X})^{-1}\mathbf{X}^{\text{T}}\mathbf{X}\hat{\boldsymbol{\beta}}&=(\mathbf{X}^{\text{T}}\mathbf{X})^{-1}\mathbf{X}^{\text{T}}\mathbf{y}\nonumber\\ \hat{\boldsymbol{\beta}}&=(\mathbf{X}^{\text{T}}\mathbf{X})^{-1}\mathbf{X}^{\text{T}}\mathbf{y}.\label{eq:betahat} \end{align} That's it, since both\mathbf{X}$and$\mathbf{y}$are known. ### Prediction If$\mathbf{X}$is full rank and spans the subspace$V\subseteq\mathbb{R}^N$, where$\mathbb{E}\mathbf{y}=\mathbf{X}\boldsymbol{\beta}\in V$. Then the predicted values of$\mathbf{y}$is given by, $$\label{eq:pred} \hat{\mathbf{y}}=\mathbb{E}\mathbf{y}=\mathbf{P}_{V}\mathbf{y}=\mathbf{X}(\mathbf{X}^{\text{T}}\mathbf{X})^{-1}\mathbf{X}^{\text{T}}\mathbf{y},$$ where$\mathbf{P}$is the projection matrix onto the space$V$. For proof of the projection matrix in Equation (\ref{eq:pred}) please refer to reference (1) below. Notice that this is equivalent to $$\label{eq:yhbh} \hat{\mathbf{y}}=\mathbb{E}\mathbf{y}=\mathbf{X}\hat{\boldsymbol{\beta}}.$$ ### Computation Let's fire up R and Python and see how we can apply those equations we derived. For purpose of illustration, we're going to simulate data from Gaussian distributed population. To do so, consider the following codes R ScriptPython ScriptHere we have two predictors x1 and x2, and our response variable y is generated by the parameters$\beta_1=3.5$and$\beta_2=2.8$, and it has Gaussian noise with variance 7. While we set the same random seeds for both R and Python, we should not expect the random values generated in both languages to be identical, instead both values are independent and identically distributed (iid). For visualization, I will use Python Plotly, you can also translate it to R Plotly. Now let's estimate the parameter$\boldsymbol{\beta}$which by default we set to$\beta_1=3.5$and$\beta_2=2.8$. We will use Equation (\ref{eq:betahat}) for estimation. So that we have R ScriptPython ScriptThat's a good estimate, and again just a reminder, the estimate in R and in Python are different because we have different random samples, the important thing is that both are iid. To proceed, we'll do prediction using Equations (\ref{eq:pred}). That is, R ScriptPython ScriptThe first column above is the data y and the second column is the prediction due to Equation (\ref{eq:pred}). Thus if we are to expand the prediction into an expectation plane, then we have You have to rotate the plot by the way to see the plane, I still can't figure out how to change it in Plotly. Anyway, at this point we can proceed computing for other statistics like the variance of the error, and so on. But I will leave it for you to explore. Our aim here is just to give us an understanding on what is happening inside the internals of our software when we try to estimate the parameters of the linear regression models. ### Reference 1. Arnold, Steven F. (1981). The Theory of Linear Models and Multivariate Analysis. Wiley. 2. OLS in Matrix Form ## May 12, 2015 ### Chris Lawrence #### That'll leave a mark Here’s a phrase you never want to see in print (in a legal decision, no less) pertaining to your academic research: “The IRB process, however, was improperly engaged by the Dartmouth researcher and ignored completely by the Stanford researchers.” Whole thing here; it’s a doozy. ## April 14, 2015 ### R you ready? #### Beautiful plots while simulating loss in two-part procrustes problem Today I was working on a two-part procrustes problem and wanted to find out why my minimization algorithm sometimes does not converge properly or renders unexpected results. The loss function to be minimized is $\displaystyle L(\mathbf{Q},c) = \| c \mathbf{A_1Q} - \mathbf{B_1} \|^2 + \| \mathbf{A_2Q} - \mathbf{B_2} \|^2 \rightarrow min$ with $\| \cdot \|$ denoting the Frobenius norm, $c$ is an unknown scalar and $\mathbf{Q}$ an unknown rotation matrix, i.e. $\mathbf{Q}^T\mathbf{Q}=\mathbf{I}$. $\;\mathbf{A_1}, \mathbf{A_2}, \mathbf{B_1}$, and $\mathbf{B_1}$ are four real valued matrices. The minimum for $c$ is easily found by setting the partial derivation of $L(\mathbf{Q},c)$ w.r.t $c$ equal to zero. $\displaystyle c = \frac {tr \; \mathbf{Q}^T \mathbf{A_1}^T \mathbf{B_1}} { \| \mathbf{A_1} \|^2 }$ By plugging $c$ into the loss function $L(\mathbf{Q},c)$ we get a new loss function $L(\mathbf{Q})$ that only depends on $\mathbf{Q}$. This is the starting situation. When trying to find out why the algorithm to minimize $L(\mathbf{Q})$ did not work as expected, I got stuck. So I decided to conduct a small simulation and generate random rotation matrices to study the relation between the parameter $c$ and the value of the loss function $L(\mathbf{Q})$. Before looking at the results for the entire two-part procrustes problem from above, let’s visualize the results for the first part of the loss function only, i.e. $\displaystyle L(\mathbf{Q},c) = \| c \mathbf{A_1Q} - \mathbf{B_1} \|^2 \rightarrow min$ Here, $c$ has the same minimum as for the whole formula above. For the simulation I used $\mathbf{A_1}= \begin{pmatrix} 0.0 & 0.4 & -0.5 \\ -0.4 & -0.8 & -0.5 \\ -0.1 & -0.5 & 0.2 \\ \end{pmatrix} \mkern18mu \qquad \text{and} \qquad \mkern36mu \mathbf{B_1}= \begin{pmatrix} -0.1 & -0.8 & -0.1 \\ 0.3 & 0.2 & -0.9 \\ 0.1 & -0.3 & -0.5 \\ \end{pmatrix}$ as input matrices. Generating many random rotation matrices $\mathbf{Q}$ and plotting $c$ against the value of the loss function yields the following plot. This is a well behaved relation, for each scaling parameter $c$ the loss is identical. Now let’s look at the full two-part loss function. As input matrices I used $\displaystyle A1= \begin{pmatrix} 0.0 & 0.4 & -0.5 \\ -0.4 & -0.8 & -0.5 \\ -0.1 & -0.5 & 0.2 \\ \end{pmatrix} \mkern18mu , \mkern36mu B1= \begin{pmatrix} -0.1 & -0.8 & -0.1 \\ 0.3 & 0.2 & -0.9 \\ 0.1 & -0.3 & -0.5 \\ \end{pmatrix}$ $A2= \begin{pmatrix} 0 & 0 & 1 \\ 1 & 0 & 0 \\ 0 & 1 & 0 \\ \end{pmatrix} \mkern18mu , \mkern36mu B2= \begin{pmatrix} 0 & 0 & 1 \\ 1 & 0 & 0 \\ 0 & 1 & 0 \\ \end{pmatrix}$ and the following R-code. # trace function tr <- function(X) sum(diag(X)) # random matrix type 1 rmat_1 <- function(n=3, p=3, min=-1, max=1){ matrix(runif(n*p, min, max), ncol=p) } # random matrix type 2, sparse rmat_2 <- function(p=3) { diag(p)[, sample(1:p, p)] } # generate random rotation matrix Q. Based on Q find # optimal scaling factor c and calculate loss function value # one_sample <- function(n=2, p=2) { Q <- mixAK::rRotationMatrix(n=1, dim=p) %*% # random rotation matrix det(Q) = 1 diag(sample(c(-1,1), p, rep=T)) # additional reflections, so det(Q) in {-1,1} s <- tr( t(Q) %*% t(A1) %*% B1 ) / norm(A1, "F")^2 # scaling factor c rss <- norm(s*A1 %*% Q - B1, "F")^2 + # get residual sum of squares norm(A2 %*% Q - B2, "F")^2 c(s=s, rss=rss) } # find c and rss or many random rotation matrices # set.seed(10) # nice case for 3 x 3 n <- 3 p <- 3 A1 <- round(rmat_1(n, p), 1) B1 <- round(rmat_1(n, p), 1) A2 <- rmat_2(p) B2 <- rmat_2(p) x <- plyr::rdply(40000, one_sample(3,3)) plot(x$s, x$rss, pch=16, cex=.4, xlab="c", ylab="L(Q)", col="#00000010")  This time the result turns out to be very different and … beautiful :) Here, we do not have a one to one relation between the scaling parameter and the loss function any more. I do not quite know what to make of this yet. But for now I am happy that it has aestethic value. Below you find some more beautiful graphics with different matrices as inputs. Cheers! ## February 24, 2015 ### Douglas Bates # RCall: Running an embedded R in Julia I have used R (and S before it) for a couple of decades. In the last few years most of my coding has been in Julia, a language for technical computing that can provide remarkable performance for a dynamically typed language via Just-In-Time (JIT) compilation of functions and via multiple dispatch. Nonetheless there are facilities in R that I would like to have access to from Julia. I created the RCall package for Julia to do exactly that. This IJulia notebook provides an introduction to RCall. This is not a novel idea by any means. Julia already has PyCall and JavaCall packages that provide access to Python and to Java. These packages are used extensively and are much more sophisticated than RCall, at present. Many other languages have facilities to run an embedded instance of R. In fact, Python has several such interfaces. The things I plan to do using RCall is to access datasets from R and R packages, to fit models that are not currently implemented in Julia and to use R graphics, especially the ggplot2 and lattice packages. Unfortunately I am not currently able to start a graphics device from the embedded R but I expect that to be fixed soon. I can tell you the most remarkable aspect of RCall although it may not mean much if you haven't tried to do this kind of thing. It is written entirely in Julia. There is absolutely no "glue" code written in a compiled language like C or C++. As I said, this may not mean much to you unless you have tried to do something like this, in which case it is astonishing. ## January 16, 2015 ### Modern Toolmaking #### caretEnsemble My package caretEnsemble, for making ensembles of caret models, is now on CRAN. Check it out, and let me know what you think! (Submit bug reports and feature requests to the issue tracker) ## January 15, 2015 ### Gregor Gorjanc #### cpumemlog: Monitor CPU and RAM usage of a process (and its children) Long time no see ... Today I pushed the cpumemlog script to GitHub https://github.com/gregorgorjanc/cpumemlog. Read more about this useful utility at the GitHub site. ## December 15, 2014 ### R you ready? #### QQ-plots in R vs. SPSS – A look at the differences We teach two software packages, R and SPSS, in Quantitative Methods 101 for psychology freshman at Bremen University (Germany). Sometimes confusion arises, when the software packages produce different results. This may be due to specifics in the implemention of a method or, as in most cases, to different default settings. One of these situations occurs when the QQ-plot is introduced. Below we see two QQ-plots, produced by SPSS and R, respectively. The data used in the plots were generated by: set.seed(0) x <- sample(0:9, 100, rep=T)  SPSS R qqnorm(x, datax=T) # uses Blom's method by default qqline(x, datax=T)  There are some obvious differences: 1. The most obvious one is that the R plot seems to contain more data points than the SPSS plot. Actually, this is not the case. Some data points are plotted on top of each in SPSS while they are spread out vertically in the R plot. The reason for this difference is that SPSS uses a different approach assigning probabilities to the values. We will expore the two approaches below. 2. The scaling of the y-axis differs. R uses quantiles from the standard normal distribution. SPSS by default rescales these values using the mean and standard deviation from the original data. This allows to directly compare the original and theoretical values. This is a simple linear transformation and will not be explained any further here. 3. The QQ-lines are not identical. R uses the 1st and 3rd quartile from both distributions to draw the line. This is different in SPSS where of a line is drawn for identical values on both axes. We will expore the differences below. # QQ-plots from scratch To get a better understanding of the difference we will build the R and SPSS-flavored QQ-plot from scratch. ## R type In order to calculate theoretical quantiles corresponding to the observed values, we first need to find a way to assign a probability to each value of the original data. A lot of different approaches exist for this purpose (for an overview see e.g. Castillo-Gutiérrez, Lozano-Aguilera, & Estudillo-Martínez, 2012b). They usually build on the ranks of the observed data points to calculate corresponding p-values, i.e. the plotting positions for each point. The qqnorm function uses two formulae for this purpose, depending on the number of observations $n$ (Blom’s mfethod, see ?qqnorm; Blom, 1958). With $r$ being the rank, for $n > 10$ it will use the formula $p = (r - 1/2) / n$, for $n \leq 10$ the formula $p = (r - 3/8) / (n + 1/4)$ to determine the probability value $p$ for each observation (see the help files for the functions qqnorm and ppoint). For simplicity reasons, we will only implement the $n > 10$ case here. n <- length(x) # number of observations r <- order(order(x)) # order of values, i.e. ranks without averaged ties p <- (r - 1/2) / n # assign to ranks using Blom's method y <- qnorm(p) # theoretical standard normal quantiles for p values plot(x, y) # plot empirical against theoretical values  Before we take at look at the code, note that our plot is identical to the plot generated by qqnorm above, except that the QQ-line is missing. The main point that makes the difference between R and SPSS is found in the command order(order(x)). The command calculates ranks for the observations using ordinal ranking. This means that all observations get different ranks and no average ranks are calculated for ties, i.e. for observations with equal values. Another approach would be to apply fractional ranking and calculate average values for ties. This is what the function rank does. The following codes shows the difference between the two approaches to assign ranks. v <- c(1,1,2,3,3) order(order(v)) # ordinal ranking used by R  ## [1] 1 2 3 4 5  rank(v) # fractional ranking used by SPSS  ## [1] 1.5 1.5 3.0 4.5 4.5  R uses ordinal ranking and SPSS uses fractional ranking by default to assign ranks to values. Thus, the positions do not overlap in R as each ordered observation is assigned a different rank and therefore a different p-value. We will pick up the second approach again later, when we reproduce the SPSS-flavored plot in R.1 The second difference between the plots concerned the scaling of the y-axis and was already clarified above. The last point to understand is how the QQ-line is drawn in R. Looking at the probs argument of qqline reveals that it uses the 1st and 3rd quartile of the original data and theoretical distribution to determine the reference points for the line. We will draw the line between the quartiles in red and overlay it with the line produced by qqline to see if our code is correct. plot(x, y) # plot empirical against theoretical values ps <- c(.25, .75) # reference probabilities a <- quantile(x, ps) # empirical quantiles b <- qnorm(ps) # theoretical quantiles lines(a, b, lwd=4, col="red") # our QQ line in red qqline(x, datax=T) # R QQ line  The reason for different lines in R and SPSS is that several approaches to fitting a straight line exist (for an overview see e.g. Castillo-Gutiérrez, Lozano-Aguilera, & Estudillo-Martínez, 2012a). Each approach has different advantages. The method used by R is more robust when we expect values to diverge from normality in the tails, and we are primarily interested in the normality of the middle range of our data. In other words, the method of fitting an adequate QQ-line depends on the purpose of the plot. An explanation of the rationale of the R approach can e.g. be found here. ## SPSS type The default SPSS approach also uses Blom’s method to assign probabilities to ranks (you may choose other methods is SPSS) and differs from the one above in the following aspects: • a) As already mentioned, SPSS uses ranks with averaged ties (fractional rankings) not the plain order ranks (ordinal ranking) as in R to derive the corresponding probabilities for each data point. The rest of the code is identical to the one above, though I am not sure if SPSS distinguishes between the $n 10$ case. • b) The theoretical quantiles are scaled to match the estimated mean and standard deviation of the original data. • c) The QQ-line goes through all quantiles with identical values on the x and y axis. n <- length(x) # number of observations r <- rank(x) # a) ranks using fractional ranking (averaging ties) p <- (r - 1/2) / n # assign to ranks using Blom's method y <- qnorm(p) # theoretical standard normal quantiles for p values y <- y * sd(x) + mean(x) # b) transform SND quantiles to mean and sd from original data plot(x, y) # plot empirical against theoretical values  Lastly, let us add the line. As the scaling of both axes is the same, the line goes through the origin with a slope of $1$. abline(0,1) # c) slope 0 through origin  The comparison to the SPSS output shows that they are (visually) identical. # Function for SPSS-type QQ-plot The whole point of this demonstration was to pinpoint and explain the differences between a QQ-plot generated in R and SPSS, so it will no longer be a reason for confusion. Note, however, that SPSS offers a whole range of options to generate the plot. For example, you can select the method to assign probabilities to ranks and decide how to treat ties. The plots above used the default setting (Blom’s method and averaging across ties). Personally I like the SPSS version. That is why I implemented the function qqnorm_spss in the ryouready package, that accompanies the course. The formulae for the different methods to assign probabilities to ranks can be found in Castillo-Gutiérrez et al. (2012b). The implentation is a preliminary version that has not yet been thoroughly tested. You can find the code here. Please report any bugs or suggestions for improvements (which are very welcome) in the github issues section. library(devtools) install_github("markheckmann/ryouready") # install from github repo library(ryouready) # load package library(ggplot2) qq <- qqnorm_spss(x, method=1, ties.method="average") # Blom's method with averaged ties plot(qq) # generate QQ-plot ggplot(qq) # use ggplot2 to generate QQ-plot  # Literature 1. Technical sidenote: Internally, qqnorm uses the function ppoints to generate the p-values. Type in stats:::qqnorm.default to the console to have a look at the code. ## October 20, 2014 ### Modern Toolmaking #### For faster R on a mac, use veclib ## Update: The links to all my github gists on blogger are broken, and I can't figure out how to fix them. If you know how to insert gitub gists on a dynamic blogger template, please let me known. In the meantime, here are instructions with links to the code: First of all, use homebrew to compile openblas. It's easy! Second of all, you can also use homebrew to install R! (But maybe stick with the CRAN version unless you really want to compile your own R binary) To use openblas with R, follow these instructions: https://gist.github.com/zachmayer/e591cf868b3a381a01d6#file-openblas-sh To use veclib with R, follow these intructions: https://gist.github.com/zachmayer/e591cf868b3a381a01d6#file-veclib-sh ## OLD POST: Inspired by this post, I decided to try using OpenBLAS for R on my mac. However, it turns out there's a simpler option, using the vecLib BLAS library, which is provided by Apple as part of the accelerate framework. If you are using R 2.15, follow these instructions to change your BLAS from the default to vecLib: However, as noted in r-sig-mac, these instructions do not work for R 3.0. You have to directly link to the accelerate framework's version of vecLib: Finally, test your new blas using this script: On my system (a retina macbook pro), the default BLAS takes 141 seconds and vecLib takes 43 seconds, which is a significant speedup. If you plan to use vecLib, note the following warning from the R development team "Although fast, it is not under our control and may possibly deliver inaccurate results." So far, I have not encountered any issues using vecLib, but it's only been a few hours :-). UPDATE: you can also install OpenBLAS on a mac: If you do this, make sure to change the directories to point to the correct location on your system (e.g. change /users/zach/source to whatever directory you clone the git repo into). On my system, the benchmark script takes ~41 seconds when using openBLAS, which is a small but significant speedup. ## September 19, 2014 ### Chris Lawrence #### What could a federal UK look like? Assuming that the “no” vote prevails in the Scottish independence referendum, the next question for the United Kingdom is to consider constitutional reform to implement a quasi-federal system and resolve the West Lothian question once and for all. In some ways, it may also provide an opportunity to resolve the stalled reform of the upper house as well. Here’s the rough outline of a proposal that might work. • Devolve identical powers to England, Northern Ireland, Scotland, and Wales, with the proviso that local self-rule can be suspended if necessary by the federal legislature (by a supermajority). • The existing House of Commons becomes the House of Commons for England, which (along with the Sovereign) shall comprise the English Parliament. This parliament would function much as the existing devolved legislatures in Scotland and Wales; the consociational structure of the Northern Ireland Assembly (requiring double majorities) would not be replicated. • The House of Lords is abolished, and replaced with a directly-elected Senate of the United Kingdom. The Senate will have authority to legislate on the non-devolved powers (in American parlance, “delegated” powers) such as foreign and European Union affairs, trade and commerce, national defense, and on matters involving Crown dependencies and territories, the authority to legislate on devolved matters in the event self-government is suspended in a constituent country, and dilatory powers including a qualified veto (requiring a supermajority) over the legislation proposed by a constituent country’s parliament. The latter power would effectively replace the review powers of the existing House of Lords; it would function much as the Council of Revision in Madison’s original plan for the U.S. Constitution. As the Senate will have relatively limited powers, it need not be as large as the existing Lords or Commons. To ensure the countries other than England have a meaningful voice, given that nearly 85% of the UK’s population is in England, two-thirds of the seats would be allocated proportionally based on population and one-third allocated equally to the four constituent countries. This would still result in a chamber with a large English majority (around 64.4%) but nonetheless would ensure the other three countries would have meaningful representation as well. ## September 12, 2014 ### R you ready? #### Using colorized PNG pictograms in R base plots Today I stumbled across a figure in an explanation on multiple factor analysis which contained pictograms. Figure 1 from Abdi & Valentin (2007), p. 8. I wanted to reproduce a similar figure in R using pictograms and additionally color them e.g. by group membership . I have almost no knowledge about image processing, so I tried out several methods of how to achieve what I want. The first thing I did was read in an PNG file and look at the data structure. The package png allows to read in PNG files. Note that all of the below may not work on Windows machines, as it does not support semi-transparency (see ?readPNG). library(png) img <- readPNG(system.file("img", "Rlogo.png", package="png")) class(img)  ## [1] "array"  dim(img)  ## [1] 76 100 4  The object is a numerical array with four layers (red, green, blue, alpha; short RGBA). Let’s have a look at the first layer (red) and replace all non-zero entries by a one and the zeros by a dot. This will show us the pattern of non-zero values and we already see the contours. l4 <- img[,,1] l4[l4 > 0] <- 1 l4[l4 == 0] <- "." d <- apply(l4, 1, function(x) { cat(paste0(x, collapse=""), "\n") })  To display the image in R one way is to raster the image (i.e. the RGBA layers are collapsed into a layer of single HEX value) and print it using rasterImage. rimg <- as.raster(img) # raster multilayer object r <- nrow(rimg) / ncol(rimg) # image ratio plot(c(0,1), c(0,r), type = "n", xlab = "", ylab = "", asp=1) rasterImage(rimg, 0, 0, 1, r)  Let’s have a look at a small part the rastered image object. It is a matrix of HEX values. rimg[40:50, 1:6]  ## [1,] "#C4C5C202" "#858981E8" "#838881FF" "#888D86FF" "#8D918AFF" "#8F938CFF" ## [2,] "#00000000" "#848881A0" "#80847CFF" "#858A83FF" "#898E87FF" "#8D918BFF" ## [3,] "#00000000" "#8B8E884C" "#7D817AFF" "#82867EFF" "#868B84FF" "#8A8E88FF" ## [4,] "#00000000" "#9FA29D04" "#7E827BE6" "#7E817AFF" "#838780FF" "#878C85FF" ## [5,] "#00000000" "#00000000" "#81857D7C" "#797E75FF" "#7F827BFF" "#838781FF" ## [6,] "#00000000" "#00000000" "#898C8510" "#787D75EE" "#797E76FF" "#7F837BFF" ## [7,] "#00000000" "#00000000" "#00000000" "#7F837C7B" "#747971FF" "#797E76FF" ## [8,] "#00000000" "#00000000" "#00000000" "#999C9608" "#767C73DB" "#747971FF" ## [9,] "#00000000" "#00000000" "#00000000" "#00000000" "#80847D40" "#71766EFD" ## [10,] "#00000000" "#00000000" "#00000000" "#00000000" "#00000000" "#787D7589" ## [11,] "#00000000" "#00000000" "#00000000" "#00000000" "#00000000" "#999C9604"  And print this small part. plot(c(0,1), c(0,.6), type = "n", xlab = "", ylab = "", asp=1) rasterImage(rimg[40:50, 1:6], 0, 0, 1, .6)  Now we have an idea of how the image object and the rastered object look like from the inside. Let’s start to modify the images to suit our needs. In order to change the color of the pictograms, my first idea was to convert the graphics to greyscale and remap the values to a color ramp of may choice. To convert to greyscale there are tons of methods around (see e.g. here). I just pick one of them I found on SO by chance. With R=Red, G=Green and B=Blue we have brightness = sqrt(0.299 * R^2 + 0.587 * G^2 + 0.114 * B^2)  This approach modifies the PNG files after they have been coerced into a raster object. # function to calculate brightness values brightness <- function(hex) { v <- col2rgb(hex) sqrt(0.299 * v[1]^2 + 0.587 * v[2]^2 + 0.114 * v[3]^2) /255 } # given a color ramp, map brightness to ramp also taking into account # the alpha level. The defaul color ramp is grey # img_to_colorramp <- function(img, ramp=grey) { cv <- as.vector(img) b <- sapply(cv, brightness) g <- ramp(b) a <- substr(cv, 8,9) # get alpha values ga <- paste0(g, a) # add alpha values to new colors img.grey <- matrix(ga, nrow(img), ncol(img), byrow=TRUE) } # read png and modify img <- readPNG(system.file("img", "Rlogo.png", package="png")) img <- as.raster(img) # raster multilayer object r <- nrow(img) / ncol(img) # image ratio s <- 3.5 # size plot(c(0,10), c(0,3.5), type = "n", xlab = "", ylab = "", asp=1) rasterImage(img, 0, 0, 0+s/r, 0+s) # original img2 <- img_to_colorramp(img) # modify using grey scale rasterImage(img2, 5, 0, 5+s/r, 0+s)  Great, it works! Now Let’s go and try out some other color palettes using colorRamp to create a color ramp. plot(c(0,10),c(0,8.5), type = "n", xlab = "", ylab = "", asp=1) img1 <- img_to_colorramp(img) rasterImage(img1, 0, 5, 0+s/r, 5+s) reds <- function(x) rgb(colorRamp(c("darkred", "white"))(x), maxColorValue = 255) img2 <- img_to_colorramp(img, reds) rasterImage(img2, 5, 5, 5+s/r, 5+s) greens <- function(x) rgb(colorRamp(c("darkgreen", "white"))(x), maxColorValue = 255) img3 <- img_to_colorramp(img, greens) rasterImage(img3, 0, 0, 0+s/r, 0+s) single_color <- function(...) "#0000BB" img4 <- img_to_colorramp(img, single_color) rasterImage(img4, 5, 0, 5+s/r, 0+s)  Okay, that basically does the job. Now we will apply it to the wine pictograms. Let’s use this wine glass from Wikimedia Commons. It’s quite big so I uploaded a reduced size version to imgur . We will use it for our purposes. # load file from web f <- tempfile() download.file("http://i.imgur.com/A14ntCt.png", f) img <- readPNG(f) img <- as.raster(img) r <- nrow(img) / ncol(img) s <- 1 # let's create a function that returns a ramp function to save typing ramp <- function(colors) function(x) rgb(colorRamp(colors)(x), maxColorValue = 255) # create dataframe with coordinates and colors set.seed(1) x <- data.frame(x=rnorm(16, c(2,2,4,4)), y=rnorm(16, c(1,3)), colors=c("black", "darkred", "garkgreen", "darkblue")) plot(c(1,6), c(0,5), type="n", xlab="", ylab="", asp=1) for (i in 1L:nrow(x)) { colorramp <- ramp(c(x[i,3], "white")) img2 <- img_to_colorramp(img, colorramp) rasterImage(img2, x[i,1], x[i,2], x[i,1]+s/r, x[i,2]+s) }  Another approach would be to modifying the RGB layers before rastering to HEX values. img <- readPNG(system.file("img", "Rlogo.png", package="png")) img2 <- img img[,,1] <- 0 # remove Red component img[,,2] <- 0 # remove Green component img[,,3] <- 1 # Set Blue to max img <- as.raster(img) r <- nrow(img) / ncol(img) # size ratio s <- 3.5 # size plot(c(0,10), c(0,3.5), type = "n", xlab = "", ylab = "", asp=1) rasterImage(img, 0, 0, 0+s/r, 0+s) img2[,,1] <- 1 # Red to max img2[,,2] <- 0 img2[,,3] <- 0 rasterImage(as.raster(img2), 5, 0, 5+s/r, 0+s)  To just colorize the image, we could weight each layer. # wrap weighting into function weight_layers <- function(img, w) { for (i in seq_along(w)) img[,,i] <- img[,,i] * w[i] img } plot(c(0,10), c(0,3.5), type = "n", xlab = "", ylab = "", asp=1) img <- readPNG(system.file("img", "Rlogo.png", package="png")) img2 <- weight_layers(img, c(.2, 1,.2)) rasterImage(img2, 0, 0, 0+s/r, 0+s) img3 <- weight_layers(img, c(1,0,0)) rasterImage(img3, 5, 0, 5+s/r, 0+s)  After playing around and hard-coding the modifications I started to search and found the EBimage package which has a lot of features for image processing that make ones life (in this case only a bit) easier. library(EBImage) f <- system.file("img", "Rlogo.png", package="png") img <- readImage(f) img2 <- img img[,,2] = 0 # zero out green layer img[,,3] = 0 # zero out blue layer img <- as.raster(img) img2[,,1] = 0 img2[,,3] = 0 img2 <- as.raster(img2) r <- nrow(img) / ncol(img) s <- 3.5 plot(c(0,10), c(0,3.5), type = "n", xlab = "", ylab = "", asp=1) rasterImage(img, 0, 0, 0+s/r, 0+s) rasterImage(img2, 5, 0, 5+s/r, 0+s)  EBImage is a good choice and fairly easy to handle. Now let’s again print the pictograms. f <- tempfile(fileext=".png") download.file("http://i.imgur.com/A14ntCt.png", f) img <- readImage(f) # will replace whole image layers by one value # only makes sense if there is a alpha layer that # gives the contours # mod_color <- function(img, col) { v <- col2rgb(col) / 255 img = channel(img, 'rgb') img[,,1] = v[1] # Red img[,,2] = v[2] # Green img[,,3] = v[3] # Blue as.raster(img) } r <- nrow(img) / ncol(img) # get image ratio s <- 1 # size # create random data set.seed(1) x <- data.frame(x=rnorm(16, c(2,2,4,4)), y=rnorm(16, c(1,3)), colors=1:4) # plot pictograms plot(c(1,6), c(0,5), type="n", xlab="", ylab="", asp=1) for (i in 1L:nrow(x)) { img2 <- mod_color(img, x[i, 3]) rasterImage(img2, x[i,1], x[i,2], x[i,1]+s*r, x[i,2]+s) }  Note, that above I did not bother to center each pictogram to position it correctly. This still needs to be done. Anyway, that’s it! Mission completed. ### Literature Abdi, H., & Valentin, D. (2007). Multiple factor analysis (MFA). In N. Salkind (Ed.), Encyclopedia of Measurement and Statistics (pp. 1–14). Thousand Oaks, CA: Sage Publications. Retrieved from https://www.utdallas.edu/~herve/Abdi-MFA2007-pretty.pdf ## June 18, 2014 ### Chris Lawrence #### Soccer queries answered Kevin Drum asks a bunch of questions about soccer: 1. Outside the penalty area there’s a hemisphere about 20 yards wide. I can’t recall ever seeing it used for anything. What’s it for? 2. On several occasions, I’ve noticed that if the ball goes out of bounds at the end of stoppage time, the referee doesn’t whistle the match over. Instead, he waits for the throw-in, and then immediately whistles the match over. What’s the point of this? 3. Speaking of stoppage time, how has it managed to last through the years? I know, I know: tradition. But seriously. Having a timekeeper who stops the clock for goals, free kicks, etc. has lots of upside and no downside. Right? It wouldn’t change the game in any way, it would just make timekeeping more accurate, more consistent, and more transparent for the fans and players. Why keep up the current pretense? 4. What’s the best way to get a better sense of what’s a foul and what’s a legal tackle? Obviously you can’t tell from the players’ reactions, since they all writhe around like landed fish if they so much as trip over their own shoelaces. Reading the rules provides the basics, but doesn’t really help a newbie very much. Maybe a video that shows a lot of different tackles and explains why each one is legal, not legal, bookable, etc.? The first one’s easy: there’s a general rule that no defensive player can be within 10 yards of the spot of a direct free kick. A penalty kick (which is a type of direct free kick) takes place in the 18-yard box, and no players other than the player taking the kick and the goalkeeper are allowed in the box. However, owing to geometry, the 18 yard box and the 10 yard exclusion zone don’t fully coincide, hence the penalty arc. (That’s also why there are two tiny hash-marks on the goal line and side line 10 yards from the corner flag. And why now referees have a can of shaving cream to mark the 10 yards for other free kicks, one of the few MLS innovations that has been a good idea.) Second one’s also easy: the half and the game cannot end while the ball is out of play. Third one’s harder. First, keeping time inexactly forestalls the silly premature celebrations that are common in most US sports. You’d never see the Stanford-Cal play happen in a soccer game. Second, it allows some slippage for short delays and doesn’t require exact timekeeping; granted, this was more valuable before instant replays and fourth officials, but most US sports require a lot of administrative record-keeping by ancillary officials. A soccer game can be played with one official (and often is, particularly at the amateur level) without having to change timing rules;* in developing countries in particular this lowers the barriers to entry for the sport (along with the low equipment requirements) without changing the nature of the game appreciably. Perhaps most importantly, if the clock was allowed to stop regularly it would create an excuse for commercial timeouts and advertising breaks, which would interrupt the flow of the game and potentially reduce the advantages of better-conditioned and more skilled athletes. (MLS tried this, along with other exciting American ideas like “no tied games,” and it was as appealing to actual soccer fans as ketchup on filet mignon would be to a foodie, and perhaps more importantly didn’t make any non-soccer fans watch.) Fourth, the key distinction is usually whether there was an obvious attempt to play the ball; in addition, in the modern game, even some attempts to play the ball are considered inherently dangerous (tackling from behind, many sliding tackles, etc.) and therefore are fouls even if they are successful in getting more ball than human. * To call offside, you’d also probably need what in my day we called a “linesman.” ## May 07, 2014 ### Chris Lawrence #### The mission and vision thing Probably the worst-kept non-secret is that the next stage of the institutional evolution of my current employer is to some ill-defined concept of “university status,” which mostly involves the establishment of some to-be-determined master’s degree programs. In the context of the University System of Georgia, it means a small jump from the “state college” prestige tier (a motley collection of schools that largely started out as two-year community colleges and transfer institutions) to the “state university” tier (which is where most of the ex-normal schools hang out these days). What is yet to be determined is how that transition will affect the broader institution that will be the University of Middle Georgia.* People on high are said to be working on these things; in any event, here are my assorted random thoughts on what might be reasonable things to pursue: • Marketing and positioning: Unlike the situation facing many of the other USG institutions, the population of the two anchor counties of our core service area (Bibb and Houston) is growing, and Houston County in particular has a statewide reputation for the quality of its public school system. Rather than conceding that the most prepared students from these schools will go to Athens or Atlanta or Valdosta, we should strongly market our institutional advantages over these more “prestigious” institutions, particularly in terms of the student experience in the first two years and the core curriculum: we have no large lecture courses, no teaching assistants, no lengthy bus rides to and from class every day, and the vast majority of the core is taught by full-time faculty with terminal degrees. Not to mention costs to students are much lower, particularly in the case of students who do not qualify for need-based aid. Even if we were to “lose” these students as transfers to the top-tier institutions after 1–4 semesters, we’d still benefit from the tuition and fees they bring in and we would not be penalized in the upcoming state performance funding formula. Dual enrollment in Warner Robins in particular is an opportunity to showcase our institution as a real alternative for better prepared students rather than a safety school. • Comprehensive offerings at the bachelor’s level: As a state university, we will need to offer a comprehensive range of options for bachelor’s students to attract and retain students, both traditional and nontraditional. In particular, B.S. degrees in political science and sociology with emphasis in applied empirical skills would meet public and private employer demand for workers who have research skills and the ability to collect, manage, understand, and use data appropriately. There are other gaps in the liberal arts and sciences as well that need to be addressed to become a truly comprehensive state university. • Create incentives to boost the residential population: The college currently has a heavy debt burden inherited from the overbuilding of dorms at the Cochran campus. We need to identify ways to encourage students to live in Cochran, which may require public-private partnerships to try to build a “college town” atmosphere in the community near campus. We also need to work with wireless providers like Sprint and T-Mobile to ensure that students from the “big city” can fully use their cell phones and tablets in Cochran and Eastman without roaming fees or changing wireless providers. • Tie the institution more closely to the communities we serve: This includes both physical ties and psychological ties. The Macon campus in particular has poor physical links to the city itself for students who might walk or ride bicycles; extending the existing bike/walking trail from Wesleyan to the Macon campus should be a priority, as should pedestrian access and bike facilities along Columbus Road. Access to the Warner Robins campus is somewhat better but still could be improved. More generally, the institution is perceived as an afterthought or alternative of last resort in the community. Improving this situation and perception among community leaders and political figures may require a physical presence in or near downtown Macon, perhaps in partnership with the GCSU Graduate Center. * There is no official name-in-waiting, but given that our former interim president seemed to believe he could will this name into existence by repeating it enough I’ll stick with it. The straw poll of faculty trivia night suggests that it’s the least bad option available, which inevitably means the regents will choose something else instead (if the last name change is anything to go by). ## February 17, 2014 ### Seth Falcon #### Have Your SHA and Bcrypt Too ## Fear I've been putting off sharing this idea because I've heard the rumors about what happens to folks who aren't security experts when they post about security on the internet. If this blog is replaced with cat photos and rainbows, you'll know what happened. ## The Sad Truth It's 2014 and chances are you have accounts on websites that are not properly handling user passwords. I did no research to produce the following list of ways passwords are mishandled in decreasing order of frequency: 1. Site uses a fast hashing algorithm, typically SHA1(salt + plain-password). 2. Site doesn't salt password hashes 3. Site stores raw passwords We know that sites should be generating secure random salts and using an established slow hashing algorithm (bcrypt, scrypt, or PBKDF2). Why are sites not doing this? While security issues deserve a top spot on any site's priority list, new features often trump addressing legacy security concerns. The immediacy of the risk is hard to quantify and it's easy to fall prey to a "nothing bad has happened yet, why should we change now" attitude. It's easy for other bugs, features, or performance issues to win out when measured by immediate impact. Fixing security or other "legacy" issues is the Right Thing To Do and often you will see no measurable benefit from the investment. It's like having insurance. You don't need it until you do. Specific to the improper storage of user password data is the issue of the impact to a site imposed by upgrading. There are two common approaches to upgrading password storage. You can switch cold turkey to the improved algorithms and force password resets on all of your users. Alternatively, you can migrate incrementally such that new users and any user who changes their password gets the increased security. The cold turkey approach is not a great user experience and sites might choose to delay an upgrade to avoid admitting to a weak security implementation and disrupting their site by forcing password resets. The incremental approach is more appealing, but the security benefit is drastically diminished for any site with a substantial set of existing users. Given the above migration choices, perhaps it's (slightly) less surprising that businesses choose to prioritize other work ahead of fixing poorly stored user password data. ## The Idea What if you could upgrade a site so that both new and existing users immediately benefited from the increased security, but without the disruption of password resets? It turns out that you can and it isn't very hard. Consider a user table with columns: userid salt hashed_pass  Where the hashed_pass column is computed using a weak fast algorithm, for example SHA1(salt + plain_pass). The core of the idea is to apply a proper algorithm on top of the data we already have. I'll use bcrypt to make the discussion concrete. Add columns to the user table as follows: userid salt hashed_pass hash_type salt2  Process the existing user table by computing bcrypt(salt2 + hashed_pass) and storing the result in the hashed_pass column (overwriting the less secure value); save the new salt value to salt2 and set hash_type to bycrpt+sha1. To verify a user where hash_type is bcrypt+sha1, compute bcrypt(salt2 + SHA1(salt + plain_pass)) and compare to the hashed_pass value. Note that bcrypt implementations encode the salt as a prefix of the hashed value so you could avoid the salt2 column, but it makes the idea easier to explain to have it there. You can take this approach further and have any user that logs in (as well as new users) upgrade to a "clean" bcrypt only algorithm since you can now support different verification algorithms using hash_type. With the proper application code changes in place, the upgrade can be done live. This scheme will also work for sites storing non-salted password hashes as well as those storing plain text passwords (THE HORROR). ## Less Sadness, Maybe Perhaps this approach makes implementing a password storage security upgrade more palatable and more likely to be prioritized. And if there's a horrible flaw in this approach, maybe you'll let me know without turning this blog into a tangle of cat photos and rainbows. ## December 26, 2013 ### Seth Falcon #### A Rebar Plugin for Locking Deps: Reproducible Erlang Project Builds For Fun and Profit ## What's this lock-deps of which you speak? If you use rebar to generate an OTP release project and want to have reproducible builds, you need the rebar_lock_deps_plugin plugin. The plugin provides a lock-deps command that will generate a rebar.config.lock file containing the complete flattened set of project dependencies each pegged to a git SHA. The lock file acts similarly to Bundler's Gemfile.lock file and allows for reproducible builds (*). Without lock-deps you might rely on the discipline of using a tag for all of your application's deps. This is insufficient if any dep depends on something not specified as a tag. It can also be a problem if a third party dep doesn't provide a tag. Generating a rebar.config.lock file solves these issues. Moreover, using lock-deps can simplify the work of putting together a release consisting of many of your own repos. If you treat the master branch as shippable, then rather than tagging each subproject and updating rebar.config throughout your project's dependency chain, you can run get-deps (without the lock file), compile, and re-lock at the latest versions throughout your project repositories. The reproducibility of builds when using lock-deps depends on the SHAs captured in rebar.config.lock. The plugin works by scanning the cloned repos in your project's deps directory and extracting the current commit SHA. This works great until a repository's history is rewritten with a force push. If you really want reproducible builds, you need to not nuke your SHAs and you'll need to fork all third party repos to ensure that someone else doesn't screw you over in this fashion either. If you make a habit of only depending on third party repos using a tag, assume that upstream maintainers are not completely bat shit crazy, and don't force push your master branch, then you'll probably be fine. ## Getting Started Install the plugin in your project by adding the following to your rebar.config file: %% Plugin dependency {deps, [ {rebar_lock_deps_plugin, ".*", {git, "git://github.com/seth/rebar_lock_deps_plugin.git", {branch, "master"}}} ]}. %% Plugin usage {plugins, [rebar_lock_deps_plugin]}.  To test it out do: rebar get-deps # the plugin has to be compiled so you can use it rebar compile rebar lock-deps  If you'd like to take a look at a project that uses the plugin, take a look at CHEF's erchef project. ## Bonus features If you are building an OTP release project using rebar generate then you can use rebar_lock_deps_plugin to enhance your build experience in three easy steps. 1. Use rebar bump-rel-version version=$BUMP to automate the process of editing rel/reltool.config to update the release version. The argument $BUMP can be major, minor, or patch (default) to increment the specified part of a semver X.Y.Z version. If $BUMP is any other value, it is used as the new version verbatim. Note that this function rewrites rel/reltool.config using ~p. I check-in the reformatted version and maintain the formatting when editing. This way, the general case of a version bump via bump-rel-version results in a minimal diff.

2. Autogenerate a change summary commit message for all project deps. Assuming you've generated a new lock file and bumped the release version, use rebar commit-release to commit the changes to rebar.config.lock and rel/reltool.config with a commit message that summarizes the changes made to each dependency between the previously locked version and the newly locked version. You can get a preview of the commit message via rebar log-changed-deps.

3. Finally, create an annotated tag for your new release with rebar tag-release which will read the current version from rel/reltool.config and create an annotated tag named with the version.

## The dependencies, they are ordered

Up to version 2.0.1 of rebar_lock_deps_plugin, the dependencies in the generated lock file were ordered alphabetically. This was a side-effect of using filelib:wildcard/1 to list the dependencies in the top-level deps directory. In most cases, the order of the full dependency set does not matter. However, if some of the code in your project uses parse transforms, then it will be important for the parse transform to be compiled and on the code path before attempting to compile code that uses the parse transform.

This issue was recently discovered by a colleague who ran into build issues using the lock file for a project that had recently integrated lager for logging. He came up with the idea of maintaining the order of deps as they appear in the various rebar.config files along with a prototype patch proving out the idea. As of rebar_lock_deps_plugin 3.0.0, the lock-deps command will (mostly) maintain the relative order of dependencies as found in the rebar.config files.

The "mostly" is that when a dep is shared across two subprojects, it will appear in the expected order for the first subproject (based on the ordering of the two subprojects). The deps for the second subproject will not be in strict rebar.config order, but the resulting order should address any compile-time dependencies and be relatively stable (only changing when project deps alter their deps with larger impact when shared deps are introduced or removed).

## Digression: fun with dependencies

There are times, as a programmer, when a real-world problem looks like a text book exercise (or an interview whiteboard question). Just the other day at work we had to design some manhole covers, but I digress.

Fixing the order of the dependencies in the generated lock file is (nearly) the same as finding an install order for a set of projects with inter-dependencies. I had some fun coding up the text book solution even though the approach doesn't handle the constraint of respecting the order provided by the rebar.config files. Onward with the digression.

We have a set of "packages" where some packages depend on others and we want to determine an install order such that a package's dependencies are always installed before the package. The set of packages and the relation "depends on" form a directed acyclic graph or DAG. The topological sort of a DAG produces an install order for such a graph. The ordering is not unique. For example, with a single package C depending on A and B, valid install orders are [A, B, C] and [B, A, C].

To setup the problem, we load all of the project dependency information into a proplist mapping each package to a list of its dependencies extracted from the package's rebar.config file.

read_all_deps(Config, Dir) ->
TopDeps = rebar_config:get(Config, deps, []),
Acc = [{top, dep_names(TopDeps)}],
DepDirs = filelib:wildcard(filename:join(Dir, "*")),
Acc ++ [
{filename:basename(D), dep_names(extract_deps(D))}
|| D <- DepDirs ].

Erlang's standard library provides the digraph and digraph_utils modules for constructing and operating on directed graphs. The digraph_utils module includes a topsort/1 function which we can make use of for our "exercise". The docs say:

Returns a topological ordering of the vertices of the digraph Digraph if such an ordering exists, false otherwise. For each vertex in the returned list, there are no out-neighbours that occur earlier in the list.

To figure out which way to point the edges when building our graph, consider two packages A and B with A depending on B. We know we want to end up with an install order of [B, A]. Rereading the topsort/1 docs, we must want an edge B => A. With that, we can build our DAG and obtain an install order with the topological sort:

load_digraph(Config, Dir) ->
G = digraph:new(),
Nodes = all_nodes(AllDeps),
[ digraph:add_vertex(G, N) || N <- Nodes ],
%% If A depends on B, then we add an edge A <= B
[
|| Dep <- DepList ]
|| {Item, DepList} <- AllDeps, Item =/= top ],
digraph_utils:topsort(G).

%% extract a sorted unique list of all deps
all_nodes(AllDeps) ->
lists:usort(lists:foldl(fun({top, L}, Acc) ->
L ++ Acc;
({K, L}, Acc) ->
[K|L] ++ Acc
end, [], AllDeps)).

The digraph module manages graphs using ETS giving it a convenient API, though one that feels un-erlang-y in its reliance on side-effects.

The above gives an install order, but doesn't take into account the relative order of deps as specified in the rebar.config files. The solution implemented in the plugin is a bit less fancy, recursing over the deps and maintaining the desired ordering. The only tricky bit being that shared deps are ignored until the end and the entire linearized list is de-duped which required a . Here's the code:

order_deps(AllDeps) ->
Top = proplists:get_value(top, AllDeps),
order_deps(lists:reverse(Top), AllDeps, []).

order_deps([], _AllDeps, Acc) ->
de_dup(Acc);
order_deps([Item|Rest], AllDeps, Acc) ->
ItemDeps = proplists:get_value(Item, AllDeps),
order_deps(lists:reverse(ItemDeps) ++ Rest, AllDeps, [Item | Acc]).

de_dup(AccIn) ->
WithIndex = lists:zip(AccIn, lists:seq(1, length(AccIn))),
UWithIndex = lists:usort(fun({A, _}, {B, _}) ->
A =< B
end, WithIndex),
Ans0 = lists:sort(fun({_, I1}, {_, I2}) ->
I1 =< I2
end, UWithIndex),
[ V || {V, _} <- Ans0 ].

## Conclusion and the end of this post

The great thing about posting to your blog is, you don't have to have a proper conclusion if you don't want to.

# Probabilistic bug hunting

Have you ever run into a bug that, no matter how careful you are trying to reproduce it, it only happens sometimes? And then, you think you've got it, and finally solved it - and tested a couple of times without any manifestation. How do you know that you have tested enough? Are you sure you were not "lucky" in your tests?

In this article we will see how to answer those questions and the math behind it without going into too much detail. This is a pragmatic guide.

## The Bug

The following program is supposed to generate two random 8-bit integer and print them on stdout:

#include <stdio.h>
#include <fcntl.h>
#include <unistd.h>

/* Returns -1 if error, other number if ok. */
int get_random_chars(char *r1, char*r2)
{
int f = open("/dev/urandom", O_RDONLY);

if (f < 0)
return -1;
if (read(f, r1, sizeof(*r1)) < 0)
return -1;
if (read(f, r2, sizeof(*r2)) < 0)
return -1;
close(f);

return *r1 & *r2;
}

int main(void)
{
char r1;
char r2;
int ret;

ret = get_random_chars(&r1, &r2);

if (ret < 0)
fprintf(stderr, "error");
else
printf("%d %d\n", r1, r2);

return ret < 0;
}

On my architecture (Linux on IA-32) it has a bug that makes it print "error" instead of the numbers sometimes.

## The Model

Every time we run the program, the bug can either show up or not. It has a non-deterministic behaviour that requires statistical analysis.

We will model a single program run as a Bernoulli trial, with success defined as "seeing the bug", as that is the event we are interested in. We have the following parameters when using this model:

• $$n$$: the number of tests made;
• $$k$$: the number of times the bug was observed in the $$n$$ tests;
• $$p$$: the unknown (and, most of the time, unknowable) probability of seeing the bug.

As a Bernoulli trial, the number of errors $$k$$ of running the program $$n$$ times follows a binomial distribution $$k \sim B(n,p)$$. We will use this model to estimate $$p$$ and to confirm the hypotheses that the bug no longer exists, after fixing the bug in whichever way we can.

By using this model we are implicitly assuming that all our tests are performed independently and identically. In order words: if the bug happens more ofter in one environment, we either test always in that environment or never; if the bug gets more and more frequent the longer the computer is running, we reset the computer after each trial. If we don't do that, we are effectively estimating the value of $$p$$ with trials from different experiments, while in truth each experiment has its own $$p$$. We will find a single value anyway, but it has no meaning and can lead us to wrong conclusions.

### Physical analogy

Another way of thinking about the model and the strategy is by creating a physical analogy with a box that has an unknown number of green and red balls:

• Bernoulli trial: taking a single ball out of the box and looking at its color - if it is red, we have observed the bug, otherwise we haven't. We then put the ball back in the box.
• $$n$$: the total number of trials we have performed.
• $$k$$: the total number of red balls seen.
• $$p$$: the total number of red balls in the box divided by the total number of green balls in the box.

• If we open the box and count the balls, we can know $$p$$, in contrast with our original problem.
• Without opening the box, we can estimate $$p$$ by repeating the trial. As $$n$$ increases, our estimate for $$p$$ improves. Mathematically: $p = \lim_{n\to\infty}\frac{k}{n}$
• Performing the trials in different conditions is like taking balls out of several different boxes. The results tell us nothing about any single box.

## Estimating $$p$$

Before we try fixing anything, we have to know more about the bug, starting by the probability $$p$$ of reproducing it. We can estimate this probability by dividing the number of times we see the bug $$k$$ by the number of times we tested for it $$n$$. Let's try that with our sample bug:

  $./hasbug 67 -68$ ./hasbug
79 -101
$./hasbug error  We know from the source code that $$p=25%$$, but let's pretend that we don't, as will be the case with practically every non-deterministic bug. We tested 3 times, so $$k=1, n=3 \Rightarrow p \sim 33%$$, right? It would be better if we tested more, but how much more, and exactly what would be better? ### $$p$$ precision Let's go back to our box analogy: imagine that there are 4 balls in the box, one red and three green. That means that $$p = 1/4$$. What are the possible results when we test three times? Red balls Green balls $$p$$ estimate 0 3 0% 1 2 33% 2 1 66% 3 0 100% The less we test, the smaller our precision is. Roughly, $$p$$ precision will be at most $$1/n$$ - in this case, 33%. That's the step of values we can find for $$p$$, and the minimal value for it. Testing more improves the precision of our estimate. ### $$p$$ likelihood Let's now approach the problem from another angle: if $$p = 1/4$$, what are the odds of seeing one error in four tests? Let's name the 4 balls as 0-red, 1-green, 2-green and 3-green: The table above has all the possible results for getting 4 balls out of the box. That's $$4^4=256$$ rows, generated by this python script. The same script counts the number of red balls in each row, and outputs the following table: k rows % 0 81 31.64% 1 108 42.19% 2 54 21.09% 3 12 4.69% 4 1 0.39% That means that, for $$p=1/4$$, we see 1 red ball and 3 green balls only 42% of the time when getting out 4 balls. What if $$p = 1/3$$ - one red ball and two green balls? We would get the following table: k rows % 0 16 19.75% 1 32 39.51% 2 24 29.63% 3 8 9.88% 4 1 1.23% What about $$p = 1/2$$? k rows % 0 1 6.25% 1 4 25.00% 2 6 37.50% 3 4 25.00% 4 1 6.25% So, let's assume that you've seen the bug once in 4 trials. What is the value of $$p$$? You know that can happen 42% of the time if $$p=1/4$$, but you also know it can happen 39% of the time if $$p=1/3$$, and 25% of the time if $$p=1/2$$. Which one is it? The graph bellow shows the discrete likelihood for all $$p$$ percentual values for getting 1 red and 3 green balls: The fact is that, given the data, the estimate for $$p$$ follows a beta distribution $$Beta(k+1, n-k+1) = Beta(2, 4)$$ (1) The graph below shows the probability distribution density of $$p$$: The R script used to generate the first plot is here, the one used for the second plot is here. ### Increasing $$n$$, narrowing down the interval What happens when we test more? We obviously increase our precision, as it is at most $$1/n$$, as we said before - there is no way to estimate that $$p=1/3$$ when we only test twice. But there is also another effect: the distribution for $$p$$ gets taller and narrower around the observed ratio $$k/n$$: ### Investigation framework So, which value will we use for $$p$$? • The smaller the value of $$p$$, the more we have to test to reach a given confidence in the bug solution. • We must, then, choose the probability of error that we want to tolerate, and take the smallest value of $$p$$ that we can. A usual value for the probability of error is 5% (2.5% on each side). • That means that we take the value of $$p$$ that leaves 2.5% of the area of the density curve out on the left side. Let's call this value $$p_{min}$$. • That way, if the observed $$k/n$$ remains somewhat constant, $$p_{min}$$ will raise, converging to the "real" $$p$$ value. • As $$p_{min}$$ raises, the amount of testing we have to do after fixing the bug decreases. By using this framework we have direct, visual and tangible incentives to test more. We can objectively measure the potential contribution of each test. In order to calculate $$p_{min}$$ with the mentioned properties, we have to solve the following equation: $\sum_{k=0}^{k}{n\choose{k}}p_{min} ^k(1-p_{min})^{n-k}=\frac{\alpha}{2}$ $$alpha$$ here is twice the error we want to tolerate: 5% for an error of 2.5%. That's not a trivial equation to solve for $$p_{min}$$. Fortunately, that's the formula for the confidence interval of the binomial distribution, and there are a lot of sites that can calculate it: ## Is the bug fixed? So, you have tested a lot and calculated $$p_{min}$$. The next step is fixing the bug. After fixing the bug, you will want to test again, in order to confirm that the bug is fixed. How much testing is enough testing? Let's say that $$t$$ is the number of times we test the bug after it is fixed. Then, if our fix is not effective and the bug still presents itself with a probability greater than the $$p_{min}$$ that we calculated, the probability of not seeing the bug after $$t$$ tests is: $\alpha = (1-p_{min})^t$ Here, $$\alpha$$ is also the probability of making a type I error, while $$1 - \alpha$$ is the statistical significance of our tests. We now have two options: • arbitrarily determining a standard statistical significance and testing enough times to assert it. • test as much as we can and report the achieved statistical significance. Both options are valid. The first one is not always feasible, as the cost of each trial can be high in time and/or other kind of resources. The standard statistical significance in the industry is 5%, we recommend either that or less. Formally, this is very similar to a statistical hypothesis testing. ## Back to the Bug ### Testing 20 times This file has the results found after running our program 5000 times. We must never throw out data, but let's pretend that we have tested our program only 20 times. The observed $$k/n$$ ration and the calculated $$p_{min}$$ evolved as shown in the following graph: After those 20 tests, our $$p_{min}$$ is about 12%. Suppose that we fix the bug and test it again. The following graph shows the statistical significance corresponding to the number of tests we do: In words: we have to test 24 times after fixing the bug to reach 95% statistical significance, and 35 to reach 99%. Now, what happens if we test more before fixing the bug? ### Testing 5000 times Let's now use all the results and assume that we tested 5000 times before fixing the bug. The graph bellow shows $$k/n$$ and $$p_{min}$$: After those 5000 tests, our $$p_{min}$$ is about 23% - much closer to the real $$p$$. The following graph shows the statistical significance corresponding to the number of tests we do after fixing the bug: We can see in that graph that after about 11 tests we reach 95%, and after about 16 we get to 99%. As we have tested more before fixing the bug, we found a higher $$p_{min}$$, and that allowed us to test less after fixing the bug. ## Optimal testing We have seen that we decrease $$t$$ as we increase $$n$$, as that can potentially increases our lower estimate for $$p$$. Of course, that value can decrease as we test, but that means that we "got lucky" in the first trials and we are getting to know the bug better - the estimate is approaching the real value in a non-deterministic way, after all. But, how much should we test before fixing the bug? Which value is an ideal value for $$n$$? To define an optimal value for $$n$$, we will minimize the sum $$n+t$$. This objective gives us the benefit of minimizing the total amount of testing without compromising our guarantees. Minimizing the testing can be fundamental if each test costs significant time and/or resources. The graph bellow shows us the evolution of the value of $$t$$ and $$t+n$$ using the data we generated for our bug: We can see clearly that there are some low values of $$n$$ and $$t$$ that give us the guarantees we need. Those values are $$n = 15$$ and $$t = 24$$, which gives us $$t+n = 39$$. While you can use this technique to minimize the total number of tests performed (even more so when testing is expensive), testing more is always a good thing, as it always improves our guarantee, be it in $$n$$ by providing us with a better $$p$$ or in $$t$$ by increasing the statistical significance of the conclusion that the bug is fixed. So, before fixing the bug, test until you see the bug at least once, and then at least the amount specified by this technique - but also test more if you can, there is no upper bound, specially after fixing the bug. You can then report a higher confidence in the solution. ## Conclusions When a programmer finds a bug that behaves in a non-deterministic way, he knows he should test enough to know more about the bug, and then even more after fixing it. In this article we have presented a framework that provides criteria to define numerically how much testing is "enough" and "even more." The same technique also provides a method to objectively measure the guarantee that the amount of testing performed provides, when it is not possible to test "enough." We have also provided a real example (even though the bug itself is artificial) where the framework is applied. As usual, the source code of this page (R scripts, etc) can be found and downloaded in https://github.com/lpenz/lpenz.github.io ## December 01, 2013 ### Gregor Gorjanc #### Read line by line of a file in R Are you using R for data manipulation for later use with other programs, i.e., a workflow something like this: 1. read data sets from a disk, 2. modify the data, and 3. write it back to a disk. All fine, but of data set is really big, then you will soon stumble on memory issues. If data processing is simple and you can read only chunks, say only line by line, then the following might be useful: ## Filefile <- "myfile.txt" ## Create connectioncon <- file(description=file, open="r") ## Hopefully you know the number of lines from some other source orcom <- paste("wc -l ", file, " | awk '{ print$1 }'", sep="")n <- system(command=com, intern=TRUE) ## Loop over a file connectionfor(i in 1:n) {  tmp <- scan(file=con, nlines=1, quiet=TRUE)  ## do something on a line of data }
Created by Pretty R at inside-R.org

## August 13, 2013

### Gregor Gorjanc

#### Setup up the inverse of additive relationship matrix in R

Additive genetic covariance between individuals is one of the key concepts in (quantitative) genetics. When doing the prediction of additive genetic values for pedigree members, we need the inverse of the so called numerator relationship matrix (NRM) or simply A. Matrix A has off-diagonal entries equal to numerator of Wright's relationship coefficient and diagonal elements equal to 1 + inbreeding coefficient. I have blogged before about setting up such inverse in R using routine from the ASReml-R program or importing the inverse from the CFC program. However, this is not the only way to "skin this cat" in R. I am aware of the following attempts to provide this feature in R for various things (the list is probably incomplete and I would grateful if you point me to other implementations):
• pedigree R package has function makeA() and makeAinv() with obvious meanings; there is also calcG() if you have a lot of marker data instead of pedigree information; there are also some other very handy functions calcInbreeding(), orderPed(), trimPed(), etc.
• pedigreemm R package does not have direct implementation to get A inverse, but has all the needed ingredients, which makes the package even more interesting
• MCMCglmm R package has function inverseA() which works with pedigree or phlyo objects; there are also handy functions such as prunePed(), rbv()sm2asreml(), etc.
• kinship and kinship2 R packages have function kinship() to setup kinship matrix, which is equal to the half of A; there are also nice functions for plotting pedigrees etc. (see also here)
As I described before, the interesting thing is that setting up inverse of A is easier and cheaper than setting up A and inverting it. This is very important for large applications. This is an old result using the following matrix theory. We can decompose symmetric positive definite matrix as A = LU = LL' (Cholesky decomposition) or as A = LDU = LDL' (Generalized Cholesky decomposition), where L (U) is lower (upper) triangular, and D is diagonal matrix. Note that L and U in previous two equations are not the same thing (L from Cholesky is not equal to L from Generalized Cholesky decomposition)! Sorry for sloppy notation. In order to confuse you even more note that Henderson usually wrote A = TDT'. We can even do A = LSSU, where S diagonal is equal to the square root of D diagonal. This can get us back to A = LU = LL' as LSSU = LSSL' = LSS'L' = LS(LS)' = L'L (be ware of sloppy notation)! The inverse rule says that inv(A) = inv(LDU) = inv(U) inv(D) inv(L) = inv(L)' inv(D) inv(L) = inv(L)' inv(S) inv(S) inv(L). I thank to Martin Maechler for pointing out to the last (obviously) bit to me. In Henderson's notation this would be inv(A) = inv(T)' inv(D) inv(T) = inv(T)' inv(S) inv(S) inv(T) Uf ... The important bit is that with NRM (aka A) inv(L) has nice simple structure - it shows the directed graph of additive genetic values in pedigree, while inv(D) tells us about the precision (inverse variance) of additive genetic values given the additive genetic values of parents and therefore depends on knowledge of parents and their inbreeding (the more they are inbred less variation can we expect in their progeny). Both inv(L) and inv(D) are easier to setup.

Packages MCMCglmm and pedigree give us inv(A) directly (we can also get inv(D) in MCMCglmm), but pedigreemm enables us to play around with the above matrix algebra and graph theory. First we need a small example pedigree. Bellow is an example with 10 members and there is also some inbreeding and some individuals have both, one, or no parents known. It is hard to see inbreeding directly from the table, but we will improve that later (see also here).

ped <- data.frame( id=c(  1,   2,   3,   4,   5,   6,   7,   8,   9,  10),                  fid=c( NA,  NA,   2,   2,   4,   2,   5,   5,  NA,   8),                  mid=c( NA,  NA,   1,  NA,   3,   3,   6,   6,  NA,   9))

Now we will create an object of a pedigree class and show the A = U'U stuff:

## install.packages(pkgs="pedigreemm")library(package="pedigreemm") ped2 <- with(ped, pedigree(sire=fid, dam=mid, label=id)) U <- relfactor(ped2)A <- crossprod(U) round(U, digits=2)## 10 x 10 sparse Matrix of class "dtCMatrix"                                     ##  [1,] 1 . 0.50 .    0.25 0.25 0.25 0.25 . 0.12##  [2,] . 1 0.50 0.50 0.50 0.75 0.62 0.62 . 0.31##  [3,] . . 0.71 .    0.35 0.35 0.35 0.35 . 0.18##  [4,] . . .    0.87 0.43 .    0.22 0.22 . 0.11##  [5,] . . .    .    0.71 .    0.35 0.35 . 0.18##  [6,] . . .    .    .    0.71 0.35 0.35 . 0.18##  [7,] . . .    .    .    .    0.64 .    . .   ##  [8,] . . .    .    .    .    .    0.64 . 0.32##  [9,] . . .    .    .    .    .    .    1 0.50## [10,] . . .    .    .    .    .    .    . 0.66

## To check
U - chol(A)
round(A, digits=2)## 10 x 10 sparse Matrix of class "dsCMatrix"                                     ##  [1,] 1.00 .    0.50 .    0.25 0.25 0.25 0.25 .   0.12##  [2,] .    1.00 0.50 0.50 0.50 0.75 0.62 0.62 .   0.31##  [3,] 0.50 0.50 1.00 0.25 0.62 0.75 0.69 0.69 .   0.34##  [4,] .    0.50 0.25 1.00 0.62 0.38 0.50 0.50 .   0.25##  [5,] 0.25 0.50 0.62 0.62 1.12 0.56 0.84 0.84 .   0.42##  [6,] 0.25 0.75 0.75 0.38 0.56 1.25 0.91 0.91 .   0.45##  [7,] 0.25 0.62 0.69 0.50 0.84 0.91 1.28 0.88 .   0.44##  [8,] 0.25 0.62 0.69 0.50 0.84 0.91 0.88 1.28 .   0.64##  [9,] .    .    .    .    .    .    .    .    1.0 0.50## [10,] 0.12 0.31 0.34 0.25 0.42 0.45 0.44 0.64 0.5
1.
0
0

N
ote tha
t
pedigreem
m package uses Matrix classes in order to store only what we need to store, e.g., matrix U is triangular (t in "dtCMatrix") and matrix A is symmetric (s in "dsCMatrix"). To show the generalized Cholesky A = LDU (or using Henderson notation A = TDT') we use gchol() from the bdsmatrix R package. Matrix T shows the "flow" of genes in pedigree.

## install.packages(pkgs="bdsmatrix")library(package="bdsmatrix")tmp <- gchol(as.matrix(A))D <- diag(tmp)(T <- as(as.matrix(tmp), "dtCMatrix"))## 10 x 10 sparse Matrix of class "dtCMatrix"                                     ##  [1,] 1.000 .      .    .     .    .    . .   .   .##  [2,] .     1.0000 .    .     .    .    . .   .   .##  [3,] 0.500 0.5000 1.00 .     .    .    . .   .   .##  [4,] .     0.5000 .    1.000 .    .    . .   .   .##  [5,] 0.250 0.5000 0.50 0.500 1.00 .    . .   .   .##  [6,] 0.250 0.7500 0.50 .     .    1.00 . .   .   .##  [7,] 0.250 0.6250 0.50 0.250 0.50 0.50 1 .   .   .##  [8,] 0.250 0.6250 0.50 0.250 0.50 0.50 . 1.0 .   .##  [9,] .     .      .    .     .    .    . .   1.0 .## [10,] 0.125 0.3125 0.25 0.125 0.25 0.25 . 0.5 0.5 1

## To chec
k
L
&
lt;
- T %*% diag(sqrt(D))
L - t(U)
Now the A inverse part (inv(A) = inv(T)' inv(D) inv(T) = inv(T)' inv(S) inv(S) inv(T) using Henderson's notation, note that ). The nice thing is that pedigreemm authors provided functions to get inv(T) and D.

(TInv <- as(ped2, "sparseMatrix"))## 10 x 10 sparse Matrix of class "dtCMatrix" (unitriangular)            ## 1   1.0  .    .    .    .    .   .  .    .   .## 2   .    1.0  .    .    .    .   .  .    .   .## 3  -0.5 -0.5  1.0  .    .    .   .  .    .   .## 4   .   -0.5  .    1.0  .    .   .  .    .   .## 5   .    .   -0.5 -0.5  1.0  .   .  .    .   .## 6   .   -0.5 -0.5  .    .    1.0 .  .    .   .## 7   .    .    .    .   -0.5 -0.5 1  .    .   .## 8   .    .    .    .   -0.5 -0.5 .  1.0  .   .## 9   .    .    .    .    .    .   .  .    1.0 .## 10  .    .    .    .    .    .   . -0.5 -0.5 1round(DInv <- Diagonal(x=1/Dmat(ped2)), digits=2)## 10 x 10 diagonal matrix of class "ddiMatrix"##       [,1] [,2] [,3] [,4] [,5] [,6] [,7] [,8] [,9] [,10]##  [1,]    1    .    .    .    .    .    .    .    .     .##  [2,]    .    1    .    .    .    .    .    .    .     .##  [3,]    .    .    2    .    .    .    .    .    .     .##  [4,]    .    .    . 1.33    .    .    .    .    .     .##  [5,]    .    .    .    .    2    .    .    .    .     .##  [6,]    .    .    .    .    .    2    .    .    .     .##  [7,]    .    .    .    .    .    . 2.46    .    .     .##  [8,]    .    .    .    .    .    .    . 2.46    .     .##  [9,]    .    .    .    .    .    .    .    .    1     .## [10,]    .    .    .    .    .    .    .    .    .  2.33 round(t(TInv) %*% DInv %*% TInv, digits=2)## 10 x 10 sparse Matrix of class "dgCMatrix"## ...round(crossprod(sqrt(DInv) %*% TInv), digits=2)
## 10 x 10 sparse Matrix of class "dsCMatrix"
##  [1,]  1.5  0.50 -1.0  .     .     .     .     .     .     .   ##  [2,]  0.5  2.33 -0.5 -0.67  .    -1.00  .     .     .     .   ##  [3,] -1.0 -0.50  3.0  0.50 -1.00 -1.00  .     .     .     .   ##  [4,]  .   -0.67  0.5  1.83 -1.00  .     .     .     .     .   ##  [5,]  .    .    -1.0 -1.00  3.23  1.23 -1.23 -1.23  .     .   ##  [6,]  .   -1.00 -1.0  .     1.23  3.23 -1.23 -1.23  .     .   ##  [7,]  .    .     .    .    -1.23 -1.23  2.46  .     .     .   ##  [8,]  .    .     .    .    -1.23 -1.23  .     3.04  0.58 -1.16##  [9,]  .    .     .    .     .     .     .     0.58  1.58 -1.16## [10,]  .    .     .    .     .     .     .    -1.16 -1.16
  2
.3
3

#
# T
o c
heck
so
l
ve
(A
) - crossprod(sqrt(DInv) %*% TInv)

The second method (using crossprod) is preferred as it leads directly to symmetric matrix (dsCMatrix), which stores only upper or lower triangle. And make sure you do not do cprod(TInv %*% sqrt(DInv)) as it is the wrong order of matrices.

As promised we will display (plot) pedigree by use of conversion functions of matrix objects to graph objects using the following code. Two examples are provided using the graph and igraph packages. The former does a very good job on this example, but otherwise igraph seems to have much nicer support for editing etc.

## source("http://www.bioconductor.org/biocLite.R")## biocLite(pkgs=c("graph", "Rgraphviz"))library(package="graph")library(package="Rgraphviz")g <- as(t(TInv), "graph")plot(g)

## install.packages(pkgs="igraph")library(package="igraph")i <- igraph.from.graphNEL(graphNEL=g)V(i)$label <- 1:10plot(i, layout=layout.kamada.kawai)## tkplot(i) ## July 02, 2013 ### Gregor Gorjanc #### Parse arguments of an R script R can be used also as a scripting tool. We just need to add shebang in the first line of a file (script): #!/usr/bin/Rscript and then the R code should follow. Often we want to pass arguments to such a script, which can be collected in the script by the commandArgs() function. Then we need to parse the arguments and conditional on them do something. I came with a rather general way of parsing these arguments using simply these few lines: ## Collect argumentsargs <- commandArgs(TRUE) ## Default setting when no arguments passedif(length(args) < 1) { args <- c("--help")} ## Help sectionif("--help" %in% args) { cat(" The R Script Arguments: --arg1=someValue - numeric, blah blah --arg2=someValue - character, blah blah --arg3=someValue - logical, blah blah --help - print this text Example: ./test.R --arg1=1 --arg2="output.txt" --arg3=TRUE \n\n") q(save="no")} ## Parse arguments (we expect the form --arg=value)parseArgs <- function(x) strsplit(sub("^--", "", x), "=")argsDF <- as.data.frame(do.call("rbind", parseArgs(args)))argsL <- as.list(as.character(argsDF$V2))names(argsL) <- argsDF$V1 ## Arg1 defaultif(is.null(args$arg1)) {  ## do something} ## Arg2 defaultif(is.null(args$arg2)) { ## do something} ## Arg3 defaultif(is.null(args$arg3)) {  ## do something}

## ... your code here ...
Created by Pretty R at inside-R.org

It is some work, but I find it pretty neat and use it for quite a while now. I do wonder what others have come up for this task. I hope I did not miss some very general solution.

## March 24, 2013

### Romain Francois

#### Moving

This blog is moving to blog.r-enthusiasts.com. The new one is powered by wordpress and gets a subdomain of r-enthusiasts.com.

See you there

## March 17, 2013

### Modern Toolmaking

#### caretEnsemble Classification example

Here's a quick demo of how to fit a binary classification model with caretEnsemble.  Please note that I haven't spent as much time debugging caretEnsemble for classification models, so there's probably more bugs than my last post.  Also note that multi class models are not yet supported.

Right now, this code fails for me if I try a model like a nnet or an SVM for stacking, so there's clearly bugs to fix.

The greedy model relies 100% on the gbm, which makes sense as the gbm has an AUC of 1 on the training set.  The linear model uses all of the models, and achieves an AUC of .5.  This is a little weird, as the gbm, rf, SVN, and knn all achieve an AUC of close to 1.0 on the training set, and I would have expected the linear model to focus on these predictions. I'm not sure if this is a bug, or a failure of my stacking model.