FU_Statistics Module

Calculates the mean value of a set of values given in a vector of any size with one dimension applying the following equation:

$$\overline{x} = \frac{\sum\limits_{i=1}^n x_{i}}{n}$$

Where:

• $x$ is a vector with real numbers.
• $n$ is how many numbers are included in $x$.

Syntax

y = mean(x)

Where:

• x = vector of rank 1 with real numbers. See this example to use an array of rank larger than 1.
• y = real number of the same kind as x with the mean value of x.

Example

The following program calculates the mean value of a vector:

 PROGRAM meanExample
    USE FU_Statistics, ONLY: mean
    IMPLICIT NONE
    REAL, DIMENSION(5) :: x = [1., 2., 3., 4., 5.]
    WRITE(*,*) mean(x)
 END PROGRAM meanExample

Calculates the geometric mean of a set of values given in a vector of any size with one dimension applying the following equation:

$$\overline{x} = \left(\prod\limits_{i=1}^{n}x_{i}\right)^\frac{1}{n} = \sqrt[n]{x_{1} \times x_{2} \times \dots \times x_{n}}$$

Where:

• $x$ is a vector with real numbers.
• $n$ is how many numbers are included in $x$.

Syntax

y = gmean(x)

Where:

• x = vector of rank 1 with real numbers. See this example to use an array of rank larger than 1.
• y = real number of the same kind as x with the geometric mean of x.

Example

The following program calculates the geometric mean value of a vector:

 PROGRAM gmeanExample
    USE FU_Statistics, ONLY: gmean
    IMPLICIT NONE
    REAL, DIMENSION(5) :: x = [1., 2., 3., 4., 5.]
    WRITE(*,*) gmean(x)
 END PROGRAM gmeanExample

Calculates the sample variance of a set of values given in a vector of any size with one dimension applying the following equation:

$$\sigma^{2} = \frac{\sum\limits_{i=1}^{n}\left(x_{i} - \overline{x}\right)^2}{n-1}$$

Where:

• $x$ is a vector with real numbers.
• $n$ is how many numbers are included in $x$.

Syntax

y = variance(x)

Where:

• x = vector of rank 1 with real numbers. See this example to use an array of rank larger than 1.
• y = real number of the same kind as x with the sample variance of x.

Example

The following program calculates the variance of a vector:

 PROGRAM varianceExample
    USE FU_Statistics, ONLY: variance
    IMPLICIT NONE
    REAL, DIMENSION(5) :: x = [1., 2., 3., 4., 5.]
    WRITE(*,*) variance(x)
 END PROGRAM varianceExample

Calculates the sample standard deviation of a set of values given in a vector of any size with one dimension applying the following equation:

$$\sigma = \sqrt{\frac{\sum\limits_{i=1}^{n}\left(x_{i} - \overline{x}\right)^2}{n-1}}$$

Where:

• $x$ is a vector with real numbers.
• $n$ is how many numbers are included in $x$.

Syntax

y = stdev(x)

Where:

• x = vector of rank 1 with real numbers. See this example to use an array of rank larger than 1.
• y = real number of the same kind as x with the sample standard deviation of x.

Example

The following program calculates the sample standard deviation of a vector:

 PROGRAM stdevExample
    USE FU_Statistics, ONLY: stdev
    IMPLICIT NONE
    REAL, DIMENSION(5) :: x = [1., 2., 3., 4., 5.]
    WRITE(*,*) stdev(x)
 END PROGRAM stdevExample

Calculates the population variance of a set of values given in a vector of any size with one dimension applying the following equation:

$$\sigma^{2} = \frac{\sum\limits_{i=1}^{n}\left(x_{i} - \overline{x}\right)^2}{n}$$

Where:

• $x$ is a vector with real numbers.
• $n$ is how many numbers are included in $x$.

Syntax

y = pvariance(x)

Where:

• x = vector of rank 1 with real numbers. See this example to use an array of rank larger than 1.
• y = real number of the same kind as x with the population variance of x.

Example

The following program calculates the population variance of a vector:

 PROGRAM pvarianceExample
    USE FU_Statistics, ONLY: pvariance
    IMPLICIT NONE
    REAL, DIMENSION(5) :: x = [1., 2., 3., 4., 5.]
    WRITE(*,*) pvariance(x)
 END PROGRAM pvarianceExample

Calculates the population standard deviation of a set of values given in a vector of any size with one dimension applying the following equation:

$$\sigma = \sqrt{\frac{\sum\limits_{i=1}^{n}\left(x_{i} - \overline{x}\right)^2}{n}}$$

Where:

• $x$ is a vector with real numbers.
• $n$ is how many numbers are included in $x$.

Syntax

y = pstdev(x)

Where:

• x = vector of rank 1 with real numbers. See this example to use an array of rank larger than 1.
• y = real number of the same kind as x with the population standard deviation of x.

Example

The following program calculates the population standard deviation of a vector:

 PROGRAM pstdevExample
    USE FU_Statistics, ONLY: pstdev
    IMPLICIT NONE
    REAL, DIMENSION(5) :: x = [1., 2., 3., 4., 5.]
    WRITE(*,*) pstdev(x)
 END PROGRAM pstdevExample

Calculates the sample covariance between two variables given in two vectors of any size with one dimension applying the following equation:

$$\sigma_{xy} = \frac{\sum\limits_{i=1}^{n}\left(x_{i} - \overline{x}\right)\left(y_{i} - \overline{y}\right)}{n-1}$$

Where:

• $x$ and $y$ are vectors with real numbers.
• $n$ is how many numbers are included in $x$ and $y$.

Syntax

z = covariance(x,y)

Where:

• x and y = vectors of rank 1 with real numbers. See this example to use an array of rank larger than 1.
• z = real number of the same kind as x and y with the sample covariance of x and y.

Example

The following program calculates the covariance between two variables:

 PROGRAM covarianceExample
    USE FU_Statistics, ONLY: covariance
    IMPLICIT NONE
    REAL, DIMENSION(5) :: x = [1., 2., 3., 4., 5.]
    REAL, DIMENSION(5) :: y = [6., 7., 8., 9., 10.]
    WRITE(*,*) covariance(x, y)
 END PROGRAM covarianceExample

Calculates the population covariance between two variables given in two vectors of any size with one dimension applying the following equation:

$$\sigma_{xy} = \frac{\sum\limits_{i=1}^{n}\left(x_{i} - \overline{x}\right)\left(y_{i} - \overline{y}\right)}{n}$$

Where:

• $x$ and $y$ are vectors with real numbers.
• $n$ is how many numbers are included in $x$ and $y$.

Syntax

z = pcovariance(x,y)

Where:

• x and y = vectors of rank 1 with real numbers. See this example to use an array of rank larger than 1.
• z = real number of the same kind as x and y with the population covariance of x and y.

Example

The following program calculates the population covariance between two variables:

 PROGRAM pcovarianceExample
    USE FU_Statistics, ONLY: pcovariance
    IMPLICIT NONE
    REAL, DIMENSION(5) :: x = [1., 2., 3., 4., 5.]
    REAL, DIMENSION(5) :: y = [6., 7., 8., 9., 10.]
    WRITE(*,*) pcovariance(x, y)
 END PROGRAM pcovarianceExample

Calculates the correlation coefficient between two variables given in two vectors of any size with one dimension applying the following equation:

$$\rho_{xy} = \frac{\sigma_{xy} }{\sigma_{x} \sigma_{y}}$$

Where:

• $x$ and $y$ are vectors with real numbers.
• $n$ is how many numbers are included in $x$ and $y$.

Syntax

z = correlation(x,y)

Where:

• x and y = vectors of rank 1 with real numbers. See this example to use an array of rank larger than 1.
• z = real number of the same kind as x and y with the correlation coefficient of x and y.

Example

The following program calculates the correlation coefficient between two variables:

 PROGRAM correlationExample
    USE FU_Statistics, ONLY: correlation
    IMPLICIT NONE
    REAL, DIMENSION(5) :: x = [1., 2., 3., 4., 5.]
    REAL, DIMENSION(5) :: y = [6., 7., 8., 9., 10.]
    WRITE(*,*) correlation(x, y)
 END PROGRAM correlationExample

Performs linear error (or uncertainties) propagation given the sensitivity coefficients and a covariance matrix. The following formula is applied:

$$\sigma^2_{y} = S \Sigma^{X} S^\intercal$$

Where:

• $y$ is the response whose uncertainty is to be calculated.
• $X$ is a set of input parameters to propagate their uncertainty to $y$.
• $S$ is the vector of sensitivity coefficients of $y$ with respect to the different parameters in $X$.
• $\Sigma^{x}$ is the covariance matrix of the parameters in $X$.

Syntax

y = lin_error_propagation(s,m)

Where:

• s = vector of rank 1 with real numbers containing the sensitivity coefficients.
• m = array of rank 2 containing the covariance matrix.
• y = real number of the same kind as s and m with the error or uncertainty propagated to this new variable.

Example

The following program calculates the linearly propagated error:

 PROGRAM lin_error_propagationExample
    USE FU_Statistics, ONLY: lin_error_propagation
    IMPLICIT NONE
    REAL, DIMENSION(3) :: s = [1., 2., 3.]
    REAL, DIMENSION(3,3) :: m = &
       RESHAPE((/0.5, 2., 3., 2., 5.4, 6., 3., 6., 3.3/),SHAPE(m))
    WRITE(*,*) lin_error_propagation(s, m)
 END PROGRAM lin_error_propagationExample

Calculates the median value. This function does not work with quadruple precision numbers because of the ordering subroutine written in C++.

Syntax

y = median(x)

Where:

• x = vector of rank 1 with real numbers. See this example to use an array of rank larger than 1.
• y = real number of the same kind as x with the median of x.

Example

The following program calculates the median of a vector:

 PROGRAM medianExample
    USE FU_Statistics, ONLY: median
    IMPLICIT NONE
    REAL, DIMENSION(5) :: x = [1., 2., 3., 4., 5.]
    WRITE(*,*) median(x)
 END PROGRAM medianExample

Calculates the sample skewness of a set of values given in a vector of any size with one dimension applying the following equation:

$$S_{x} = \frac{n}{\left( n-1 \right)\left( n-2 \right) }\sum\limits_{i=1}^{n}\left( \frac{x_{i}-\overline{x} }{\sigma_{x}} \right)^3$$

Where:

• $x$ is a vector with real numbers.
• $n$ is how many numbers are included in $x$.

Syntax

y = skewness(x)

Where:

• x = vector of rank 1 with real numbers. See this example to use an array of rank larger than 1.
• y = real number of the same kind as x with the sample skewness of x.

Example

The following program calculates the sample skewness of a vector:

 PROGRAM skewnessExample
    USE FU_Statistics, ONLY: skewness
    IMPLICIT NONE
    REAL, DIMENSION(5) :: x = [1., 2., 3., 4., 5.]
    WRITE(*,*) skewness(x)
 END PROGRAM skewnessExample

Calculates the population skewness of a set of values given in a vector of any size with one dimension applying the following equation:

$$S_{x} = \frac{1}{n }\sum\limits_{i=1}^{n}\left( \frac{x_{i}-\overline{x} }{\sigma_{x}} \right)^3$$

Where:

• $x$ is a vector with real numbers.
• $n$ is how many numbers are included in $x$.

Syntax

y = pskewness(x)

Where:

• x = vector of rank 1 with real numbers. See this example to use an array of rank larger than 1.
• y = real number of the same kind as x with the population skewness of x.

Example

The following program calculates the population skewness of a vector:

 PROGRAM pskewnessExample
    USE FU_Statistics, ONLY: pskewness
    IMPLICIT NONE
    REAL, DIMENSION(5) :: x = [1., 2., 3., 4., 5.]
    WRITE(*,*) pskewness(x)
 END PROGRAM pskewnessExample

Performs linear regression between two sets of values, obtaining parameters $a$ and $b$ of the following equation.

$$y = a \cdot x+b$$

Where:

• $x$ and $y$ are vectors with real numbers.
• $a$ and $b$ are the regression coefficients.

Parameter $R^2$ is also calculated to measure the goodness of fit.

Syntax

CALL linreg(x,y,a,b,R2)

Where:

• x and y = vectors of rank 1 with real numbers. See this example to use an array of rank larger than 1.
• a, b = regression coefficients calculated by the subroutine.
• R2 = the determination coefficient to measure the goodness of fit, calculated by the subroutine.

Example

The following program performs a linear regression of two variables:

 PROGRAM linregExample
    USE FU_Statistics, ONLY: linreg
    IMPLICIT NONE
    REAL, DIMENSION(5) :: x = [1., 2., 3., 4., 5.]
    REAL, DIMENSION(5) :: y = [3., 4., 5., 6., 7.]
    REAL :: a, b, R2
    CALL linreg(x, y, a, b, R2)
    WRITE(*,*) a, b, R2
 END PROGRAM linregExample

Performs logarithmic regression between two sets of values, obtaining parameters $a$ and $b$ of the following equation.

$$y = a \cdot ln(x)+b$$

Where:

• $x$ and $y$ are vectors with real numbers.
• $a$ and $b$ are the regression coefficients.

Parameter $R^2$ is also calculated to determine the goodness of fit.

Syntax

CALL logreg(x,y,a,b,R2)

Where:

• x and y = vectors of rank 1 with real numbers. See this example to use an array of rank larger than 1.
• a, b = regression coefficients calculated by the subroutine.
• R2 = the determination coefficient to measure the goodness of fit, calculated by the subroutine.

Example

The following program performs a logarithmic regression of two variables:

 PROGRAM logregExample
    USE FU_Statistics, ONLY: logreg
    IMPLICIT NONE
    REAL, DIMENSION(5) :: x = [1., 2., 3., 4., 5.]
    REAL, DIMENSION(5) :: y = [3., 4., 5., 6., 7.]
    REAL :: a, b, R2
    CALL logreg(x, y, a, b, R2)
    WRITE(*,*) a, b, R2
 END PROGRAM logregExample

Performs exponential regression between two sets of values, obtaining parameters $a$ and $b$ of the following equation.

$$y = b \cdot e^{(a \cdot x)}$$

Where:

• $x$ and $y$ are vectors with real numbers.
• $a$ and $b$ are the regression coefficients.

Parameter $R^2$ is also calculated to determine the goodness of fit.

Syntax

CALL expreg(x,y,a,b,R2)

Where:

• x and y = vectors of rank 1 with real numbers. See this example to use an array of rank larger than 1.
• a, b = regression coefficients calculated by the subroutine.
• R2 = the determination coefficient to measure the goodness of fit, calculated by the subroutine.

Example

The following program performs a exponential regression of two variables:

 PROGRAM expregExample
    USE FU_Statistics, ONLY: expreg
    IMPLICIT NONE
    REAL, DIMENSION(5) :: x = [1., 2., 3., 4., 5.]
    REAL, DIMENSION(5) :: y = [3., 4., 5., 6., 7.]
    REAL :: a, b, R2
    CALL expreg(x, y, a, b, R2)
    WRITE(*,*) a, b, R2
 END PROGRAM expregExample

Performs potential regression between two sets of values, obtaining parameters $a$ and $b$ of the following equation.

$$y = b \cdot x^a$$

Where:

• $x$ and $y$ are vectors with real numbers.
• $a$ and $b$ are the regression coefficients.

Parameter $R^2$ is also calculated to determine the goodness of fit.

Syntax

CALL potreg(x,y,a,b,R2)

Where:

• x and y = vectors of rank 1 with real numbers. See this example to use an array of rank larger than 1.
• a, b = regression coefficients calculated by the subroutine.
• R2 = the determination coefficient to measure the goodness of fit, calculated by the subroutine.

Example

The following program performs a potential regression of two variables:

 PROGRAM potregExample
    USE FU_Statistics, ONLY: potreg
    IMPLICIT NONE
    REAL, DIMENSION(5) :: x = [1., 2., 3., 4., 5.]
    REAL, DIMENSION(5) :: y = [3., 4., 5., 6., 7.]
    REAL :: a, b, R2
    CALL potreg(x, y, a, b, R2)
    WRITE(*,*) a, b, R2
 END PROGRAM potregExample