Spearman Rank Correlation and Kendall Tau

/**
 * Fintech Math Bootcamp · Lesson 084 of 120
 * Spearman Rank Correlation and Kendall Tau
 * Module 09: Dependence, Regression, and Model Foundations
 *
 * Scenario: Comparing monotone rankings and pair ordering
 * Rule:     Spearman = Pearson(ranks); Kendall tau-a=(C−D)/choose(n,2)
 *
 * Try it:   Is the Kendall convention in this course tau-b?
 *
 * Lesson article: https://thefintechbuilder.com/financial-mathematics-statistics-and-data-foundations/dependence-regression-and-model-foundations/spearman-rank-correlation-and-kendall-tau/
 * Free course:    https://courses.thefintechbuilder.com
 * Synthetic teaching example, not financial advice or a production library.
 */

export function lesson084() {
  const x=[1,2,3,4,5], y=[2,4,5,4,5];
  const ranks=(a:number[])=>a.map(v=>1+a.filter(z=>z<v).length+(a.filter(z=>z===v).length-1)/2);
  const n=x.length, rx=ranks(x), ry=ranks(y), m=(n+1)/2; // mean rank
  const cross=rx.reduce((s,v,i)=>s+(v-m)*(ry[i]-m),0);
  const ss=(a:number[])=>a.reduce((s,v)=>s+(v-m)**2,0);
  let signedPairs=0;
  for(let i=0;i<n;i++) for(let j=i+1;j<n;j++)
    signedPairs+=Math.sign((x[j]-x[i])*(y[j]-y[i]));
  const result={rx,ry,spearman:cross/Math.sqrt(ss(rx)*ss(ry)),tauA:signedPairs/(n*(n-1)/2)};
  return result;
}

export const checkedResult = {"rx":[1,2,3,4,5],"ry":[1,2.5,4.5,2.5,4.5],"spearman":0.7378647873726218,"tauA":0.6};

// Run this file directly: npx tsx lessons/09-dependence-regression-and-model-foundations/084-spearman-rank-correlation-and-kendall-tau.ts
if (process.argv[1] && import.meta.url.endsWith(process.argv[1].replace(/\\/g, "/").split("/").pop()!)) {
  console.log(JSON.stringify(lesson084(), null, 2));
}

"""
Fintech Math Bootcamp · Lesson 084 of 120
Spearman Rank Correlation and Kendall Tau
Module 09: Dependence, Regression, and Model Foundations

Scenario: Comparing monotone rankings and pair ordering
Rule:     Spearman = Pearson(ranks); Kendall tau-a=(C−D)/choose(n,2)

Try it:   Is the Kendall convention in this course tau-b?

Lesson article: https://thefintechbuilder.com/financial-mathematics-statistics-and-data-foundations/dependence-regression-and-model-foundations/spearman-rank-correlation-and-kendall-tau/
Free course:    https://courses.thefintechbuilder.com
Synthetic teaching example, not financial advice or a production library.
"""

import json
import math


def sign(v):
    return (v > 0) - (v < 0)


def ranks(a):
    """Average ranks: ties share the mean of the positions they occupy."""
    return [1 + sum(1 for z in a if z < v) + (sum(1 for z in a if z == v) - 1) / 2 for v in a]


def lesson_084():
    x, y = [1, 2, 3, 4, 5], [2, 4, 5, 4, 5]
    n, rx, ry = len(x), ranks(x), ranks(y)
    m = (n + 1) / 2  # mean rank
    cross = 0
    for a, b in zip(rx, ry):
        cross += (a - m) * (b - m)

    def ss(a):
        total = 0
        for v in a:
            total += (v - m) ** 2
        return total

    signed_pairs = 0
    for i in range(n):
        for j in range(i + 1, n):
            signed_pairs += sign((x[j] - x[i]) * (y[j] - y[i]))
    return {
        "rx": rx,
        "ry": ry,
        "spearman": cross / math.sqrt(ss(rx) * ss(ry)),
        "tauA": signed_pairs / (n * (n - 1) / 2),
    }


if __name__ == "__main__":
    print(json.dumps(lesson_084(), indent=2))

// Fintech Math Bootcamp - Lesson 084 of 120
// Spearman Rank Correlation and Kendall Tau
// Module 09: Dependence, Regression, and Model Foundations
//
// Scenario: Comparing monotone rankings and pair ordering
// Rule:     Spearman = Pearson(ranks); Kendall tau-a=(C-D)/choose(n,2)
//
// Try it:   Is the Kendall convention in this course tau-b?
//
// Lesson article: https://thefintechbuilder.com/financial-mathematics-statistics-and-data-foundations/dependence-regression-and-model-foundations/spearman-rank-correlation-and-kendall-tau/
// Free course:    https://courses.thefintechbuilder.com
// Synthetic teaching example, not financial advice or a production library.
import java.util.ArrayList;
import java.util.Arrays;
import java.util.LinkedHashMap;
import java.util.List;
import java.util.Map;

public class Main {
    // Average ranks: ties share the mean of the positions they occupy.
    static double[] ranks(double[] a) {
        double[] out = new double[a.length];
        for (int i = 0; i < a.length; i++) {
            int below = 0, equal = 0;
            for (double z : a) {
                if (z < a[i]) below++;
                if (z == a[i]) equal++;
            }
            out[i] = 1 + below + (equal - 1) / 2.0;
        }
        return out;
    }

    static double sumOfSquares(double[] a, double center) {
        double total = 0;
        for (double v : a) total += Math.pow(v - center, 2);
        return total;
    }

    static Map<String, Object> lesson084() {
        double[] x = {1, 2, 3, 4, 5}, y = {2, 4, 5, 4, 5};
        int n = x.length;
        double[] rx = ranks(x), ry = ranks(y);
        double m = (n + 1) / 2.0; // mean rank
        double cross = 0;
        for (int i = 0; i < n; i++) cross += (rx[i] - m) * (ry[i] - m);
        double signedPairs = 0;
        for (int i = 0; i < n; i++) {
            for (int j = i + 1; j < n; j++) {
                signedPairs += Math.signum((x[j] - x[i]) * (y[j] - y[i]));
            }
        }
        Map<String, Object> result = new LinkedHashMap<String, Object>();
        result.put("rx", rx);
        result.put("ry", ry);
        result.put("spearman", cross / Math.sqrt(sumOfSquares(rx, m) * sumOfSquares(ry, m)));
        result.put("tauA", signedPairs / (n * (n - 1) / 2.0));
        return result;
    }

    public static void main(String[] args) {
        System.out.println(toJson(lesson084(), ""));
    }

    // Minimal JSON writer: two-space indent, whole numbers without a decimal point, NaN as null.
    static String toJson(Object value, String indent) {
        if (value == null) return "null";
        if (value instanceof Boolean) return value.toString();
        if (value instanceof Number) return formatNumber(((Number) value).doubleValue());
        if (value instanceof String) return quote((String) value);
        if (value instanceof double[]) {
            List<Object> boxed = new ArrayList<Object>();
            for (double d : (double[]) value) boxed.add(d);
            return toJson(boxed, indent);
        }
        if (value instanceof Object[]) return toJson(Arrays.asList((Object[]) value), indent);
        String inner = indent + "  ";
        StringBuilder out = new StringBuilder();
        if (value instanceof Map) {
            Map<?, ?> map = (Map<?, ?>) value;
            if (map.isEmpty()) return "{}";
            out.append("{\n");
            int i = 0;
            for (Map.Entry<?, ?> entry : map.entrySet()) {
                out.append(inner).append(quote(entry.getKey().toString())).append(": ")
                   .append(toJson(entry.getValue(), inner));
                out.append(++i < map.size() ? ",\n" : "\n");
            }
            return out.append(indent).append("}").toString();
        }
        List<?> list = (List<?>) value;
        if (list.isEmpty()) return "[]";
        out.append("[\n");
        for (int i = 0; i < list.size(); i++) {
            out.append(inner).append(toJson(list.get(i), inner));
            out.append(i + 1 < list.size() ? ",\n" : "\n");
        }
        return out.append(indent).append("]").toString();
    }

    static String formatNumber(double x) {
        if (Double.isNaN(x) || Double.isInfinite(x)) return "null";
        if (x == Math.rint(x) && Math.abs(x) < 1e15) return Long.toString((long) x);
        return Double.toString(x);
    }

    static String quote(String s) {
        StringBuilder out = new StringBuilder("\"");
        for (char c : s.toCharArray()) {
            if (c == '"' || c == '\\') out.append('\\').append(c);
            else if (c == '\n') out.append("\\n");
            else if (c < 0x20) out.append(String.format("\\u%04x", (int) c));
            else out.append(c);
        }
        return out.append('"').toString();
    }
}

// Fintech Math Bootcamp · Lesson 084 of 120
// Spearman Rank Correlation and Kendall Tau
// Module 09: Dependence, Regression, and Model Foundations
//
// Scenario: Comparing monotone rankings and pair ordering
// Rule:     Spearman = Pearson(ranks); Kendall tau-a=(C−D)/choose(n,2)
//
// Try it:   Is the Kendall convention in this course tau-b?
//
// Lesson article: https://thefintechbuilder.com/financial-mathematics-statistics-and-data-foundations/dependence-regression-and-model-foundations/spearman-rank-correlation-and-kendall-tau/
// Free course:    https://courses.thefintechbuilder.com
// Synthetic teaching example, not financial advice or a production library.
package main

import (
	"encoding/json"
	"fmt"
	"math"
)

type Lesson084Result struct {
	Rx       []float64 `json:"rx"`
	Ry       []float64 `json:"ry"`
	Spearman float64   `json:"spearman"`
	TauA     float64   `json:"tauA"`
}

// ranks assigns average ranks: ties share the mean of the positions they occupy.
func ranks(a []float64) []float64 {
	out := make([]float64, len(a))
	for i, v := range a {
		below, equal := 0, 0
		for _, z := range a {
			if z < v {
				below++
			}
			if z == v {
				equal++
			}
		}
		out[i] = 1 + float64(below) + float64(equal-1)/2
	}
	return out
}

func sign(v float64) float64 {
	switch {
	case v > 0:
		return 1
	case v < 0:
		return -1
	}
	return 0
}

func lesson084() Lesson084Result {
	x := []float64{1, 2, 3, 4, 5}
	y := []float64{2, 4, 5, 4, 5}
	n := len(x)
	rx, ry := ranks(x), ranks(y)
	m := float64(n+1) / 2 // mean rank
	cross := 0.0
	for i := range rx {
		cross += (rx[i] - m) * (ry[i] - m)
	}
	ss := func(a []float64) float64 {
		total := 0.0
		for _, v := range a {
			total += math.Pow(v-m, 2)
		}
		return total
	}
	signedPairs := 0.0
	for i := 0; i < n; i++ {
		for j := i + 1; j < n; j++ {
			signedPairs += sign((x[j] - x[i]) * (y[j] - y[i]))
		}
	}
	return Lesson084Result{
		Rx:       rx,
		Ry:       ry,
		Spearman: cross / math.Sqrt(ss(rx)*ss(ry)),
		TauA:     signedPairs / float64(n*(n-1)/2),
	}
}

func main() {
	out, err := json.MarshalIndent(lesson084(), "", "  ")
	if err != nil {
		panic(err)
	}
	fmt.Println(string(out))
}

// Fintech Math Bootcamp · Lesson 084 of 120
// Spearman Rank Correlation and Kendall Tau
// Module 09: Dependence, Regression, and Model Foundations
//
// Scenario: Comparing monotone rankings and pair ordering
// Rule:     Spearman = Pearson(ranks); Kendall tau-a=(C−D)/choose(n,2)
//
// Try it:   Is the Kendall convention in this course tau-b?
//
// Lesson article: https://thefintechbuilder.com/financial-mathematics-statistics-and-data-foundations/dependence-regression-and-model-foundations/spearman-rank-correlation-and-kendall-tau/
// Free course:    https://courses.thefintechbuilder.com
// Synthetic teaching example, not financial advice or a production library.
#include <cmath>
#include <cstdio>
#include <cstdlib>
#include <iostream>
#include <optional>
#include <stdexcept>
#include <string>
#include <utility>
#include <vector>

// A minimal JSON value, enough to print this lesson's result.
struct Json {
    enum class Kind { Null, Bool, Number, String, Array, Object };
    Kind kind = Kind::Null;
    bool flag = false;
    double number = 0.0;
    std::string text;
    std::vector<std::string> keys;  // object keys, parallel to items
    std::vector<Json> items;        // array elements or object values

    Json() = default;
    Json(bool value) : kind(Kind::Bool), flag(value) {}
    Json(int value) : kind(Kind::Number), number(value) {}
    Json(double value) : kind(Kind::Number), number(value) {}
    Json(const char* value) : kind(Kind::String), text(value) {}
    Json(const std::string& value) : kind(Kind::String), text(value) {}
    Json(const std::vector<double>& values) : kind(Kind::Array) {
        for (double v : values) items.push_back(Json(v));
    }
};

Json jsonArray(const std::vector<Json>& values) {
    Json array;
    array.kind = Json::Kind::Array;
    array.items = values;
    return array;
}

Json jsonObject(const std::vector<std::pair<std::string, Json>>& fields) {
    Json object;
    object.kind = Json::Kind::Object;
    for (const auto& field : fields) {
        object.keys.push_back(field.first);
        object.items.push_back(field.second);
    }
    return object;
}

// Shortest decimal form that reads back as the same double.
std::string formatNumber(double x) {
    if (!std::isfinite(x)) return "null";
    char buffer[32];
    if (x == std::floor(x) && std::fabs(x) < 1e15) {
        std::snprintf(buffer, sizeof buffer, "%.0f", x);
        return buffer;
    }
    for (int precision = 1; precision <= 17; ++precision) {
        std::snprintf(buffer, sizeof buffer, "%.*g", precision, x);
        if (std::strtod(buffer, nullptr) == x) break;
    }
    return buffer;
}

std::string quote(const std::string& s) {
    std::string out = "\"";
    for (char c : s) {
        if (c == '"' || c == '\\') { out += '\\'; out += c; }
        else if (c == '\n') out += "\\n";
        else out += c;
    }
    return out + "\"";
}

std::string toJson(const Json& value, const std::string& indent = "") {
    switch (value.kind) {
        case Json::Kind::Null: return "null";
        case Json::Kind::Bool: return value.flag ? "true" : "false";
        case Json::Kind::Number: return formatNumber(value.number);
        case Json::Kind::String: return quote(value.text);
        default: break;
    }
    const bool isObject = value.kind == Json::Kind::Object;
    if (value.items.empty()) return isObject ? "{}" : "[]";
    const std::string inner = indent + "  ";
    std::string out = isObject ? "{\n" : "[\n";
    for (std::size_t i = 0; i < value.items.size(); ++i) {
        out += inner;
        if (isObject) out += quote(value.keys[i]) + ": ";
        out += toJson(value.items[i], inner);
        out += i + 1 < value.items.size() ? ",\n" : "\n";
    }
    return out + indent + (isObject ? "}" : "]");
}

// Average ranks: ties share the mean of the positions they occupy.
std::vector<double> ranks(const std::vector<double>& a) {
    std::vector<double> out;
    for (double v : a) {
        int below = 0, equal = 0;
        for (double z : a) {
            if (z < v) ++below;
            if (z == v) ++equal;
        }
        out.push_back(1 + below + (equal - 1) / 2.0);
    }
    return out;
}

double sign(double v) { return (v > 0) - (v < 0); }

Json lesson084() {
    const std::vector<double> x = {1, 2, 3, 4, 5}, y = {2, 4, 5, 4, 5};
    const int n = static_cast<int>(x.size());
    const std::vector<double> rx = ranks(x), ry = ranks(y);
    const double m = (n + 1) / 2.0;  // mean rank
    double cross = 0.0;
    for (int i = 0; i < n; ++i) cross += (rx[i] - m) * (ry[i] - m);
    auto ss = [m](const std::vector<double>& a) {
        double total = 0.0;
        for (double v : a) total += std::pow(v - m, 2);
        return total;
    };
    double signedPairs = 0.0;
    for (int i = 0; i < n; ++i) {
        for (int j = i + 1; j < n; ++j) signedPairs += sign((x[j] - x[i]) * (y[j] - y[i]));
    }
    return jsonObject({
        {"rx", rx},
        {"ry", ry},
        {"spearman", cross / std::sqrt(ss(rx) * ss(ry))},
        {"tauA", signedPairs / (n * (n - 1) / 2.0)},
    });
}

int main() {
    std::cout << toJson(lesson084()) << '\n';
    return 0;
}

// Fintech Math Bootcamp · Lesson 084 of 120
// Spearman Rank Correlation and Kendall Tau
// Module 09: Dependence, Regression, and Model Foundations
//
// Scenario: Comparing monotone rankings and pair ordering
// Rule:     Spearman = Pearson(ranks); Kendall tau-a=(C−D)/choose(n,2)
//
// Try it:   Is the Kendall convention in this course tau-b?
//
// Lesson article: https://thefintechbuilder.com/financial-mathematics-statistics-and-data-foundations/dependence-regression-and-model-foundations/spearman-rank-correlation-and-kendall-tau/
// Free course:    https://courses.thefintechbuilder.com
// Synthetic teaching example, not financial advice or a production library.

/// A minimal JSON value, enough to print this lesson's result.
#[allow(dead_code)]
enum Json {
    Null,
    Bool(bool),
    Num(f64),
    Str(String),
    Arr(Vec<Json>),
    Obj(Vec<(String, Json)>),
}

#[allow(dead_code)]
impl Json {
    fn obj(fields: Vec<(&str, Json)>) -> Json {
        Json::Obj(fields.into_iter().map(|(k, v)| (k.to_string(), v)).collect())
    }

    fn nums(values: &[f64]) -> Json {
        Json::Arr(values.iter().map(|&v| Json::Num(v)).collect())
    }

    /// Pretty-prints with two-space indentation.
    fn pretty(&self, indent: &str) -> String {
        let inner = format!("{}  ", indent);
        match self {
            Json::Null => "null".to_string(),
            Json::Bool(b) => b.to_string(),
            Json::Num(x) => format_number(*x),
            Json::Str(s) => quote(s),
            Json::Arr(items) if items.is_empty() => "[]".to_string(),
            Json::Obj(fields) if fields.is_empty() => "{}".to_string(),
            Json::Arr(items) => {
                let body: Vec<String> = items
                    .iter()
                    .map(|v| format!("{}{}", inner, v.pretty(&inner)))
                    .collect();
                format!("[\n{}\n{}]", body.join(",\n"), indent)
            }
            Json::Obj(fields) => {
                let body: Vec<String> = fields
                    .iter()
                    .map(|(k, v)| format!("{}{}: {}", inner, quote(k), v.pretty(&inner)))
                    .collect();
                format!("{{\n{}\n{}}}", body.join(",\n"), indent)
            }
        }
    }
}

fn format_number(x: f64) -> String {
    if !x.is_finite() {
        "null".to_string()
    } else if x == x.trunc() && x.abs() < 1e15 {
        format!("{}", x as i64)
    } else {
        format!("{}", x)
    }
}

fn quote(s: &str) -> String {
    let mut out = String::from("\"");
    for c in s.chars() {
        match c {
            '"' => out.push_str("\\\""),
            '\\' => out.push_str("\\\\"),
            '\n' => out.push_str("\\n"),
            c => out.push(c),
        }
    }
    out.push('"');
    out
}

/// Average ranks: ties share the mean of the positions they occupy.
fn ranks(a: &[f64]) -> Vec<f64> {
    a.iter()
        .map(|&v| {
            let below = a.iter().filter(|&&z| z < v).count() as f64;
            let equal = a.iter().filter(|&&z| z == v).count() as f64;
            1.0 + below + (equal - 1.0) / 2.0
        })
        .collect()
}

/// Sign like JavaScript's Math.sign: zero stays zero.
fn sign(v: f64) -> f64 {
    if v > 0.0 {
        1.0
    } else if v < 0.0 {
        -1.0
    } else {
        0.0
    }
}

fn lesson_084() -> Json {
    let x = [1.0_f64, 2.0, 3.0, 4.0, 5.0];
    let y = [2.0_f64, 4.0, 5.0, 4.0, 5.0];
    let n = x.len();
    let (rx, ry) = (ranks(&x), ranks(&y));
    let m = (n + 1) as f64 / 2.0; // mean rank
    let cross = rx
        .iter()
        .zip(ry.iter())
        .fold(0.0_f64, |s, (a, b)| s + (a - m) * (b - m));
    let ss = |a: &[f64]| a.iter().fold(0.0_f64, |s, v| s + (v - m).powf(2.0));
    let mut signed_pairs = 0.0_f64;
    for i in 0..n {
        for j in i + 1..n {
            signed_pairs += sign((x[j] - x[i]) * (y[j] - y[i]));
        }
    }
    Json::obj(vec![
        ("rx", Json::nums(&rx)),
        ("ry", Json::nums(&ry)),
        ("spearman", Json::Num(cross / (ss(&rx[..]) * ss(&ry[..])).sqrt())),
        ("tauA", Json::Num(signed_pairs / (n * (n - 1) / 2) as f64)),
    ])
}

fn main() {
    println!("{}", lesson_084().pretty(""));
}

// Fintech Math Bootcamp · Lesson 084 of 120
// Spearman Rank Correlation and Kendall Tau
// Module 09: Dependence, Regression, and Model Foundations
//
// Scenario: Comparing monotone rankings and pair ordering
// Rule:     Spearman = Pearson(ranks); Kendall tau-a=(C−D)/choose(n,2)
//
// Try it:   Is the Kendall convention in this course tau-b?
//
// Lesson article: https://thefintechbuilder.com/financial-mathematics-statistics-and-data-foundations/dependence-regression-and-model-foundations/spearman-rank-correlation-and-kendall-tau/
// Free course:    https://courses.thefintechbuilder.com
// Synthetic teaching example, not financial advice or a production library.
using System;
using System.Collections.Generic;
using System.Linq;
using System.Text.Json;

var options = new JsonSerializerOptions { WriteIndented = true };
Console.WriteLine(JsonSerializer.Serialize(Lesson084(), options));

// Average ranks: ties share the mean of the positions they occupy.
static double[] Ranks(double[] a) =>
    a.Select(v => 1 + a.Count(z => z < v) + (a.Count(z => z == v) - 1) / 2.0).ToArray();

static object Lesson084()
{
    double[] x = { 1, 2, 3, 4, 5 }, y = { 2, 4, 5, 4, 5 };
    int n = x.Length;
    double[] rx = Ranks(x), ry = Ranks(y);
    double m = (n + 1) / 2.0; // mean rank
    double cross = rx.Select((v, i) => (v - m) * (ry[i] - m)).Aggregate(0.0, (s, p) => s + p);
    double Ss(double[] a) => a.Aggregate(0.0, (s, v) => s + Math.Pow(v - m, 2));
    double signedPairs = 0;
    for (int i = 0; i < n; i++)
        for (int j = i + 1; j < n; j++)
            signedPairs += Math.Sign((x[j] - x[i]) * (y[j] - y[i]));
    return new
    {
        rx,
        ry,
        spearman = cross / Math.Sqrt(Ss(rx) * Ss(ry)),
        tauA = signedPairs / (n * (n - 1) / 2.0),
    };
}