Stable Online Variance and Welford’s Algorithm

/**
 * Fintech Math Bootcamp · Lesson 113 of 120
 * Stable Online Variance and Welford’s Algorithm
 * Module 12: Statistical Computing and Reproducibility
 *
 * Scenario: Updating variance as records arrive without unstable raw subtraction
 * Rule:     n → mean → M₂; sample variance=M₂/(n−1)
 *
 * Try it:   Does ordinary Welford automatically expire the oldest rolling-window observation?
 *
 * Lesson article: https://thefintechbuilder.com/financial-mathematics-statistics-and-data-foundations/statistical-computing-and-reproducibility/stable-online-variance-and-welford-s-algorithm/
 * Free course:    https://courses.thefintechbuilder.com
 * Synthetic teaching example, not financial advice or a production library.
 */

export function lesson113() {
  const values=[1,2,3,4,5];
  let count=0,mean=0,m2=0;
  const states:{count:number;mean:number;m2:number}[]=[];
  for(const x of values){count++;const delta=x-mean;
    mean+=delta/count;m2+=delta*(x-mean);
    states.push({count,mean,m2});}
  const result={count,mean,m2,sampleVariance:m2/(count-1),states};
  return result;
}

export const checkedResult = {"count":5,"mean":3,"m2":10,"sampleVariance":2.5,"states":[{"count":1,"mean":1,"m2":0},{"count":2,"mean":1.5,"m2":0.5},{"count":3,"mean":2,"m2":2},{"count":4,"mean":2.5,"m2":5},{"count":5,"mean":3,"m2":10}]};

// Run this file directly: npx tsx lessons/12-statistical-computing-and-reproducibility/113-stable-online-variance-and-welfords-algorithm.ts
if (process.argv[1] && import.meta.url.endsWith(process.argv[1].replace(/\\/g, "/").split("/").pop()!)) {
  console.log(JSON.stringify(lesson113(), null, 2));
}

# Fintech Math Bootcamp · Lesson 113 of 120
# Stable Online Variance and Welford’s Algorithm
# Module 12: Statistical Computing and Reproducibility
#
# Scenario: Updating variance as records arrive without unstable raw subtraction
# Rule:     n → mean → M₂; sample variance=M₂/(n−1)
#
# Try it:   Does ordinary Welford automatically expire the oldest rolling-window observation?
#
# Lesson article: https://thefintechbuilder.com/financial-mathematics-statistics-and-data-foundations/statistical-computing-and-reproducibility/stable-online-variance-and-welford-s-algorithm/
# Free course:    https://courses.thefintechbuilder.com
# Synthetic teaching example, not financial advice or a production library.
#
# Run: python main.py

import json


def lesson_113():
    values = [1, 2, 3, 4, 5]
    count, mean, m2 = 0, 0.0, 0.0
    states = []
    for x in values:
        count += 1
        delta = x - mean
        mean += delta / count
        m2 += delta * (x - mean)
        states.append({"count": count, "mean": mean, "m2": m2})
    return {"count": count, "mean": mean, "m2": m2, "sampleVariance": m2 / (count - 1), "states": states}


if __name__ == "__main__":
    print(json.dumps(lesson_113(), indent=2))

/*
 * Fintech Math Bootcamp - Lesson 113 of 120
 * Stable Online Variance and Welford's Algorithm
 * Module 12: Statistical Computing and Reproducibility
 *
 * Scenario: Updating variance as records arrive without unstable raw subtraction
 * Rule:     n -> mean -> M_2; sample variance=M_2/(n-1)
 *
 * Try it:   Does ordinary Welford automatically expire the oldest rolling-window observation?
 *
 * Lesson article: https://thefintechbuilder.com/financial-mathematics-statistics-and-data-foundations/statistical-computing-and-reproducibility/stable-online-variance-and-welford-s-algorithm/
 * 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 Map<String, Object> lesson113() {
        double[] values = {1, 2, 3, 4, 5};
        int count = 0;
        double mean = 0, m2 = 0;
        List<Object> states = new ArrayList<Object>();
        for (double x : values) {
            count++;
            double delta = x - mean;
            mean += delta / count;
            m2 += delta * (x - mean);
            states.add(obj("count", count, "mean", mean, "m2", m2));
        }
        return obj("count", count, "mean", mean, "m2", m2, "sampleVariance", m2 / (count - 1), "states", states);
    }

    public static void main(String[] args) {
        System.out.println(toJson(lesson113(), ""));
    }

    // 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 113 of 120
// Stable Online Variance and Welford’s Algorithm
// Module 12: Statistical Computing and Reproducibility
//
// Scenario: Updating variance as records arrive without unstable raw subtraction
// Rule:     n → mean → M₂; sample variance=M₂/(n−1)
//
// Try it:   Does ordinary Welford automatically expire the oldest rolling-window observation?
//
// Lesson article: https://thefintechbuilder.com/financial-mathematics-statistics-and-data-foundations/statistical-computing-and-reproducibility/stable-online-variance-and-welford-s-algorithm/
// 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 State struct {
	Count int     `json:"count"`
	Mean  float64 `json:"mean"`
	M2    float64 `json:"m2"`
}

type Result struct {
	Count          int     `json:"count"`
	Mean           float64 `json:"mean"`
	M2             float64 `json:"m2"`
	SampleVariance float64 `json:"sampleVariance"`
	States         []State `json:"states"`
}

func lesson113() Result {
	values := []float64{1, 2, 3, 4, 5}
	count, mean, m2 := 0, 0.0, 0.0
	var states []State
	for _, x := range values {
		count++
		delta := x - mean
		mean += delta / float64(count)
		m2 += delta * (x - mean)
		states = append(states, State{Count: count, Mean: mean, M2: m2})
	}
	return Result{Count: count, Mean: mean, M2: m2, SampleVariance: m2 / float64(count-1), States: states}
}

func main() {
	out, err := json.MarshalIndent(lesson113(), "", "  ")
	if err != nil {
		panic(err)
	}
	fmt.Println(string(out))
}

// Fintech Math Bootcamp · Lesson 113 of 120
// Stable Online Variance and Welford’s Algorithm
// Module 12: Statistical Computing and Reproducibility
//
// Scenario: Updating variance as records arrive without unstable raw subtraction
// Rule:     n → mean → M₂; sample variance=M₂/(n−1)
//
// Try it:   Does ordinary Welford automatically expire the oldest rolling-window observation?
//
// Lesson article: https://thefintechbuilder.com/financial-mathematics-statistics-and-data-foundations/statistical-computing-and-reproducibility/stable-online-variance-and-welford-s-algorithm/
// 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 lesson113() {
    const std::vector<double> values{1, 2, 3, 4, 5};
    int count = 0;
    double mean = 0, m2 = 0;
    std::vector<Json> states;
    for (double x : values) {
        ++count;
        const double delta = x - mean;
        mean += delta / count;
        m2 += delta * (x - mean);
        states.push_back(jobj({{"count", jnum(count)}, {"mean", jnum(mean)}, {"m2", jnum(m2)}}));
    }
    return jobj({{"count", jnum(count)},
                 {"mean", jnum(mean)},
                 {"m2", jnum(m2)},
                 {"sampleVariance", jnum(m2 / (count - 1))},
                 {"states", jarr(states)}});
}

int main() {
    writeJson(std::cout, lesson113());
    std::cout << "\n";
    return 0;
}

// Fintech Math Bootcamp · Lesson 113 of 120
// Stable Online Variance and Welford’s Algorithm
// Module 12: Statistical Computing and Reproducibility
//
// Scenario: Updating variance as records arrive without unstable raw subtraction
// Rule:     n → mean → M₂; sample variance=M₂/(n−1)
//
// Try it:   Does ordinary Welford automatically expire the oldest rolling-window observation?
//
// Lesson article: https://thefintechbuilder.com/financial-mathematics-statistics-and-data-foundations/statistical-computing-and-reproducibility/stable-online-variance-and-welford-s-algorithm/
// 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_113() -> Json {
    let values: [f64; 5] = [1.0, 2.0, 3.0, 4.0, 5.0];
    let (mut count, mut mean, mut m2) = (0_u32, 0.0_f64, 0.0_f64);
    let mut states = Vec::new();
    for &x in &values {
        count += 1;
        let delta = x - mean;
        mean += delta / count as f64;
        m2 += delta * (x - mean);
        states.push(obj(vec![
            ("count", Json::Num(count as f64)),
            ("mean", Json::Num(mean)),
            ("m2", Json::Num(m2)),
        ]));
    }
    obj(vec![
        ("count", Json::Num(count as f64)),
        ("mean", Json::Num(mean)),
        ("m2", Json::Num(m2)),
        ("sampleVariance", Json::Num(m2 / (count - 1) as f64)),
        ("states", Json::Arr(states)),
    ])
}

fn main() {
    let mut out = String::new();
    write_json(&lesson_113(), "", &mut out);
    println!("{}", out);
}

// Fintech Math Bootcamp · Lesson 113 of 120
// Stable Online Variance and Welford’s Algorithm
// Module 12: Statistical Computing and Reproducibility
//
// Scenario: Updating variance as records arrive without unstable raw subtraction
// Rule:     n → mean → M₂; sample variance=M₂/(n−1)
//
// Try it:   Does ordinary Welford automatically expire the oldest rolling-window observation?
//
// Lesson article: https://thefintechbuilder.com/financial-mathematics-statistics-and-data-foundations/statistical-computing-and-reproducibility/stable-online-variance-and-welford-s-algorithm/
// 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(Lesson113(), options));

static object Lesson113()
{
    double[] values = { 1, 2, 3, 4, 5 };
    int count = 0;
    double mean = 0, m2 = 0;
    var states = new List<object>();
    foreach (double x in values)
    {
        count++;
        double delta = x - mean;
        mean += delta / count;
        m2 += delta * (x - mean);
        states.Add(new { count, mean, m2 });
    }
    return new { count, mean, m2, sampleVariance = m2 / (count - 1), states };
}