Autocovariance and Autocorrelation

/**
 * Fintech Math Bootcamp · Lesson 097 of 120
 * Autocovariance and Autocorrelation
 * Module 10: Financial Time-Series Foundations
 *
 * Scenario: Measuring lagged similarity on a declared finite-sample convention
 * Rule:     γₖ=(1/n)Σ(xₜ−x̄)(xₜ₋ₖ−x̄); ρₖ=γₖ/γ₀
 *
 * Try it:   What denominator does this source convention use at lag one?
 *
 * Lesson article: https://thefintechbuilder.com/financial-mathematics-statistics-and-data-foundations/financial-time-series-foundations/autocovariance-and-autocorrelation/
 * Free course:    https://courses.thefintechbuilder.com
 * Synthetic teaching example, not financial advice or a production library.
 */

export function lesson097() {
  const x=[1,2,3,4];
  const mean=x.reduce((s,v)=>s+v,0)/x.length;
  const gamma=(k:number)=>x.slice(k).reduce((s,v,j)=>s+(v-mean)*(x[j]-mean),0)/x.length;
  const result={gamma0:gamma(0),gamma1:gamma(1),rho1:gamma(1)/gamma(0)};
  return result;
}

export const checkedResult = {"gamma0":1.25,"gamma1":0.3125,"rho1":0.25};

// Run this file directly: npx tsx lessons/10-financial-time-series-foundations/097-autocovariance-and-autocorrelation.ts
if (process.argv[1] && import.meta.url.endsWith(process.argv[1].replace(/\\/g, "/").split("/").pop()!)) {
  console.log(JSON.stringify(lesson097(), null, 2));
}

# Fintech Math Bootcamp · Lesson 097 of 120
# Autocovariance and Autocorrelation
# Module 10: Financial Time-Series Foundations
#
# Scenario: Measuring lagged similarity on a declared finite-sample convention
# Rule:     γₖ=(1/n)Σ(xₜ−x̄)(xₜ₋ₖ−x̄); ρₖ=γₖ/γ₀
#
# Try it:   What denominator does this source convention use at lag one?
#
# Lesson article: https://thefintechbuilder.com/financial-mathematics-statistics-and-data-foundations/financial-time-series-foundations/autocovariance-and-autocorrelation/
# Free course:    https://courses.thefintechbuilder.com
# Synthetic teaching example, not financial advice or a production library.
#
# Run: python main.py

import json


def lesson_097():
    x = [1, 2, 3, 4]
    mean = sum(x) / len(x)

    def gamma(k):
        total = 0.0
        for j, v in enumerate(x[k:]):
            total += (v - mean) * (x[j] - mean)
        return total / len(x)

    return {"gamma0": gamma(0), "gamma1": gamma(1), "rho1": gamma(1) / gamma(0)}


if __name__ == "__main__":
    print(json.dumps(lesson_097(), indent=2))

/*
 * Fintech Math Bootcamp - Lesson 097 of 120
 * Autocovariance and Autocorrelation
 * Module 10: Financial Time-Series Foundations
 *
 * Scenario: Measuring lagged similarity on a declared finite-sample convention
 * Rule:     gamma_k=(1/n)Sigma(x_t-xbar)(x_t-k-xbar); rho_k=gamma_k/gamma_0
 *
 * Try it:   What denominator does this source convention use at lag one?
 *
 * Lesson article: https://thefintechbuilder.com/financial-mathematics-statistics-and-data-foundations/financial-time-series-foundations/autocovariance-and-autocorrelation/
 * Free course:    https://courses.thefintechbuilder.com
 * Synthetic teaching example, not financial advice or a production library.
 *
 * Run: javac Main.java && java Main
 */

import java.util.*;

public class Main {
    static double[] x = {1, 2, 3, 4};

    static double gamma(double mean, int k) {
        double s = 0;
        for (int j = 0; j + k < x.length; j++) s += (x[j + k] - mean) * (x[j] - mean);
        return s / x.length; // divide by n at every lag
    }

    static Map<String, Object> lesson097() {
        double mean = 0;
        for (double v : x) mean += v;
        mean /= x.length;
        return obj("gamma0", gamma(mean, 0), "gamma1", gamma(mean, 1), "rho1", gamma(mean, 1) / gamma(mean, 0));
    }

    public static void main(String[] args) {
        System.out.println(toJson(lesson097(), ""));
    }

    // Minimal JSON writer: insertion-ordered Map, List, Number, Boolean, String and null.
    static String toJson(Object value, String indent) {
        if (value == null) return "null";
        if (value instanceof String) return quote((String) value);
        if (value instanceof Boolean) return value.toString();
        if (value instanceof Double) return formatNumber((Double) value);
        if (value instanceof Number) return value.toString();
        String inner = indent + "  ";
        StringBuilder sb = new StringBuilder();
        if (value instanceof Map) {
            Map<?, ?> map = (Map<?, ?>) value;
            if (map.isEmpty()) return "{}";
            sb.append("{\n");
            int i = 0;
            for (Map.Entry<?, ?> e : map.entrySet()) {
                sb.append(inner).append(quote(e.getKey().toString())).append(": ").append(toJson(e.getValue(), inner));
                sb.append(++i < map.size() ? ",\n" : "\n");
            }
            return sb.append(indent).append("}").toString();
        }
        List<?> list = (List<?>) value;
        if (list.isEmpty()) return "[]";
        sb.append("[\n");
        for (int i = 0; i < list.size(); i++) {
            sb.append(inner).append(toJson(list.get(i), inner)).append(i + 1 < list.size() ? ",\n" : "\n");
        }
        return sb.append(indent).append("]").toString();
    }

    // Integral doubles print without ".0", as JavaScript does; others use Java's round-trip form.
    static String formatNumber(double x) {
        if (x == Math.rint(x) && Math.abs(x) < 1e15) return Long.toString((long) x);
        return Double.toString(x);
    }

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

    // Builds an insertion-ordered object from alternating keys and values.
    static Map<String, Object> obj(Object... keysAndValues) {
        Map<String, Object> map = new LinkedHashMap<String, Object>();
        for (int i = 0; i < keysAndValues.length; i += 2) map.put((String) keysAndValues[i], keysAndValues[i + 1]);
        return map;
    }

    static List<Object> list(double... values) {
        List<Object> out = new ArrayList<Object>();
        for (double v : values) out.add(v);
        return out;
    }
}

// Fintech Math Bootcamp · Lesson 097 of 120
// Autocovariance and Autocorrelation
// Module 10: Financial Time-Series Foundations
//
// Scenario: Measuring lagged similarity on a declared finite-sample convention
// Rule:     γₖ=(1/n)Σ(xₜ−x̄)(xₜ₋ₖ−x̄); ρₖ=γₖ/γ₀
//
// Try it:   What denominator does this source convention use at lag one?
//
// Lesson article: https://thefintechbuilder.com/financial-mathematics-statistics-and-data-foundations/financial-time-series-foundations/autocovariance-and-autocorrelation/
// Free course:    https://courses.thefintechbuilder.com
// Synthetic teaching example, not financial advice or a production library.
//
// Run: go run main.go

package main

import (
	"encoding/json"
	"fmt"
)

type Result struct {
	Gamma0 float64 `json:"gamma0"`
	Gamma1 float64 `json:"gamma1"`
	Rho1   float64 `json:"rho1"`
}

func lesson097() Result {
	x := []float64{1, 2, 3, 4}
	mean := 0.0
	for _, v := range x {
		mean += v
	}
	mean /= float64(len(x))
	gamma := func(k int) float64 {
		s := 0.0
		for j, v := range x[k:] {
			s += (v - mean) * (x[j] - mean)
		}
		return s / float64(len(x)) // divide by n at every lag
	}
	return Result{Gamma0: gamma(0), Gamma1: gamma(1), Rho1: gamma(1) / gamma(0)}
}

func main() {
	out, err := json.MarshalIndent(lesson097(), "", "  ")
	if err != nil {
		panic(err)
	}
	fmt.Println(string(out))
}

// Fintech Math Bootcamp · Lesson 097 of 120
// Autocovariance and Autocorrelation
// Module 10: Financial Time-Series Foundations
//
// Scenario: Measuring lagged similarity on a declared finite-sample convention
// Rule:     γₖ=(1/n)Σ(xₜ−x̄)(xₜ₋ₖ−x̄); ρₖ=γₖ/γ₀
//
// Try it:   What denominator does this source convention use at lag one?
//
// Lesson article: https://thefintechbuilder.com/financial-mathematics-statistics-and-data-foundations/financial-time-series-foundations/autocovariance-and-autocorrelation/
// Free course:    https://courses.thefintechbuilder.com
// Synthetic teaching example, not financial advice or a production library.
//
// Run: g++ -std=c++17 -O1 -o lesson main.cpp && ./lesson

#include <algorithm>
#include <charconv>
#include <cmath>
#include <cstdint>
#include <cstdio>
#include <deque>
#include <iostream>
#include <limits>
#include <map>
#include <optional>
#include <stdexcept>
#include <string>
#include <utility>
#include <variant>
#include <vector>

// Minimal JSON value: objects keep insertion order; numbers print in shortest round-trip form.
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 items, or object values
};

Json jnull() { return Json{}; }
Json jbool(bool value) { Json j; j.kind = Json::Kind::Bool; j.flag = value; return j; }
Json jnum(double value) { Json j; j.kind = Json::Kind::Number; j.number = value; return j; }
Json jstr(const std::string& value) { Json j; j.kind = Json::Kind::String; j.text = value; return j; }
Json jarr(std::vector<Json> items) { Json j; j.kind = Json::Kind::Array; j.items = std::move(items); return j; }

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

Json jnums(const std::vector<double>& values) {
    std::vector<Json> items;
    for (double v : values) items.push_back(jnum(v));
    return jarr(std::move(items));
}

Json jnums(const std::vector<std::optional<double>>& values) {
    std::vector<Json> items;
    for (const auto& v : values) items.push_back(v ? jnum(*v) : jnull());
    return jarr(std::move(items));
}

std::string formatNumber(double value) {
    char buffer[64];
    auto result = std::to_chars(buffer, buffer + sizeof buffer, value);  // shortest round-trip form
    return std::string(buffer, result.ptr);
}

std::string quote(const std::string& text) {
    std::string out = "\"";
    for (char c : text) {
        if (c == '"' || c == '\\') {
            out += '\\';
            out += c;
        } else if (static_cast<unsigned char>(c) < 0x20) {
            char buffer[8];
            std::snprintf(buffer, sizeof buffer, "\\u%04x", static_cast<unsigned>(c));
            out += buffer;
        } else {
            out += c;
        }
    }
    return out + "\"";
}

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

Json lesson097() {
    const std::vector<double> x{1, 2, 3, 4};
    double mean = 0;
    for (double v : x) mean += v;
    mean /= static_cast<double>(x.size());
    auto gamma = [&](std::size_t k) {
        double s = 0;
        for (std::size_t j = 0; j + k < x.size(); ++j) s += (x[j + k] - mean) * (x[j] - mean);
        return s / static_cast<double>(x.size());  // divide by n at every lag
    };
    return jobj({{"gamma0", jnum(gamma(0))}, {"gamma1", jnum(gamma(1))}, {"rho1", jnum(gamma(1) / gamma(0))}});
}

int main() {
    writeJson(std::cout, lesson097());
    std::cout << "\n";
    return 0;
}

// Fintech Math Bootcamp · Lesson 097 of 120
// Autocovariance and Autocorrelation
// Module 10: Financial Time-Series Foundations
//
// Scenario: Measuring lagged similarity on a declared finite-sample convention
// Rule:     γₖ=(1/n)Σ(xₜ−x̄)(xₜ₋ₖ−x̄); ρₖ=γₖ/γ₀
//
// Try it:   What denominator does this source convention use at lag one?
//
// Lesson article: https://thefintechbuilder.com/financial-mathematics-statistics-and-data-foundations/financial-time-series-foundations/autocovariance-and-autocorrelation/
// Free course:    https://courses.thefintechbuilder.com
// Synthetic teaching example, not financial advice or a production library.
//
// Run: rustc -O main.rs && ./main

#![allow(dead_code)]

/// Minimal JSON value; objects keep insertion order.
enum Json {
    Null,
    Bool(bool),
    Num(f64),
    Str(String),
    Arr(Vec<Json>),
    Obj(Vec<(String, 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())
}

fn optional_nums(values: &[Option<f64>]) -> Json {
    Json::Arr(values.iter().map(|v| v.map_or(Json::Null, Json::Num)).collect())
}

fn quote(text: &str) -> String {
    let mut out = String::from("\"");
    for c in text.chars() {
        match c {
            '"' => out.push_str("\\\""),
            '\\' => out.push_str("\\\\"),
            c if (c as u32) < 0x20 => out.push_str(&format!("\\u{:04x}", c as u32)),
            c => out.push(c),
        }
    }
    out.push('"');
    out
}

// Display for f64 prints the shortest string that round-trips, as JavaScript does.
fn write_json(value: &Json, indent: &str, out: &mut String) {
    let inner = format!("{}  ", indent);
    match value {
        Json::Null => out.push_str("null"),
        Json::Bool(b) => out.push_str(if *b { "true" } else { "false" }),
        Json::Num(n) => out.push_str(&format!("{}", n)),
        Json::Str(s) => out.push_str(&quote(s)),
        Json::Arr(items) if items.is_empty() => out.push_str("[]"),
        Json::Obj(fields) if fields.is_empty() => out.push_str("{}"),
        Json::Arr(items) => {
            out.push_str("[\n");
            for (i, item) in items.iter().enumerate() {
                out.push_str(&inner);
                write_json(item, &inner, out);
                out.push_str(if i + 1 < items.len() { ",\n" } else { "\n" });
            }
            out.push_str(indent);
            out.push(']');
        }
        Json::Obj(fields) => {
            out.push_str("{\n");
            for (i, (key, item)) in fields.iter().enumerate() {
                out.push_str(&inner);
                out.push_str(&quote(key));
                out.push_str(": ");
                write_json(item, &inner, out);
                out.push_str(if i + 1 < fields.len() { ",\n" } else { "\n" });
            }
            out.push_str(indent);
            out.push('}');
        }
    }
}

fn lesson_097() -> Json {
    let x: [f64; 4] = [1.0, 2.0, 3.0, 4.0];
    let n = x.len() as f64;
    let mean = x.iter().fold(0.0, |s, v| s + v) / n;
    // Divide by n at every lag.
    let gamma = |k: usize| x[k..].iter().enumerate().fold(0.0, |s, (j, v)| s + (v - mean) * (x[j] - mean)) / n;
    obj(vec![
        ("gamma0", Json::Num(gamma(0))),
        ("gamma1", Json::Num(gamma(1))),
        ("rho1", Json::Num(gamma(1) / gamma(0))),
    ])
}

fn main() {
    let mut out = String::new();
    write_json(&lesson_097(), "", &mut out);
    println!("{}", out);
}

// Fintech Math Bootcamp · Lesson 097 of 120
// Autocovariance and Autocorrelation
// Module 10: Financial Time-Series Foundations
//
// Scenario: Measuring lagged similarity on a declared finite-sample convention
// Rule:     γₖ=(1/n)Σ(xₜ−x̄)(xₜ₋ₖ−x̄); ρₖ=γₖ/γ₀
//
// Try it:   What denominator does this source convention use at lag one?
//
// Lesson article: https://thefintechbuilder.com/financial-mathematics-statistics-and-data-foundations/financial-time-series-foundations/autocovariance-and-autocorrelation/
// Free course:    https://courses.thefintechbuilder.com
// Synthetic teaching example, not financial advice or a production library.
//
// Run: dotnet run

using System;
using System.Collections.Generic;
using System.Globalization;
using System.Linq;
using System.Text.Encodings.Web;
using System.Text.Json;

var options = new JsonSerializerOptions { WriteIndented = true, Encoder = JavaScriptEncoder.UnsafeRelaxedJsonEscaping };
Console.WriteLine(JsonSerializer.Serialize(Lesson097(), options));

static object Lesson097()
{
    double[] x = { 1, 2, 3, 4 };
    double mean = x.Aggregate(0.0, (s, v) => s + v) / x.Length;
    // Divide by n at every lag.
    double Gamma(int k) => x.Skip(k).Select((v, j) => (v, j)).Aggregate(0.0, (s, p) => s + (p.v - mean) * (x[p.j] - mean)) / x.Length;
    return new { gamma0 = Gamma(0), gamma1 = Gamma(1), rho1 = Gamma(1) / Gamma(0) };
}