Probability.h File Reference

#include "../Matrix.h"
#include "../matrix_utils.h"

Go to the source code of this file.

Classes
struct	LinearModel

Enumerations
enum	P_NORM { Inf , One , Eukl }

Functions
double	getExponent (double x)
double	round (double x, int precision)
double	abs (double x)
double	pow (double x, int exponent)
double	norm (const Matrix< double > &vec, P_NORM pNorm)
double	cov (const Matrix< double > &x, const Matrix< double > &y)
double	var (const Matrix< double > &x)
Matrix< double >	sd (const Matrix< double > &x, int axis=0)
LinearModel	lm (const Matrix< double > &x, const Matrix< double > &y)
double	coefficientOfDetermination (const Matrix< double > &y, const Matrix< double > &yHat)
double	likelihood (const Matrix< double > &)
Matrix< double >	Regression (const Matrix< double > &A)
Matrix< double >	corr (const Matrix< double > &A, const Matrix< double > &B)
Matrix< double >	expected_value (const Matrix< double > &A)

Detailed Description

Contains methods of the field of probability calculation.

Enumeration Type Documentation

P_NORM

enum P_NORM

enum to chose a norm to compute

Enumerator
Inf	$$||x||_\infty$$
One	$$||x||_1$$
Eukl	$$||x||_2$$

Function Documentation

abs()

double abs

(

double

x

)

Absolute value $$|x|$$

Parameters

x

a value

Returns

$$|x|$$

coefficientOfDetermination()

double coefficientOfDetermination	(	const Matrix< double > &	y,
		const Matrix< double > &	yHat
	)

The coefficient of determination ($R^2$) is the proportion of the variation in the dependent variable that is predictable from independent variables.

$$R^2 = \frac{\text{var}(y)}{\text{var}(\hat{y})}$$

Parameters

y	$$y$$
yHat	$$\hat{y}$$

Returns

proportion of the variation

corr()

Matrix< double > corr	(	const Matrix< double > &	A,
		const Matrix< double > &	B
	)

Computes correlation matrix of two matrices

$$ \text{corr}(X, Y) = \frac{\text{cov}(X, Y)}{\sigma_X\sigma_Y} $$ with $\sigma$ the standard deviations of $X$ and $Y$.

The correlation matrix contains correlation coefficients $\text{corr}_{x_i, y_i} \in [-1, 1]$.

• negative correlation coefficients: The values negatively influence each other
• positive correlation coefficients: The values positively influence each other
• neutral correlation coefficients: The values don't influence each other

  Parameters

  A	the first matrix of values
  B	second matrix of values in same shape as A

  Returns

  The correlation matrix of A and B

Examples

statistics/TestProbability.cpp.

cov()

double cov	(	const Matrix< double > &	x,
		const Matrix< double > &	y
	)

Computes covariance of two given vectors $$\text{cov}(x, y) = E[(x - E[x])(y - E[y])]$$

The covariance is a measure of joint variability of two random variables.

• Positive covariance -> variables have same tendencies, i.e. high and low values in both variables at the same time
• Negative covariance -> variables have different tendencies, i.e. high values in variable $x$ correspond to low values in variable $y$ and vice versa

Parameters

x	a vector
y	another vector

Returns

resulting covariance

expected_value()

Matrix< double > expected_value

(

const Matrix< double > &

A

)

Computes the expected value of a given vector.

$$ E[X] = \sum x_i P(x_i) $$

The expected value is a generalization of the weighted average. Given a set of random values, the expected value is the value that is most likely to occur within given distribution of values.

Parameters

A	the matrix to caclulate the expected value(s) of, each row will receive a corresponding expected value

Returns

The expected value(s) of A

getExponent()

double getExponent

(

double

x

)

regular helpers Method to calculate exponent of given value

Parameters

x	value to calculate exponent of

Returns

exponent

likelihood()

double likelihood

(

const Matrix< double > &

)

Computes likelihood

Returns

lm()

LinearModel lm	(	const Matrix< double > &	x,
		const Matrix< double > &	y
	)

Computes linear (regression) model for given random sample $$ (Y_i; X_i) $$ $$ Y_i = \beta_0 + \beta_1\phi_1(x_i) + \epsilon_i $$

Parameters

x	$$X_i$$
y	$$Y_i$$

Returns

constructed linear model

Examples

statistics/TestProbability.cpp.

norm()

double norm	(	const Matrix< double > &	vec,
		P_NORM	pNorm
	)

Calculates the Norm of a given vector

Parameters

vec	vector to calculate the norm of
pNorm	desired norm to compute

Returns

computed norm value

pow()

double pow	(	double	x,
		int	exponent
	)

calculates $$x^{exponent}$$

Parameters

x	a value
exponent	desired exponent

Returns

$$x^{exponent}$$

Examples

statistics/TestProbability.cpp, and symb/TestSymbolic.cpp.

Regression()

Matrix< double > Regression

(

const Matrix< double > &

A

)

Computes orthogonal-linear regression for given values A, mapped onto points $$P(i, a_i)$$.

$$Y_i = \beta_0 + \beta_1 * a_i$$

Parameters

A	vector of y-values

Returns

A vector of x-value approximations

Examples

statistics/TestProbability.cpp.

round()

double round	(	double	x,
		int	precision
	)

Rounds double value to given precision, cuts off other elements

Parameters

x	value to round
precision	precision in resulting value

Returns

rounded value

Examples

statistics/TestProbability.cpp.

sd()

Matrix< double > sd	(	const Matrix< double > &	x,
		int	axis = 0
	)

Computes standard deviation of given vector $$ \text{sd}(x) = \sigma(x) = \sqrt{E[(X-\mu)^2]} = \sqrt{E[X^2] - (E[X])^2} $$

The standard deviation is a measure of variance or dispersion of a set of values.

Parameters

x	set of values to compute standard deviation from @returnss standard deviation of given x

Examples

statistics/TestProbability.cpp.

var()

double var

(

const Matrix< double > &

x

)

Computes variance of given vector $$\text{var}(x)=E[(X - \mu)^2] = \text{cov}(X,X)$$

The variance is the expectation of the square deviation of a random variable from its population mean or sample mean.

Parameters

x

a vector

Returns

$$\text{var}(x)$$