Statistical correlation. More...

Static Public Member Functions
static	covariance (array $X, array $Y, bool $population=false)
	Covariance Convenience method to access population and sample covariance. More...

static	populationCovariance (array $X, array $Y)
	Population Covariance A measure of how much two random variables change together. More...

static	sampleCovariance (array $X, array $Y)
	Sample covariance A measure of how much two random variables change together. More...

static	weightedCovariance (array $X, array $Y, array $w)
	Weighted covariance A measure of how much two random variables change together with weights. More...

static	r (array $X, array $Y, bool $population=false)
	r - correlation coefficient Pearson product-moment correlation coefficient (PPMCC or PCC or Pearson's r) More...

static	populationCorrelationCoefficient (array $X, array $Y)
	Population correlation coefficient Pearson product-moment correlation coefficient (PPMCC or PCC or Pearson's r) More...

static	sampleCorrelationCoefficient (array $X, array $Y)
	Sample correlation coefficient Pearson product-moment correlation coefficient (PPMCC or PCC or Pearson's r) More...

static	r2 (array $X, array $Y, bool $popluation=false)
	R² - coefficient of determination Convenience wrapper for coefficientOfDetermination. More...

static	coefficientOfDetermination (array $X, array $Y, bool $popluation=false)
	R² - coefficient of determination. More...

static	weightedCorrelationCoefficient (array $X, array $Y, array $w)
	Weighted correlation coefficient Pearson product-moment correlation coefficient (PPMCC or PCC or Pearson's r) width weighted values. More...

static	kendallsTau (array $X, array $Y)
	τ - Kendall rank correlation coefficient (Kendall's tau) More...

static	spearmansRho (array $X, array $Y)
	ρ - Spearman's rank correlation coefficient (Spearman's rho) More...

static	describe (array $X, array $Y, bool $population=false)
	Descriptive correlation report about two random variables. More...

static	confidenceEllipse (array $X, array $Y, float $z, int $num_points=11)
	Confidence ellipse (error ellipse) Given the data in $X and $Y, create an ellipse surrounding the data at $z standard deviations. More...

Public Attributes
const	X = 0

const	Y = 1

Detailed Description

Statistical correlation.

covariance
correlation coefficient (r)
coefficient of determination (R²)
Kendall's tau (τ)
Spearman's rho (ρ)
confidence ellipse

Member Function Documentation

◆ coefficientOfDetermination()

static MathPHP\Statistics\Correlation::coefficientOfDetermination	(	array	$X,
		array	$Y,
		bool	$popluation = `false`
	)

static

R² - coefficient of determination.

Indicates the proportion of the variance in the dependent variable that is predictable from the independent variable. Range of 0 - 1. Close to 1 means the regression line is a good fit https://en.wikipedia.org/wiki/Coefficient_of_determination

Parameters

array	$X	values for random variable X
array	$Y	values for random variable Y
bool	$popluation

Returns: float

Exceptions

Exception

◆ confidenceEllipse()

static MathPHP\Statistics\Correlation::confidenceEllipse	(	array	$X,
		array	$Y,
		float	$z,
		int	$num_points = `11`
	)

static

Confidence ellipse (error ellipse) Given the data in $X and $Y, create an ellipse surrounding the data at $z standard deviations.

The function will return $num_points pairs of X,Y data http://stackoverflow.com/questions/3417028/ellipse-around-the-data-in-matlab

Parameters

array	$X	an array of independent data
array	$Y	an array of dependent data
float	$z	the number of standard deviations to encompass
int	$num_points	the number of points to include around the ellipse. The actual array will be one larger because the first point and last will be repeated to ease display.

Returns: array paired x and y points on an ellipse aligned with the data provided

Exceptions

Exception

◆ covariance()

static MathPHP\Statistics\Correlation::covariance	(	array	$X,
		array	$Y,
		bool	$population = `false`
	)

static

Covariance Convenience method to access population and sample covariance.

A measure of how much two random variables change together. Average product of their deviations from their respective means. The population covariance is defined in terms of the sample means x, y https://en.wikipedia.org/wiki/Covariance

Parameters

array	$X	values for random variable X
array	$Y	values for random variable Y
bool	$population	Optional flag for population or sample covariance

Returns: float

Exceptions

Exception

◆ describe()

static MathPHP\Statistics\Correlation::describe	(	array	$X,
		array	$Y,
		bool	$population = `false`
	)

static

Descriptive correlation report about two random variables.

Parameters

array	$X	values for random variable X
array	$Y	values for random variable Y
bool	$population	Optional flag if all samples of a population are present

Returns: array [cov, r, R2, tau, rho]

Exceptions

Exception

◆ kendallsTau()

static MathPHP\Statistics\Correlation::kendallsTau	(	array	$X,
		array	$Y
	)

static

τ - Kendall rank correlation coefficient (Kendall's tau)

A statistic used to measure the ordinal association between two measured quantities. It is a measure of rank correlation: the similarity of the orderings of the data when ranked by each of the quantities. https://en.wikipedia.org/wiki/Kendall_rank_correlation_coefficient https://onlinecourses.science.psu.edu/stat509/node/158

tau-a (no rank ties):

   nc - nd

τ = -------— n(n - 1)/2

Where nc: number of concordant pairs nd: number of discordant pairs

tau-b (rank ties exist):

            nc - nd

τ = --------------------------— √(nc + nd + X₀)(nc + nd + Y₀)

Where X₀: number of pairs tied only on the X variable Y₀: number of pairs tied only on the Y variable

Parameters

array	$X	values for random variable X
array	$Y	values for random variable Y

Todo:: Implement with algorithm faster than O(n²)

Returns: float

Exceptions

Exception

◆ populationCorrelationCoefficient()

static MathPHP\Statistics\Correlation::populationCorrelationCoefficient	(	array	$X,
		array	$Y
	)

static

Population correlation coefficient Pearson product-moment correlation coefficient (PPMCC or PCC or Pearson's r)

A normalized measure of the linear correlation between two variables X and Y, giving a value between +1 and −1 inclusive, where 1 is total positive correlation, 0 is no correlation, and −1 is total negative correlation. It is widely used in the sciences as a measure of the degree of linear dependence between two variables. https://en.wikipedia.org/wiki/Pearson_product-moment_correlation_coefficient

The correlation coefficient of two variables in a data sample is their covariance divided by the product of their individual standard deviations.

   cov(X,Y)

ρxy = -------— σx σy

conv(X,Y) is the population covariance σx is the population standard deviation of X σy is the population standard deviation of Y

Parameters

array	$X	values for random variable X
array	$Y	values for random variable Y

Returns: float

Exceptions

Exception

◆ populationCovariance()

static MathPHP\Statistics\Correlation::populationCovariance	(	array	$X,
		array	$Y
	)

static

Population Covariance A measure of how much two random variables change together.

Average product of their deviations from their respective means. The population covariance is defined in terms of the population means μx, μy https://en.wikipedia.org/wiki/Covariance

cov(X, Y) = σxy = E[⟮X - μx⟯⟮Y - μy⟯]

              ∑⟮xᵢ - μₓ⟯⟮yᵢ - μy⟯

cov(X, Y) = σxy = --------------— N

Parameters

array	$X	values for random variable X
array	$Y	values for random variable Y

Returns: float

Exceptions

Exception

◆ r()

static MathPHP\Statistics\Correlation::r	(	array	$X,
		array	$Y,
		bool	$population = `false`
	)

static

r - correlation coefficient Pearson product-moment correlation coefficient (PPMCC or PCC or Pearson's r)

Convenience method for population and sample correlationCoefficient

Parameters

array	$X	values for random variable X
array	$Y	values for random variable Y
bool	$population	Optional flag for population or sample covariance

Returns: float

Exceptions

Exception

◆ r2()

static MathPHP\Statistics\Correlation::r2	(	array	$X,
		array	$Y,
		bool	$popluation = `false`
	)

static

R² - coefficient of determination Convenience wrapper for coefficientOfDetermination.

Parameters

array	$X	values for random variable X
array	$Y	values for random variable Y
bool	$popluation

Returns: float

Exceptions

Exception

◆ sampleCorrelationCoefficient()

static MathPHP\Statistics\Correlation::sampleCorrelationCoefficient	(	array	$X,
		array	$Y
	)

static

Sample correlation coefficient Pearson product-moment correlation coefficient (PPMCC or PCC or Pearson's r)

A normalized measure of the linear correlation between two variables X and Y, giving a value between +1 and −1 inclusive, where 1 is total positive correlation, 0 is no correlation, and −1 is total negative correlation. It is widely used in the sciences as a measure of the degree of linear dependence between two variables. https://en.wikipedia.org/wiki/Pearson_product-moment_correlation_coefficient

The correlation coefficient of two variables in a data sample is their covariance divided by the product of their individual standard deviations.

Sxy

rxy = -------— sx sy

Sxy is the sample covariance σx is the sample standard deviation of X σy is the sample standard deviation of Y

Parameters

array	$X	values for random variable X
array	$Y	values for random variable Y

Returns: float

Exceptions

Exception

◆ sampleCovariance()

static MathPHP\Statistics\Correlation::sampleCovariance	(	array	$X,
		array	$Y
	)

static

Sample covariance A measure of how much two random variables change together.

Average product of their deviations from their respective means. The population covariance is defined in terms of the sample means x, y https://en.wikipedia.org/wiki/Covariance

cov(X, Y) = Sxy = E[⟮X - x⟯⟮Y - y⟯]

              ∑⟮xᵢ - x⟯⟮yᵢ - y⟯

cov(X, Y) = Sxy = ------------— n - 1

Parameters

array	$X	values for random variable X
array	$Y	values for random variable Y

Returns: float

Exceptions

Exception

◆ spearmansRho()

static MathPHP\Statistics\Correlation::spearmansRho	(	array	$X,
		array	$Y
	)

static

ρ - Spearman's rank correlation coefficient (Spearman's rho)

https://en.wikipedia.org/wiki/Spearman%27s_rank_correlation_coefficient

     6 ∑ dᵢ²

ρ = 1 - ------— n(n² − 1)

Where dᵢ: the difference between the two ranks of each observation

Parameters

array	$X	values for random variable X
array	$Y	values for random variable Y

Returns: float

Exceptions

Exception

◆ weightedCorrelationCoefficient()

static MathPHP\Statistics\Correlation::weightedCorrelationCoefficient	(	array	$X,
		array	$Y,
		array	$w
	)

static

Weighted correlation coefficient Pearson product-moment correlation coefficient (PPMCC or PCC or Pearson's r) width weighted values.

A normalized measure of the linear correlation between two variables X and Y, giving a value between +1 and −1 inclusive, where 1 is total positive correlation, 0 is no correlation, and −1 is total negative correlation. It is widely used in the sciences as a measure of the degree of linear dependence between two variables. https://en.wikipedia.org/wiki/Pearson_correlation_coefficient#Weighted_correlation_coefficient

The weighted correlation coefficient of two variables in a data sample is their covariance divided by the product of their individual standard deviations.

     cov(X,Y,w)

ρxyw = ----------— √(sxw syw)

conv(X,Y, w) is the weighted covariance sxw is the weighted variance of X syw is the weighted variance of Y

Parameters

array	$X	values for random variable X
array	$Y	values for random variable Y
array	$w	values for weights

Returns: float

Exceptions

Exception

◆ weightedCovariance()

static MathPHP\Statistics\Correlation::weightedCovariance	(	array	$X,
		array	$Y,
		array	$w
	)

static

Weighted covariance A measure of how much two random variables change together with weights.

https://en.wikipedia.org/wiki/Pearson_correlation_coefficient#Weighted_correlation_coefficient

                  ∑wᵢ⟮xᵢ - μₓ⟯⟮yᵢ - μy⟯

cov(X, Y, w) = sxyw = -----------------— ∑wᵢ

Parameters

array	$X	values for random variable X
array	$Y	values for random variable Y
array	$w	values for weights

Returns: float

Exceptions

Exception

The documentation for this class was generated from the following file:

src/Statistics/Correlation.php

Static Public Member Functions

Public Attributes

Detailed Description

Member Function Documentation

◆ coefficientOfDetermination()

◆ confidenceEllipse()

◆ covariance()

◆ describe()

◆ kendallsTau()

◆ populationCorrelationCoefficient()

◆ populationCovariance()

◆ r()

◆ r2()

◆ sampleCorrelationCoefficient()

◆ sampleCovariance()

◆ spearmansRho()

◆ weightedCorrelationCoefficient()

◆ weightedCovariance()