Regression Assumptions, Heteroskedasticity, and Multicollinearity

/**
 * Fintech Math Bootcamp · Lesson 090 of 120
 * Regression Assumptions, Heteroskedasticity, and Multicollinearity
 * Module 09: Dependence, Regression, and Model Foundations
 *
 * Scenario: Diagnosing unstable uncertainty and redundant predictors
 * Rule:     nonconstant error variance; collinearity creates unstable coefficient identification
 *
 * Try it:   Does a robust standard error automatically solve omitted-variable confounding?
 *
 * Lesson article: https://thefintechbuilder.com/financial-mathematics-statistics-and-data-foundations/dependence-regression-and-model-foundations/regression-assumptions-heteroskedasticity-and-multicollinearity/
 * Free course:    https://courses.thefintechbuilder.com
 * Synthetic teaching example, not financial advice or a production library.
 */

export function lesson090() {
  const lowErrors=[-1,1],highErrors=[-4,4];
  const meanSquare=(a:number[])=>a.reduce((s,x)=>s+x*x,0)/a.length;
  const x=[1,2,3,4],duplicate=x.map(v=>2*v);
  const avg=(a:number[])=>a.reduce((s,v)=>s+v,0)/a.length;
  const dx=x.map(v=>v-avg(x)),dd=duplicate.map(v=>v-avg(duplicate));
  const dot=(a:number[],b:number[])=>a.reduce((s,v,i)=>s+v*b[i],0);
  const r2Between=dot(dx,dd)**2/(dot(dx,dx)*dot(dd,dd)); // 1 means VIF is infinite
  const result={lowSpread:meanSquare(lowErrors),highSpread:meanSquare(highErrors),
    exactRedundancy:r2Between===1};
  return result;
}

export const checkedResult = {"lowSpread":1,"highSpread":16,"exactRedundancy":true};

// Run this file directly: npx tsx lessons/09-dependence-regression-and-model-foundations/090-regression-assumptions-heteroskedasticity-and-multicollinearity.ts
if (process.argv[1] && import.meta.url.endsWith(process.argv[1].replace(/\\/g, "/").split("/").pop()!)) {
  console.log(JSON.stringify(lesson090(), null, 2));
}

"""
Fintech Math Bootcamp · Lesson 090 of 120
Regression Assumptions, Heteroskedasticity, and Multicollinearity
Module 09: Dependence, Regression, and Model Foundations

Scenario: Diagnosing unstable uncertainty and redundant predictors
Rule:     nonconstant error variance; collinearity creates unstable coefficient identification

Try it:   Does a robust standard error automatically solve omitted-variable confounding?

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

import json


def mean_square(a):
    total = 0
    for x in a:
        total += x * x
    return total / len(a)


def average(a):
    total = 0
    for v in a:
        total += v
    return total / len(a)


def dot(a, b):
    total = 0
    for ai, bi in zip(a, b):
        total += ai * bi
    return total


def lesson_090():
    low_errors, high_errors = [-1, 1], [-4, 4]
    x = [1, 2, 3, 4]
    duplicate = [2 * v for v in x]
    dx = [v - average(x) for v in x]
    dd = [v - average(duplicate) for v in duplicate]
    r2_between = dot(dx, dd) ** 2 / (dot(dx, dx) * dot(dd, dd))  # 1 means VIF is infinite
    return {
        "lowSpread": mean_square(low_errors),
        "highSpread": mean_square(high_errors),
        "exactRedundancy": r2_between == 1,
    }


if __name__ == "__main__":
    print(json.dumps(lesson_090(), indent=2))

// Fintech Math Bootcamp - Lesson 090 of 120
// Regression Assumptions, Heteroskedasticity, and Multicollinearity
// Module 09: Dependence, Regression, and Model Foundations
//
// Scenario: Diagnosing unstable uncertainty and redundant predictors
// Rule:     nonconstant error variance; collinearity creates unstable coefficient identification
//
// Try it:   Does a robust standard error automatically solve omitted-variable confounding?
//
// Lesson article: https://thefintechbuilder.com/financial-mathematics-statistics-and-data-foundations/dependence-regression-and-model-foundations/regression-assumptions-heteroskedasticity-and-multicollinearity/
// 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 {
    static double meanSquare(double[] a) {
        double total = 0;
        for (double x : a) total += x * x;
        return total / a.length;
    }

    static double[] deviations(double[] values) {
        double total = 0;
        for (double v : values) total += v;
        double m = total / values.length;
        double[] out = new double[values.length];
        for (int i = 0; i < values.length; i++) out[i] = values[i] - m;
        return out;
    }

    static double dot(double[] a, double[] b) {
        double total = 0;
        for (int i = 0; i < a.length; i++) total += a[i] * b[i];
        return total;
    }

    static Map<String, Object> lesson090() {
        double[] lowErrors = {-1, 1}, highErrors = {-4, 4};
        double[] x = {1, 2, 3, 4};
        double[] duplicate = new double[x.length];
        for (int i = 0; i < x.length; i++) duplicate[i] = 2 * x[i];
        double[] dx = deviations(x), dd = deviations(duplicate);
        double r2Between = Math.pow(dot(dx, dd), 2) / (dot(dx, dx) * dot(dd, dd)); // 1 means VIF is infinite
        Map<String, Object> result = new LinkedHashMap<String, Object>();
        result.put("lowSpread", meanSquare(lowErrors));
        result.put("highSpread", meanSquare(highErrors));
        result.put("exactRedundancy", r2Between == 1);
        return result;
    }

    public static void main(String[] args) {
        System.out.println(toJson(lesson090(), ""));
    }

    // 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 090 of 120
// Regression Assumptions, Heteroskedasticity, and Multicollinearity
// Module 09: Dependence, Regression, and Model Foundations
//
// Scenario: Diagnosing unstable uncertainty and redundant predictors
// Rule:     nonconstant error variance; collinearity creates unstable coefficient identification
//
// Try it:   Does a robust standard error automatically solve omitted-variable confounding?
//
// Lesson article: https://thefintechbuilder.com/financial-mathematics-statistics-and-data-foundations/dependence-regression-and-model-foundations/regression-assumptions-heteroskedasticity-and-multicollinearity/
// Free course:    https://courses.thefintechbuilder.com
// Synthetic teaching example, not financial advice or a production library.
package main

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

type Lesson090Result struct {
	LowSpread       float64 `json:"lowSpread"`
	HighSpread      float64 `json:"highSpread"`
	ExactRedundancy bool    `json:"exactRedundancy"`
}

func meanSquare(a []float64) float64 {
	total := 0.0
	for _, x := range a {
		total += x * x
	}
	return total / float64(len(a))
}

func deviations(values []float64) []float64 {
	total := 0.0
	for _, v := range values {
		total += v
	}
	m := total / float64(len(values))
	out := make([]float64, len(values))
	for i, v := range values {
		out[i] = v - m
	}
	return out
}

func dot(a, b []float64) float64 {
	total := 0.0
	for i := range a {
		total += a[i] * b[i]
	}
	return total
}

func lesson090() Lesson090Result {
	lowErrors := []float64{-1, 1}
	highErrors := []float64{-4, 4}
	x := []float64{1, 2, 3, 4}
	duplicate := make([]float64, len(x))
	for i, v := range x {
		duplicate[i] = 2 * v
	}
	dx, dd := deviations(x), deviations(duplicate)
	r2Between := math.Pow(dot(dx, dd), 2) / (dot(dx, dx) * dot(dd, dd)) // 1 means VIF is infinite
	return Lesson090Result{
		LowSpread:       meanSquare(lowErrors),
		HighSpread:      meanSquare(highErrors),
		ExactRedundancy: r2Between == 1,
	}
}

func main() {
	out, err := json.MarshalIndent(lesson090(), "", "  ")
	if err != nil {
		panic(err)
	}
	fmt.Println(string(out))
}

// Fintech Math Bootcamp · Lesson 090 of 120
// Regression Assumptions, Heteroskedasticity, and Multicollinearity
// Module 09: Dependence, Regression, and Model Foundations
//
// Scenario: Diagnosing unstable uncertainty and redundant predictors
// Rule:     nonconstant error variance; collinearity creates unstable coefficient identification
//
// Try it:   Does a robust standard error automatically solve omitted-variable confounding?
//
// Lesson article: https://thefintechbuilder.com/financial-mathematics-statistics-and-data-foundations/dependence-regression-and-model-foundations/regression-assumptions-heteroskedasticity-and-multicollinearity/
// 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 ? "}" : "]");
}

double meanSquare(const std::vector<double>& a) {
    double total = 0.0;
    for (double x : a) total += x * x;
    return total / a.size();
}

std::vector<double> deviations(const std::vector<double>& values) {
    double total = 0.0;
    for (double v : values) total += v;
    const double m = total / values.size();
    std::vector<double> out;
    for (double v : values) out.push_back(v - m);
    return out;
}

double dot(const std::vector<double>& a, const std::vector<double>& b) {
    double total = 0.0;
    for (std::size_t i = 0; i < a.size(); ++i) total += a[i] * b[i];
    return total;
}

Json lesson090() {
    const std::vector<double> lowErrors = {-1, 1}, highErrors = {-4, 4};
    const std::vector<double> x = {1, 2, 3, 4};
    std::vector<double> duplicate;
    for (double v : x) duplicate.push_back(2 * v);
    const std::vector<double> dx = deviations(x), dd = deviations(duplicate);
    const double r2Between = std::pow(dot(dx, dd), 2) / (dot(dx, dx) * dot(dd, dd));  // 1 means VIF is infinite
    return jsonObject({
        {"lowSpread", meanSquare(lowErrors)},
        {"highSpread", meanSquare(highErrors)},
        {"exactRedundancy", r2Between == 1},
    });
}

int main() {
    std::cout << toJson(lesson090()) << '\n';
    return 0;
}

// Fintech Math Bootcamp · Lesson 090 of 120
// Regression Assumptions, Heteroskedasticity, and Multicollinearity
// Module 09: Dependence, Regression, and Model Foundations
//
// Scenario: Diagnosing unstable uncertainty and redundant predictors
// Rule:     nonconstant error variance; collinearity creates unstable coefficient identification
//
// Try it:   Does a robust standard error automatically solve omitted-variable confounding?
//
// Lesson article: https://thefintechbuilder.com/financial-mathematics-statistics-and-data-foundations/dependence-regression-and-model-foundations/regression-assumptions-heteroskedasticity-and-multicollinearity/
// 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
}

fn mean_square(a: &[f64]) -> f64 {
    a.iter().fold(0.0_f64, |s, x| s + x * x) / a.len() as f64
}

fn deviations(values: &[f64]) -> Vec<f64> {
    let m = values.iter().fold(0.0_f64, |s, v| s + v) / values.len() as f64;
    values.iter().map(|v| v - m).collect()
}

fn dot(a: &[f64], b: &[f64]) -> f64 {
    a.iter().zip(b.iter()).fold(0.0_f64, |s, (x, y)| s + x * y)
}

fn lesson_090() -> Json {
    let low_errors = [-1.0_f64, 1.0];
    let high_errors = [-4.0_f64, 4.0];
    let x = [1.0_f64, 2.0, 3.0, 4.0];
    let duplicate: Vec<f64> = x.iter().map(|v| 2.0 * v).collect();
    let (dx, dd) = (deviations(&x), deviations(&duplicate));
    let r2_between = dot(&dx, &dd).powf(2.0) / (dot(&dx, &dx) * dot(&dd, &dd)); // 1 means VIF is infinite
    Json::obj(vec![
        ("lowSpread", Json::Num(mean_square(&low_errors))),
        ("highSpread", Json::Num(mean_square(&high_errors))),
        ("exactRedundancy", Json::Bool(r2_between == 1.0)),
    ])
}

fn main() {
    println!("{}", lesson_090().pretty(""));
}

// Fintech Math Bootcamp · Lesson 090 of 120
// Regression Assumptions, Heteroskedasticity, and Multicollinearity
// Module 09: Dependence, Regression, and Model Foundations
//
// Scenario: Diagnosing unstable uncertainty and redundant predictors
// Rule:     nonconstant error variance; collinearity creates unstable coefficient identification
//
// Try it:   Does a robust standard error automatically solve omitted-variable confounding?
//
// Lesson article: https://thefintechbuilder.com/financial-mathematics-statistics-and-data-foundations/dependence-regression-and-model-foundations/regression-assumptions-heteroskedasticity-and-multicollinearity/
// 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(Lesson090(), options));

static double MeanSquare(double[] a) => a.Aggregate(0.0, (s, x) => s + x * x) / a.Length;

static double Average(double[] a) => a.Aggregate(0.0, (s, v) => s + v) / a.Length;

static double Dot(double[] a, double[] b) => a.Select((v, i) => v * b[i]).Aggregate(0.0, (s, p) => s + p);

static object Lesson090()
{
    double[] lowErrors = { -1, 1 }, highErrors = { -4, 4 };
    double[] x = { 1, 2, 3, 4 };
    double[] duplicate = x.Select(v => 2 * v).ToArray();
    double[] dx = x.Select(v => v - Average(x)).ToArray();
    double[] dd = duplicate.Select(v => v - Average(duplicate)).ToArray();
    double r2Between = Math.Pow(Dot(dx, dd), 2) / (Dot(dx, dx) * Dot(dd, dd)); // 1 means VIF is infinite
    return new
    {
        lowSpread = MeanSquare(lowErrors),
        highSpread = MeanSquare(highErrors),
        exactRedundancy = r2Between == 1,
    };
}