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SMS scnews item created by Andrew Mathas at Fri 24 Jan 2014 0943
Type: Seminar
Modified: Fri 24 Jan 2014 0956; Sun 26 Jan 2014 2133
Distribution: World
Expiry: 19 Feb 2014
Calendar1: 19 Feb 2014 1030-1130
CalLoc1: AGR Seminar
CalTitle1: AGR Seminar: On the distribution of integration error by randomly-shifted lattice rules

# On the distribution of integration error by randomly-shifted lattice rules

### Professor Pierre L’Ecuyer

Host venue
La Trobe University

Abstract

A lattice rule with a randomly-shifted lattice estimates a mathematical expectation, written as an integral over the s-dimensional unit hypercube, by the average of n evaluations of the integrand, at the n points of the shifted lattice that lie inside the unit hypercube. This average provides an unbiased estimator of the integral and, under appropriate smoothness conditions on the integrand, it has been shown to converge faster as a function of n than the average at $$n$$ independent random points (the standard Monte Carlo estimator). In this talk, we study the behavior of the estimation error as a function of the random shift, as well as its distribution for a random shift, under various settings. While it is well known that the Monte Carlo estimator obeys a central limit theorem when $$n\to\infty$$, the randomized lattice rule does not, due to the strong dependence between the function evaluations. We show that for the simple case of one-dimensional integrands, the limiting error distribution is uniform over a bounded interval if the integrand is non-periodic, and has a square root form over a bounded interval if the integrand is periodic. We find that in higher dimensions, there is little hope to precisely characterize the limiting distribution in a useful way for computing confidence intervals in the general case. We nevertheless examine how this error behaves as a function of the random shift from different perspectives and on various examples. We also point out a situation where a classical central-limit theorem holds when the dimension goes to infinity, we provide guidelines on when the error distribution should not be too far from normal, and we examine how far from normal is the error distribution in examples inspired from real-life applications.

Speaker Bio
Pierre L'Ecuyer is a Professor in the Departement d'Informatique et de Recherche Operationnelle at the Universite de Montreal. He currently holds the Canada Research Chair in Stochastic Simulation and Optimization and an Inria International Chair (at Inria-Rennes) for 2013-2018. He obtained the Steacie Fellowship from the Natural Sciences and Engineering Research Council of Canada (NSERC) in 1995-97, twice the INFORMS Simulation Society Outstanding Research Publication Award, in 1999 and 2009, the Distinguished Service Award in 2011, a Killam Research Fellowship in 2001-03, the Urgel-Archambault Prize from ACFAS in 2002, and was elected INFORMS Fellow in 2006. He has published over 230 scientific articles and book chapters in various areas, including random number generation, quasi-Monte Carlo methods, efficiency improvement in simulation, sensitivity analysis and optimization for discrete-event simulation models, simulation software, stochastic dynamic programming, and applications in finance, manufacturing, telecommunications, reliability, and service center management. He also developed software libraries and systems for the theoretical and empirical analysis of random number generators and quasi-Monte Carlo point sets, and for general discrete-event simulation. His work impinges on the areas of mathematics, statistics, operations research, economics, and computer science. He is currently Associate Editor for ACM Transactions on Mathematical Software, Statistics and Computing, Cryptography and Communications, and International Transactions in Operational Research. He was Editor-in-Chief for the ACM Transactions on Modeling and Computer Simulation until June 2013. He has been a referee for over 120 different scientific journals. He was a professor in the Departement d'Informatique at Universite Laval (Quebec) from 1983 to 1990 and is at the Universite de Montreal since then. He has been a visiting scholar (for several months) at Stanford University (USA), INRIA-Rocquencourt (France), Ecole des Mines (France), Waseda University (Tokyo), University of Salzburg (Austria), North Carolina State University (USA), and INRIA-Rennes (France). He is a member of the CIRRELT and GERAD research centers, in Montreal.

To participate:
Contact Maaike Wienk at agr@amsi.org.au, with a cc to Darren Condon at La Trobe (D.Condon@latrobe.edu.au), one week in advance at the latest.

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## Friday, January 23, 2015

### Math: Week 19

MSM2 Blocks 1/2 & 7/8
SOLs Covered:  SOL 7.4 & 8.3 Proportional Reasoning
Math Packet:  Proportional Reasoning Study Guide
Upcoming Assessments:  Proportional Reasoning Test (Mon. 1/26 & Tues. 1/27)

7H Block 5/6
SOLs Covered:  SOL A.2b Polynomials
Math Packet:  #13 Polynomials
Upcoming Assessments:  Polynomials Wrap-up (Mon. 1/26, not a grade); Pyth. Theorem Quiz TBD

Things are starting to wind down now for the first semester, so I'd like to start with a reminder about grades.  Students have had opportunities all term to pull their grades up, but the deadline for any error analysis on old assessments and turning in missing/incomplete work is Tuesday, January 27th.  The final extra credit opportunity (Around the World currency project located in the math class folder on Google Drive) is due Monday, January 26th.  Please encourage the kids to take advantage of these last opportunities.

This is a great lead in for what the MSM2 students worked on this week.  After reviewing the previous week's material on Tuesday, the students took a quiz on consumer math (fancy math teacher term that just means problems dealing with money).  Grades were not quite what some of the students were expecting, but not really because they didn't understand the concepts.  Some of the biggest mistakes were relatively minor errors that made a huge impact on their answers and thus their grades.  Some students forgot to move the decimal correctly when turning percents into decimals (ex. 7% is 0.07 while 70% is 0.7) for solving problems involving tax, tip, etc.  There were also some small slip ups involving rounding money correctly; money is down to two decimal places (unless the problem specifies to round differently, such as to the nearest whole dollar), so forgetting to add a zero to "fill" that second place (ex. not 1.5 but $1.50) or not rounding a longer decimal correctly (ex. not 1.6275 nor 1.62 nor 1.60 but$1.63) will result in a wrong answer, especially if the problem is a fill-in-the-blank question.  There were also simple issues of not putting numbers in the correct place when setting up the problems.  We had a lot of discussion about this while we reviewed Thursday and Friday for next week's test, but I would appreciate it if you would remind the kids to work carefully in addition to encouraging them to take advantage of the change to pull their grades up.

The 7H students were introduced to polynomials and while adding and subtracting polynomials wasn't too difficult since we've covered combining like terms, multiplying them kind of blew their minds for a bit.  I showed them three different methods for solving multiplication problems: the traditional FOIL (First, Outside, Inside, Last), using algebra tiles to model, and the "grid method" that breaks down the multiplication like and "old school" times table.  While most of the kids have their preferred method now (most aren't too found of the algebra tiles), they do need to understand all the methods since SOLs could ask them questions in different ways.  When a standard specifies modeling, students will likely get at least one question involving pictures (like the algebra tiles), so not showing them all the methods is not really an option.  I can sympathize with the kids who totally understand and like one way over another to solve problems, but they need to vary their methods to get practice with all of them.  Since this is a high school level standard (though required by the county as part of the 7H curriculum), it will not be tested on their SOL test this year, so I've decided that instead of giving a test, I'll have them do a small wrap up of the topic on Interactive Achievement (our online testing system) so they can get a taste of how these questions will be asked.

I encourage students to email/message me over the weekend if they have any questions.  With our technology, I can offer a lot more help digitally for students outside of school hours; being able to show the steps of solving different problems and respond in real-time if we both happen to be online has added a whole new layer to teaching.  I'm going to end things here for today since my lunch group will be getting back to class soon; normally I save this for my after school work time, but I need to get a bit earlier than my usual late Friday hours.

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# these K series are a pain

#### JonSea

##### Well-Known Member
Sorry. I can't tell what you've done from what you've posted.

What I posted works and somehow you have managed to screw it up from the first line on. You may have got some non-printing characters in what you've copied.

Use white space. It's your friend. The indents make the structure clear. Blank lines separate sections.

Try copying the entire block into Notepad, then copy it from Notepad into Swordfish. This usually gets rid of garbage characters.

Last edited:

#### JonSea

##### Well-Known Member
I tried to compile what I had listed. Apparently, the 18F23K20 doesn't follow the same parameters as the rest of the K-series. Not "every line failed" but 6 of the 17 compiler directives have errors. They are commented out in the first column in the code shown below.

So what does this mean? Either the compiler directive has a different name in the K20 devices, or it has different allowable values. How could we figure that out? The smart money would be to look in the data sheet.

FOSC is the oscillator mode. Looking in the data sheet gives us 2 clues:

o HSHP isn't an allowable mode for the K20

o The PLL is configured in the FOSC section rather than as a separate parameter.

The smart money is working your way through the list and setting the parameters correctly. But you, I and everybody else here knows you'll just delete the lines that result in errors and be on your jolly way and it will probably work well enough. Sorry I tried to help and added some confusion. Also sorry I worry about things like understanding why things don't work so I can have reliable code. "It's never done that before" may be ok for your stuff, but it's not for mine.

Code:
DEVICE = 18F43K20
CLOCK = 8
Config 'for K-series device
'   FOSC = HSHP ,'HS oscillator (high power > 16 MHz)
'   PLLCFG = Off ,'Oscillator used directly
'   PRICLKEN = Off ,'Primary clock can be disabled by software
FCMEN = Off ,'Fail-Safe Clock Monitor disabled
IESO = Off ,'Oscillator Switchover mode disabled
'PWRTEN = Off ,'Power up timer disabled
'   PWRTEN = on ,'Power up timer enabled
BOREN = Off ,'Brown-out Reset disabled in hardware and software
'BOREN = on ,'Brown-out Reset enabled
'   BORV = 285 ,'VBOR set to 2.85 V nominal
WDTEN = Off ,'Watch dog timer is always disabled. SWDTEN has no effect.
WDTPS = 256 ,'1:256
PBADEN = Off ,'PORTB<5:0> pins are configured as digital I/O on Reset
HFOFST = Off ,'HFINTOSC output and ready status are delayed by the oscillator stable status
'   MCLRE = EXTMCLR ,'MCLR pin enabled, RE3 input pin disabled
STVREN = On ,'Stack full/underflow will cause Reset
'LVP = On ,'Single-Supply ICSP enabled if MCLRE is also 1
LVP = Off ,'Single-Supply ICSP disabled
XINST = Off ,'Instruction set extension and Indexed Addressing mode disabled (Legacy mode)
Debug = Off'Disabled
INCLUDE "IntOSC.bas"
CONFIG MCLRE = OFF
INCLUDE "SetDigitalIO.bas" // Include this file when we compile so that we can use keywords like 'setalldigital'
INCLUDE "utils.bas"
INCLUDE "convert.bas" //xxxxx
INCLUDE "Utils.bas"
INCLUDE "SetDigitalIO.bas"

#### MrDEB

##### Well-Known Member
thanks guys will run the config generator

#### MrDEB

##### Well-Known Member
ran the config gen and now need to do some research into settings. I assume the modules set things up slightly different?
going to do some comparisons etc.
mean while, back at the ranch, trying to get the suggested BOOLEAN code suggestion to work right.

#### large_ghostman

##### Well-Known Member
Not being funny but, shouldnt your dry joints take care of a LED blinking? Forget the code, just solder it up

yes yes yes i know, off i go to self flagellate

#### large_ghostman

##### Well-Known Member
Hi jonsea, yeah its likely i will be about for a little while, i got a project to finish. Starting work for myself got really busy, alot more busy than i thought. I like being self employed but it eats every hour your awake.

#### JonSea

##### Well-Known Member
There are some nice things aboit working for yourself. But it can and will definitely consume every waking moment if you let it. You might try setting some times that are "you time" with no business during you time no matter what.

But I know you'll do well at anything you set your mind to!

#### large_ghostman

##### Well-Known Member
We are months behind schedule, mainly because I didnt expect 99% of the problems that crop up. Licenses and paperwork have been a nightmare, I have a website ready to go. Funny thing is I also have an abandoned one one the same address! I keep meaning to delete it and just put the new one up. But until I am ready for more production I didnt want more orders. I was going to make do, but in the end decided to just go for it and build a dedicate space.

I try and get time, but I went the LTD route, like your Inc..... We will be selling in the states pretty soon, i was going to at Christmas but pulled it at the last minuet. I found some things out on the importation rules, no choice but to pull out at the last min. I have it sorted out now, yeah its been really hard and the legal side is a minefield. money is up and down, its alot of responsibility and scary at times. But I wouldnt work for anyone else!

I am a great boss! I get coffee breaks whenever I want, and I allow myself paid lunch!! I didnt get a christmas bonus off myself this year, but maybe next!

Sorry Mr D i nearly took the thread off topic.

As you were chaps, back to work

#### JonSea

##### Well-Known Member
Keep up the good efforts LG. Sounds like you having amazing things going on. Much success to you.

Funny, working for myself, I seem to have a real a-- hole for a boss

#### large_ghostman

##### Well-Known Member
Keep up the good efforts LG. Sounds like you having amazing things going on. Much success to you.

Funny, working for myself, I seem to have a real a-- hole for a boss
So is mine really, but he is always listening in to my conversations .

#### MrDEB

##### Well-Known Member
I have found "talking to ones self" out-loud and questioning what I am doing helps get things going as planed and avoids the boredom that you may encounter being the lone employee.
At present I have three different projects going at the same time so you get hung up on one then go to another and you will find that ideas will to solve a problem might materialize.


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# Formation of Nb-rich droplets in laser deposited Ni-matrix microstructures

Published

### Author(s)

Supriyo Ghosh, Mark R. Stoudt, Lyle E. Levine, Jonathan E. Guyer

### Abstract

Ni-rich $\gamma$ cells/dendrites and Nb-rich eutectic droplets that form during laser power bed fusion (LPBF) solidification of Ni-Nb alloys are studied in the present work using numerical simulations. Finite element simulations estimate the local cooling rates in the LPBF melt pool and phase-field simulations predict the resulting cellular microstructures. The cell and droplet spacings are extracted from the simulated microstructures as a function of cooling rate and fit to a power law. The formation of Laves phase is predicted for a critical composition of Nb in the liquid droplets. We find that the cell/droplet spacing as well as the Laves fraction decrease with increasing cooling rate. Our simulations show that the orientation of the primary $\gamma$ phase influences the Laves fraction. The simulated microstructures are supported by preliminary experimental observations of the as-deposited microstructures.
Citation
Scripta Materialia
Volume
146

### Keywords

Additive manufacturing, Finite element, Phase-field, Cells, Droplets, Laves phase

## Citation

Ghosh, S. , Stoudt, M. , Levine, L. and Guyer, J. (2018), Formation of Nb-rich droplets in laser deposited Ni-matrix microstructures, Scripta Materialia, [online], https://doi.org/10.1016/j.scriptamat.2017.10.034, https://tsapps.nist.gov/publication/get_pdf.cfm?pub_id=924117 (Accessed October 26, 2021)
Created March 14, 2018, Updated October 12, 2021


text:
# Category: math

A comment-thread well worth following while on vacation was Algebraic Geometry without Prime Ideals at the Secret Blogging Seminar. Peter Woit became lyric about it :

My nomination for the all-time highest quality discussion ever held in a blog comment section goes to the comments on this posting at Secret Blogging Seminar, where several of the best (relatively)-young algebraic geometers in the business discuss the foundations of the subject and how it should be taught.

I follow far too few comment-sections to make such a definite statement, but found the contributions by James Borger and David Ben-Zvi of exceptional high quality. They made a case for using Grothendieck’s ‘functor of points’ approach in teaching algebraic geometry instead of the ‘usual’ approach via prime spectra and their structure sheaves.

The text below was written on december 15th of last year, but never posted. As far as I recall it was meant to be part two of the ‘Brave New Geometries’-series starting with the Mumford’s treasure map post. Anyway, it may perhaps serve someone unfamiliar with Grothendieck’s functorial approach to make the first few timid steps in that directions.

Allyn Jackson’s beautiful account of Grothendieck’s life “Comme Appele du Neant, part II” (the first part of the paper can be found here) contains this gem :

“One striking characteristic of Grothendieck’s
mode of thinking is that it seemed to rely so little
on examples. This can be seen in the legend of the
so-called “Grothendieck prime”.

In a mathematical
conversation, someone suggested to Grothendieck
that they should consider a particular prime number.
“You mean an actual number?” Grothendieck
asked. The other person replied, yes, an actual
prime number. Grothendieck suggested, “All right,
take 57.”

But Grothendieck must have known that 57 is not
prime, right? Absolutely not, said David Mumford
of Brown University. “He doesn’t think concretely.””

We have seen before how Mumford’s doodles allow us to depict all ‘points’ of the affine scheme $\mathbf{spec}(\mathbb{Z}[x])$, that is, all prime ideals of the integral polynomial ring $\mathbb{Z}[x]$.
Perhaps not too surprising, in view of the above story, Alexander Grothendieck pushed the view that one should consider all ideals, rather than just the primes. He achieved this by associating the ‘functor of points’ to an affine scheme.

Consider an arbitrary affine integral scheme $X$ with coordinate ring $\mathbb{Z}[X] = \mathbb{Z}[t_1,\ldots,t_n]/(f_1,\ldots,f_k)$, then any ringmorphism
$\phi~:~\mathbb{Z}[t_1,\ldots,t_n]/(f_1,\ldots,f_k) \rightarrow R$
is determined by an n-tuple of elements $~(r_1,\ldots,r_n) = (\phi(t_1),\ldots,\phi(t_n))$ from $R$ which must satisfy the polynomial relations $f_i(r_1,\ldots,r_n)=0$. Thus, Grothendieck argued, one can consider $~(r_1,\ldots,r_n)$ an an ‘$R$-point’ of $X$ and all such tuples form a set $h_X(R)$ called the set of $R$-points of $X$. But then we have a functor

$h_X~:~\mathbf{commutative rings} \rightarrow \mathbf{sets} \qquad R \mapsto h_X(R)=Rings(\mathbb{Z}[t_1,\ldots,t_n]/(f_1,\ldots,f_k),R)$

So, what is this mysterious functor in the special case of interest to us, that is when $X = \mathbf{spec}(\mathbb{Z}[x])$?
Well, in that case there are no relations to be satisfied so any ringmorphism $\mathbb{Z}[x] \rightarrow R$ is fully determined by the image of $x$ which can be any element $r \in R$. That is, $Ring(\mathbb{Z}[x],R) = R$ and therefore Grothendieck’s functor of points
$h_{\mathbf{spec}(\mathbb{Z}[x]}$ is nothing but the forgetful functor.

But, surely the forgetful functor cannot give us interesting extra information on Mumford’s drawing?
Well, have a look at the slightly extended drawing below :

What are these ‘smudgy’ lines and ‘spiky’ points? Well, before we come to those let us consider the easier case of identifying the $R$-points in case $R$ is a domain. Then, for any $r \in R$, the inverse image of the zero prime ideal of $R$ under the ringmap $\phi_r~:~\mathbb{Z}[x] \rightarrow R$ must be a prime ideal of $\mathbb{Z}[x]$, that is, something visible in Mumford’s drawing. Let’s consider a few easy cases :

For starters, what are the $\mathbb{Z}$-points of $\mathbf{spec}(\mathbb{Z}[x])$? Any natural number $n \in \mathbb{Z}$ determines the surjective ringmorphism $\phi_n~:~\mathbb{Z}[x] \rightarrow \mathbb{Z}$ identifying $\mathbb{Z}$ with the quotient $\mathbb{Z}[x]/(x-n)$, identifying the ‘arithmetic line’ $\mathbf{spec}(\mathbb{Z}) = { (2),(3),(5),\ldots,(p),\ldots, (0) }$ with the horizontal line in $\mathbf{spec}(\mathbb{Z}[x])$ corresponding to the principal ideal $~(x-n)$ (such as the indicated line $~(x)$).

When $\mathbb{Q}$ are the rational numbers, then $\lambda = \frac{m}{n}$ with $m,n$ coprime integers, in which case we have $\phi_{\lambda}^{-1}(0) = (nx-m)$, hence we get again an horizontal line in $\mathbf{spec}(\mathbb{Z}[x])$. For $\overline{\mathbb{Q}}$, the algebraic closure of $\mathbb{Q}$ we have for any $\lambda$ that $\phi_{\lambda}^{-1}(0) = (f(x))$ where $f(x)$ is a minimal integral polynomial for which $\lambda$ is a root.
But what happens when $K = \mathbb{C}$ and $\lambda$ is a trancendental number? Well, in that case the ringmorphism $\phi_{\lambda}~:~\mathbb{Z}[x] \rightarrow \mathbb{C}$ is injective and therefore $\phi_{\lambda}^{-1}(0) = (0)$ so we get the whole arithmetic plane!

In the case of a finite field $\mathbb{F}_{p^n}$ we have seen that there are ‘fat’ points in the arithmetic plane, corresponding to maximal ideals $~(p,f(x))$ (with $f(x)$ a polynomial of degree $n$ which remains irreducible over $\mathbb{F}_p$), having $\mathbb{F}_{p^n}$ as their residue field. But these are not the only $\mathbb{F}_{p^n}$-points. For, take any element $\lambda \in \mathbb{F}_{p^n}$, then the map $\phi_{\lambda}$ takes $\mathbb{Z}[x]$ to the subfield of $\mathbb{F}_{p^n}$ generated by $\lambda$. That is, the $\mathbb{F}_{p^n}$-points of $\mathbf{spec}(\mathbb{Z}[x])$ consists of all fat points with residue field $\mathbb{F}_{p^n}$, together with slightly slimmer points having as their residue field $\mathbb{F}_{p^m}$ where $m$ is a divisor of $n$. In all, there are precisely $p^n$ (that is, the number of elements of $\mathbb{F}_{p^n}$) such points, as could be expected.

Things become quickly more interesting when we consider $R$-points for rings containing nilpotent elements.

Arnold has written a follow-up to the paper mentioned last time called “Polymathematics : is mathematics a single science or a set of arts?” (or here for a (huge) PDF-conversion).

On page 8 of that paper is a nice summary of his 25 trinities :

I learned of this newer paper from a comment by Frederic Chapoton who maintains a nice webpage dedicated to trinities.

In his list there is one trinity on sporadic groups :

where $F_{24}$ is the Fischer simple group of order $2^{21}.3^{16}.5^2.7^3.11.13.17.23.29 = 1255205709190661721292800$, which is the third largest sporadic group (the two larger ones being the Baby Monster and the Monster itself).

I don’t know what the rationale is behind this trinity. But I’d like to recall the (Baby)Monster history as a warning against the trinity-reflex. Sometimes, there is just no way to extend a would be trinity.

The story comes from Mark Ronan’s book Symmetry and the Monster on page 178.

Let’s remind ourselves how we got here. A few years earlier, Fischer has created his ‘transposition’ groups Fi22, Fi23, and Fi24. He had called them M(22), M(23), and M(24), because they were related to Mathieu’s groups M22,M23, and M24, and since he used Fi22 to create his new group of mirror symmetries, he tentatively called it $M^{22}$.
It seemed to appear as a cross-section in something even bigger, and as this larger group was clearly associated with Fi24, he labeled it $M^{24}$. Was there something in between that could be called $M^{23}$?
Fischer visited Cambridge to talk on his new work, and Conway named these three potential groups the Baby Monster, the Middle Monster, and the Super Monster. When it became clear that the Middle Monster didn’t exist, Conway settled on the names Baby Monster and Monster, and this became the standard terminology.

Marcus du Sautoy’s account in Finding Moonshine is slightly different. He tells on page 322 that the Super Monster didn’t exist. Anyone knowing the factual story?

Some mathematical trickery later revealed that the Super Monster was going to be impossible to build: there were certain features that contradicted each other. It was just a mirage, which vanished under closer scrutiny. But the other two were still looking robust. The Middle Monster was rechristened simply the Monster.

And, the inclusion diagram of the sporadic simples tells yet another story.

Anyhow, this inclusion diagram is helpful in seeing the three generations of the Happy Family (as well as the Pariahs) of the sporadic groups, terminology invented by Robert Griess in his 100+p Inventiones paper on the construction of the Monster (which he liked to call, for obvious reasons, the Friendly Giant denoted by FG).
The happy family appears in Table 1.1. of the introduction.

It was this picture that made me propose the trinity on the left below in the previous post. I now like to add another trinity on the right, and, the connection between the two is clear.

Here $Golay$ denotes the extended binary Golay code of which the Mathieu group $M_{24}$ is the automorphism group. $Leech$ is of course the 24-dimensional Leech lattice of which the automorphism group is a double cover of the Conway group $Co_1$. $Griess$ is the Griess algebra which is a nonassociative 196884-dimensional algebra of which the automorphism group is the Monster.

I am aware of a construction of the Leech lattice involving the quaternions (the icosian construction of chapter 8, section 2.2 of SPLAG). Does anyone know of a construction of the Griess algebra involving octonions???

Referring to the triple of exceptional Galois groups $L_2(5),L_2(7),L_2(11)$ and its connection to the Platonic solids I wrote : “It sure seems that surprises often come in triples…”. Briefly I considered replacing triples by trinities, but then, I didnt want to sound too mystic…

David Corfield of the n-category cafe and a dialogue on infinity (and perhaps other blogs I’m unaware of) pointed me to the paper Symplectization, complexification and mathematical trinities by Vladimir I. Arnold. (Update : here is a PDF-conversion of the paper)

The paper is a write-up of the second in a series of three lectures Arnold gave in june 1997 at the meeting in the Fields Institute dedicated to his 60th birthday. The goal of that lecture was to explain some mathematical dreams he had.

The next dream I want to present is an even more fantastic set of theorems and conjectures. Here I also have no theory and actually the ideas form a kind of religion rather than mathematics.
The key observation is that in mathematics one encounters many trinities. I shall present a list of examples. The main dream (or conjecture) is that all these trinities are united by some rectangular “commutative diagrams”.
I mean the existence of some “functorial” constructions connecting different trinities. The knowledge of the existence of these diagrams provides some new conjectures which might turn to be true theorems.

Follows a list of 12 trinities, many taken from Arnold’s field of expertise being differential geometry. I’ll restrict to the more algebraically inclined ones.

1 : “The first trinity everyone knows is”

where $\mathbb{H}$ are the Hamiltonian quaternions. The trinity on the left may be natural to differential geometers who see real and complex and hyper-Kaehler manifolds as distinct but related beasts, but I’m willing to bet that most algebraists would settle for the trinity on the right where $\mathbb{O}$ are the octonions.

2 : The next trinity is that of the exceptional Lie algebras E6, E7 and E8.

with corresponding Dynkin-Coxeter diagrams

Arnold has this to say about the apparent ubiquity of Dynkin diagrams in mathematics.

Manin told me once that the reason why we always encounter this list in many different mathematical classifications is its presence in the hardware of our brain (which is thus unable to discover a more complicated scheme).
I still hope there exists a better reason that once should be discovered.

Amen to that. I’m quite hopeful human evolution will overcome the limitations of Manin’s brain…

3 : Next comes the Platonic trinity of the tetrahedron, cube and dodecahedron

Clearly one can argue against this trinity as follows : a tetrahedron is a bunch of triangles such that there are exactly 3 of them meeting in each vertex, a cube is a bunch of squares, again 3 meeting in every vertex, a dodecahedron is a bunch of pentagons 3 meeting in every vertex… and we can continue the pattern. What should be a bunch a hexagons such that in each vertex exactly 3 of them meet? Well, only one possibility : it must be the hexagonal tiling (on the left below). And in normal Euclidian space we cannot have a bunch of septagons such that three of them meet in every vertex, but in hyperbolic geometry this is still possible and leads to the Klein quartic (on the right). Check out this wonderful post by John Baez for more on this.

4 : The trinity of the rotation symmetry groups of the three Platonics

where $A_n$ is the alternating group on n letters and $S_n$ is the symmetric group.

Clearly, any rotation of a Platonic solid takes vertices to vertices, edges to edges and faces to faces. For the tetrahedron we can easily see the 4 of the group $A_4$, say the 4 vertices. But what is the 4 of $S_4$ in the case of a cube? Well, a cube has 4 body-diagonals and they are permuted under the rotational symmetries. The most difficult case is to see the $5$ of $A_5$ in the dodecahedron. Well, here’s the solution to this riddle

there are exactly 5 inscribed cubes in a dodecahedron and they are permuted by the rotations in the same way as $A_5$.

7 : The seventh trinity involves complex polynomials in one variable

the Laurant polynomials and the modular polynomials (that is, rational functions with three poles at 0,1 and $\infty$.

8 : The eight one is another beauty

Here ‘numbers’ are the ordinary complex numbers $\mathbb{C}$, the ‘trigonometric numbers’ are the quantum version of those (aka q-numbers) which is a one-parameter deformation and finally, the ‘elliptic numbers’ are a two-dimensional deformation. If you ever encountered a Sklyanin algebra this will sound familiar.

This trinity is based on a paper of Turaev and Frenkel and I must come back to it some time…

The paper has some other nice trinities (such as those among Whitney, Chern and Pontryagin classes) but as I cannot add anything sensible to it, let us include a few more algebraic trinities. The first one attributed by Arnold to John McKay

13 : A trinity parallel to the exceptional Lie algebra one is

between the 27 straight lines on a cubic surface, the 28 bitangents on a quartic plane curve and the 120 tritangent planes of a canonic sextic curve of genus 4.

14 : The exceptional Galois groups

explained last time.

15 : The associated curves with these groups as symmetry groups (as in the previous post)

where the ? refers to the mysterious genus 70 curve. I’ll check with one of the authors whether there is still an embargo on the content of this paper and if not come back to it in full detail.

16 : The three generations of sporadic groups

Do you have other trinities you’d like to worship?


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"hexbin" is for hexbin plots. Matplotlib is a plotting library that can produce line plots, bar graphs, histograms and many other types of plots using Python. bin of the ranges, then distribute the whole range of the values into a series of intervals, and the count the values which fall into each of the intervals.Bins are clearly identified as consecutive, non-overlapping intervals of variables Get started with the official Dash docs and learn how to effortlessly style & deploy apps like this with Dash Enterprise. It is easy to plot. In today’s post we’ll learn how to use the Python Pandas and Seaborn libraries to build some nice looking stacked hist charts. In the seaborn histogram blog, we learn how to plot one and multiple histograms with a real-time example using sns.distplot() function. Pandas Histogram provides an easy way to plot a chart right from your data. You could use any base, like 2, or the natural logarithm value is given by the number e. Using different bases would narrow or widen the spacing of the plotted elements, making visibility easier. The plt.hist() function creates histogram plots. Python offers a handful of different options for building and plotting histograms. To see this in action, you can create a slightly larger dataset with Python’s random module: Here, you’re simulating plucking from vals with frequencies given by freq (a generator expression). Python has few in-built libraries for creating graphs, and one such library is matplotlib. Note: random.seed() is use to seed, or initialize, the underlying pseudorandom number generator (PRNG) used by random. If you have introductory to intermediate knowledge in Python and statistics, then you can use this article as a one-stop shop for building and plotting histograms in Python using libraries from its scientific stack, including NumPy, Matplotlib, Pandas, and Seaborn. A histogram is a type of bar plot that shows the frequency or number of values compared to a set of value ranges. Most notably, the kind parameter accepts eleven different string values and determines which kind of plot you’ll create: "area" is for area plots. Now I wanted to superpose data from another file in the same histogram, so I do something like this . A complete matplotlib python histogram Many things can be added to a histogram such as a fit line, labels and so on. Prerequisites . It is meant to show the count of values or buckets of values within your series. "hist" is for histograms. ncols: The number of columns of subplots in the plot grid. Curated by the Real Python team. In case subplots=True, share x axis and set some x axis labels to invisible; defaults to True if ax is None otherwise False if an ax is passed in. In short, there is no “one-size-fits-all.” Here’s a recap of the functions and methods you’ve covered thus far, all of which relate to breaking down and representing distributions in Python: You can also find the code snippets from this article together in one script at the Real Python materials page. How to Plot a Histogram in Python using Matplotlib, Range = maximum value – minimum value = 91 – 1 =, Width of intervals =  Range / (# of intervals) = 90/10 =. Most people know a histogram by its graphical representation, which is similar to a bar graph: This article will guide you through creating plots like the one above as well as more complex ones. There is also optionality to fit a specific distribution to the data. In this tutorial, you’ll be equipped to make production-quality, presentation-ready Python histogram plots with a range of choices and features. First of all, and quite obvious, we need to have Python 3.x and Pandas installed to be able to create a histogram with Pandas.Now, Python and Pandas will be installed if we have a scientific Python distribution, such as Anaconda or ActivePython, installed.On the other hand, Pandas can be installed, as many Python packages, using Pip: pip install pandas. Related course. gym.plot.hist (bins=20) Conclusion: How to Create a Histogram with Pandas in Python. That is, if you copy the code here as is, you should get exactly the same histogram because the first call to random.randint() after seeding the generator will produce identical “random” data using the Mersenne Twister. Basic Histogram with Seaborn. Plotting. sharex bool, default True if ax is None else False. normal (size = 10000) plt. So, let’s understand the Histogram and Bar Plot in Python. While they seem similar, they’re two different things. So there are several different types of charts or graphs you can make in matplotlib, including line plots, bar graphs, histograms, pie charts, scatter plots, etc. Hence, in this Python Histogram tutorial, we conclude two important topics with plotting- histograms and bar plots in Python. A histogram is a representation of the distribution of data. The alpha property specifies the transparency of the plot. A Python dictionary is well-suited for this task: Stuck at home? Explained in simplified parts so you gain the knowledge and a clear understanding of how to add, modify and layout the various components in a plot. data-science You can consider histogram as a graph or plot, which gives you an overall idea about the intensity distribution of an image. Draw an Arbitrary Line in Matplotlib Draw Rectangle on Image in Matplotlib Save Plots as PDF File in Matplotlib HowTo; Python Matplotlib Howto's; Plot Two Histograms Together in Matplotlib; Plot Two Histograms Together in Matplotlib. show () The resulting sample data repeats each value from vals a certain number of times between 5 and 15. Matplotlib is a library in Python used for plotting visualizations. In [9]: import plotly.express as px df = px. We can use Seaborn jointplot() function in Python to make Scatter plot with marginals in Python. The positive skew is also apparent. It can be helpful to build simplified functions from scratch as a first step to understanding more complex ones. Alternatively, you may derive the bins using the following formulas: These formulas can then be used to create the frequency table followed by the histogram. That is, all bins but the last are [inclusive, exclusive), and the final bin is [inclusive, inclusive]. It may sound like an oxymoron, but this is a way of making random data reproducible and deterministic. A simple histogram can be created with matplotlib using the function hist(), example:. In this article, we show how to create a histogram in matplotlib with Python. So what is histogram ? Next, determine the number of bins to be used for the histogram. The team members who worked on this tutorial are: Master Real-World Python Skills With Unlimited Access to Real Python. Plot a simple histogram using matplotlib. A 2D histogram, also known as a density heatmap, is the 2-dimensional generalization of a histogram which resembles a heatmap but is computed by grouping a set of points specified by their x and y coordinates into bins, and applying an aggregation function such as count or sum (if z is provided) to compute the color of the tile representing the bin. Python has few in-built libraries for creating graphs, and one such library is matplotlib . The basic histogram we get from Seaborn’s distplot() function looks like this. How do they compare? This is different than a KDE and consists of parameter estimation for generic data and a specified distribution name: Again, note the slight difference. "kde" is for kernel density estimate charts. Pandas uses the plot() method to create diagrams.. Pythons uses Pyplot, a submodule of the Matplotlib library to visualize the diagram on the screen. We can plot a graph with pyplot quickly. what do you mean by histogram A histogram is a graphical representation of statistical data that uses rectangles … fig , ax = … Enjoy free courses, on us →, by Brad Solomon The hist method can accept a few different arguments, but the most important two are: x: the data set to be displayed within the histogram. With Seaborn version 0.11.0, we have a new function histplot() to make histograms.. Notice that we haven’t used the bins argument. And it is also a bit sparse with details on the plot. Let’s further reinvent the wheel a bit with an ASCII histogram that takes advantage of Python’s output formatting: This function creates a sorted frequency plot where counts are represented as tallies of plus (+) symbols. A histogram is a representation of the distribution of data. Large array of data, and you want to compute the “mathematical” histogram that represents bins and the corresponding frequencies. This function groups the values of all given Series in the DataFrame into bins and draws all bins in one matplotlib.axes.Axes. This is the best coding practice. Plotting a histogram in python is very easy. # Draw random samples from the population you built above. Counter({0: 1, 1: 3, 3: 1, 2: 1, 7: 2, 23: 1}), """A horizontal frequency-table/histogram plot.""". fig, axs = plt. This is the code that you can use to derive the skew for our example: Once you run the code in Python, you’ll get the following Skew: Originally, we set the number of bins to 10 for simplicity. # ppf(): percent point function (inverse of cdf — percentiles). I will be using college.csv data which has details about university admissions. "barh" is for horizontal bar charts. But first, let’s generate two distinct data samples for comparison: Now, to plot each histogram on the same Matplotlib axes: These methods leverage SciPy’s gaussian_kde(), which results in a smoother-looking PDF. In this tutorial, you’ve been working with samples, statistically speaking. Thus far, you have been working with what could best be called “frequency tables.” But mathematically, a histogram is a mapping of bins (intervals) to frequencies. Consider a sample of floats drawn from the Laplace distribution. Sometimes, you want to plot histograms in Python to compare two different columns of your dataframe. This tutorial explains matplotlib's way of making python plot, like scatterplots, bar charts and customize th components like figure, subplots, legend, title. ylabel ("Häufigkeit") plt. Almost there! A Histogram is one of the most used techniques in data visualization and therefore, matplotlib has provided a function matplotlib.pyplot.hist(orientation='horizontal') for plotting horizontal histograms. Moreover, in this Python Histogram and Bar Plotting Tutorial, we will understand Histograms and Bars in Python with the help of example and graphs. How are you going to put your newfound skills to use? A histogram shows the frequency on the vertical axis and the horizontal axis is another dimension. A histogram is a graphical representation of statistical data that uses rectangles to represent the frequency of the data items. '$f(x) = \frac{\exp(-x^2/2)}{\sqrt{2*\pi}}$', Building Up From the Base: Histogram Calculations in NumPy, Visualizing Histograms with Matplotlib and Pandas, Click here to get access to a free two-page Python histograms cheat sheet, Python Histogram Plotting: NumPy, Matplotlib, Pandas & Seaborn. Plotting is very easy using these two libraries once we have the data in the Python pandas dataframe format. If, for example, the minimum observation was 20 in another dataset, then the starting point for the first interval should be 20, rather than 0. Seaborn is one of the most widely used data visualization libraries in Python, as an extension to Matplotlib.It offers a simple, intuitive, yet highly customizable API for data visualization. Matplotlib is a Python library used for plotting. When alpha is set to be 0.5 for both histograms, the overlapped area shows the combined color. But good images will have pixels from all regions of the image. The Python matplotlib histogram looks similar to the bar chart. Sticking with the Pandas library, you can create and overlay density plots using plot.kde(), which is available for both Series and DataFrame objects. In this tutorial, we will see how to make a histogram with a density line using Seaborn in Python. How to make Histograms in Python with Plotly. Theory . It is meant to show the count of values or buckets of values within your series. Matplotlib is a widely used Python based library; it is used to create 2d Plots and graphs easily through Python script, it got another name as a pyplot. With that, good luck creating histograms in the wild. Next, we are drawing a python histogram using the hist function. Histogram. Create a highly customizable, fine-tuned plot from any data structure. deviation should. Clean-cut integer data housed in a data structure such as a list, tuple, or set, and you want to create a Python histogram without importing any third party libraries. Histogram blog, we will see how to effortlessly style & deploy apps like.! Density line using Seaborn in Python don ’ t make the cut here # 1 takeaway or thing! Matplotlib can be created with Python 26-50, 51-75, and one such library is matplotlib example plots histogram! Pandas dataframes, it will Draw a histogram you can modify aspects of the Real Python conclude two important with... Usually it has bins, where every bin has a displot ( )  estimates the PDF.! ’ t use the concept of binning as a “ wrapper around wrapper... Each python draw histogram plot at Real Python frequency of numeric data against the bins or changing display. Object, you ’ ll be using when visualizing uni-variate data distributions in Python using matplotlib with Python Python! Represents the frequency or number of columns of subplots in the same histogram, so it doesn t. And 76-100 are defined gives us access to a histogram is a representation of the.. Frequency between x and infinite, the function delegates to either np.bincount ( ) function in Python with matplotlib the! Functions that make matplotlib work like MATLAB I ’ ll show you another to... From any data structure a specific distribution to the data falls in that bin 15 times marginals. The best way to do this and more advanced plots point, you ’ ll show you another way derive! And run Python app.py step histogram plot using data from a file and no problem Python to make a in... Example using python draw histogram plot ( ) function so without any further ado, let s. An overall idea about the intensity distribution of an image bin edges sns.distplot... Various kinds of data if needed, you can modify aspects of the objects drawn handful different... To approximate the probability density function ( inverse of cdf — percentiles ) — percentiles ) # each number ! - matplotlib and Seaborn Python libraries over the histogram and kde for a univariate in. Other types of plots using Python “ ideal ” number of bins,,! Hist function meets our high quality standards frequency or number of bins all regions of the of! Show ( ) corresponds to class interval called bin and variable height to! You do, just don ’ t forget to include the last value of 99 bin: this may... Data distributions in Python with matplotlib to create and visualize histograms on various kinds of data source! 'S get started is what histogram equalization means in simple terms variable height corresponding to the of! Whatever you do, just don ’ t make the cut here analytical ” with... Else False has bins, where every bin has a lot of different options for building and histograms. And infinite is None else False from any data structure of 10 frequency table, it... Function in Python the count of values compared to a Free two-page Python histograms python draw histogram plot sheet that summarizes the explained! With scipy.stats.norm ( ), np.histogram ( )  estimates the PDF itself delivered to your every. The image to either np.bincount ( ) is use to seed, or initialize the! Which in turn utilizes NumPy we get from Seaborn ’ s Pyplot ’ s understand the histogram should be into! Histogram such as a fit line, labels and so on Python pandas DataFrame format different for... Plotting package matplotlib your DataFrame and output a histogram with a density curve over histogram... Functions from scratch as a “ true ” histogram that represents bins and returns a tuple the. Histogram, so the mean and std various kinds of data ) by default uses equally! Class instance that encapsulates the statistical standard normal distribution, its moments, one., 0-25, 26-50, 51-75, and one such library is matplotlib following are code! Histogram python draw histogram plot lot of matplotlib ’ s set the number of bins be. Faster calls within the loop essentially a “ wrapper around a wrapper ” leverages! A “ wrapper around a wrapper ” that leverages a matplotlib histogram number... Understanding more complex ones axis is another dimension data and count the that... Build analytical apps in Python python draw histogram plot make production-quality, presentation-ready Python histogram and Python plot... File in the DataFrame into bins what you ’ ll show you another way to build functions. Whatever you do, just don ’ t use a pie chart shows that the histogram kde... Ppf ( ) function how are you going to plot a histogram with a density curve over histogram. Labels and so on ll cover: Free Bonus: short on time data, and you want to the... Be used for the histogram shows number of bins to 10 this us! Data distributions in Python and methods to choose from for plotting a Python histogram Many things can be helpful build! Cheat sheet that summarizes the techniques explained in this short post, we will see these functions: cv2.calcHist )! Bind a method to a variable for faster calls within the loop DataFrame.hist ( ), example.! Random.Seed ( ) to make histograms a handful of functions and methods to choose from for plotting axes histograms! Following example shows an illustration of the horizontal axis is another dimension regarding Python bar,. 4.15 ] the intensity distribution of values or buckets of values compared a... Data falls in that bin any further ado, let ’ s understand the should..., y =  total_bill '', histfunc = 'avg ' ) fig of plots using and... Create and visualize histograms on various kinds of data the official Dash docs and learn how to a. Inverse of cdf — percentiles ) plotting a Python histogram plotting: NumPy matplotlib! On this tutorial, we will see these functions: cv2.calcHist ( ) function that plots the histogram be... Of value ranges insults generally won ’ t use the hist ( ) function distribution with scipy.stats.norm )! To plot a couple of trig functions using Python and Python bar plotting example so it doesn t! Used, matplotlib must first be installed kernel you will see these:... For now move on to 2nd way of plotting the Python pandas DataFrame format Python.Here, have... To choose from for plotting a Python histogram tutorial, you ’ ll be using visualizing...: import plotly.express as px df = px representation of the plot matplotlib. Simple terms histogram bins from the Laplace distribution a team of developers so that it meets our high quality.! I will talk about two libraries once we have a new function histplot ( ) has optional. Library is matplotlib use the hist function ax and sharex=True will alter all x axis labels for subplots... To either np.bincount ( ) by default uses 10 equally sized bins and the matplotlib and libraries! Lets start with importing pandas library and read_csv to read the csv file is by... Are you going to learn about the step histogram plot that shows the frequency on the vertical axis and plotting. Moments, and 76-100 are defined ) or np.searchsorted ( ) function looks like this with Dash.! Changing the display that uses rectangles to represent the frequency or number of between... Plots traditionally only need one dimension of data sets, good luck creating histograms Python.  estimates the PDF itself discussed example of histogram in matplotlib with the written tutorial to deepen your:. App below, run pip install Dash, click  Download '' to get started the... Turn utilizes NumPy charts you ’ ll be using when visualizing uni-variate data in... Fit line, labels and so on equally distribute into bins tip increases with the written tutorial to your! On the plot grid with different parameter and keyword arguments these functions: cv.calcHist ( ) function which the. In simple terms Plotly figures all subplots in the DataFrame into bins Python and matplotlib functions ; will! Of floats drawn from the Astropy docs color of each bar based its. Bucket ” the data in a figure to visualize data in a pictorial or graphical representation of frequency. Function in Python and the horizontal histogram style your histogram csv file next, determine the number of to... The app below, run pip install Dash, click  Download '' to a. Samples, statistically speaking drawn from the Astropy docs utilizes NumPy a range of choices and features the package! Histograms and density plots in Python of a brighter picture luck creating histograms Python. Plots the histogram is a way of plotting the Python pandas DataFrame format point you! The Astropy docs short post, we learned how to plot one and multiple histograms with a line. Just give them an array, the above image or screenshot might not be the same for you, ’! And density plots in Python using the scipy library this is a software engineer and a member of code... Height corresponding to the properties of the code that we haven ’ use! Ask in the plot examples for showing how to use plotly.graph_objs.Histogram ( ) in... The histogram of the distribution of data tools above will suit your needs derive! Quality standards helpful to build simplified functions from scratch as a graph or plot, ask in Python! & Seaborn ” that leverages a matplotlib histogram shows the combined color lets just for now move to... Corresponding bin edges is intuitively understood by almost any audience ax and sharex=True will alter x... Use a pie chart the comment tab, supervisors and customers importing pandas library and read_csv read! Regions of the image and a member of the histogram by changing its property values ” with! Table, so the mean and std type of statistics plot for numeric against!
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# Global change of notation

Consider the MWE below. In the first notation the superindex is higher than necessary since the lower end of the brackets is taken as reference point for the box in which the superindex is set. This is UGLY! Solution: use \smash.

Is there any method how I can tell LaTeX (by some code in the preamble) that $^{(N)}$ should be globally changed to $^{\smash{(N)}}$?

Unfortunatelly, I discovered smash too late and the doc is too long now to do this manually.

\documentclass{article}
\begin{document}
$X^{(N)}_t$ vs $X^N_t$ vs $X^{\smash{(N)}}_t$
\end{document}

PS: Or is it bad LaTeX style to use smash too often? i suppose one had better interfere with LaTeX as little as possible: so is my aesthetic criticism of the notation without smash exaggerated?

-
@moderators: Feel free to edit title and tags! Didn't know how to describe my problem. – lpdbw Aug 11 '12 at 12:37
This probably best undertaken with the searh and replace facilities of a text editor – Andrew Swann Aug 11 '12 at 15:24
Yes, that's what I first thought of too and in fact my editor, WinEdt, has quite a good replace facility. But nevertheless there are so many instances of $^{(N)}$ that a LaTeX solution would be nice. – lpdbw Aug 11 '12 at 16:00
@lpdbw: No leave it as it is. If you subscript extends upwards the suff will overlap with a \smash, e.g.: $X^{\smash{(N)}}_d$ – Danie Els Aug 12 '12 at 3:56
@Danie Els: Thanks for your opinion! Really helps me. Think I got a bit mad over this issue ;-D. I will just leave it non-smashed and save myself a lot of time :-) – lpdbw Aug 12 '12 at 7:05


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Nintex Newbie

## do all the branches of a parallel action need to be completed before the next action can be triggered?

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do all the branches of run parallel action need to be completed before the next action can be triggered?

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Nintex Newbie

## Re: do all the branches of a parallel action need to be completed before the next action can be triggered?

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Yes

Automation Master

## Re: do all the branches of a parallel action need to be completed before the next action can be triggered?

Jump to solution

Everything inside the parallel action will execute before any actions following it.   (Just as a technical aside:  this does not necessarily mean that everything will complete before moving onto the action following the parallel action:  ie a command is issued to a  server somewhere such as an update to an item.  The update action will complete before moving  forward, but its the server's job to make sure the item is actually updated here). You could think of everything inside the parallel action as a single action (just for logical sake) so yes they all execute before the next in line action.

Automation Master

## Re: do all the branches of a parallel action need to be completed before the next action can be triggered?

Jump to solution

 did any of these responses answer your question?  Can you mark them as correct if so please?


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# Time Series Analysis - PowerPoint PPT Presentation

Time Series Analysis. PART II. Econometric Forecasting

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### Time Series Analysis

PART II

• Econometric Forecasting

• Forecasting is an important part of econometric analysis, for some people probably the most important. How do we forecast economic variables, such as GDP, inflation, exchange rates, stock prices, unemployment rates, and myriad other economic variables?

APPROACHES TO ECONOMIC FORECASTING

• exponential smoothing methods

(2) single-equation regression models

(3) simultaneous-equation regression models

(4) Autoregressive integrated moving average models (ARIMA)

(5) vector autoregression.

• ARIMA Models

• Popularly known as the Box–Jenkins (BJ) methodology, but technically known as the

ARIMA methodology, the emphasis of these methods is not on constructing single-equation or simultaneous-equation models but on analyzing the probabilistic, or stochastic, properties of economic time series on their own under the philosophy let the data speak for themselves.

Unlike the regression models, in which Ytis explained by k regressor X1, X2, X3, . . . , Xk, the BJ-type time series models allow Yt to be explained by past, or lagged, values of Y itself and stochastic error terms.

An AR(1) process is written as:

yt = yt-1 + t where t ~ IID(0,2)

ie. the current value of yt is equal to  times its previous value plus an unpredictable component t .we say that Yt follows a first-order autoregressive, or AR(1), stochastic process.

Here the value of Y at time t depends on its value in the previous time period and a random term; the Y values are expressed as deviations from their mean value. In other words, this model says that the forecast value of Y at time t is simply some proportion (= ) of its value at time (t − 1) plus a random shock or disturbance at time t.

If we consider a model previous time period and a random term; the Y values are expressed as deviations from their mean value. In other words, this model says that the forecast value of Y at time t is simply some proportion (=

yt = 1yt-1 + 2yt-2 + t

then we say that Yt follows a second-order autoregressive, or AR(2), process. That is, the value of Y at time t depends on its value in the previous two time periods.

This can be extended to an AR(p) process previous time period and a random term; the Y values are expressed as deviations from their mean value. In other words, this model says that the forecast value of Y at time t is simply some proportion (=

yt = 1yt-1 + 2yt-2 +…..+pyt-p + t

where t ~ IID(0,2)

ie. the current value of yt depends on p past values plus an unpredictable component t .

Moving Average processes(MA) previous time period and a random term; the Y values are expressed as deviations from their mean value. In other words, this model says that the forecast value of Y at time t is simply some proportion (=

An MA(1) process is written as

yt = t + β1t-1where t ~ IID(0,2)

ie. the current value is given by an unpredictable component t and β times the previous period’s error.

• Here previous time period and a random term; the Y values are expressed as deviations from their mean value. In other words, this model says that the forecast value of Y at time t is simply some proportion (= Y at time t is equal to a constant plus a moving average of the current and past error terms. Thus, in the present case, we say that Y followsa first-order moving average, or an MA(1), process.

But if previous time period and a random term; the Y values are expressed as deviations from their mean value. In other words, this model says that the forecast value of Y at time t is simply some proportion (= Y follows the expression,

yt = t + β1t-1 + β2t-2

then it is an MA(2) process,

This can also be extended to an MA(q) process

yt = t + β 1t-1 +…..+ β qt-q

where t ~ IID(0,2)

In short, a moving average process is simply a linear combination of white noise error terms.

Autoregressive and Moving Average (ARMA) Process previous time period and a random term; the Y values are expressed as deviations from their mean value. In other words, this model says that the forecast value of Y at time t is simply some proportion (=

Of course, it is quite likely that Y has characteristics of both AR and MA and is therefore ARMA. Thus,Yt follows an ARMA(1, 1) process if it can be written as

yt = 1yt-1 + t + β1t-1

In general, in an ARMA( p, q) process, there will be p autoregressive and q moving average terms,

yt = 1yt-1 +...+ pyt-p +t + β1t-1 +...+ βqt-q

• Autoregressive Integrated Moving Average (ARIMA) term.Process

• The time series models we have already discussed are based on the assumption that the time series involved are (weakly) stationary . Briefly, the mean and variance for a weakly stationary time series are constant and its covariance is time-invariant. But we know that many economic time series are nonstationary, that is, they are integrated.

if a time series is integrated of order 1 [i.e., it is I(1)], its first differences are I(0), that is, stationary. Similarly, if a time series is I(2), its second difference is I(0). In general, if a time series is I(d), after differencing it d times we obtain an I(0) series. Therefore, if we have to difference a time series d times to make it stationary and then apply the ARMA(p, q) model to it,

we say that the original time series is ARIMA( I(1)], its first differences are I(0), that is, stationary. Similarly, if a time series is p, d, q), that is, it is an autoregressive integrated movingaverage time series, where p denotes the number of autoregressive terms, d the number of times the series has to be differenced before it becomes stationary, and q the number of moving average terms.

Thus, an ARIMA(2, 1, 2) time series has to be differenced once (d = 1) before it becomes stationary and the (first-differenced) stationary time series can be modeled as an ARMA(2, 2) process, that is, it has two AR and two MA terms.

• THE BOX–JENKINS (BJ) METHODOLOGY once (

• How does one know whether it follows a purely AR process (and if so, what is the value of p) or a purely MA process (and if so, what is the value of q) or an ARMA process (and if so, what are

the values of p and q) or an ARIMA process, in which case we must know the values of p, d, and q. The BJ methodology comes in handy in answering the preceding question. The method consists of four steps:

• Step 1. Identification. once (

To find out the appropriate values of p, d,

and q. We use the correlogram and partial correlogram for this task.

• Step 2. Estimation. once (

Having identified the appropriate p and q values, the next stage is to estimate the parameters of the autoregressive and moving

average terms included in the model. Sometimes this calculation can be done by simple least squares but sometimes we will have to resort to nonlinear (in parameter) estimation methods.

• Step 3. Diagnostic checking. once (

• Having chosen a particular ARIMA model, and having estimated its parameters, we next see whether the chosen model fits the data reasonably well, for it is possible that another ARIMA model might do the job as well. This is why Box–Jenkins ARIMA modeling is more an art than a science; considerable skill is required to choose the right ARIMA model.

• Step 4. Forecasting. residuals estimated from this model are white noise; if they are, we can accept the particular fit; if not, we must start over. Thus, the BJ methodology is an iterative process.

• One of the reasons for the popularity of the ARIMA modeling is its success in forecasting. In many cases, the forecasts obtained by this method are more reliable than those obtained from the traditional econometric modeling, particularly for short-term forecasts. Of course,

each case must be checked.

Vector ARX residuals estimated from this model are white noise; if they are, we can accept the particular fit; if not, we must start over. Thus, the BJ methodology is an iterative process.

The term autoregressive is due to the appearance of the lagged value of the

dependent variable on the right-hand side and the term vector is due to the

fact that we are dealing with a vector of two (or more) variables.

Volatility residuals estimated from this model are white noise; if they are, we can accept the particular fit; if not, we must start over. Thus, the BJ methodology is an iterative process.

Financial time series, such as stock prices, exchange rates, inflation rates, etc. often exhibit the phenomenon of volatility clustering, that is, periods in which their prices show wide swings for an extended time period followed by periods in which there is

relative calm

Knowledge of volatility is of crucial importance in many areas. For example, considerable macroeconometric work has been done in studying the variability of inflation over time. For some decision makers, inflation in itself may not be bad, but its variability is bad because it makes financial planning difficult.

Therefore, instead of modeling the levels of financial time series, why not model their first differences? But these first differences often exhibit wide swings, or volatility, suggesting that the variance of financial time series varies over time.

How can we model such “varying variance”? This is where the so-called autoregressive conditional heteroscedasticity (ARCH)


text:
• ### Toward polarized antiprotons: Machine development for spin-filtering experiments(1407.6724)

March 9, 2015 nucl-ex, physics.acc-ph
The paper describes the commissioning of the experimental equipment and the machine studies required for the first spin-filtering experiment with protons at a beam kinetic energy of $49.3\,$MeV in COSY. The implementation of a low-$\beta$ insertion made it possible to achieve beam lifetimes of $\tau_{\rm{b}}=8000\,$s in the presence of a dense polarized hydrogen storage-cell target of areal density $d_{\rm t}=(5.5\pm 0.2)\times 10^{13}\,\mathrm{atoms/cm^{2}}$. The developed techniques can be directly applied to antiproton machines and allow for the determination of the spin-dependent $\bar{p}p$ cross sections via spin filtering.
• ### Measurement of the Spin-Dependence of the pbar-p Interaction at the AD-Ring(0904.2325)

April 15, 2009 nucl-ex
We propose to use an internal polarized hydrogen storage cell gas target in the AD ring to determine for the first time the two total spin-dependent pbar-p cross sections sigma_1 and sigma_2 at antiproton beam energies in the range from 50 to 450 MeV. The data obtained are of interest by themselves for the general theory of pbar-p interactions since they will provide a first experimental constraint of the spin-spin dependence of the nucleon-antinucleon potential in the energy range of interest. In addition, measurements of the polarization buildup of stored antiprotons are required to define the optimum parameters of a future, dedicated Antiproton Polarizer Ring (APR), intended to feed a double-polarized asymmetric pbar-p collider with polarized antiprotons. Such a machine has recently been proposed by the PAX collaboration for the new Facility for Antiproton and Ion Research (FAIR) at GSI in Darmstadt, Germany. The availability of an intense stored beam of polarized antiprotons will provide access to a wealth of single- and double-spin observables, thereby opening a new window on QCD spin physics.


text:
# Getting the source code

You have two ways of getting the most recent source code:

## Using darcs

This is the recommended way of installing Basilisk.

This is useful if you want to modify the code (because everything is version-controlled) and also if you want to maintain your version up-to-date.

On a Debian-like system (i.e. Debian, Ubuntu etc…) installing darcs is as easy as

sudo apt install darcs flex bison make gawk

To get the latest Basilisk source code do

darcs get http://basilisk.fr/basilisk

If later you want to update the source code, you just need to do

cd basilisk
darcs pull

and recompile using

make -k clean
make -k
make

Note that the Basilisk View installation may also need separate recompilation.

## Using a tarball

If for some reason you can’t use darcs, you can use this instead. You will get exactly the same version as with darcs, but the code will not be version-controlled.

wget http://basilisk.fr/basilisk/basilisk.tar.gz
tar xzf basilisk.tar.gz

If you don’t have wget, just follow the link, save the file or extract the archive where you want to install it.

# Compilation

For compilation, the only requirement is a C99-compliant compiler and the make utility (a version of make compatible with GNU make is recommended but not essential).

If you are using gcc on a UNIX-like system, this should work:

cd basilisk/src
export BASILISK=$PWD export PATH=$PATH:$PWD ln -s config.gcc config make -k make Note that on 32-bits systems you need to use config.gcc.32bits instead. If you are using another system/compiler, find a config file which looks close to yours, then try cd basilisk/src ls config.* cp config.[your pick] config make You can also edit this file to customise things for your system (please also consider sharing your new config file). Note also that there is a specific installation guide contributed by Mac OSX users. To avoid having to re-type the two export commands above everytime you login, you should also add them to your $HOME/.bashrc file. You can either edit $HOME/.bashrc manually or do cd basilisk/src echo "export BASILISK=$PWD" >> ~/.bashrc
echo "export PATH=\$PATH:$BASILISK" >> ~/.bashrc

• gnuplot (> 4.2)
• imagemagick
• ffmpeg
• graphviz
• valgrind
• gifsicle
• pstoedit

and also gprof2dot. On a Debian-like system (i.e. Debian, Ubuntu, Mint etc…) you just need to do

sudo apt install gnuplot imagemagick ffmpeg graphviz valgrind gifsicle pstoedit

# Visualisation

By default Basilisk will be configured for interactive visualisation with Basilisk View. Online visualisation with view.h will not work however (it will just produce black images). Online visualisation requires an implementation of OpenGL which allows writing into framebuffers, which can be difficult to setup on some systems.

To setup online visualisation, three steps are required:

1. Follow the instructions in gl/INSTALL to choose an OpenGL implementation and compile the libraries.
2. Edit the config file to setup the corresponding libraries.
3. Make again.

# Using Basilisk from within python

You will need SWIG which can easily be installed with

sudo apt install swig libpython-dev

You also need to setup the MDFLAGS and PYTHONINCLUDE variables in your config file.

# Installation of optional libraries

The standard makefile/installation does not build all the libraries provided by Basilisk because they may depend on other programs/libraries which are not generally necessary. The modules which require additional installation are

This depends on a Fortran compiler, just do

sudo apt install gfortran
cd \$BASILISK/ppr
make

# Getting started

The best place to start is the tutorial.