correlation – internal product equal to 1? (where X has a mean of zero and a variance of 1)

I was looking at the test of two random variables, each correlated with a third.

I cannot understand some parts of the test regarding the internal product. For example, in the test, he mentions that we assume that X, a random variable, has an average of 0 and a variance of 1, so does Y. Why is that? < XX > = 1 and that < XY > is the correlation
coefficient of X and Y? by < XX >, I think Var (X) = 1 = E (X ^ 2) – 0 = < XX >/ n but no < XX >.

pr.probability: concentration of the scaled norm $ l_p $ of a correlation matrix

Background:

Among Hermitan's random matrices, the correlation matrix has many applications in statistics. People have studied the "empirical spectral distribution (ESD)" of a correlation matrix, the highest value of a correlation matrix, the logarithmic determinant of a correlation matrix, the largest entries outside the diagonal of a matrix of correlation, all with applications to statistics. However, it seems that there is no work on the scale $ l_p $ norm of a correlation matrix. the $ l_p $ The norm of a correlation matrix provides a criterion on whether the "strong law of large numbers (SLLN)" associated with the classical problem of normal means is valid.

Problem Statement:

Leave $ x_i, i = 1, ldots, n $ be $ n $ i.i.d. observations of a $ m $three-dimensional random vector $ x in mathbb {R} ^ m $ such that $ x $ has a correlation matrix $ Sigma in mathbb {R} ^ {m times m} $, leave $ R = (r_ {ij}) in mathbb {R} ^ {m times m} $ be the correlation matrix obtained from $ left {x_i right } _ {i = 1} ^ n $ such that each $ r_ {ij} $ is a Pearson correlation coefficient, and let the $ l_p $ rule of $ R $ be $$ Vert R Vert_p = sum_ {i, j = m} ^ p vert r_ {ij} vert ^ p $$ for $ 0 <p < infty $.

Consider the case $ p = 1 $. So $ Vert R Vert_1 ge m $. Suppose for some $ alpha> 0 $, the scaling $ l_p $ rule $ m ^ {- alpha} Vert R Vert_1 $ almost surely or likely tending to $ 1 $ lies in a compact interval, say, $ (a, b) $. What can you say about $$ Pr ( vert m ^ {- alpha} Vert R Vert_1 – mathbb {E} (m ^ {- alpha} Vert R Vert_1) vert> t) $$ for a fixed $ t> 0 $ when $ m $ is big and $ n $ satisfies any relationship with $ m $), where $ mathbb {E} $ denotes expectation?

Gauss specialization $ x $:

Yes $ x $ follow a Gaussian distribution, then each entry $ r_ {ij} $, being a Pearson correlation coefficient, it has an explicit marginal distribution and covariance between $ r_ {ij} $ Y $ r_ {jk} $ It is given by JH Steiger's document here (https://psycnet.apa.org/record/1980-08757-001). These results were also provided by Pearson and Filon (and Steiger's article cited these results). However, Pearson and Filon's notations are very outdated (compared to modern notes) and are difficult to digest.

For study $ m ^ {- alpha} Vert R Vert_1 $ and its concentration properties, and in particular, to show the SLLN for $ m ^ {- alpha} Vert R Vert_1 $ (using triangular matrix techniques), we need covariance between the absolute values ​​of $ r_ {ij} $ Y $ r_ {jk} $that is, we need $$ mathsf {cov} ( vert r_ {ij} vert, vert r_ {jk} vert) $$ This requires the joint distribution of $ r_ {ij} $ Y $ r_ {jk} $, that I cannot find an existing formula (and whose derivation will be very tedious and cumbersome).

Specialized questions and attempts:

Does anyone know the joint distribution of $ r_ {ij} $ Y $ r_ {jk} $? Caution: no attempt should be made to obtain the joint distribution of $ r_ {ij} $ Y $ r_ {jk} $ by using the distribution of $ R $ when $ x $ is Gaussian (although in this case $ R $ it has an analytical density that involves its determinant) since it induces more chaos than directly calculating covariance using a normal trivariate distribution.

Can not solve the coupled differential equations and how to find the correlation of solutions?

I'm trying to solve the equations.

$ x & # 39;

$ and & # 39;

with the initial conditions $ x[0]= 0, and[0]= v_0 $

This is what I have tried.

  1. $[Omega] = Sqrt[w^2 + 4*[Gamma]^ 2]$
  2. System = {x & # 39;
  3. Dsolve[{X&#39;[{x'[{X'[{x'
    Y

But Mathematica 11.3 gives output. Dsolve[{-Y[{-Y[{-y[{-y
0, (w ^ 2 + 4 [Gamma]^ 2) x
Derivative[1][y]

I would like expressions from $ x

Partial and semi-partial correlation.

I am interested in knowing if there are incorporated. Mathematics Functions that perform partial and partial calculations.

If not, what would be a simple and efficient way to calculate them from the results of a LinearModelFit?

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Classification: 5

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dnd 5e: What is the correlation between the CA and the cost of the armor?

The "Armor and shields" section of the Player's Manual lists the CA and the cost of the armor and shields. The upper end of each type of armor (Light, Medium, Heavy) is substantially more expensive than the rest.

Has anyone devised a formula to determine the cost of the armor using the variables of type, AC, Requirements, Stealth, and Weight? This would be useful when preparing a new armor (for example, a heavy armor of CA 15).

pr.probability – Correlation between the square of normal random variables

Suppose I have $ X, Y $ bivariable normal with correlation coefficient $ rho in (0.1) $ . So, what is the correlation between $ X ^ 2 $ Y $ Y ^ 2 $ ?

I am aware of the fact that the square of normal follows a chi-squared distribution. Then, I can find out $ Var (X ^ 2) $ Y $ Var (Y ^ 2) $ . However, I am unable to calculate the covariance between $ X ^ 2 $ Y $ Y ^ 2 $ .

Statistics: How to measure the deviation of a value from a correlation?

Suppose I have a table with 20 columns and 1000 lines. The lines represent different objects and the columns represent different characteristics of the objects. Suppose that the first column is a characteristic with a special meaning.

Suppose I would like to measure how much the value of the i-th characteristic of the n-th object deviates from its expected value, in a scenario where the correlation of the i-th characteristic has a perfect correlation with the first column. How can I measure this deviation?

beginner – Statistical methods using PHP (mean, co / variance, standard deviation, bias, correlation)

Functionality

This class has a list of basic statistical functions such as the mean, variance, standard deviation, asymmetry, etc., and it works well, executing these functions in the last 30 days.

It extends to a parent class that estimates future prices very close to a list of actions using an API delayed in 60-second data.

Would you be so kind and possibly revise it to know performance, efficiency, mathematics or coding best practices?

Code

// Config class for route and other constants
require_once __DIR__. "/ConstEQ.php";

/ **
* This is an extended class with basic statistical method.
* Values
* /
The ST class extends the ConstEQ equalization tools
{

/ **
*
* @return a number equal to the average of the values ​​of a matrix
* /
Public static function getMean ($ array)
{
if (count ($ array) == 0) {
returns ConstEQ: NEAR_ZERO_NUMBER;
} else {
return array_sum ($ array) / count ($ array);
}
}

/ **
*
* @ Return a normalized number between 0 and 1
* /
static public function getNormalize ($ value, $ min, $ max)
{
yes ($ max - $ min! = 0) {
$ normalized = 2 * (($ value - $ min) / ($ max - $ min)) - 1;
} else {
$ normalized = 2 * (($ value - $ min)) - 1;
}
returns $ normalized;
}

/ **
*
* @ Return a normalized number between 0.0 to 1 from any input -inf to inf
* /
Static public function getSigmoid ($ t)
{
returns 1 / (1 + pow (M_EULER, - $ t));
}

/ **
*
* @return a number equal to the square of the average value
* /
static public function getMeanSquare ($ x, $ average)
{return pow ($ x - $ mean, 2);}

/ **
*
* @return a number equal to the standard deviation of the values ​​of a matrix
* /
Static public function getStandardDeviation ($ array)
{
yes (account ($ array) <2) {
returns ConstEQ: NEAR_ZERO_NUMBER;
} else {
return sqrt (array_sum (array_map ("ST :: getMeanSquare", $ array, array_fill (0, count ($ array), (array_sum ($ array) / count ($ array)))))) / (count ($ array ) - one));
}
}

/ **
*
* @return a number equal to the covariance of values ​​of two matrices
* /
Public static function getCovariance ($ valuesA, $ valuesB)
{
// size both equal matrices, if they are of different sizes
$ no_keys = min (count ($ valuesA), count ($ valuesB));
$ valuesA = array_slice ($ valuesA, 0, $ no_keys);
$ valuesB = array_slice ($ valuesB, 0, $ no_keys);

// if the size of the matrices is too small
if ($ no_keys <2) {return ConstEQ :: NEAR_ZERO_NUMBER;}

// Use the library function if available
if (function_exists (& # 39; stats_covariance & # 39;)) {return stats_covariance ($ valuesA, $ valuesB);}

$ meanA = array_sum ($ valuesA) / $ no_keys;
$ meanB = array_sum ($ valuesB) / $ no_keys;
$ add = 0.0;

for ($ pos = 0; $ pos <$ no_keys; $ pos ++) {
$ valueA = $ valuesA[$pos];
if (! is_numeric ($ valueA)) {
trigger_error (& # 39; Non-numeric value in matrix A in position & # 39 ;. $ pos. & # 39 ;, value = & # 39 ;. $ valueA, E_USER_WARNING);
false return;
}

$ valueB = $ valuesB[$pos];
if (! is_numeric ($ valueB)) {
trigger_error (& # 39; Non-numeric value in matrix B at position & # 39 ;. $ pos. & # 39 ;, value = & # 39 ;. $ valueB, E_USER_WARNING);
false return;
}

$ difA = $ valueA - $ meanA;
$ difB = $ valueB - $ meanB;
$ add + = ($ difA * $ difB);
}

returns $ add / $ no_keys;
}

/ **
*
* @Serve a number equal to the bias of the values ​​of the matrix
* /
Static public function getSkewness ($ values)
{
$ numValues ​​= count ($ values);
if ($ numValues ​​== 0) {return 0.0;}

// Use the function of the php_stats library if it is available
if (function_exists (& # 39; stats_skew & # 39;)) {return stats_skew ($ values);}

$ media = array_sum ($ values) / floatval ($ numValues);

$ add2 = 0.0;
$ add3 = 0.0;

foreach ($ values ​​as $ value) {
if (! is_numeric ($ value)) {return false;}

$ dif = $ value - $ average;
$ add2 + = ($ dif * $ dif);
$ add3 + = ($ dif * $ dif * $ dif);

}

$ variance = $ add2 / floatval ($ numValues);

if ($ variance == 0) {return ConstEQ :: NEAR_ZERO_NUMBER;} else {return ($ add3 / floatval ($ numValues)) / pow ($ variance, 3 / 2.0);}
}

/ **
*
* @serving a number equal to the kurtosis of the matrix values
* /
Public static function getKurtosis ($ values)
{
$ numValues ​​= count ($ values);
if ($ numValues ​​== 0) {return 0.0;}

// Use the function of the php_stats library if it is available
if (function_exists (& # 39; stats_kurtosis & # 39;)) {return stats_kurtosis ($ values);}

$ media = array_sum ($ values) / floatval ($ numValues);
$ add2 = 0.0;
$ add4 = 0.0;

foreach ($ values ​​as $ value) {
if (! is_numeric ($ value)) {return false;}
$ dif = $ value - $ average;
$ dif2 = $ dif * $ dif;
$ add2 + = $ dif2;
$ add4 + = ($ dif2 * $ dif2);
}

$ variance = $ add2 / floatval ($ numValues);
if ($ variance == 0) {return ConstEQ :: NEAR_ZERO_NUMBER;} else {return ($ add4 * $ numValues) / ($ add2 * $ add2) - 3.0;}
}

/ **
*
* @ returns a number equal to the correlation of two matrices
* /
Public static function getCorrelation ($ arr1, $ arr2)
{
$ correlation = 0;

$ k = ST :: sumProductMeanDeviation ($ arr1, $ arr2);
$ ssmd1 = ST :: sumSquareMeanDeviation ($ arr1);
$ ssmd2 = ST :: sumSquareMeanDeviation ($ arr2);

$ product = $ ssmd1 * $ ssmd2;

$ res = sqrt ($ product);
if ($ res == 0) {return ConstEQ :: NEAR_ZERO_NUMBER;}
$ correlation = $ k / $ res;

if ($ correlation == 0) {return ConstEQ :: NEAR_ZERO_NUMBER;} else {return $ correlation;}
}

/ **
*
* @detain a number equal to the sum of the average product deviation of each matrix value
* /
public static function sumProductMeanDeviation ($ arr1, $ arr2)
{
$ sum = 0;
$ num = count ($ arr1);

for ($ i = 0; $ i <$ num; $ i ++) {$ sum = $ sum + ST :: productMeanDeviation ($ arr1, $ arr2, $ i);}
returns $ sum;
}

/ **
*
* @serving a number equal to the average product deviation of each matrix value
* /
public static function productMeanDeviation ($ arr1, $ arr2, $ item)
{return (ST :: meanDeviation ($ arr1, $ item) * ST :: meanDeviation ($ arr2, $ item));}

/ **
*
* @return a number equal to the sum of the mean square deviation of the values ​​of each matrix
* /
Public static function sumSquareMeanDeviation ($ arr)
{
$ sum = 0;
$ num = count ($ arr);

for ($ i = 0; $ i <$ num; $ i ++) {$ sum = $ sum + ST :: squareMeanDeviation ($ arr, $ i);}
returns $ sum;
}

/ **
*
* @return a number equal to the mean square deviation of the values ​​of each matrix
* /
public static function squareMeanDeviation ($ arr, $ item)
{
return ST :: meanDeviation ($ arr, $ item) * ST :: meanDeviation ($ arr, $ item);
}

/ **
*
* @return a number equal to the sum of the average deviation of the values ​​of each matrix
* /
public static function sumMeanDeviation ($ arr)
{
$ sum = 0;
$ num = count ($ arr);

for ($ i = 0; $ i <$ num; $ i ++) {$ sum = $ sum + ST :: meanDeviation ($ arr, $ i);}
returns $ sum;
}

/ **
*
* @ returns a number equal to the average deviation of the values ​​of each matrix
* /
public static function meanDeviation ($ arr, $ item)
{
$ average = ST :: average ($ arr); return $ arr[$item] - average $;
}

/ **
*
* @remits a number equal to the average of the values ​​in the matrix
* /
average public static function ($ arr)
{
$ sum = ST :: sum ($ arr);
$ num = count ($ arr); returns $ sum / $ num;}

/ **
*
* @return a number equal to the sum of a matrix
* /
sum of the public static function ($ arr)
{return array_sum ($ arr);}

/ **
*
* @Back a series of coefficients for 7 levels of volatilities
* /
public static function getCoefParams ($ overall_market_coeff)
{
$ daily_coef = 0.9 + ($ overall_market_coeff / 10);

$ coefs = array (
ConstEQ :: LEVEL_VOLATILITY_COEF_1 * $ daily_coef,
ConstEQ :: LEVEL_VOLATILITY_COEF_2 * $ daily_coef,
ConstEQ :: LEVEL_VOLATILITY_COEF_3 * $ daily_coef,
ConstEQ :: LEVEL_VOLATILITY_COEF_4 * $ daily_coef,
ConstEQ :: LEVEL_VOLATILITY_COEF_5 * $ daily_coef,
ConstEQ :: LEVEL_VOLATILITY_COEF_6 * $ daily_coef,
ConstEQ :: LEVEL_VOLATILITY_COEF_7 * $ daily_coef,
);

returns $ coefs;
}

/ **
* @return a true or false binary for the is_numeric test of a string
* /
The public static function is Number ($ arr)
{
foreach ($ arr as $ b) {
yes (! is_numeric ($ b)) {
false return;
}
}
true returns
}

}

Functional analysis. Measurement and characterization of local correlation matrices?

Definition: a matrix $ C in mathbb R ^ {m} n} $ is the local correlation matrix if there are real random variables $ x_1, dots, x_m, y_1, dots, and_n $ defined in a common probability space that takes values ​​in $[-1,+1]$ such that $$ C_ {ij} = mathbb E[x_iy_j]$$ Holds on each $ (i, j) in {1, dots, m } times {1, dots, n } $.

Denote set of all $ m times n $ local correlation matrices by $ mathsf {LC} _ {m, n} $.

  1. What is the measure of $ mathsf {LC} _ {m, n} cap[-1,+1]^ {m} n} $ in $[-1,+1]^ {m} n} $?

  2. Is there any non-trivial geometric or functional characterization of $ mathsf {LC} _ {m, n} $?

  3. They are the explicit families of matrices in $[-1,+1]^ {m} n} $ that are not$ mathsf {LC} _ {m, n} $?

  4. Given a $ m times n $ matrix there is simple test is a member of $ mathsf {LC} _ {m, n} $?