Scatter Plots, Association, and Nonlinear Patterns

/**
 * Fintech Math Bootcamp · Lesson 081 of 120
 * Scatter Plots, Association, and Nonlinear Patterns
 * Module 09: Dependence, Regression, and Model Foundations
 *
 * Scenario: Looking at paired observations before compressing them
 * Rule:     a scatter plot preserves paired coordinates (xᵢ,yᵢ)
 *
 * Try it:   Can zero sample covariance hide an obvious pattern?
 *
 * Lesson article: https://thefintechbuilder.com/financial-mathematics-statistics-and-data-foundations/dependence-regression-and-model-foundations/scatter-plots-association-and-nonlinear-patterns/
 * Free course:    https://courses.thefintechbuilder.com
 * Synthetic teaching example, not financial advice or a production library.
 */

export function lesson081() {
  const x=[-2,-1,0,1,2];
  const y=x.map(v=>v*v);
  const mean=(a:number[])=>a.reduce((s,v)=>s+v,0)/a.length;
  const mx=mean(x),my=mean(y);
  const covariance=x.reduce((s,v,i)=>s+(v-mx)*(y[i]-my),0)/(x.length-1);
  const result={points:x.map((v,i)=>[v,y[i]]),covariance};
  return result;
}

export const checkedResult = {"points":[[-2,4],[-1,1],[0,0],[1,1],[2,4]],"covariance":0};

// Run this file directly: npx tsx lessons/09-dependence-regression-and-model-foundations/081-scatter-plots-association-and-nonlinear-patterns.ts
if (process.argv[1] && import.meta.url.endsWith(process.argv[1].replace(/\\/g, "/").split("/").pop()!)) {
  console.log(JSON.stringify(lesson081(), null, 2));
}

"""
Fintech Math Bootcamp · Lesson 081 of 120
Scatter Plots, Association, and Nonlinear Patterns
Module 09: Dependence, Regression, and Model Foundations

Scenario: Looking at paired observations before compressing them
Rule:     a scatter plot preserves paired coordinates (xᵢ,yᵢ)

Try it:   Can zero sample covariance hide an obvious pattern?

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

import json


def mean(values):
    total = 0
    for v in values:
        total += v
    return total / len(values)


def lesson_081():
    x = [-2, -1, 0, 1, 2]
    y = [v * v for v in x]
    mx, my = mean(x), mean(y)
    cross = 0
    for xi, yi in zip(x, y):
        cross += (xi - mx) * (yi - my)
    covariance = cross / (len(x) - 1)
    return {"points": [[xi, yi] for xi, yi in zip(x, y)], "covariance": covariance}


if __name__ == "__main__":
    print(json.dumps(lesson_081(), indent=2))

// Fintech Math Bootcamp - Lesson 081 of 120
// Scatter Plots, Association, and Nonlinear Patterns
// Module 09: Dependence, Regression, and Model Foundations
//
// Scenario: Looking at paired observations before compressing them
// Rule:     a scatter plot preserves paired coordinates (x_i,y_i)
//
// Try it:   Can zero sample covariance hide an obvious pattern?
//
// Lesson article: https://thefintechbuilder.com/financial-mathematics-statistics-and-data-foundations/dependence-regression-and-model-foundations/scatter-plots-association-and-nonlinear-patterns/
// 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 mean(double[] values) {
        double total = 0;
        for (double v : values) total += v;
        return total / values.length;
    }

    static Map<String, Object> lesson081() {
        double[] x = {-2, -1, 0, 1, 2};
        double[] y = new double[x.length];
        for (int i = 0; i < x.length; i++) y[i] = x[i] * x[i];
        double mx = mean(x), my = mean(y);
        double cross = 0;
        double[][] points = new double[x.length][];
        for (int i = 0; i < x.length; i++) {
            cross += (x[i] - mx) * (y[i] - my);
            points[i] = new double[] {x[i], y[i]};
        }
        Map<String, Object> result = new LinkedHashMap<String, Object>();
        result.put("points", points);
        result.put("covariance", cross / (x.length - 1));
        return result;
    }

    public static void main(String[] args) {
        System.out.println(toJson(lesson081(), ""));
    }

    // 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 081 of 120
// Scatter Plots, Association, and Nonlinear Patterns
// Module 09: Dependence, Regression, and Model Foundations
//
// Scenario: Looking at paired observations before compressing them
// Rule:     a scatter plot preserves paired coordinates (xᵢ,yᵢ)
//
// Try it:   Can zero sample covariance hide an obvious pattern?
//
// Lesson article: https://thefintechbuilder.com/financial-mathematics-statistics-and-data-foundations/dependence-regression-and-model-foundations/scatter-plots-association-and-nonlinear-patterns/
// Free course:    https://courses.thefintechbuilder.com
// Synthetic teaching example, not financial advice or a production library.
package main

import (
	"encoding/json"
	"fmt"
)

type Lesson081Result struct {
	Points     [][]float64 `json:"points"`
	Covariance float64     `json:"covariance"`
}

func mean(values []float64) float64 {
	total := 0.0
	for _, v := range values {
		total += v
	}
	return total / float64(len(values))
}

func lesson081() Lesson081Result {
	x := []float64{-2, -1, 0, 1, 2}
	y := make([]float64, len(x))
	for i, v := range x {
		y[i] = v * v
	}
	mx, my := mean(x), mean(y)
	cross := 0.0
	points := make([][]float64, len(x))
	for i, v := range x {
		cross += (v - mx) * (y[i] - my)
		points[i] = []float64{v, y[i]}
	}
	return Lesson081Result{Points: points, Covariance: cross / float64(len(x)-1)}
}

func main() {
	out, err := json.MarshalIndent(lesson081(), "", "  ")
	if err != nil {
		panic(err)
	}
	fmt.Println(string(out))
}

// Fintech Math Bootcamp · Lesson 081 of 120
// Scatter Plots, Association, and Nonlinear Patterns
// Module 09: Dependence, Regression, and Model Foundations
//
// Scenario: Looking at paired observations before compressing them
// Rule:     a scatter plot preserves paired coordinates (xᵢ,yᵢ)
//
// Try it:   Can zero sample covariance hide an obvious pattern?
//
// Lesson article: https://thefintechbuilder.com/financial-mathematics-statistics-and-data-foundations/dependence-regression-and-model-foundations/scatter-plots-association-and-nonlinear-patterns/
// 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 mean(const std::vector<double>& values) {
    double total = 0.0;
    for (double v : values) total += v;
    return total / values.size();
}

Json lesson081() {
    const std::vector<double> x = {-2, -1, 0, 1, 2};
    std::vector<double> y;
    for (double v : x) y.push_back(v * v);
    const double mx = mean(x), my = mean(y);
    double cross = 0.0;
    std::vector<Json> points;
    for (std::size_t i = 0; i < x.size(); ++i) {
        cross += (x[i] - mx) * (y[i] - my);
        points.push_back(std::vector<double>{x[i], y[i]});
    }
    return jsonObject({
        {"points", jsonArray(points)},
        {"covariance", cross / (x.size() - 1)},
    });
}

int main() {
    std::cout << toJson(lesson081()) << '\n';
    return 0;
}

// Fintech Math Bootcamp · Lesson 081 of 120
// Scatter Plots, Association, and Nonlinear Patterns
// Module 09: Dependence, Regression, and Model Foundations
//
// Scenario: Looking at paired observations before compressing them
// Rule:     a scatter plot preserves paired coordinates (xᵢ,yᵢ)
//
// Try it:   Can zero sample covariance hide an obvious pattern?
//
// Lesson article: https://thefintechbuilder.com/financial-mathematics-statistics-and-data-foundations/dependence-regression-and-model-foundations/scatter-plots-association-and-nonlinear-patterns/
// 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(values: &[f64]) -> f64 {
    values.iter().fold(0.0_f64, |s, v| s + v) / values.len() as f64
}

fn lesson_081() -> Json {
    let x = [-2.0_f64, -1.0, 0.0, 1.0, 2.0];
    let y: Vec<f64> = x.iter().map(|v| v * v).collect();
    let (mx, my) = (mean(&x), mean(&y));
    let cross = x
        .iter()
        .zip(y.iter())
        .fold(0.0_f64, |s, (xi, yi)| s + (xi - mx) * (yi - my));
    let covariance = cross / (x.len() - 1) as f64;
    let points = x.iter().zip(y.iter()).map(|(&xi, &yi)| Json::nums(&[xi, yi])).collect();
    Json::obj(vec![
        ("points", Json::Arr(points)),
        ("covariance", Json::Num(covariance)),
    ])
}

fn main() {
    println!("{}", lesson_081().pretty(""));
}

// Fintech Math Bootcamp · Lesson 081 of 120
// Scatter Plots, Association, and Nonlinear Patterns
// Module 09: Dependence, Regression, and Model Foundations
//
// Scenario: Looking at paired observations before compressing them
// Rule:     a scatter plot preserves paired coordinates (xᵢ,yᵢ)
//
// Try it:   Can zero sample covariance hide an obvious pattern?
//
// Lesson article: https://thefintechbuilder.com/financial-mathematics-statistics-and-data-foundations/dependence-regression-and-model-foundations/scatter-plots-association-and-nonlinear-patterns/
// 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(Lesson081(), options));

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

static object Lesson081()
{
    double[] x = { -2, -1, 0, 1, 2 };
    double[] y = x.Select(v => v * v).ToArray();
    double mx = Mean(x), my = Mean(y);
    double covariance = x.Select((v, i) => (v - mx) * (y[i] - my)).Aggregate(0.0, (s, p) => s + p) / (x.Length - 1);
    double[][] points = x.Select((v, i) => new[] { v, y[i] }).ToArray();
    return new { points, covariance };
}