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 (lower). 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_lin_mahalanobis_core.

Compute Mahalanobis distance between vectors x and y using covariance matrix cov if provided; otherwise estimate covariance from the two-sample dataset formed by x and y. NOTE: check if cov matrix is positive definite.

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_median_core.

Computes median using s_utl_rank for tie-aware ranking

Impure wrapper function for f_sts_trend_core.

Computes Pearson correlation coefficient.

Impure wrapper for f_sts_scc_core.

Computes Spearman rank correlation coefficient between x and y. Uses f_sts_pcc_core on ranks.

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.

Empirical Orthogonal Function (EOF) analysis is a procedure to reduce the dimensionality of multivariate data by identifying a set of orthogonal vectors (EOFs or eigenvectores) that represent directions of maximum variance in the dataset. The term EOF analysis is often used interchangably with the geographically weighted principal component analysis (PCA). The procedures are mathematically equivalent, but procedures for EOF analysis offer some additional options that are mostly relevant for geoscience. The procedure fsml_pca is a wrapper for fsml_eof that offers a simpler, more familiar interface for non-geoscientists.

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 procedure is an implementation of the agglomerative hierarchical clustering method that groups data points into clusters by iteratively merging the most similar clusters. The procedure uses centroid linkage and the Mahalanobis distance as a measure of similarity.

The procedure implements a hybrid clustering approach combining agglomerative hierarchical clustering and k-means clustering, both using the Mahalanobis distance as the similarity measure. The hierarchical step first partitions the data into nc clusters by iteratively merging the most similar clusters. The resulting centroids from are then used as initial centroids (cm_in) for the k-means procedure, which refines them iteratively.

The procedure implements the K-means clustering algorithm using the Mahalanobis distance as the similarity measure. It accepts initial centroids (cm_in), refines them iteratively, and returns the final centroids (cm).

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).

interface fsml_lda_2class The 2-class multivariate Linear Discriminant Analysis (LDA) is a statistical procedure for classification and the investigation and explanation of differences between two groups (or classes) with regard to their attribute variables. It quantifies the discriminability of the groups and the contribution of each of the attribute variables to this discriminability.

Computes the Mahalanobis distance between two input feature vectors x and y. If a covariance matrix cov is provided, it is used directly in the calculation. Otherwise, the procedure estimates the covariance matrix from the two-sample dataset formed by x and y. A Cholesky-based solver is used to perform the distance calculation.

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.

Computes median of vector x. The procedures can handle tied ranks.

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.

The multiple linear Ordinary Least Squares (OLS) regression models the relationship or linear dependence between a dependent (predictand) variable and and one or more independent (predictor) variables. The procedure estimates the linear regression coefficients by minimising the sum of squared residuals.

Principal Component Analysis (PCA) is a procedure to reduce the dimensionality of multivariate data by identifying a set of orthogonal vectors (eigenvectores) that represent directions of maximum variance in the dataset.

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 multiple linear Ridge regression models the relationship or linear dependence between a dependent (predictand) variable and one or more independent (predictor) variables, incorporating a penalty term on the size of the regression coefficients to reduce multicollinearity and overfitting.

Computes the Spearman rank correlation coefficient (SCC). The procedure gets the ranks of cectors x and y, then calculates the Pearson correlation coefficient on these ranks.

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

2-class multivariate Linear Discriminant Analysis (LDA)

Multiple Linear Ordinary Least Squares (OLS) Regression with intercept. NOTE: OLS could be wrapper for ridge (with lambda = 0 or presence checks if mande an optional argument). However, it would increase computation slightly and make code less readable. OLS is often used in teaching and therefore, an easily readable standalone is kept.

Principal Component Analysis (PCA). It is a special (simplified) case of EOF analysis offered as a separate procedure for clarity/familiarity. It calls s_lin_eof with equal weights.

Multiple Linear Ridge Regression (λ >= 0) with intercept.

Impure wrapper procedure for s_nlp_hclust_core.

Perform agglomerative hierarchical clustering using centroid linkage and the Mahalanobis distance. NOTE: The procedure is exact, but slow for large nd. For most pracitcal purposes, using Lance–Williams algorithm and other distances is advised. TODO: Implement distance switch and L-W algorithm + approx. distance updates.

Impure wrapper procedure for s_nlp_hkmeans_core.

Perform agglomerative hierarchical clustering using centroid linkage and the Mahalanobis distance, then passes cluster centroids and covariance matrix to kmeans cluster procedure for refinement.

Impure wrapper procedure for s_nlp_kmeans_core.

K-means clustering using Mahalanobis distance. NOTE: think about only accepting standardised data to avoid redundant computation in successive calls of procedure. This and repeated Cholesky fractionisation are potential performance bottlenecks.

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 and uses the same pure procedure (s_tst_ttest_1s_core).

Solve a * x = b for x using Cholesky factor returned by stdlib's chol(). a : (n,n) symmetric positive-definite b : (n)

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.

Sort real array in ascending (mode=1) or descending (mode=2) order. Preserves the input array. Outputs sorted array and index mapping.