Procedures

Computes the regularised incomplete beta function. beta_inc and beta_cf algorithms are based on several public domain Fortran and C code, Lentz's algorithm (1976), and modified to use 2008+ intrinsics.

Impure wrapper function for f_dst_chi2_cdf_core. Handles optional arguments and invalid values for arguments.

Cumulative distribution function for the chi-squared distribution.

Impure wrapper function for f_dst_chi2_pdf_core. Handles optional arguments and invalid values for arguments.

Probability density function for the chi-squared distribution.

Impure wrapper function for f_dst_chi2_ppf_core. Handles optional arguments and invalid values for arguments.

Percent point function/quantile functionfor the chi-squared distribution.

Impure wrapper function for f_dst_exp_cdf_core. Handles optional arguments and invalid values for arguments.

Cumulative distribution function for exponential distribution.

Impure wrapper function for f_dst_exp_pdf_core. Handles optional arguments and invalid values for arguments.

Probability density function for exponential distribution.

Impure wrapper function for f_dst_exp_ppf_core. Handles optional arguments and invalid values for arguments.

Percent point function/quantile function for exponential distribution.

Impure wrapper function for f_dst_f_cdf_core. Handles optional arguments and invalid values for arguments.

Cumulative density function for the F distribution.

Impure wrapper function for f_dst_f_pdf_core. Handles optional arguments and invalid values for arguments.

Probability density function for the F distribution.

Impure wrapper function for f_dst_f_ppf_core. Handles optional arguments and invalid values for arguments.

Percent point function / quantile function for the F distribution.

Impure wrapper function for f_dst_gamma_cdf_core. Handles optional arguments and invalid values for arguments.

Cumulative distribution function for gamma distribution.

Impure wrapper function for f_dst_gamma_pdf_core. Handles optional arguments and invalid values for arguments.

Probability density function for gamma distribution.

Impure wrapper function for f_dst_gamma_ppf_core. Handles optional arguments and invalid values for arguments.

Percent point function/quantile function for gamma distribution.

Incomplete gamma function. Needed by gamma and chi-squared cdf. Uses Fortran 2008+ intrinsics.

Impure wrapper function for f_dst_gpd_cdf_core. Handles optional arguments and invalid values for arguments.

Cumulative distribution function for generalised pareto distribution.

Impure wrapper function for f_dst_gpd_pdf_core. Handles optional arguments and invalid values for arguments.

Probability density function for generalised pareto distribution.

Impure wrapper function for f_dst_gpd_ppf_core. Handles optional arguments and invalid values for arguments.

Percent point function/quantile function for generalised pareto distribution.

Impure wrapper function for f_dst_norm_cdf_core. Handles optional arguments and invalid values for arguments.

Cumulative distribution function for normal distribution.

Impure wrapper function for f_dst_norm_pdf_core. Handles optional arguments and invalid values for arguments.

Probability density function for normal distribution.

Impure wrapper function for f_dst_norm_ppf_core. Handles optional arguments and invalid values for arguments.

Percent point function/quantile function for normal distribution.

Impure wrapper function for f_dst_t_cdf_core. Handles optional arguments and invalid values for arguments.

Cumulative distribution function for student t distribution.

Impure wrapper function for f_dst_t_pdf_core. Handles optional arguments and invalid values for arguments.

Probability density function for student t distribution.

Impure wrapper function for f_dst_t_ppf_core. Handles optional arguments and invalid values for arguments.

Percent point function/quantile function for t distribution.

Impure wrapper function for f_sts_cov_core.

Computes covariance.

Impure wrapper function for f_sts_mean_core.

Computes arithmetic mean.

Impure wrapper function for f_sts_trend_core.

Computes Pearson correlation coefficient.

Impure wrapper function for f_sts_std_core.

Computes standard deviation.

Impure wrapper function for f_sts_trend_core.

Computes regression coefficient/trend.

Impure wrapper function for f_sts_var_core.

Computes (sample) variance.

Converts char to real.

Convert integer to char.

Convert real to char.

The one-way ANOVA (Analysis of Variance) tests whether three or more population means $\mu_1, \mu_2, \dots, \mu_k$ are equal.

Cumulative distribution function $F(x) = \mathbb{P}(X \leq x)$ for the chi-squared distribution.

Probability density function for the chi-squared distribution. Uses intrinsic exp and gamma function. $$f(x) = \frac{x^{\frac{k}{2} - 1} \cdot e^{ - \frac{x}{2} }}{2^{\frac{k}{2}} \cdot \Gamma\left(\frac{k}{2}\right)}, \quad x \geq 0, \ k > 0$$ where $k$ = degrees of freedom (df) and $\Gamma$ is the gamma function.

Percent point function/quantile function $Q(p) = {F}_{x}^{-1}(p)$ for the chi-squared distribution. Uses the bisection method for numerical inversion of the CDF.

Computes the population or sample covariance (depending on passed arguments). $$\operatorname{cov}(x, y) = \frac{1}{n - \nu} \cdot \sum_{i=1}^{n} (x_i - \bar{x}) \cdot (y_i - \bar{y})$$ where $n$ is the size of (or number of observations in) vectors x and y, $x_i$ and $y_i$ are individual elements in x and y, $\nu$ (ddof) is a degrees of freedom adjustment (ddof = 0.0 for population variance, ddof = 1.0 for sample variance), and $\bar{x}$ and $\bar{y}$ are the arithmetic means of x and y.

Cumulative distribution function $F(x) = \mathbb{P}(X \leq x)$ for exponential distribution.

Probability density function for exponential distribution. Uses intrinsic exp function. $$f(x) = \lambda \cdot e^{-\lambda \cdot x}, \quad x \geq 0, \ \lambda > 0$$

Percent point function/quantile function $Q(p) = {F}_{x}^{-1}(p)$ for exponential distribution. Procedure uses bisection method. p should be between 0.0 and 1.0.

Cumulative density function $F(x) = \mathbb{P}(X \leq x)$ for the F distribution.

Probability density function for the F distribution. $$f(x) = \frac{1}{\mathrm{B}\left(\frac{d_1}{2}, \frac{d_2}{2}\right)} \cdot \left( \frac{d_1}{d_2} \right)^{ \frac{d_1}{2} } \cdot \frac{x^{ \frac{d_1}{2} - 1}}{\left(1 + \frac{d_1}{d_2} x \right)^{ \frac{d_1 + d_2}{2} }}, \quad x > 0$$ where $d_1$ = numerator degrees of freedom, $d_2$ = denominator degrees of freedom and $B$ is the complete beta function. (Uses intrinsic gamma function for beta.)

Percent point function / quantile function $Q(p) = F^{-1}(p)$ for the F distribution. Uses the bisection method to numerically invert the CDF.

Cumulative distribution function $F(x) = \mathbb{P}(X \leq x)$ for gamma distribution.

Probability density function for gamma distribution. Uses intrinsic exp function. $$f(x) = \frac{\lambda^\alpha}{\Gamma(\alpha)} \cdot x^{\alpha - 1} \cdot e^{-\lambda \cdot x}, \quad x > 0, \ \alpha > 0, \ \lambda > 0$$

Percent point function/quantile function $Q(p) = {F}_{x}^{-1}(p)$ for gamma distribution. Procedure uses bisection method. p should be between 0.0 and 1.0.

Cumulative distribution function $F(x) = \mathbb{P}(X \leq x)$ for generalised pareto distribution.

Probability density function for generalised pareto distribution. $$f(x) = \frac{1}{\sigma} \cdot \left ( 1 + \frac{\xi \cdot (x - \mu)}{\sigma} \right)^{-\frac{1}{\xi} - 1}, \quad x \geq \mu, \ \sigma > 0, \ \xi \in \mathbb{R}$$ where $\xi$ is a shape parameter (xi), $\sigma$ is the scale parameter (sigma), $\mu$ (mu) is the location (not mean).

Percent point function/quantile function $Q(p) = {F}_{x}^{-1}(p)$ for generalised pareto distribution. Procedure uses bisection method. p must be between 0.0 and 1.0.

The Kruskal-Wallis H-test is used to determine whether samples originate from the same distribution without assuming normality. It is therefore considered a nonparametric alternative to the one-way ANOVA (Analysis of Variance).

Computes arithmetic mean. $$\bar{x} = \frac{1}{n} \cdot \sum_{i=1}^{n} x_i$$ where $n$ is the size of (or number of observations in) vector x, $x_i$ are individual elements in x, and $\bar{x}$ is the arithmetic mean of x.

Cumulative distribution function $F(x) = \mathbb{P}(X \leq x)$ for normal distribution.

Probability density function for normal distribution. $$f(x) = \frac{1}{\sigma \cdot \sqrt{2 \cdot \pi}} e^{ -\frac{1}{2} \cdot \left( \frac{x - \mu}{\sigma} \right)^2 }$$

Percent point function/quantile function $Q(p) = {F}_{x}^{-1}(p)$ for normal distribution.

Principal Component Analysis (PCA) or Empirical Orthogonal Function (EOF) analysis is a procedure that reduces the dimensionality of multivariate data by identifying a set of orthogonal vectors (eigenvectors or EOFs) that represent directions of maximum variance in the dataset. EOF analysis is often used interchangably with the geographically weighted PCA. As they are mathematically identical, a single pca procedure is offered with optional arguments and outputs that also makes it usable as a classic EOF analysis.

Computes Pearson correlation coefficient (PCC). $$\rho_{x,y} = \frac{\operatorname{cov}(x, y)}{\sigma_x \cdot \sigma_y}$$ where $\rho_{x,y}$ is the Pearson correlation coefficient for vectors x and y, $\operatorname{cov}(x, y)$ is the covariance of x and y, and $\sigma_{x}$ and $\sigma_{y}$ are the standard deviations of x and y.

Ranks all samples such that the smallest value obtains rank 1 and the largest rank n. Handles tied ranks and assigns average rank to tied elements within one group of tied elements.

The ranks sum test (Wilcoxon rank-sum test or Mann–Whitney U test) is a non-parametric test to determine if two independent samples $x_1$ and $x_2$ are have the same distribution. It can be regarded as the non-parametric equivalent of the 2-sample t-test.

Read CSV file directly into dataframe.

The 1-sample Wilcoxon signed rank test is a non-parametric test that determines if data comes from a symmetric population with centre $mu_0$. It can be regarded as a non-parametric version of the 1-sample t-test.

The Wilcoxon signed rank test is a non-parametric test that determines if two related paired samples come from the same distribution. It can be regarded as a non-parametric version of the paired t-test.

Computes the population or sample standard deviation (depending on passed arguments). $$\sigma = \sqrt{\operatorname{var}(x)}$$ where $\operatorname{var}(x)$ is the variance of vector x. $\nu$ (ddof) can also be passed and serves as a degrees of freedom adjustment when the variance is caulculated. (ddof = 0.0 for population standard deviation, ddof = 1.0 for sample standard deviation)

Cumulative distribution function $F(x) = \mathbb{P}(X \leq x)$ for student t distribution.

Probability density function for student t distribution. Uses intrinsic gamma function (Fortran 2008 and later). $$f(x) = \frac{\Gamma\left(\frac{\nu + 1}{2}\right)}{\sqrt{\nu \cdot \pi}\, \cdot \Gamma\left(\frac{\nu}{2}\right)} \left(1 + \frac{x^2}{\nu}\right)^{-\frac{\nu + 1}{2}}$$ where $v$ = degrees of freedom (df) and $\Gamma$ is the gamma function.

Percent point function/quantile function $Q(p) = {F}_{x}^{-1}(p)$ for t distribution.

Computes regression coefficient/trend. $$m = \frac{\operatorname{cov}(x, y)}{\operatorname{var}(x)}$$ where $m$ is the slope of the regression line (linear trend), $\operatorname{cov}(x, y)$ is the covariance of x and y, and $\operatorname{var}(x)$ is the variance of x.

The 1-sample t-test determines if the sample mean has the value specified in the null hypothesis.

The 2-sample t-test determines if two population means $\mu_1$ and $\mu_2$ are the same. The procedure can handle 2-sample t-tests for equal variances and Welch's t-tests for unequal variances.

The paired sample t-test (or dependent sample t-test) determines if the mean difference between two sample sets are zero. It is mathematically equivalent to the 1-sample t-test conducted on the difference vector $d$ with $\mu_0 = 0$.

Computes the population or sample variance (depending on passed arguments). $$\operatorname{var}(x) = \frac{1}{n - \nu} \cdot \sum_{i=1}^{n} (x_i - \bar{x})^2$$ where $n$ is the size of (or number of observations in) vector x, $x_i$ are individual elements in x, $\nu$ (ddof) is a degrees of freedom adjustment (ddof = 0.0 for population variance, ddof = 1.0 for sample variance), and $\bar{x}$ is the arithmetic mean of x.

Read CSV file directly into dataframe.

Prints error message in specific format.

Prints warning message in specific format.

Empirical Orthogonal Function (EOF) analysis / Principal Component Analysis (PCA)

Impure wrapper procedure for s_tst_anova_1w_core.

One-way ANOVA.

Impure wrapper procedure for s_tst_kruskalwallis_core.

Kruskal-Wallis H-test for independent samples. No tie correction.

Impure wrapper procedure for s_tst_ranksum_core.

The ranks sum test (Wilcoxon rank-sum test or Mann–Whitney U test).

Impure wrapper procedure for s_tst_signedrank_1s_core.

The 1-sample Wilcoxon signed rank test.

Impure wrapper procedure for s_tst_signedrank_2s_core.

The Wilcoxon signed rank test.

Impure wrapper procedure for s_tst_ttest_1s_core.

The 1-sample t-test.

Impure wrapper procedure for s_tst_ttest_2s_core.

The 2-sample t-test.

Impure wrapper procedure for s_tst_ttest_paired_core.

The paired sample t-test (or dependent sample t-test). It is a special case of s_tst_ttest_1s offered and uses the same pure procedure (s_tst_ttest_1s_core).

Ranks all samples such that the smallest value obtains rank 1 and the largest rank n. Handles tied ranks and assigns average rank to tied elements within one group of tied elements.