text:
Volker Roth

#### Abstract

The problem of detecting "atypical objects" or "outliers" is one of the classical topics in (robust) statistics. Recently, it has been proposed to address this problem by means of one-class SVM classifiers. The main conceptual shortcoming of most one-class approaches, however, is that in a strict sense they are unable to detect outliers, since the expected fraction of outliers has to be specified in advance. The method presented in this paper overcomes this problem by relating kernelized one-class classifica- tion to Gaussian density estimation in the induced feature space. Having established this relation, it is possible to identify "atypical objects" by quantifying their deviations from the Gaussian model. For RBF kernels it is shown that the Gaussian model is "rich enough" in the sense that it asymptotically provides an unbiased estimator for the true density. In or- der to overcome the inherent model selection problem, a cross-validated likelihood criterion for selecting all free model parameters is applied.

1 Introduction

A one-class-classifier attempts to find a separating boundary between a data set and the rest of the feature space. A natural application of such a classifier is estimating a contour line of the underlying data density for a certain quantile value. Such contour lines may be used to separate "typical" objects from "atypical" ones. Objects that look "sufficiently atypical" are often considered to be outliers, for which one rejects the hypothesis that they come from the same distribution as the majority of the objects. Thus, a useful application scenario would be to find a boundary which separates the jointly distributed objects from the outliers. Finding such a boundary defines a classification problem in which, however, usually only sufficiently many labeled samples from one class are available. Usually no labeled samples from the outlier class are available at all, and it is even unknown if there are any outliers present.

It is interesting to notice that the approach of directly estimating a boundary, as opposed to first estimating the whole density, follows one of the main ideas in learning theory which states that one should avoid solving a too hard intermediate problem. While this line of rea- soning seems to be appealing from a theoretical point of view, it leads to a severe problem in practical applications: when it comes to detecting outliers, the restriction to estimating only a boundary makes it impossible to derive a formal characterization of outliers with- out prior assumptions on the expected fraction of outliers or even on their distribution. In practice, however, any such prior assumptions can hardly be justified. The fundamental

problem of the one-class approach lies in the fact that outlier detection is a (partially) unsu- pervised task which has been "squeezed" into a classification framework. The missing part of information has been shifted to prior assumptions which can probably only be justified, if the solution of the original problem was known in advance.

This paper aims at overcoming this problem by linking kernel-based one-class classifiers to Gaussian density estimation in the induced feature space. Objects which have an "unex- pected" high Mahalanobis distance to the sample mean are considered as "atypical objects" or outliers. A particular Mahalanobis distance is considered to be unexpected, if it is very unlikely to observe an object that far away from the mean vector in a random sample of a certain size. We will formalize this concept in section 3 by way of fitting linear models in quantile-quantile plots. The main technical ingredient of our method is the one-class kernel Fisher discriminant classifier (OC-KFD), for which the relation to Gaussian density estimation is shown. From the classification side, the OC-KFD-based model inherits the simple complexity control mechanism by using regularization techniques. The explicit re- lation to Gaussian density estimation, on the other hand, makes it possible to formalize the notion of atypical objects by observing deviations from the Gaussian model. It is clear that these deviations will heavily depend on the chosen model parameters. In order to derive an objective characterization of atypical objects it is, thus, necessary to select a suitable model in advance. This model-selection problem is overcome by using a likelihood-based cross-validation framework for inferring the free parameters.

2 Gaussian density estimation and one-class LDA

Let X denote the n d data matrix which contains the n input vectors xi Rd as rows. It has been proposed to estimate a one-class decision boundary by separating the dataset from the origin [12], which effectively coincides with replicating all xi with opposite sign and separating X and -X. Typically, a -SVM classifier with RBF kernel function is used. The parameter bounds the expected number of outliers and must be selected a priori. The method proposed here follows the same idea of separating the data from their negatively replicated counterparts. Instead of a SVM, however, a Kernel Fisher Discriminant (KFD) classifier is used [7, 10]. The latter has the advantage that is is closely related to Gaussian density estimation in the induced feature space. By making this relation explicit, outliers can be identified without specifying the expected fraction of outliers in advance. We start with a linear discriminant analysis (LDA) model, and then kernels will be introduced.

Let X = (X, -X) denote the augmented (2n d) data matrix which also contains the negative samples -xi. Without loss of generality we assume that the sample mean + x i i > 0, so that the sample means of the positive data and the negative data differ: + = -. Let us now assume that our data are realizations of a normally distributed random variable in d dimensions: X Nd(, ). Denoting by Xc the centered data matrix, the estimator for takes the form ^ W = (1/n)Xc Xc.

The LDA solution B maximizes the between-class scatter with B = ++ + W = 1 - - under the constraint on the within-class scatter . Note that in our special case with X = (X, -X) the usual pooled within-class matrix W simply reduces to the above defined W = (1/n)Xc Xc. Denoting by y = (2, . . . , 2, -2, . . . , -2) a 2n-indicator vector for class membership in class "+" or "-", it is well-known (see e.g. [1]) that the LDA solution (up to a scaling factor) can be found by minimizing a least-squares functional: ^ = arg min y -X 2. In [3] a slightly more general form of the problem is described where the above functional is minimized under a constrained on , which in the simplest case amounts to adding a term to the functional. Such a ridge regression model assumes a penalized total covariance of the form T = (1/(2n)) X X + I = (1/n) X X + I. Defining an n-vector of ones y = (1, . . . , 1) , the solution vector ^

reads ^ = (X X + I)-1X y = (X X + I)-1X y. (1)

According to [3], an appropriate scaling factor is defied in terms of the quantity s2 = (1/n) y ^ y = (1/n) y X ^ , which leads us to the correctly scaled LDA vector = s-1(1 - s2)-1/2 ^ that fulfills the normalization condition W = 1 .

One further derives from [3] that the mean vector of X, projected onto the 1-dimensional LDA-subspace has the coordinate value m+ = s(1 - s2)-1/2, and that the Mahalanobis distance from a vector x to the sample mean + is the sum of the squared Euclidean distance in the projected space and an orthogonal distance term:

 D(x, +) = ( x - m                                                x)2 + x T -1x.                              +)2 + D with D = -(1 - s2)(                                  (2)

Note that it is the term D which makes the density estimation model essentially different from OC-classification: while the latter considers only distances in the direction of the projection vector , the true density model additionally takes into account the distances in the orthogonal subspace.

Since the assumption X Nd(, ) is very restrictive, we propose to relax it by assuming that we have found a suitable transformation of our input data : Rd Rp, x (x), such that the transformed data are Gaussian in p dimensions. If the transformation is carried out implicitly by introducing a Mercer kernel k(xi, xj), we arrive at an equivalent problem in terms of the kernel matrix K = and the expansion coefficients :

                                    ^                                         = (K + I)-1y.                                        (3)

From [11] it follows that the mapped vectors can be represented in Rn as (x) = K-1/2k(x), where k(x) denotes the kernel vector k(x) = (k(x, x1), . . . , k(x, xn)) . Finally we derive the following form of the Mahalanobis distances which again consist of the Euclidean distance in the classification subspace plus an orthogonal term:

        D(x, +) = ( k(x) - m                                 k(x))2 + n(x),                                              +)2 - (1 - s2)(                                (4)

where (x) = (x)( + I)-1(x) = k (x)(K + I)-1K-1k(x), m+ = s(1 - s2)-1/2, s2 = (1/n) y ^ y = (1/n) y K ^ , and = s-1(1 - s2)-1/2 ^ .

Equation (4) establishes the desired link between OC-KFD and Gaussian density estima- tion, since for our outlier detection mechanism only Mahalanobis distances are needed. While it seems to be rather complicated to estimate a density by the above procedure, the main benefit over directly estimating the mean and the covariance lies in the inherent com- plexity regulation properties of ridge regression. Such a complexity control mechanism is of particular importance in highly nonlinear kernel models. Moreover, for ridge-regression models it is possible to analytically calculate the effective degrees of freedom, a quantity that will be of particular interest when it comes to detecting outliers.

3 Detecting outliers

Let us assume that the model is completely specified, i.e. both the kernel function k(, ) and the regularization parameter are fixed. The central lemma that helps us to detect outliers can be found in most statistical textbooks:

Lemma 1. Let X be a Gaussian random variable X Nd(, ). Then (X - ) -1(X - ) follows a chi-square (2) distribution on d degrees of freedom.

For the penalized regression models, it might be more appropriate to use the effective de- grees of freedom df instead of d in the above lemma. In the case of one-class LDA with ridge penalties we can easily estimate it as df = trace(X(X X + I)-1X ), [8], which

for a kernel model translates into df = trace(K(K + I)-1). The intuitive interpretation of the quantity df is the following: denoting by V the matrix of eigenvectors of K and by {i}ni=1 the corresponding eigenvalues, the fitted values ^ y read

                        ^                            y = V diag {i = i/(i + )} V y.                                 (5)

It follows that compared to the unpenalized case, where all eigenvectors vi are constantly weighted by 1, the contribution of the i-th eigenvector vi is down-weighted by a factor i/1 = i. If the ordered eigenvalues decrease rapidly, however, the values i are either close to zero or close to one, and df determines the number of terms that are "essentially different" from zero. The same is true for the orthogonal distance term in eq. (4): note that

(x) = k (x)(K + I)-1K-1k(x) = k V diag i = ((i + )i)-1 V k(x). (6)

Compared to the unpenalized case (the contribution of vi is weighted by -2 i ), the contri- bution of vi is down-weighted by the same factor i/-2 = i i .

From lemma 1 we conclude that if the data are well described by a Gaussian model in the kernel feature space, the observed Mahalanobis distances should look like a sample from a 2-distribution with df degrees of freedom. A graphical way to test this hypothesis is to plot the observed quantiles against the theoretical 2 quantiles, which in the ideal case gives a straight line. Such a quantile-quantile plot is constructed as follows: Let (i) denote the observed Mahalanobis distances ordered from lowest to highest, and pi the cumulative proportion before each (i) given by pi = (i - 1/2)/n. Let further zi = F -1pi denote the theoretical quantile at position pi, where F is the cumulative 2-distribution function. The quantile-quantile plot is then obtained by plotting (i) against zi. Deviations from linearity can be formalized by fitting a linear model on the observed quantiles and calculating confidence intervals around the fit. Observations falling outside the confidence interval are then treated as outliers. A potential problem of this approach is that the outliers themselves heavily influence the quantile-quantile fit. In order to overcome this problem, the use of robust fitting procedures has been proposed in the literature, see e.g. [4]. In the experiments below we use an M-estimator with Huber loss function. For estimating confidence intervals around the fit we use the standard formula (see [2, 5])

                    ((i)) = b  (2(zi))-1 (pi(1 - pi))/n,                              (7)

which can be intuitively understood as the product of the slope b and the standard error of the quantiles. A 100(1 - )% envelope around the fit is then defined as (i) z/2((i)) where z/2 is the 1 - (1 - )/2 quantile of the standard normal distribution.

The choice of the confidence level is somewhat arbitrary, and from a conceptual viewpoint one might even argue that the problem of specifying one free parameter (i.e. the expected fraction of outliers) has simply been transferred into the problem of specifying another one. In practice, however, selecting is a much more intuitive procedure than guessing the fraction of outliers. Whereas the latter requires problem-specific prior knowledge which is hardly available in practice, the former depends only on the variance of a linear model fit. Thus, can be specified in a problem independent way.

4 Model selection

In our model the data are first mapped into some feature space, in which then a Gaussian model is fitted. Mahalanobis distances to the mean of this Gaussian are computed by evaluating (4). The feature space mapping is implicitly defined by the kernel function, for which we assume that it is parametrized by a kernel parameter . For selecting all free parameters in (4), we are, thus, left with the problem of selecting = (, ) .

The idea is now to select by maximizing the cross-validated likelihood. From a theoret- ical viewpoint, the cross-validated (CV) likelihood framework is appealing, since in [13]

the CV likelihood selector has been shown to asymptotically perform as well as the opti- mal benchmark selector which characterizes the best possible model (in terms of Kullback- Leibler divergence to the true distribution) contained in the parametric family.

For kernels that map into a space with dimension p > n, however, two problems arise: (i) the subspace spanned by the mapped samples varies with different sample sizes; (ii) not the whole feature space is accessible for vectors in the input space. As a consequence, it is difficult to find a "proper" normalization of the Gaussian density in the induced feature space. We propose to avoid this problem by considering the likelihood in the input space rather than in the feature space, i.e. we are looking for a properly normalized density model p(x|) in Rd such that p(x|) has the same contour lines as the Gaussian model in the feature space: p(xi|) = p(xi|) p((xi)|) = p((xj)|). Denoting by Xn = {xi}ni=1 a sample from p(x) from which the kernel matrix K is built, a natural input space model is

         pn(x|Xn, ) = Z-1 exp{- 1 D(x; X                                p                                       2          n, )}, with Z =      Rd         n(x|Xn, ) dx,            (8) where D(x; Xn, ) denotes the (parametrized) Mahalanobis distances (4) of a Gaussian model in the feature space. Note that this density model in the input space has the same form as our Gaussian model in the feature space, except for the different normalization constant Z. Computing this constant Z requires us to solve a normalization integral over the whole d-dimensional input space. Since in general this integral is not analytically tractable for nonlinear kernel models, we propose to approximate Z by a Monte Carlo sampling method. In our experiments, for instance, the VEGAS algorithm [6], which implements a mixed importance-stratified sampling approach, showed to be a reasonable method for up to 10 input dimensions.

By using the CV likelihood framework we are guaranteed to (asymptotically) perform as well as the best model in the parametrized family. Thus, the question arises whether the family of densities defined by a Gaussian model in a kernel-induced feature space is "rich enough" such that no systematic errors occur. For RBF kernels, the following lemma pro- vides a positive answer to this question. Lemma 2. Let k(x 2 i, xj ) = exp(- xi - xj /). As 0 , pn(x|Xn, ) converges to a Parzen window with vanishing kernel width: p n n(x|Xn, ) 1 (x - x n i=1 i).

A formal proof is omitted due to space limitations. The basic ingredients of the proof are: (i) In the limit 0 the expansion coefficients approach ^ 1/(1 + )1. Thus, ^ y = K ^ 1/(1 + )1 and s2 1/(1 + ). (ii) D(x; , ) C(x) < , if x {x n i}n i=1, and D(x; , ) , else. Finally pn(x|Xn, , ) 1 (x - x n i=1 i).

Note that in the limit 0 a Parzen window becomes an unbiased estimator for any continuous density, which provides an asymptotic justification for our approach: the cross- validated likelihood framework guarantees us to convergence to a model that performs as well as the best model in our model class as n . The latter, however, is "rich enough" in the sense that it contains models which in the limit 0 converge to an unbiased estimator for every continuous p(x). Since contour lines of pn(x) are contour lines of a Gaussian model in the feature space, the Mahalanobis distances are expected to follow a 2 distribution, and atypical objects can be detected by observing the distribution of the empirical Mahalanobis distances as described in the last section.

It remains to show that describing the data as a Gaussian in a kernel-induced feature space is a statistically sound model. This is actually the case, since there exist decay rates for the kernel width such that n grows at a higher rate as the effective degrees of freedom df : Lemma 3. Let k(x 2 i, xj ) = exp(- xi - xj /) and pn(x|Xn, , ) defined by (8). If 1 decays like O(n-1/2), and for fixed 1, the ratio df /n 0 as n .

A formal proof is omitted due to space limitations. The basic ingredients of the proof are: (i) the eigenvalues i of (1/n)K converge to i as n , (ii) the eigenvalue spectrum of

a Gaussian RBF kernel decays at an exponential-quadratic rate: i exp(-i2), (iii) for n sufficiently large, it holds that n 1/[1 + (/n) exp(n-1/2i2)] n1/2-1 log(n/) i=1 (proof by induction, using the fact that ln(n + 1) - ln(n) 1/(n2 + n) which follows from a Taylor expansion of the logarithm) df (n)/n 0.


text:
How do you find all the solutions between 0 and 2π for 4sin^2x-3 = 0?

Aug 17, 2017

$x \in \left\{\frac{\pi}{3} , \frac{2 \pi}{3} , \frac{5 \pi}{3} , \frac{7 \pi}{3}\right\}$

Explanation:

If $4 {\sin}^{2} x - 3 = 0$
then
$\textcolor{w h i t e}{\text{XXX}} {\sin}^{2} x = \frac{3}{4}$
and
$\textcolor{w h i t e}{\text{XXX}} \sin x = \pm \frac{\sqrt{3}}{2}$

This is one of the standard reference angles $= \frac{\pi}{3}$

Between 0 and $2 \pi$, the possibilities for this reference angle are
$\textcolor{w h i t e}{\text{XXX}} \frac{\pi}{3} , \frac{2 \pi}{3} , \frac{5 \pi}{3} , \frac{7 \pi}{3}$

format: plain


text:
<img src="https://d5nxst8fruw4z.cloudfront.net/atrk.gif?account=iA1Pi1a8Dy00ym" style="display:none" height="1" width="1" alt="" />
You are viewing an older version of this Concept. Go to the latest version.

# Simplify Variable Expressions Involving Integer Division

## Divide integers with and without variables to create a new term.

Estimated5 minsto complete
%
Progress
Practice Simplify Variable Expressions Involving Integer Division
Progress
Estimated5 minsto complete
%
Simplify Variable Expressions Involving Integer Division

Have you ever had to do homework while on vacation?

Molly was working on her homework in the ski lodge. She had one more problem to do before she could go back out for another run on the mountain.

Here is the problem.

\begin{align*}\frac{-12x}{-3}\end{align*}

Molly isn't sure how to work with this problem. She knows that it involves integers, variables and division.

This Concept will teach you how to help Molly with her dilemma.

### Guidance

Previously we used variable expressions with addition, subtraction and multiplication. Now we are going to apply division of integers with variable expressions. Remember that a variable expression is a math phrase that uses numbers, variables and operations.

Find the value of this expression \begin{align*}-18x \div (-2)\end{align*}

It may help you to rewrite the problem like this using a fraction bar to divide. Now you can see which values can be divided.

\begin{align*}\frac{-18x}{-2}\end{align*}

Then separate out the integers like this.

\begin{align*}\frac{-18x}{-2} = \frac{-18 \cdot x}{-2} = \frac{-18}{-2} \cdot x\end{align*}

Notice that we can divide the integers. The \begin{align*}x\end{align*} remains alone because there isn’t another \begin{align*}x\end{align*}. We separate it out. Then we divide the integer part and add the \begin{align*}x\end{align*} to the answer.

Since \begin{align*}-18 \div (-2) = 18 \div 2 = 9\end{align*}, we know that \begin{align*}\frac{-18}{-2} \cdot x= 9 \cdot x = 9x\end{align*}

So, the value of the expression is \begin{align*}9x\end{align*}.

Let’s look at another problem where there is a matching variable too.

\begin{align*}-24y \div 2y\end{align*}

Next, we rewrite the expression using a fraction bar.

\begin{align*}\frac{-24y}{2y}\end{align*}

Now, we can separate the terms.

\begin{align*}&\frac{-24}{2} \cdot \frac{y}{y}\\ &-24 \div 2 = -12\end{align*}

\begin{align*}y \div y =1\end{align*} because the \begin{align*}y\end{align*}’s cancel each other out

\begin{align*}-12(1) = -12\end{align*}

The value of the expression is -12.

Now it's time for you to try a few on your own.

#### Example A

\begin{align*}-14a \div -7\end{align*}

Solution:\begin{align*}2a\end{align*}

#### Example B

\begin{align*}28ab \div -7b\end{align*}

Solution:\begin{align*}-4a\end{align*}

#### Example C

\begin{align*}-6x \div -2\end{align*}

Solution:\begin{align*}3x\end{align*}

Here is the original problem once again.

Molly was working on her homework in the ski lodge. She had one more problem to do before she could go back out for another run on the mountain.

Here is the problem.

\begin{align*}\frac{-12x}{-3}\end{align*}

Molly isn't sure how to work with this problem. She knows that it involves integers, variables and division.

Molly is correct.

First, let's complete the division of the numbers. We can follow the rules for dividing integers and divide negative 12 by negative 3. That answer is 4.

Because there isn't another \begin{align*}x\end{align*} to work with, we can simply add that to our quotient.

\begin{align*}4x\end{align*}

### Vocabulary

Quotient
the answer in a division problem.
Undefined
when an integer is divided by 0, the answer of that is undefined.
Variable Expression
a math phrase with numbers, operations and variables

### Guided Practice

Here is one for you to try on your own.

\begin{align*}-18ab \div 9b\end{align*}

First, rewrite the expression using a fraction bar.

\begin{align*}\frac{-18ab}{-9b}\end{align*}

Next, separate out the terms.

\begin{align*}&\frac{-18}{-9} \cdot \frac{ab}{b}\\ &-18 \div -9 = 2\end{align*}

\begin{align*}ab \div b= a\end{align*} Notice that the \begin{align*}b\end{align*}’s cancel, but the a doesn’t. It is left as part of the final expression.

Our final answer is \begin{align*}2a\end{align*}.

### Practice

Directions: Find each quotient with variable expressions.

1. \begin{align*}36t \div (-9)\end{align*}

2. \begin{align*}-56n \div (-7)\end{align*}

3. \begin{align*}-22n \div -11n\end{align*}

4. \begin{align*}-28n \div 7\end{align*}

5. \begin{align*}18xy \div 2x\end{align*}

6. \begin{align*}72t \div (-9t)\end{align*}

7. \begin{align*}48xy \div (-8y)\end{align*}

8. \begin{align*}54xy \div (-9xy)\end{align*}

9. \begin{align*}16a \div (4a)\end{align*}

10. \begin{align*}-16ab \div (-4b)\end{align*}

11. \begin{align*}-99xy \div (-9x)\end{align*}

12. \begin{align*}121a \div (11b)\end{align*}

13. \begin{align*}-144xy \div (-12)\end{align*}

14. \begin{align*}-169y \div (-13x)\end{align*}

15. \begin{align*}-225xy \div (5z)\end{align*}

### Vocabulary Language: English

Quotient

Quotient

The quotient is the result after two amounts have been divided.
Undefined

Undefined

An expression in mathematics is undefined when it has no meaning. Division by zero is undefined.
Variable Expression

Variable Expression

A variable expression is a mathematical phrase that contains at least one variable or unknown quantity.


text:
# Revision history [back]

There is no big surprise here. Elements of QR have distinct monomials $\alpha_1^{d_1}\dots \alpha_n^{d_n}$, where $d_i\in[0,2]$. That is, your polynomials will contain as many as $3^n$ distinct monomials. Btw, by using g^(k.order() - 1) - 1 rather than g^k.order() - g you disallow zero coefficients (that is, you'll get a mapping from $(k^*)^n$ to $E$ rather than from $k^n$). If you use g^k.order() - g, the number of monomials becomes $4^n$.

Anyway, the most direct way to get the required mapping is via companion matrix:

k = GF(4, 'a')
n = 8
R = PolynomialRing(k, ["alpha_%d" % i for i in range(n)])
QR = R.quotient([r^k.order()-r for r in R.gens()])
P = PolynomialRing(k, 'x')
f = P.irreducible_element(n)

v = sum( QR.gen(i)*companion_matrix(f)^i for i in range(n) )

print('Coefficients of v:',v[:,0].T)
print('Coefficients of v^65537:',(v^65537)[:,0].T)

It works quite fast for degrees around powers of 2 (like 65537), but it'll be slow for something random simply because there will be polynomials with about $4^n$ terms.

There is no big surprise here. Elements of QR have distinct monomials $\alpha_1^{d_1}\dots \alpha_n^{d_n}$, where $d_i\in[0,2]$. That is, your polynomials will contain as many as $3^n$ distinct monomials. Btw, by using g^(k.order() - 1) - 1 rather than g^k.order() - g you disallow zero coefficients (that is, you'll get a mapping from $(k^*)^n$ to $E$ rather than from $k^n$). If you use g^k.order() - g, the number of monomials becomes $4^n$.

Anyway, the most direct way to get the required mapping is via companion matrix:

k = GF(4, 'a')
n = 8
R = PolynomialRing(k, ["alpha_%d" % i for i in range(n)])
QR = R.quotient([r^k.order()-r for r in R.gens()])
P = PolynomialRing(k, 'x')
f = P.irreducible_element(n)

v = sum( QR.gen(i)*companion_matrix(f)^i for i in range(n) )

print('Coefficients of v:',v[:,0].T)
v:', v[:,0].T)
print('Coefficients of v^65537:',(v^65537)[:,0].T)
v^65537:', (v^65537)[:,0].T)

It works quite fast for degrees exponents around powers of 2 (like 65537), but it'll be slow for something random simply because there will be polynomials with about $4^n$ terms.

There is no big surprise here. Elements of QR have distinct monomials $\alpha_1^{d_1}\dots \alpha_n^{d_n}$, where $d_i\in[0,2]$. That is, your polynomials will contain as many as $3^n$ distinct monomials. Btw, by using g^(k.order() - 1) - 1 rather than g^k.order() - g you disallow zero coefficients (that is, you'll get a mapping from $(k^*)^n$ to $E$ rather than from $k^n$). If you use g^k.order() - g, the number of monomials becomes $4^n$.

Anyway, the most direct way to get reduce an overhead, the required mapping is can be obtained via companion matrix: as follows:

k = GF(4, 'a')
n = 8
R = PolynomialRing(k, ["alpha_%d" % i for i in range(n)])
QR = R.quotient([r^k.order()-r for r in R.gens()])
P = PolynomialRing(k, 'x')
f = P.irreducible_element(n)

v = sum( QR.gen(i)*companion_matrix(f)^i for i in range(n) )

print('Coefficients of v:', v[:,0].T)
print('Coefficients of v^65537:', (v^65537)[:,0].T)

It works quite fast for exponents around powers of 2 (like 65537), but it'll be slow for something random simply because there will be polynomials with about $4^n$ terms.


text:
## Calculus (3rd Edition)

$f(x)=5^{x}$ and $a=0$
Compare the difference quotient $$\frac{5^{h}-1}{h}$$ with $$\frac{f(a+h)-f(a)}{h}$$ Then we get $f(x)=5^{x}$ and $a=0$.


text:
FACTOID # 25: If you're tired of sitting in traffic on your way to work, move to North Dakota.

 Home Encyclopedia Statistics States A-Z Flags Maps FAQ About

 WHAT'S NEW

SEARCH ALL

Search encyclopedia, statistics and forums:

(* = Graphable)

Encyclopedia > W. Edwards Deming
W. Edwards Deming
Born October 14, 1900
Sioux City, Iowa, USA
Died December 20, 1993 (aged 93)
Washington DC, USA
Occupation Statistician

Ford Motor Company was simultaneously manufacturing a car model with transmissions made in Japan and the United States. Soon after the car model was on the market, Ford customers were requesting the model with Japanese transmission over the USA-made transmission, and they were willing to wait for the Japanese model. As both transmissions were made to the same specifications, Ford engineers could not understand the customer preference for the model with Japanese transmission. It delivered smoother performance with a lower defect rate. Finally, Ford engineers decided to take apart the two different transmissions. The American-made car parts were all within specified tolerance levels. On the other hand, the Japanese car parts had much closer tolerances than the USA-made parts - i.e. if a part was supposed to be one foot long, plus or minus 1/8 of an inch - then the Japanese parts were within 1/16 of an inch. This made the Japanese cars run more smoothly and customers experienced fewer problems. This is an example of Dr. Deming's teachings, having been adopted by the Japanese, delivering better quality products [3].

Deming was the author of Out of the Crisis (1982–1986) and The New Economics for Industry, Government, Education (1993), which includes his System of Profound Knowledge™ and the 14 Points for Management (described below). Deming played flute & drums and composed music throughout his life, including sacred choral compositions and an arrangement of The Star Spangled Banner.[4]

In 1993, Deming founded the W. Edwards Deming Institute in Washington, D.C., where the Deming Collection at the U.S. Library of Congress includes an extensive audiotape and videotape archive. The aim of the W. Edwards Deming Institute is to foster understanding of The Deming System of Profound Knowledge™ to advance commerce, prosperity and peace.[5] For other uses, see Washington, D.C. (disambiguation). ... Construction of the Thomas Jefferson Building, from July 8, 1888 to May 15, 1894. ...

## Early life and work

In 1927, Deming was introduced to Walter A. Shewhart of the Bell Telephone Laboratories by Dr. C.H. Kunsman of the United States Department of Agriculture (USDA). Deming found great inspiration in the work of Shewhart, the originator of the concepts of statistical control of processes and the related technical tool of the control chart, as Deming began to move toward the application of statistical methods to industrial production and management. Shewhart's idea of common and special causes of variation led directly to Deming's theory of management. Deming saw that these ideas could be applied not only to manufacturing processes but also to the processes by which enterprises are led and managed. This key insight made possible his enormous influence on the economics of the industrialized world after 1950.[7] Walter Andrew Shewhart (March 18, 1891 - March 11, 1967) was a physicist, engineer and statistician, sometimes known as the father of statistical quality control. ... Bell Telephone Laboratories or Bell Labs was originally the research and development arm of the United States Bell System, and was the premier corporate facility of its type, developing a range of revolutionary technologies from telephone switches to specialized coverings for telephone cables, to the transistor. ... USDA redirects here. ... In statistical process control, the control chart, also known as the Shewhart chart or process-behaviour chart is a tool to determine whether a manufacturing or business process is in a state of statistical control or not. ...

Deming edited a series of lectures delivered by Shewhart at USDA, Statistical Method from the Viewpoint of Quality Control, into a book published in 1939. One reason he learned so much from Shewhart, Deming remarked in a videotaped interview, was that, while brilliant, Shewhart had an "uncanny ability to make things difficult." Deming thus spent a great deal of time both copying Shewhart's ideas and devising ways to present them with his own twist.[8]

Deming developed the sampling techniques that were used for the first time during the 1940 U.S. Census. During World War II, Deming was a member of the five-man Emergency Technical Committee. He worked with H.F. Dodge, A.G. Ashcroft, Leslie E. Simon, R.E. Wareham, and John Gaillard in the compilation of the American War Standards (American Standards Association ZI.1-3 published in 1942)[9] and taught statistical process control (SPC) techniques to workers engaged in wartime production. Statistical methods were widely applied during World War II, but faded into disuse a few years later in the face of huge overseas demand for American mass-produced products. The United States Census Bureau (officially Bureau of the Census as defined in Title ) is a part of the United States Department of Commerce. ... The American National Standards Institute (ANSI) is a private, non-profit United States. ... Statistical process control (SPC) is a method for achieving quality control in manufacturing processes. ...

### Work in Japan

In 1947, Deming was involved in early planning for the 1951 Japanese Census. The Allied powers were occupying Japan, and he was asked by the U.S. United States Department of the Army to assist with the census. While Deming was there, his expertise in quality control techniques, combined with his involvement in Japanese society, led to his receiving an invitation from the Japanese Union of Scientists and Engineers (JUSE).[6] This article is about the independent states that comprised the Allies. ... Capital Tokyo Language(s) Japanese Political structure Military occupation Military Governor  - 1945-1951 Douglas MacArthur  - 1951-1952 Matthew Ridgway Emperor  - 1926-1989 Hirohito Historical era Post-WWII  - Surrender of Japan August 15, 1945  - San Francisco Treaty April 28, 1952 At the end of the Second World War, Japan was occupied... Seal The Department of the Army is one of the three military departments in the United States Department of Defense. ...

JUSE members had studied Shewhart's techniques, and as part of Japan's reconstruction efforts, they sought an expert to teach statistical control. During June–August 1950, Deming trained hundreds of engineers, managers, and scholars in statistical process control (SPC) and concepts of quality. He also conducted at least one session for top management.[10] Deming's message to Japan's chief executives: improving quality will reduce expenses while increasing productivity and market share.[1] Perhaps the best known of these management lectures was delivered at the Mt. Hakone Conference Center in August of 1950. Statistical process control (SPC) is a method for achieving quality control in manufacturing processes. ... For the Talib Kweli album Quality (album) Quality can refer to a. ... --158. ...

A number of Japanese manufacturers applied his techniques widely and experienced theretofore unheard of levels of quality and productivity. The improved quality combined with the lowered cost created new international demand for Japanese products.

Deming declined to receive royalties from the transcripts of his 1950 lectures, so JUSE's board of directors established the Deming Prize (December 1950) to repay him for his friendship and kindness.[10] The Deming Prize—especially the Deming Application Prize, which is given to companies—has exerted an immeasurable influence directly or indirectly on the development of quality control and quality management in Japan.[11][12] The Deming prize, established in December 1950 in honor of W. Edwards Deming, was originally designed to reward Japanese companies for major advances in quality improvement. ...

### Honors

In 1960, the Prime Minister of Japan (Nobusuke Kishi), acting on behalf of Emperor Hirohito, awarded Dr. Deming Japan’s Order of the Sacred Treasure, Second Class.[13] The citation on the medal recognizes Deming's contributions to Japan’s industrial rebirth and its worldwide success. The first section of the meritorious service record describes his work in Japan:[10] Nobusuke Kishi Nobusuke Kishi (å²¸ ä¿¡ä»‹ Kishi Nobusuke, November 13, 1896â€“August 7, 1987) was a Japanese politician and the 56th and 57th Prime Minister of Japan from February 25, 1957 to June 12, 1958 and from then to July 19, 1960. ... Hirohito (裕仁), the Shōwa Emperor (昭和天皇), (April 29, 1901 - January 7, 1989) reigned over Japan from 1926 to 1989. ... The Order of the Sacred Treasures ) is a Japanese Order (decoration), established on January 4, 1888 by Emperor Meiji of Japan. ...

The second half of the record lists his service to private enterprise through the introduction of epochal ideas, such as quality control and market survey techniques. Supreme Commander of the Allied Powers (SCAP) was the title for Douglas MacArthur during the Occupation of Japan following WWII. The title did belong to Dwight David Eisenhower during WWII, however, he had nothing to do with the attacks on Japan. ...

### Later work in the U.S.

David Salsburg wrote: David Salsburg is the author of The Lady Tasting Tea, subtitled How Statistics Revolutionized Science in the Twentieth Century. ...

"He was known for his kindness to and consideration for those he worked with, for his robust, if very subtle, humor, and for his interest in music. He sang in a choir, played drums and flute, and published several original pieces of sacred music." (page 254, The Lady Tasting Tea)[14]

Ford Motor Company was one of the first American corporations to seek help from Deming. In 1981, Ford recruited Deming to help jump-start its quality movement. Ford's sales were falling. Between 1979 and 1982, Ford had incurred \$3 billion in losses. Deming questioned the company's culture and the way its managers operated. To Ford's surprise, Deming talked not about quality but about management. He told Ford that management actions were responsible for 85% of all problems in developing better cars. After 1982, Ford came out with a profitable line of cars, the Taurus-Sable line. In a letter to Autoweek Magazine, Donald Petersen, then Ford Chairman, said, "We are moving toward building a quality culture at Ford and the many changes that have been taking place here have their roots directly in Dr. Deming's teachings."[15] By 1986, Ford had become the most profitable American auto company. For the first time since the 1920s, its earnings had exceeded those of arch rival General Motors (GM). Ford had come to lead the American automobile industry in improvements. Ford's following years' earnings confirmed that its success was not a fluke, for its earnings continued to exceed GM and Chrysler's. â€œFordâ€ redirects here. ... Donald E. Petersen (born 1926) is an American businessman, who worked at the Ford Motor Company for forty years, and served as its Chief Executive Officer from 1985 to 1989. ... General Motors Corporation, also known as GM, is a multinational corporation headquartered in the United States and has been the worlds most dominant automaker since 1931. ...

In 1982, Dr. Deming, as author, had his book published by the MIT Center for Advanced Engineering as Quality, Productivity, and Competitive Position, which was renamed Out of the Crisis in 1986. Deming offers a theory of management based on his famous 14 Points for Management. Management's failure to plan for the future brings about loss of market, which brings about loss of jobs. Management must be judged not only by the quarterly dividend, but by innovative plans to stay in business, protect investment, ensure future dividends, and provide more jobs through improved products and services. "Long-term commitment to new learning and new philosophy is required of any management that seeks transformation. The timid and the fainthearted, and the people that expect quick results, are doomed to disappointment." â€œMITâ€ redirects here. ...

Over the course of his career, Deming received dozens of academic awards, including another, honorary, Ph.D. from Oregon State University. In 1987 he was awarded the National Medal of Technology: "For his forceful promotion of statistical methodology, for his contributions to sampling theory, and for his advocacy to corporations and nations of a general management philosophy that has resulted in improved product quality." In 1988, he received the Distinguished Career in Science award from the National Academy of Sciences.[6] Oregon State University (OSU) is a four-year research and degree-granting public university, located in Corvallis, Oregon (USA). ... The National Medal of Technology is an honor granted by the President of the United States to inventors and innovators that have made significant contributions to the development of new and important technology. ... President Harding and the National Academy of Sciences at the White House, Washington, DC, April 1921 The National Academy of Sciences (NAS) is a corporation in the United States whose members serve pro bono as advisers to the nation on science, engineering, and medicine. ...

In 1993, Dr. Deming published his final book, The New Economics for Industry, Government, Education, which included the System of Profound Knowledge™ and the 14 Points for Management. It also contained educational concepts involving group-based teaching without grades, as well as management without individual merit or performance reviews.

In December 1993, W. Edwards Deming died in his sleep at his Washington home at about 3 a.m. due to "natural causes." His family was by his side when he died.[16]

## Deming philosophy synopsis

The philosophy of W. Edwards Deming has been summarized as follows:

"Dr. W. Edwards Deming taught that by adopting appropriate principles of management, organizations can increase quality and simultaneously reduce costs (by reducing waste, rework, staff attrition and litigation while increasing customer loyalty). The key is to practice continual improvement and think of manufacturing as a system, not as bits and pieces."[17]

In the 1970s, Dr. Deming's philosophy was summarized by some of his Japanese proponents with the following 'a'-versus-'b' comparison:

(a) When people and organizations focus primarily on quality, defined by the following ratio,
$Quality = frac{Results of work efforts}{Total costs}$
quality tends to increase and costs fall over time.
(b) However, when people and organizations focus primarily on costs (often dominant/typical human behavior), costs (due to not minimizing waste, ignoring amount of rework occurring, taking staff for granted, not rapidly resolving disputes, and failing to notice lack of product improvement—plus, over time, loss of customer loyalty) tend to rise and quality declines over time.

### The Deming System of Profound Knowledge™

"The prevailing style of management must undergo transformation. A system cannot understand itself. The transformation requires a view from outside. The aim of this chapter is to provide an outside view—a lens—that I call a system of profound knowledge. It provides a map of theory by which to understand the organizations that we work in.

"The first step is transformation of the individual. This transformation is discontinuous. It comes from understanding of the system of profound knowledge. The individual, transformed, will perceive new meaning to his life, to events, to numbers, to interactions between people.

"Once the individual understands the system of profound knowledge, he will apply its principles in every kind of relationship with other people. He will have a basis for judgment of his own decisions and for transformation of the organizations that he belongs to. The individual, once transformed, will:

• Set an example;
• Be a good listener, but will not compromise;
• Continually teach other people; and
• Help people to pull away from their current practices and beliefs and move into the new philosophy without a feeling of guilt about the past."

Deming advocated that all managers need to have what he called a System of Profound Knowledge, consisting of four parts:

1. Appreciation of a system: understanding the overall processes involving suppliers, producers, and customers (or recipients) of goods and services (explained below);
2. Knowledge of variation: the range and causes of variation in quality, and use of statistical sampling in measurements;
3. Theory of knowledge: the concepts explaining knowledge and the limits of what can be known (see also: epistemology);
4. Knowledge of psychology: concepts of human nature.

Deming explained, "One need not be eminent in any part nor in all four parts in order to understand it and to apply it. The 14 points for management in industry, education, and government follow naturally as application of this outside knowledge, for transformation from the present style of Western management to one of optimization." Theory of knowledge redirects here: for other uses, see theory of knowledge (disambiguation) According to Plato, knowledge is a subset of that which is both true and believed Epistemology or theory of knowledge is the branch of philosophy that studies the nature, methods, limitations, and validity of knowledge and belief. ...

"The various segments of the system of profound knowledge proposed here cannot be separated. They interact with each other. Thus, knowledge of psychology is incomplete without knowledge of variation.

"A manager of people needs to understand that all people are different. This is not ranking people. He needs to understand that the performance of anyone is governed largely by the system that he works in, the responsibility of management. A psychologist that possesses even a crude understanding of variation as will be learned in the experiment with the Red Beads (Ch. 7) could no longer participate in refinement of a plan for ranking people."[18]

The Appreciation of a system involves understanding how interactions (i.e. feedback) between the elements of a system can result in internal restrictions that force the system to behave as a single organism that automatically seeks a steady state. It is this steady state that determines the output of the system rather than the individual elements. Thus it is the structure of the organization rather than the employees, alone, which holds the key to improving the quality of output. HELLO EVERYONE!! Steady state is a more general situation than Dynamic equilibrium. ...

The Knowledge of variation involves understanding that everything measured consists of both "normal" variation due to the flexibility of the system and of "special causes" that create defects. Quality involves recognizing the difference in order to eliminate "special causes" while controlling normal variation. Deming taught that making changes in response to "normal" variation would only make the system perform worse. Understanding variation includes the mathematical certainty that variation will normally occur within six standard deviations (thus six sigma: the symbol for standard deviation) of the mean. In probability and statistics, the standard deviation of a probability distribution, random variable, or population or multiset of values is a measure of the spread of its values. ... The often-used six sigma symbol. ...

The System of Profound Knowledge™ is the basis for application of Deming's famous 14 Points for Management, described below.

### Deming's 14 points

Deming offered fourteen key principles for management for transforming business effectiveness. In summary: For other uses, see Management (disambiguation). ...

1. Create constancy of purpose toward improvement of a product and service with a plan to become competitive and stay in business. Decide to whom top management is responsible.
2. Adopt the new philosophy. We are in a new economic age. We can no longer live with commonly accepted levels of delays, mistakes, defective materials, and defective workmanship.
3. Cease dependence on mass inspection. Require, instead, statistical evidence that quality is built in. (prevent defects instead of detect defects.)
4. End of the practice of awarding business on the basis of price tag. Instead, depend on meaningful measures of quality along with price. Eliminate suppliers that cannot qualify with statistical evidence of quality.
5. Find Problems. It is a management’s job to work continually on the system (design, incoming materials, composition of material, maintenance, improvement of machine, training, supervision, retraining)
6. Institute modern methods of training on the job
7. The responsibility of the foreman must be to change from sheer numbers to quality… [which] will automatically improve productivity. Management must prepare to take immediate action on reports from the foremen concerning barriers such as inherent defects, machines not maintained, poor tools, and fuzzy operational definitions.
8. Drive out fear, so that everyone may work effectively for the company.
9. Break down barriers between departments. People in research, design, sales and production must work as a team to foresee problems of production that may be encountered with various materials and specifications.
10. Eliminate numerical goals, posters, slogans for the workforce, asking for new levels of productivity without providing methods.
11. Eliminate work standards that prescribe numerical quotas.
12. Remove barriers that stand between the hourly worker and his right of pride of workmanship.
13. Institute a vigorous program of education and retraining.
14. Create a structure in top management that will push every day on the above 13pts.

1. Lack of constancy of purpose.
2. Emphasis on short-term profits.
3. Evaluation by performance, merit rating, or annual review of performance.
4. Mobility of management.
5. Running a company on visible figures alone.
6. Excessive medical costs.
7. Excessive costs of warranty, fueled by lawyers who work for contingency fees.

A Lesser Category of Obstacles:

1. Neglecting long-range planning.
2. Relying on technology to solve problems.
3. Seeking examples to follow rather than developing solutions.
4. Excuses, such as "Our problems are different."

## Quotations and concepts

In his later years, Dr. Deming taught many concepts, which he emphasized by key sayings or quotations that he repeated. A number of these quotes have been recorded.[19] Some of the concepts might seem to be oxymorons or contradictory to each other; however, the student is given each concept to ponder its meaning in the whole system, over time. Some phrases below might appear repetitious, but consider not only the phrase, but also the situation when applied: Look up oxymoron in Wiktionary, the free dictionary. ...

• "There is no substitute for knowledge." This statement emphasizes the need to know more, about everything in the system. It is considered as a contrast to the old statement, "There is no substitute for hard work" by Thomas Alva Edison (1847–1931). Instead, a small amount of knowledge could save many hours of hard work, just as knowing a password could save hours trying to remember or getting the password reauthorized or reset.
• "The most important things cannot be measured." The issues that are most important, long term, cannot be measured in advance. However, they might be among the factors that an organization is measuring, just not understood as most important at the time.
• "The most important things are unknown or unknowable." The factors that have the greatest impact, long term, can be quite surprising. Analogous to an earthquake that disrupts service, other "earth-shattering" events that most affect an organization will be unknown or unknowable, in advance. Other examples of important things would be: a drastic change in technology, or new investment capital.
• "Experience by itself teaches nothing."[19] This statement emphasizes the need to interpret and apply information against a theory or framework of concepts that is the basis for knowledge about a system. It is considered as a contrast to the old statement, "Experience is the best teacher" (Dr. Deming disagreed with that). To Dr. Deming, knowledge is best taught by a master who explains the overall system through which experience is judged; experience, without understanding the underlying system, is just raw data that can be misinterpreted against a flawed theory of reality. Deming's view of experience is related to Shewhart's concept, "Data has no meaning apart from its context" (see Walter A. Shewhart, "Later Work").
• "By what method?... Only the method counts."[19] When information is obtained, or data is measured, the method, or process used to gather information, greatly affects the results. For example, the "Hawthorne effect" showed that people just asking frequently for opinions seemed to affect the resulting outcome, since some people felt better just being asked for their opinion. Dr. Deming warned that basing judgments on customer complaints alone ignored the general population of other opinions, which should be judged together, such as in a statistical sample of the whole, not just isolated complaints: survey the entire group about their likes and dislikes. The extreme complaints might not represent the attitudes of the whole group. Similarly, measuring or counting data depends on the instrument or method used.
• "You can expect what you inspect." Dr. Deming emphasized the importance of measuring and testing to predict typical results. If a phase consists of inputs + process + outputs, all 3 are inspected to some extent. Problems with inputs are a major source of trouble, but the process using those inputs can also have problems. By inspecting the inputs and the process more, the outputs can be better predicted, and inspected less. Rather than use mass inspection of every output product, the output can be statistically sampled in a cause-effect relationship through the process.
• "Special Causes and Common Causes": Dr. Deming considered anomalies in quality to be variations outside the control limits of a process. Such variations could be attributed to one-time events called "special causes" or to repeated events called "common causes" that hinder quality.
• Acceptable Defects: Rather than waste efforts on zero-defect goals, Dr. Deming stressed the importance of establishing a level of variation, or anomalies, acceptable to the recipient (or customer) in the next phase of a process. Often, some defects are quite acceptable, and efforts to remove all defects would be an excessive waste of time and money.
• The Deming Cycle (or Shewhart Cycle): As a repetitive process to determine the next action, the Deming Cycle describes a simple method to test information before making a major decision. The 4 steps in the Deming Cycle are: Plan-Do-Check-Act (PDCA), also known as Plan-Do-Study-Act or PDSA. Dr. Deming called the cycle the Shewhart Cycle, after Walter A. Shewhart. The cycle can be used in various ways, such as running an experiment: PLAN (design) the experiment; DO the experiment by performing the steps; CHECK the results by testing information; and ACT on the decisions based on those results.
• Semi-Automated, not Fully Automated: Dr. Deming lamented the problem of automation gone awry ("robots painting robots"): instead, he advocated human-assisted semi-automation, which allows people to change the semi-automated or computer-assisted processes, based on new knowledge. Compare to Japanese term 'jidoka' (which can be loosely translated as "automation with a human touch").
• "The problem is at the top; management is the problem." [18] Dr. Deming emphasized that the top-level management had to change to produce significant differences, in a long-term, continuous manner. As a consultant, Deming would offer advice to top-level managers, if asked repeatedly, in a continuous manner.
• "What is a system? A system is a network of interdependent components that work together to try to accomplish the aim of the system. A system must have an aim. Without an aim, there is no system. The aim of the system must be clear to everyone in the system. The aim must include plans for the future. The aim is a value judgment. (We are of course talking here about a man-made system.)" [18]
• "A system must be managed. It will not manage itself. Left to themselves in the Western world, components become selfish, competitive. We can not afford the destructive effect of competition." [18]
• "To successfully respond to the myriad of changes that shake the world, transformation into a new style of management is required. The route to take is what I call profound knowledge—knowledge for leadership of transformation." [18]
• "The worker is not the problem. The problem is at the top! Management!" [20] Management’s job. It is management’s job to direct the efforts of all components toward the aim of the system. The first step is clarification: everyone in the organization must understand the aim of the system, and how to direct his efforts toward it. Everyone must understand the damage and loss to the whole organization from a team that seeks to become a selfish, independent, profit centre." [18]
• "They realized that the gains that you get by statistical methods are gains that you get without new machinery, without new people. Anybody can produce quality if he lowers his production rate. That is not what I am talking about. Statistical thinking and statistical methods are to Japanese production workers, foremen, and all the way through the company, a second language. In statistical control, you have a reproducible product hour after hour, day after day. And see how comforting that is to management, they now know what they can produce, they know what their costs are going to be." [21]
• "I think that people here expect miracles. American management thinks that they can just copy from Japan—but they don't know what to copy!" [21]
• "What is the variation trying to tell us about a process, about the people in the process?" [18] Dr. Shewhart created the basis for the control chart and the concept of a state of statistical control by carefully designed experiments. While Dr. Shewhart drew from pure mathematical statistical theories, he understood that data from physical processes never produce a "normal distribution curve" (a Gaussian distribution, also commonly referred to as a "bell curve"). He discovered that observed variation in manufacturing data did not always behave the same way as data in nature (Brownian motion of particles). Dr. Shewhart concluded that while every process displays variation, some processes display controlled variation that is natural to the process, while others display uncontrolled variation that is not present in the process causal system at all times.[22] Dr. Deming renamed these distinctions "common cause" for chance causes and "special cause" for assignable causes. He did this so the focus would be placed on those responsible for doing something about the variation, rather than the source of the variation. It is top management’s responsibility to address "common cause" variation, and therefore it is management’s responsibility to make improvements to the whole system. Because "special cause" variation is assignable, workers, supervisors or middle managers that have direct knowledge of the assignable cause best address this type of specific intervention.[7]
• (Deming on Quality Circles) "That's all window dressing. That's not fundamental. That's not getting at change and the transformation that must take place. Sure we have to solve problems. Certainly stamp out the fire. Stamp out the fire and get nowhere. Stamp out the fires puts us back to where we were in the first place. Taking action on the basis of results without theory of knowledge, without theory of variation, without knowledge about a system. Anything goes wrong, do something about it, overreacting; acting without knowledge, the effect is to make things worse. With the best of intentions and best efforts, managing by results is, in effect, exactly the same, as Dr. Myron Tribus put it, while driving your automobile, keeping your eye on the rear view mirror, what would happen? And that's what management by results is, keeping your eye on results." [2]
• "Knowledge is theory. We should be thankful if action of management is based on theory. Knowledge has temporal spread. Information is not knowledge. The world is drowning in information but is slow in acquisition of knowledge. There is no substitute for knowledge." [18] This statement emphasizes the need for theory of knowledge (see: epistemology, Shewhart cycle, C. I. Lewis).
• "Experience by itself teaches nothing." "Without theory, experience has no meaning. Without theory, one has no questions to ask. Hence, without theory, there is no learning."[18] These statements emphasize the need to interpret information using a theory or framework of concepts for learning to take place, theory of knowledge. Deming's view of experience is related to Shewhart's concept, "Data has no meaning apart from its context" (see Walter A. Shewhart, "Later work").
• "The most important figures that one needs for management are unknown or unknowable (Lloyd S. Nelson, director of statistical methods for the Nashua corporation), but successful management must nevertheless take account of them." [23] Deming realized that many important things that must be managed couldn’t be measured. Both points are important. One, not everything of importance to management can be measured. And two, you must still manage those important things. Spend \$20,000 training 10 people in a special skill. What's the benefit? "You'll never know," answered Deming. "You'll never be able to measure it. Why did you do it? Because you believed it would pay off. Theory." Dr. Deming is often incorrectly quoted as saying, "You can't manage what you can't measure." In fact, he stated that one of the seven deadly diseases of management is running a company on visible figures alone.
• "By what method?" [19] When information is obtained, or data is measured, the method, or process used to gather information, affects the results. Dr. Deming warned that basing judgments on customer complaints alone ignored the general population of other opinions, which should be judged together, such as in a statistical sample of the whole (Sampling (statistics)). Changing the method changes the results. Aim and method are essential. An aim without a method is useless. A method without an aim is dangerous. It leads to action without direction and without constancy of purpose. Deming used an illustration of washing a table to teach a lesson about the relationship between purpose and method. If you tell someone to wash a table, but not the reason for washing it, they cannot do the job properly (will the table be used for chopping food or potting plants?). That does not mean just giving the explanation without an operational definition. The information about why the table needs to be washed, and what is to be done with it, makes it possible to do the job intelligently.

Edison redirects here. ... Walter Andrew Shewhart (March 18, 1891 - March 11, 1967) was a physicist, engineer and statistician, sometimes known as the father of statistical quality control. ... PDCA (aka the Deming Cycle, Shewhart cycle, or Deming Wheel) is an iterative four-step quality control strategy. ... Walter Andrew Shewhart (March 18, 1891 - March 11, 1967) was a physicist, engineer and statistician, sometimes known as the father of statistical quality control. ... In statistical process control, the control chart, also known as the Shewhart chart or process-behaviour chart is a tool to determine whether a manufacturing or business process is in a state of statistical control or not. ... The normal distribution, also called the Gaussian distribution, is an important family of continuous probability distributions, applicable in many fields. ... Probability density function of Gaussian distribution (bell curve). ... The graph of the probability density function of the normal distribution is sometimes called the bell curve or the bell-shaped curve; see normal distribution. ... Three different views of Brownian motion, with 32 steps, 256 steps, and 2048 steps denoted by progressively lighter colors. ... A quality circle is a volunteer group composed of workers who meet together to discuss workplace improvement, and make presentations to management with their ideas. ... Doctor Myron Tribus was at the origin or promoted many concepts with often strong and sometimes less successful results. ... Theory of knowledge redirects here: for other uses, see theory of knowledge (disambiguation) According to Plato, knowledge is a subset of that which is both true and believed Epistemology or theory of knowledge is the branch of philosophy that studies the nature, methods, limitations, and validity of knowledge and belief. ... It has been suggested that PDCA be merged into this article or section. ... Clarence Irving Lewis (April 12, 1883 _ February 3, 1964) was a pragmatist philosopher. ... Walter Andrew Shewhart (March 18, 1891 - March 11, 1967) was a physicist, engineer and statistician, sometimes known as the father of statistical quality control. ... Sampling is that part of statistical practice concerned with the selection of individual observations intended to yield some knowledge about a population of concern, especially for the purposes of statistical inference. ... An operational definition is a showing of somethingâ€”such as a variable, term, or objectâ€”in terms of the specific process or set of validation tests used to determine its presence and quantity. ...

Walter Andrew Shewhart (March 18, 1891 - March 11, 1967) was a physicist, engineer and statistician, sometimes known as the father of statistical quality control. ... Special cause Common- and special-causes are the two distinct origins of variation, in a process that features in the statistical thinking and methods of Walter A. Shewhart and W. Edwards Deming. ... In Some Theory of Sampling (1950, Chapter 7), W. Edwards Deming introduced concepts he labeled Analytic and enumerative statistical studies. ... Statistical process control (SPC) is a method for achieving quality control in manufacturing processes. ... In statistical process control, the control chart, also known as the Shewhart chart or process-behaviour chart is a tool to determine whether a manufacturing or business process is in a state of statistical control or not. ... It has been suggested that PDCA be merged into this article or section. ... Joseph Moses Juran (b. ... Continuous improvement is a phrase suggesting that a process or product should always get better as knowledge about it and experience with it accumulates over time. ... A Quality storyboard is a visual method for displaying a Quality Control story (QC story). ... The Toyota Production System (TPS) (ãƒˆãƒ¨ã‚¿ç”Ÿç”£æ–¹å¼) is the philosophy which organizes manufacturing and logistics at Toyota, including the interaction with suppliers and customers. ... Doctor Myron Tribus was at the origin or promoted many concepts with often strong and sometimes less successful results. ... Alfie Kohn is an American lecturer and author in the fields of education, psychology and parenting, residing in Belmont, Massachusetts. ... The Deming prize, established in December 1950 in honor of W. Edwards Deming, was originally designed to reward Japanese companies for major advances in quality improvement. ...

## Notes

1. ^ a b Deming's 1950 Lecture to Japanese Management. Translation by Teruhide Haga. Accessed: 2006-06-16.
2. ^ a b The Deming of America Petty Consulting/Productions 1991, Documentary broadcast on the PBS network
3. ^ (Reference : Dr. Deming by Rafael Aguayo - page 40 & 41)
4. ^ The Man: His Music. W. Edwards Deming Institute. Accessed: 2006-06-16.
5. ^ Institute History. W. Edwards Deming Institute. Accessed: 2006-06-16.
6. ^ a b c The Man: Biography W. Edwards Deming Institute. Accessed: 2006-06-17.
7. ^ a b A Brief History of Dr. W. Edwards Deming British Deming Association SPC Press, Inc. 1992
8. ^ The Man: Articles: "The Three Careers of W. Edwards Deming." W. Edwards Deming Institute. Accessed: 2006-06-17.
9. ^ Editor's Preface Elementary Principles of Statistical Control Quality The Union of Japanese Scientists and Engineers (transcript of Deming's 1950 lectures in Japan)
10. ^ a b c Junji Noguchi, "The Legacy of W. Edwards Deming" Quality Progress October 1995
11. ^ The Deming Prize and the Development of Quality Control Management in Japan. Union of Japanese Scientists and Engineers. Accessed: 2006-06-18.
12. ^ How was the Deming Prize Established? Union of Japanese Scientists and Engineers. Accessed: 2006-06-18.
13. ^ L'Harmattan web site (in French)
14. ^ Deming and his statistical methods are profiled in Chapter 24 of The Lady Tasting Tea by David Salsburg
15. ^ Ford Embraces Six-Sigma Quality Goals. Accessed: 2006-07-31.
16. ^ Quality Control Pioneer W. Edwards Deming Dead at 93 Knight Ridder/Tribune Business News; 1993-12-20.
17. ^ Dr. Deming's Management Training. Accessed: 2006-06-18.
18. ^ a b c d e f g h i Deming, W. Edwards. 1993. The New Economics for Industry, Government, Education, second edition.
19. ^ a b c d W. Edwards Deming Institute, webpage: The Man: Articles: "Four Days with W. Edwards Deming.", Accessed: 2006-06-18.
20. ^ Cultural Transformation Discussion Guide. The Deming Library. Accessed 2006-06-18.
21. ^ a b If Japan Can...Why Can't We (white paper), broadcast by NBC in 1980.
22. ^ "Why SPC?," British Deming Association SPC Press, Inc., 1992
23. ^ Deming, W. Edwards Out of the Crisis

Not to be confused with Public Broadcasting Services in Malta. ... The Lady Tasting Tea is a book by David Salsburg about the history of modern statistics and the role it played in the development of science and industry. ... David Salsburg is the author of The Lady Tasting Tea, subtitled How Statistics Revolutionized Science in the Twentieth Century. ... A white paper is an authoritative report. ... This article is about the television network. ...

## Bibliography

• Aguayo, Rafael (1991). Dr. Deming: The American Who Taught the Japanese About Quality. Fireside edition. ISBN 0-671-74621-9.

• Baker, Edward Martin (1999). Scoring a Whole in One: People in Enterprise Playing in Concert. Crisp Learning. ISBN 1-56052-549-5.
• Delavigne Kenneth T. and J. Daniel Robertson, "Deming's Profound Changes: When Will the Sleeping Giant Awaken?" (PTR Prentice Hall, 1994), ISBN 0-13-292690-3

• Deming, W. Edwards (1986). Out of the Crisis. MIT Press. ISBN 0-911379-01-0.

• Deming, W. Edwards (2000). The New Economics for Industry, Government, Education - 2nd Edition. MIT Press. ISBN 0-262-54116-5.

• Deming, W. Edwards (1966). Some Theory of Sampling. Dover Publications. ISBN 0-486-64684-X.

• Gabor, Andrea (1992). The Man Who Discovered Quality: How W. Edwards Deming Brought the Quality Revolution to America. Penguin. ISBN 0-14-016528-2.
• Gitlow, Howard S., Shelly J. Gitlow, "The Deming Guide to Quality and Competitive Position" Prentice Hall Trade (January 1987) ISBN 0-13-198441-1
• Perry Gluckman, Diana Reynolds Roome, "Everyday Heroes: From Taylor to Deming: The Journey to Higher Productivity" SPC Press, Inc. (March 1990) ISBN 0-945320-07-8

• Haller, Harold S. (1993). Managing with profound knowledge: A management process based on the Deming management theory. Harold S. Haller & Company. OCLC 40764811.

The Online Computer Library Center (OCLC) was founded in 1967 and originally named the Ohio College Library Center. ...

• Joiner, Brian L. (1994). Fourth Generation Management: The New Business Consciousness. McGraw-Hill. ISBN 0-07-032715-7.

• Kilian, Cecelia S. (1992). The World of W. Edwards Deming - 2nd Edition. SPC Press, Inc. ISBN 0-945320-29-9.

• Kohn, Alfie (1992). No Contest: The Case Against Competition; Revised edition. Mariner Books. ISBN 0-395-63125-4.

• Kohn, Alfie (1999). Punished By Rewards: The Trouble with Gold Stars, Incentive Plans, A's, Praise, and Other Bribes. Mariner Books. ISBN 0-618-00181-6.
• William J. Latzko, David M. Saunders, "Four Days with Dr. Deming: A Strategy for Modern Methods of Management" Prentice Hall PTR (January 26, 1995) ISBN 0-201-63366-3
• Langley, Gerald J., Kevin M. Nolan, Clifford L. Norman, Lloyd P. Provost, Thomas W. Nolan, "The Improvement Guide: A Practical Approach to Enhancing Organizational Performance" Jossey-Bass (July 26, 1996) ISBN 0-7879-0257-8

• Mann, Nancy (1989). Keys to Excellence: The Story of the Deming Philosophy - 3rd Edition. Prestwick Books. ISBN 1-85251-097-8.

• Neave, Henry R. (1990). The Deming Dimension. SPC Press, Inc. ISBN 0-945320-08-6.

• Scherkenbach, William W. (1991). Demings Road to Continual Improvement. SPC Press, Inc. ISBN 0-945320-10-8.

• Scholtes, Peter R. (1997). The Leader's Handbook: Making Things Happen, Getting Things Done. McGraw-Hill. ISBN 0-07-058028-6.

• Shewhart, Walter A. (1939). Statistical Method from the Viewpoint of Quality Control. Dover Publications December 1, 1986. ISBN 0-486-65232-7.

• Shewhart, Walter A. (1930). Economic Control of Quality of Manufactured Product/50th Anniversary Commemorative Issue. American Society for Quality December 1980. ISBN 0-87389-076-0.

• Tribus, Myron (1992). Quality First: Selected Papers on Quality and Productivity Improvement -4th Edition. National Society of Professional Engineers. ISBN 99938-53-23-2.

• Walton, Mary (1986). The Deming Management Method. The Putnam Publishing Group. ISBN 0-399-55000-3.

• Wheeler, Donald J. (1999). Understanding Variation: The Key to Managing Chaos - 2nd Edition. SPC Press, Inc. ISBN 0-945320-53-1.

Share your thoughts, questions and commentary here


text:
org.apache.commons.math3.ode.nonstiff

## Class DormandPrince853Integrator

• All Implemented Interfaces:
FirstOrderIntegrator, ODEIntegrator

public class DormandPrince853Integrator
extends EmbeddedRungeKuttaIntegrator
This class implements the 8(5,3) Dormand-Prince integrator for Ordinary Differential Equations.

This integrator is an embedded Runge-Kutta integrator of order 8(5,3) used in local extrapolation mode (i.e. the solution is computed using the high order formula) with stepsize control (and automatic step initialization) and continuous output. This method uses 12 functions evaluations per step for integration and 4 evaluations for interpolation. However, since the first interpolation evaluation is the same as the first integration evaluation of the next step, we have included it in the integrator rather than in the interpolator and specified the method was an fsal. Hence, despite we have 13 stages here, the cost is really 12 evaluations per step even if no interpolation is done, and the overcost of interpolation is only 3 evaluations.

This method is based on an 8(6) method by Dormand and Prince (i.e. order 8 for the integration and order 6 for error estimation) modified by Hairer and Wanner to use a 5th order error estimator with 3rd order correction. This modification was introduced because the original method failed in some cases (wrong steps can be accepted when step size is too large, for example in the Brusselator problem) and also had severe difficulties when applied to problems with discontinuities. This modification is explained in the second edition of the first volume (Nonstiff Problems) of the reference book by Hairer, Norsett and Wanner: Solving Ordinary Differential Equations (Springer-Verlag, ISBN 3-540-56670-8).

Since:
1.2
• ### Constructor Detail

• #### DormandPrince853Integrator

public DormandPrince853Integrator(double minStep,
double maxStep,
double scalAbsoluteTolerance,
double scalRelativeTolerance)
Simple constructor. Build an eighth order Dormand-Prince integrator with the given step bounds
Parameters:
minStep - minimal step (sign is irrelevant, regardless of integration direction, forward or backward), the last step can be smaller than this
maxStep - maximal step (sign is irrelevant, regardless of integration direction, forward or backward), the last step can be smaller than this
scalAbsoluteTolerance - allowed absolute error
scalRelativeTolerance - allowed relative error
• #### DormandPrince853Integrator

public DormandPrince853Integrator(double minStep,
double maxStep,
double[] vecAbsoluteTolerance,
double[] vecRelativeTolerance)
Simple constructor. Build an eighth order Dormand-Prince integrator with the given step bounds
Parameters:
minStep - minimal step (sign is irrelevant, regardless of integration direction, forward or backward), the last step can be smaller than this
maxStep - maximal step (sign is irrelevant, regardless of integration direction, forward or backward), the last step can be smaller than this
vecAbsoluteTolerance - allowed absolute error
vecRelativeTolerance - allowed relative error
• ### Method Detail

• #### estimateError

protected double estimateError(double[][] yDotK,
double[] y0,
double[] y1,
double h)
Compute the error ratio.
Specified by:
estimateError in class EmbeddedRungeKuttaIntegrator
Parameters:
yDotK - derivatives computed during the first stages
y0 - estimate of the step at the start of the step
y1 - estimate of the step at the end of the step
h - current step
Returns:
error ratio, greater than 1 if step should be rejected


text:
# Fitting when data is a density

I have fluorescence data that looks like this:

There is strong theoretical support for representing it as coming from a mixture of five normals. I was wondering how to specify such a model in Stan. This is what I had in mind:

model {
cum_value = p[1]*Phi((mu[1]-x[i])/s[1])+...+p[5]*Phi((mu[5]-x[i])/s[5])
}

Obviously this won’t work as is since cum_value is data, but that’s the gist of the model I’d like to estimate. I’m particularly interested in the mixture proportions.

EDIT: it probably makes more sense to use the pdf as the expectation for a positive-support random variable to capture the measurement error. I’ll be trying that for now.

Can you tell us a bit more about the data generating process that gave rise to the data in the figure?

We excite a solution of a certain protein with a UV laser and measure the intensity of light it emits back over a range of wavelengths. It is known that the components of the protein that emit light do so at one of five specific wavelengths (and approx normally around that value) but we don’t know a priori the proportions of these “emitters” for our particular protein.

The proportions actually also vary depending on the conformational state of the protein, which means we have a mixture over the five-proportion mixture, but just having the idea of what the average proportions are would be fine for now.

If you’re modeling the process as a linear combination of normal distributions, can’t you just normalize your data so that it actually gives the density of emissions, and estimate the five parameters giving the coefficients of this linear combination (with the constraint that they sum to 1)?
Maybe that could be formulated as a multivariate normal with a diagonal covariance matrix, since that will be normalized to unity without additional constraints - and the \mathbf{\mu} vector components would give you the relative intensities, I’m not sure about that, but check, maybe.

There’s a chapter in the manual that explains how to code up multi-component mixture models in Stan. We don’t have convenient built-ins beyond two components, but it’s not that hard to code directly.

So this is more like a curve fitting than a mixture model, but conveniently it’ll have the mixture model problems I guess haha.

So you have this process that generates a curve. Presumably we’re assuming this is deterministic?

Like the ideal version of the thing you plotted is

f(\vec{\alpha}, \vec{\text{loc}}, \vec{\text{scale}}) = \sum_i^N \alpha_i e^{-(x - \text{loc}_i)^2 / \text{scale}_i}

and for whatever reason this is a sum of squared exponential-looking terms. All you need to do is decide on a measurement process and you’re ready for a Stan model.

For continuous, the standard thing is to just fit a normal if you don’t have a reason to think anything else.

So the simplest curve fit you might try is:

parameters {
vector[5] alpha;
vector[5] loc;
vector<lower=0.0>[5] scale;
real<lower=0.0> sigma;
}
model {
y ~ normal(f(alpha, loc, scale), sigma);
}

So it’s easy enough to write that down, but you’re probably going to have a bunch of problems with this model. I’ll list em’ in order of what I think is most importance, but they’re all pretty rough.

1. The model I wrote down has 16 parameters. I do not think the curve you plotted will identify 16 parameters reliably. Just my opinion – I didn’t actually try it, but that curve looks really simple. There are usually a deceptively large number of ways to put quadratic exponentials together and fit curves. Sampling will help you find some of them, but probably not all of them, and the diagnostics will be sad.

2. You say there is strong evidence to support that this data is the sum of a bunch of squared exponentials. I believe that maybe it is a good fit, but your data looks really clean. When your data doesn’t have much noise, it is important that you have the right model. Bayesian posteriors are only reliable when you have a good model, but if you have really clean data it raises the bar on what a good enough model is. If you don’t have a good model then the posterior is going to tell you more about model mispecification than anything reliable about the parameters your system. It’s all about that generative process.

3. There’s lots of problems with the posteriors of things that are mixed together. For instance, re-orderings of the means. You can put ordering constraints on the means but it’ll still be rough.

Since I answered so negatively, here’s a similar question from awhile ago: Mixture of Gaussian functions . You can have a look at that and see if there’s anything useful.

I probably made the outlook sound really bleak, haha :P, but feel free to ask more questions. I could be super wrong. Just give things a try and see what works and what doesn’t, and post questions if you have em’! Hope that helps at all!

Gaussian mixture models (in one or more dimension) are a general non-parametric density estimation technique as the number of components grows. Andrew et al. discuss it in BDA.

But the @bmfazio’s problem is not density estimation but curve fittting as @bbbales2 worte.

I also assume it’s still going tobe difficult to estimate five normals even if the noise is quite small.

I would definitely constrain the means of normals to be ordered.

## EDIT: I’m dumb and used the wrong curve for reference. Stan actually did great.

For reference:

Thanks for the help so far - I suppose I will be back later as I try to extend this further with the rest of my data.

Hello all,

I followed the approach suggested by @bbbales2. I actually found a better parametrization for the underlying components which requires a single parameter for each one and after some exploration, it seems that only three of the five hypothesized components are present here. Regardless, I’ve been getting some odd results.

The fit that HMC settled for looks quite off:

(the intensity is normalized here and it isn’t the same curve I initially showed, but this one should actually follow the model more closely given how the experiments were conducted)

The results were consistent over a few sets of reasonable initial values. I decided to see if R’s optim() could do better, and it seemingly did:

However, this fit was very susceptible to the choice of initial values.

I thought that using rstan::optimizing() might achieve a compromise of stabiltiy and good fits, but it gave essentially identical results to those from HMC.

Since Stan’s solutions seem obviously suboptimal, I’m wondering if there might be something wrong with my specification. But the unstable fits from optim() makes me think that maybe I need to add some additional restrictions? Not sure how to proceed from here. I’ve got ~60 more curves where the one I’m trying to recover here is present mixed with more components, but if I can’t reliably recover the “pure” signal I don’t suppose using the entire dataset makes sense.

Here’s the model I’ve currently set up (the underlying components are parametrized solely in terms of their maxima):

data {
int<lower=0> n;   // n# observations
int<lower=0> k;   // n# components (tried 2-5, but 3 works best)

vector[k] lm0;    // prior maxima
vector[k] s0;     // how much to trust priors

vector[n] I;      // intensity (measured signal)
vector[n] v;      // wavenumber (reciprocal of wavelength shown in graph)
}
parameters {
// Maxima for the components
ordered[k] lm;
// Relative intensities
real<lower=0,upper=1> pu[k];
// Noise
real<lower=0> error;
}
model {
// Intermediate values to ease writing the component formula
vector[k] a;
vector[k] r;
vector[k] vm;
vector[k] vneg;
vector[k] vpos;
// Signals
vector[k] Im[n]; // Latent singal contributed per component
vector[n] mI;    // Expected total signal

// Component formula
// Place prior on maxima and invert it
lm ~ normal(lm0,s0);
for(i in 1:k){
vm[i] = (10^7)/lm[i];
}
// Intermediate calculations
vneg = 1.177*vm - 7780;
vpos = 0.831*vm + 7070;
// Per-component emission formula
for(i in 1:n){
mI[i] = 0;
}
for(j in 1:n){
for(i in 1:k){
r[i] = (vm[i] - vneg[i])/(vpos[i] - vm[i]);
a[i] = vm[i] +r[i]*(vpos[i] - vneg[i])/(r[i]^2 - 1);
Im[j][i] = exp(-log(2)*(log(a[i] - v[j])-log(a[i] - vm[i]))^2/(log(r[i])^2));
mI[j] = mI[j] + pu[i]*Im[j][i];
}
}

// Prior for noise
error ~ normal(0, 10);
// Sampling
I ~ normal(mI, error);
}


text:
# The radii of the nearby K5V and K7V stars 61 Cygni A & B. CHARA/FLUOR interferometry and CESAM2k modeling

@article{Kervella2008TheRO,
title={The radii of the nearby K5V and K7V stars 61 Cygni A \& B. CHARA/FLUOR interferometry and CESAM2k modeling},
author={Pierre Kervella and Antoine M'erand and Bernard Pichon and Fr'ed'eric Th'evenin and Ulrike Heiter and Lionel Bigot and Theo A. ten Brummelaar and Harold A. Mcalister and Stephen T. Ridgway and Nils H. Turner and Judit Sturmann and Laszlo Sturmann and P. J. Goldfinger and Christopher D. Farrington},
journal={Astronomy and Astrophysics},
year={2008},
volume={488},
pages={667-674}
}
• Published 25 June 2008
• Physics, Geology
• Astronomy and Astrophysics
Context: The main sequence binary star 61 Cyg (K5V+K7V) is our nearest stellar neighbour in the northern hemisphere. This proximity makes it a particularly well suited system for very high accuracy interferometric radius measurements. Aims: Our goal is to constrain the poorly known evolutionary status and age of this bright binary star. Methods: We obtained high accuracy interferometric observations in the infrared K' band, using the CHARA/FLUOR instrument. We then computed evolutionary models…
58 Citations

## Figures and Tables from this paper

A near-infrared interferometric survey of debris-disc stars - III. First statistics based on 42 stars observed with CHARA/FLUOR
• Physics, Geology
• 2013
Context. Dust is expected to be ubiquitous in extrasolar planetary systems owing to the dynamical activity of minor bodies. Inner dust populations are, however, still poorly known because of the high
Fundamental properties of the Population II fiducial stars HD 122563 and Gmb 1830 from CHARA interferometric observations
• Physics
• 2012
We have determined the angular diameters of two metal-poor stars, HD 122563 and Gmb 1830, using CHARA and Palomar Testbed Interferometer observations. For the giant star HD 122563, we derive an
Benchmark stars for Gaia Fundamental properties of the Population II star HD 140283 from interferometric, spectroscopic, and photometric data
• Physics
• 2015
Metal-poor halo stars are important astrophysical laboratories that allow us to unravel details about many aspects of astrophysics, including the chemical conditions at the formation of our Galaxy,
Stellar Diameters and Temperatures. II. Main-sequence K- and M-stars
• Physics
• 2012
We present interferometric angular diameter measurements of 21 low-mass, K- and M-dwarfs made with the CHARA Array. This sample is enhanced by adding a collection of radius measurements published in
Lyα Observations of High Radial Velocity Low-mass Stars Ross 1044 and Ross 825
• Physics, Geology
The Astrophysical Journal
• 2019
The discovery of habitable zone (HZ) planets around low-mass stars has highlighted the need for a comprehensive understanding of the radiation environments in which such planets reside. Of particular
Towards stellar effective temperatures and diameters at 1 per cent accuracy for future surveys
• Physics, Geology
• 2014
The apparent size of stars is a crucial benchmark for fundamental stellar properties such as effective temperatures, radii and surface gravities. While interferometric measurements of stellar angular
Characterizing the Cool Kepler Objects of Interests. New Effective Temperatures, Metallicities, Masses, and Radii of Low-mass Kepler Planet-candidate Host Stars
• Physics, Geology
• 2011
We report stellar parameters for late-K and M-type planet-candidate host stars announced by the Kepler Mission. We obtained medium-resolution, K-band spectra of 84 cool (T_eff ≲ 4400 K) Kepler
Mass-radius relation of low and very low-mass stars revisited with the VLTI
• Physics
• 2009
We measured the radii of 7 low and very low-mass stars using long baseline interferometry with the VLTI interferometer and its VINCI and AMBER near-infrared recombiners. We use these new data,
Asteroseismology and interferometry of the red giant star Ophiuchi
• Physics, Geology
• 2009
The GIII red giant starOph has been found to exhibit several modes of oscillation by the MOST mission. We interpret the observed frequencies of oscillation in terms of theoretical radial p-mode
Coronal activity cycles in nearby G and K stars - XMM-Newton monitoring of 61 Cygni and α Centauri
• Physics
• 2012
We use X-ray observations of the nearby binaries 61 Cyg A/B (K5V and K7V) and Alpha Cen A/B (G2V and K1V) to study the long-term evolution of magnetic activity in weakly to moderately active G + K

## References

SHOWING 1-10 OF 63 REFERENCES
The interferometric diameter and internal structure of Sirius A
• Physics
• 2003
The interferometric observations of dwarf stars in the solar neighbourhood, combined with Hipparcos parallaxes provide very precise values of their linear diameters. In this paper, we report the
The diameters of $\alpha$ Centauri A and B - A comparison of the asteroseismic and VINCI/VLTI views
• Physics
• 2003
We compare the first direct angular diameter measurements obtained on our closest stellar neighbour, Centauri, to recent model diameters constrained by asteroseismic observations. Using the VINCI
The limb darkening of $\mathsf{\alpha}$ Centauri B - Matching 3D hydrodynamical models with interferometric measurements
• Physics, Geology
• 2006
For the nearby dwarf star α  Cen B (K1 V), we present limb-darkening predictions from a 3D hydrodynamical radiative transfer model of its atmosphere. We first compared the results of this model to a
Metallicity and photospheric abundances in field K and M dwarfs
• Physics
• 1998
Elemental abundances in late-type stars are of interest in several ways: they determine the location of the stars in the HR diagram and therefore their ages, as well as the atmospheric structure in
Spectroscopic determination of photospheric parameters and chemical abundances of 6 K-type stars
• Physics
• 2005
High resolution, high -S /N- ratio optical spectra have been obtained for a sample of 6 K-type dwarf and subgiant stars, and have been analysed with three different LTE methods in order to derive
Improved Color-Temperature Relations and Bolometric Corrections for Cool Stars
• Physics, Geology
• 1999
We present new grids of colors and bolometric corrections for F-K stars having 4000 K ≤ Teff ≤ 6500 K, 0.0 ≤ log g ≤ 4.5, and -3.0 ≤ [Fe/H] ≤ 0.0. A companion paper extends these calculations into
A grid of MARCS model atmospheres for late-type stars. I. Methods and general properties
• Physics
• 2008
Context. In analyses of stellar spectra and colours, and for the analysis of integrated light from galaxies, a homogeneous grid of model atmospheres of late-type stars and corresponding flux spectra
A catalog of bright calibrator stars for 200-m baseline near-infrared stellar interferometry
• Physics
• 2005
We present in this paper a catalog of reference stars suitable for calibrating infrared interferometric observations. In the K band, visibilities can be calibrated with a precision of 1% on baselines
On the limb darkening, spectral energy distribution, and temperature structure of procyon
We have fit synthetic visibilities from three-dimensional ((COBOLD)-B-5+PHOENIX) and one-dimensional (PHOENIX, ATLAS 12) model stellar atmospheres of Procyon (F5 IV) to high-precision interferometric
Dwarfs in the Local Region
• Physics, Geology
• 2006
We present lithium, carbon, and oxygen abundance data for a sample of nearby dwarfs—a total of 216 stars—including samples within 15 pc of the Sun, as well as a sample of local close giant planet


text:
# 2016 is absolutely awesome 2

$$a_0, a_1,...$$ is a sequence of positive integers where $$a_0=1; a_1=1; a_2=2; a_3=6$$.

Moreover, for all $$n \ge 4$$, $$a_n$$ is the smallest positive integer such that:

$$\dfrac{a_n}{a_ia_{n-i}}$$ is an integer for all integers $$i, 0 \le i \le n$$.

Find $$\sqrt[2016]{a_{2016}}$$.

×


text:
[Note: This course is still being developed. Any feedback would be greatly appreciated and can be sent to tessler@mit.edu]

Learning statistics is like learning pottery. With pottery, you can learn how to make different shapes (e.g. a bowl, a vase, a spoon) without understanding general principles. The other way is to learn the basic strokes of forming pottery (e.g. how to mold a curved surface, a flat surface, long pointy things). In this course, we are going to learn the basic strokes of statistics, and compose those strokes to make shapes you’ve seen before (e.g. a t-test), some shapes you’ve probably never seen before, and develop ideas how you would make new shapes if you needed to. We won’t learn what tests apply to what data types but instead foster the ability to reason through data analysis. We will do this through the lens of Bayesian statistics, though the basic ideas will aid your understanding of classical (frequentist) statistics as well.

## Chapters

1. Why analyze data
Course overview

2. Probabilistic programming
A brief introduction.

Models formalize hypotheses

4. Comparing hypotheses
Hypothesis testing is model comparison

5. Causal models
Reasoning with structured knowledge

6. Elaborating models
A pinch of sophistication and elegance

7. Inference Algorithms
The various approximate inference algorithms WebPPL provides and the classes of programs for which they are each best suited.

8. Analyzing Bayesian cognitive models
The fully Bayesian treatment

## Appendix

1. Coming up with priors
Systematically interrogating one’s knowledge

2. Bayesian inference in a probabilistic program
Understanding observe, condition, and factor via rejection sampling

## Citation

M. H. Tessler (in prep). Bayesian data analysis: An introduction using probabilistic programs. Retrieved from https://mhtess.github.io/bdappl/


text:
To illustrate the look and feel I am after, please take a look at this image:

I would like to understand how to do the same on a full ToC rather than, as it is the case in the linked image, on a "mini-ToC".

-

For the standard classes, you can use the tocloft package; using \cftaftertoctitle you can place some material (a rule and some vertical space, in this case) after the ToC title and before the entries; after the ToC is typeset a rule is inserted using \hrulefill:

\documentclass{book}
\usepackage{tocloft}

\renewcommand\cftaftertoctitle{\par\noindent\hrulefill\par\vskip-4.3em}

\begin{document}

\tableofcontents
\noindent\hrulefill

\chapter{Test Chapter One}
\section{Test Section One One}
\subsection{Test Subsection One One One}
\section{Test Section One Two}
\subsection{Test Subsection One Two One}
\chapter{Test Chapter Two}
\section{Test Section Two One}
\subsection{Test Subsection Two One One}
\section{Test Section Two Two}
\subsection{Test Subsection Two Two One}

\end{document}

Without additional packages, you can redefine the \tableofcontents command as implemented in the document class used. For example, using book, such a redefinition might look like this (the changes were marked with %NEW):

\documentclass{book}

\makeatletter
\renewcommand\tableofcontents{%
\if@twocolumn
\@restonecoltrue\onecolumn
\else
\@restonecolfalse
\fi
\chapter*{\contentsname
\@mkboth{%
\MakeUppercase\contentsname}{\MakeUppercase\contentsname}}%
\par\noindent\hrulefill\par%NEW
\@starttoc{toc}%
\noindent\hrulefill%NEW
\if@restonecol\twocolumn\fi
}
\makeatother

\begin{document}

\tableofcontents

\chapter{Test Chapter One}
\section{Test Section One One}
\subsection{Test Subsection One One One}
\section{Test Section One Two}
\subsection{Test Subsection One Two One}
\chapter{Test Chapter Two}
\section{Test Section Two One}
\subsection{Test Subsection Two One One}
\section{Test Section Two Two}
\subsection{Test Subsection Two Two One}

\end{document}

The memoir document class offers you the built-in command \aftertoctitle to add material after the ToC title:

\documentclass{memoir}

\renewcommand\aftertoctitle{\par\noindent\hrulefill\par}

\begin{document}

\tableofcontents*
\noindent\hrulefill

\chapter{Test Chapter One}
\section{Test Section One One}
\subsection{Test Subsection One One One}
\section{Test Section One Two}
\subsection{Test Subsection One Two One}
\chapter{Test Chapter Two}
\section{Test Section Two One}
\subsection{Test Subsection Two One One}
\section{Test Section Two Two}
\subsection{Test Subsection Two Two One}

\end{document}

-
in the visual, that's an awfully big space after the rule above the contents list. –  barbara beeton Jul 27 '12 at 12:49
@barbarabeeton is it better now? (I am not sure if I corrected the right space.) –  Gonzalo Medina Jul 27 '12 at 14:09
oh, that's very much nicer! i like the fact that the spaces around the top rule and before the bottom rule look to be the same size; this balance is especially appropriate for such a short list. –  barbara beeton Jul 27 '12 at 15:05
You forgot to put \subsubsection to get the "Test Section Two One One" hierarchy, you are just using all \section commands in your code. –  Robottinosino Jul 31 '12 at 14:01
@Robottinosino you're right. I've updated my answer. –  Gonzalo Medina Jul 31 '12 at 14:26