Correlation & Regression

kstats-correlation covers two related tasks: measuring the strength of association between variables, and modeling a linear relationship. The module is split into two sections reflecting this distinction.

Correlation

Pearson Correlation

Measures the strength and direction of the linear association between two numeric variables. The coefficient ranges from -1 (perfect negative) to +1 (perfect positive).

val x = doubleArrayOf(1.0, 2.0, 3.0, 4.0, 5.0)
val y = doubleArrayOf(2.1, 3.9, 6.2, 7.8, 10.1)

val r = pearsonCorrelation(x, y)
r.coefficient            // 0.9987
r.pValue                 // 0.0001
r.n                      // 5

Use Pearson when both variables are continuous and the relationship is approximately linear. Sensitive to outliers.

Spearman Correlation

Applies Pearson correlation to the ranks of the data. Measures monotonic association — whether the variables tend to increase together, regardless of linearity.

val x = doubleArrayOf(1.0, 2.0, 3.0, 4.0, 5.0, 6.0, 7.0, 8.0)
val y = doubleArrayOf(2.0, 4.0, 8.0, 16.0, 32.0, 64.0, 128.0, 256.0)

val r = spearmanCorrelation(x, y)
r.coefficient            // 1.0 — perfect monotonic relationship
r.pValue                 // 0.0

Use Spearman when the relationship is monotonic but not necessarily linear, or when the data contains outliers.

Kendall Tau

Counts concordant and discordant pairs to measure ordinal association. More robust than Spearman for small samples and heavy ties.

val x = doubleArrayOf(1.0, 2.0, 3.0, 4.0, 5.0)
val y = doubleArrayOf(1.0, 3.0, 2.0, 5.0, 4.0)

val tau = kendallTau(x, y)
tau.coefficient          // 0.6
tau.pValue               // p-value for tau

The Kendall tau implementation runs in

O(n \log n)

time using a merge-sort–based algorithm, making it efficient for large datasets.

Point-Biserial Correlation

Measures the association between a binary variable (coded as 0/1 integers) and a continuous variable. Equivalent to Pearson correlation when one variable is dichotomous.

val binary     = intArrayOf(0, 0, 0, 1, 1, 1, 1)
val continuous = doubleArrayOf(1.0, 2.0, 1.5, 4.0, 5.0, 4.5, 3.5)

val r = pointBiserialCorrelation(binary, continuous)
r.coefficient            // positive — group 1 has higher values
r.pValue                 // p-value

Use when one variable is naturally binary: treatment/control, pass/fail, male/female.

Partial Correlation

Measures the association between two variables after controlling for a third variable. Removes the effect of the confounding variable.

val x = doubleArrayOf(1.0, 2.0, 3.0, 4.0, 5.0)
val y = doubleArrayOf(2.0, 4.0, 5.0, 4.0, 5.0)
val z = doubleArrayOf(1.0, 1.0, 2.0, 3.0, 3.0)

val r = partialCorrelation(x, y, z)
r.coefficient            // correlation between x and y, controlling for z
r.pValue                 // p-value

Use when a third variable might explain the apparent relationship between the first two.

Correlation and Covariance Matrices

For multi-variable analysis, build pairwise matrices. Each cell

(i, j)

contains the correlation (or covariance) between variables

i

and

j

val x = doubleArrayOf(1.0, 2.0, 3.0, 4.0, 5.0)
val y = doubleArrayOf(2.1, 3.9, 6.2, 7.8, 10.1)
val z = doubleArrayOf(5.0, 4.0, 3.0, 2.0, 1.0)

val corr = correlationMatrix(x, y, z)
corr[0][1]               // Pearson r between x and y ≈ 0.9987
corr[0][2]               // Pearson r between x and z = -1.0

val cov = covarianceMatrix(x, y, z)
cov[0][0]                // variance of x = 2.5
cov[0][1]                // covariance of x and y

Choosing a Correlation Method

Use case	Function
Linear association between two numeric variables	`pearsonCorrelation()`
Monotonic association, robust to outliers	`spearmanCorrelation()`
Ordinal association with explicit tie handling	`kendallTau()`
Binary vs continuous variable	`pointBiserialCorrelation()`
Association after removing the effect of a third variable	`partialCorrelation()`
Pairwise summaries for many variables	`correlationMatrix()`, `covarianceMatrix()`

Math details

Pearson:

r = \frac{\sum_i (x_i - \bar{x})(y_i - \bar{y})}{\sqrt{\sum_i (x_i - \bar{x})^2 \sum_i (y_i - \bar{y})^2}}

Spearman: Pearson correlation applied to ranks.Kendall tau-b:

\tau_b = \frac{C - D}{\sqrt{(C + D + T_X)(C + D + T_Y)}}

where

C

= concordant pairs,

D

= discordant pairs,

T_X

= pairs tied only on X,

T_Y

= pairs tied only on Y.Partial correlation:

r_{xy \cdot z} = \frac{r_{xy} - r_{xz} \cdot r_{yz}}{\sqrt{(1 - r_{xz}^2)(1 - r_{yz}^2)}}

Regression

Simple Linear Regression

Fits the line

\hat{y} = \beta_0 + \beta_1 x

to the data using ordinary least squares. The result includes the slope, intercept, goodness-of-fit (

R^2

), standard errors, residuals, and a prediction function.

val x = doubleArrayOf(1.0, 2.0, 3.0, 4.0, 5.0)
val y = doubleArrayOf(2.1, 3.9, 6.2, 7.8, 10.1)

val model = simpleLinearRegression(x, y)
model.slope                  // 1.99
model.intercept              // 0.06
model.rSquared               // 0.9973
model.standardErrorSlope     // standard error of the slope estimate
model.standardErrorIntercept // standard error of the intercept estimate
model.n                      // 5
model.residuals              // [0.05, -0.07, 0.15, -0.17, 0.05]

// Prediction
model.predict(6.0)           // 11.99
model.predict(doubleArrayOf(6.0, 7.0, 8.0)) // batch prediction

Math details

\hat{\beta}_1 = \frac{\sum_i (x_i - \bar{x})(y_i - \bar{y})}{\sum_i (x_i - \bar{x})^2}, \qquad \hat{\beta}_0 = \bar{y} - \hat{\beta}_1 \bar{x}

R^2 = 1 - \frac{\sum_i (y_i - \hat{y}_i)^2}{\sum_i (y_i - \bar{y})^2}

API Reference

Full API Reference

Browse all correlation functions, result types, and parameter overloads in the Dokka-generated reference.

Getting Started

Modules

Correlation & Regression

Correlation

Pearson Correlation

Spearman Correlation

Kendall Tau

Point-Biserial Correlation

Partial Correlation

Correlation and Covariance Matrices

Choosing a Correlation Method

Regression

Simple Linear Regression

API Reference

Full API Reference

​Correlation

​Pearson Correlation

​Spearman Correlation

​Kendall Tau

​Point-Biserial Correlation

​Partial Correlation

​Correlation and Covariance Matrices

​Choosing a Correlation Method

​Regression

​Simple Linear Regression

​API Reference

Full API Reference

Correlation

Pearson Correlation

Spearman Correlation

Kendall Tau

Point-Biserial Correlation

Partial Correlation

Correlation and Covariance Matrices

Choosing a Correlation Method

Regression

Simple Linear Regression

API Reference