Advanced Sequence Predictive w/ Linear Regression

README.md

Advanced Sequence Prediction

This guide offers examples into predicting sequences with JS, Python, and C implementations, with refining prediction accuracy through advanced methodologies.

$$m = \frac{n\sum xy - \sum x\sum y}{n\sum x^2 - (\sum x)^2} b = \frac{\sum y - m\sum x}{n}$$

Introduction

Linear Regression is an analytical technique employed to discern the relationship between two continuous variables. It does so by fitting a linear equation to observed data. The equation manifests as $y=mx+b$, where $y$ denotes the dependent variable, $x$ the independent variable, $m$ the slope of the line, and $b$ the y-intercept.

Table of Contents

• Introduction
• Entropy
• Fundamentals
• Implementations
  • JavaScript
  • Python
  • C
• Advanced Analytical Methods
• References

Fundamentals

Linear regression's core lies in deriving the slope $m$ and y-intercept $b$ from a dataset, minimizing prediction errors. The formulas are as follows:

To deduce the slope and y-intercept from a dataset, the subsequent formulas are applied:

$$m = \frac{n\sum xy - \sum x\sum y}{n\sum x^2 - (\sum x)^2} b = \frac{\sum y - m\sum x}{n}$$

Where:

• $n$ signifies the quantity of data points.
• $x$ and $y$ are vectors denoting the coordinates of the data points in the independent and dependent dimensions, respectively.
• $∑x$ and $∑y$ represent the cumulative sums of the $x$ and $y$ values within the dataset.
• $∑xy$ is the aggregate of the products of corresponding $x$ and $y$ pairs
• $∑x^2$ encapsulates the total of the squares of the $x$ values.
• $m$ is derived as the slope of the line, calculated per the linear regression slope
• $b$, the y-intercept of the line, is determined using the linear regression intercept formula.

$$\begin{align*} n &= 5 \\\ x &= [0, 1, 2, 3, 4] \\\ y &= [3, 6, 9, 12, 15] \\\ \sum x &= 10 \\\ \sum y &= 45 \\\ \sum xy &= 90 \\\ \sum x^2 &= 30 \\\ m &= \frac{5 \cdot 90 - 10 \cdot 45}{5 \cdot 30 - 10^2} = 3 \\\ b &= \frac{45 - 3 \cdot 10}{5} = 0 \end{align*}$$

Hence, the equation for the optimal line of fit is $y = 3x + 0$.

For predicting subsequent values in the sequence, employ the equation $y=mx+b$ with = $x=5$ (the ensuing sequence value)

Implementations

JavaScript

function forecastNextValue(series) {
  const count = series.length;
  let sumX = 0, sumY = 0, sumXY = 0, sumXX = 0;
  for (let index = 0; index < count; index++) {
    sumX += index;
    sumY += series[index];
    sumXY += index * series[index];
    sumXX += index * index;
  }
  const m = (count * sumXY - sumX * sumY) / (count * sumXX - sumX * sumX);
  const b = (sumY - m * sumX) / count;
  return m * count + b;
}

const sequence = [1, 2, 4, 7, 11];
const prediction = forecastNextValue(sequence);
console.log(prediction); // Expected output: 16.2

Python

def predict_next_value(numbers):
    n = len(numbers)
    x_sum = sum(range(n))
    y_sum = sum(numbers)
    xy_sum = sum(x * y for x, y in zip(range(n), numbers))
    xx_sum = sum(x ** 2 for x in range(n))
    
    slope = (n * xy_sum - x_sum * y_sum) / (n * xx_sum - x_sum ** 2)
    intercept = (y_sum - slope * x_sum) / n
    return slope * n + intercept

numbers = [1, 2, 4, 7, 11]
next_number = predict_next_value(numbers)
print(next_number)

C

#include <stdio.h>

double predict_next_value(int *numbers, int n) {
    int x_sum = 0, y_sum = 0, xy_sum = 0, xx_sum = 0;
    for (int i = 0; i < n; i++) {
        x_sum += i;
        y_sum += numbers[i];
        xy_sum += i * numbers[i];
        xx_sum += i * i;
    }
    
    double slope = (n * xy_sum - x_sum * y_sum) / (double)(n * xx_sum - x_sum * x_sum);
    double intercept = (y_sum - slope * x_sum) / (double)n;
    return slope * n + intercept;
}

int main() {
    int numbers[] = {1, 2, 4, 7, 11};
    int n = sizeof(numbers) / sizeof(numbers[0]);
    double next_number = predict_next_value(numbers, n);
    printf("%f\n", next_number);
    return 0;
}

Advanced Analytical Methods

Advanced techniques like Polynomial Regression, Regularization (Lasso and Ridge), and Moving Averages improve the model's accuracy, especially in non-linear datasets or to prevent overfitting. Incorporating external variables and continuous model updating are essential for adapting to new data.

References

• Linear Regression - Wikipedia
• Linear Regression - Math is Fun