Correlation function for discrete variable correlation (#7326)

* Correlation function for discrete variable correlation

* Add unit tests for Correlation class

Added unit tests for the Correlation class to validate correlation calculations under various scenarios, including linear dependence and constant values.

* Added missing bracket

* Refactor variable initialization in correlation method

* Remove unused imports and clean up CorrelationTest

* Fix formatting of variable declarations in correlation method

* Update Correlation.java

* Fix formatting in CorrelationTest.java

* Enhance comments in correlation function

Added detailed comments to the correlation function for better understanding.

* Add correlation tests for various scenarios

* Format comments for clarity in correlation method

* Fix formatting and comments in correlation method

Author: kvadrik

src/main/java/com/thealgorithms/maths/Correlation.java

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+package com.thealgorithms.maths;
+
+/**
+ * Class for correlation of two discrete variables
+ */
+
+public final class Correlation {
+    private Correlation() {
+    }
+
+    public static final double DELTA = 1e-9;
+
+    /**
+     * Discrete correlation function.
+     * Correlation between two discrete variables is calculated
+     * according to the formula: Cor(x, y)=Cov(x, y)/sqrt(Var(x)*Var(y)).
+     * Correlation with a constant variable is taken to be zero.
+     *
+     * @param x The first discrete variable
+     * @param y The second discrete variable
+     * @param n The number of values for each variable
+     * @return The result of the correlation of variables x,y.
+     */
+    public static double correlation(double[] x, double[] y, int n) {
+        double exy = 0; // E(XY)
+        double ex = 0; // E(X)
+        double exx = 0; // E(X^2)
+        double ey = 0; // E(Y)
+        double eyy = 0; // E(Y^2)
+        for (int i = 0; i < n; i++) {
+            exy += x[i] * y[i];
+            ex += x[i];
+            exx += x[i] * x[i];
+            ey += y[i];
+            eyy += y[i] * y[i];
+        }
+        exy /= n;
+        ex /= n;
+        exx /= n;
+        ey /= n;
+        eyy /= n;
+        double cov = exy - ex * ey; // Cov(X, Y) = E(XY)-E(X)E(Y)
+        double varx = Math.sqrt(exx - ex * ex); // Var(X) = sqrt(E(X^2)-E(X)^2)
+        double vary = Math.sqrt(eyy - ey * ey); // Var(Y) = sqrt(E(Y^2)-E(Y)^2)
+        if (varx * vary < DELTA) { // Var(X) = 0 means X = const, the same about Y
+            return 0;
+        } else {
+            return cov / Math.sqrt(varx * vary);
+        }
+    }
+}

src/test/java/com/thealgorithms/maths/CorrelationTest.java

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+package com.thealgorithms.maths;
+
+import static org.junit.jupiter.api.Assertions.assertEquals;
+
+/**
+ * Test class for Correlation class
+ */
+public class CorrelationTest {
+
+    public static final double DELTA = 1e-9;
+
+    // Regular correlation test
+    public void testCorrelationFirst() {
+        double[] x = {1, 2, 3, 4};
+        double[] y = {7, 1, 4, 9};
+        int n = 4;
+        assertEquals(0.3319700011, Correlation.correlation(x, y, n), DELTA);
+    }
+
+    // Regular correlation test (zero correlation)
+    public void testCorrelationSecond() {
+        double[] x = {1, 2, 3, 4};
+        double[] y = {5, 0, 9, 2};
+        int n = 4;
+        assertEquals(0, Correlation.correlation(x, y, n), DELTA);
+    }
+
+    // Correlation with a constant variable is taken to be zero
+    public void testCorrelationConstant() {
+        double[] x = {1, 2, 3};
+        double[] y = {4, 4, 4};
+        int n = 3;
+        assertEquals(0, Correlation.correlation(x, y, n), DELTA);
+    }
+
+    // Linear dependence gives correlation 1
+    public void testCorrelationLinearDependence() {
+        double[] x = {1, 2, 3, 4};
+        double[] y = {6, 8, 10, 12};
+        int n = 4;
+        assertEquals(1, Correlation.correlation(x, y, n), DELTA);
+    }
+
+    // Inverse linear dependence gives correlation -1
+    public void testCorrelationInverseLinearDependence() {
+        double[] x = {1, 2, 3, 4, 5};
+        double[] y = {18, 15, 12, 9, 6};
+        int n = 5;
+        assertEquals(-1, Correlation.correlation(x, y, n), DELTA);
+    }
+}