Autocorrelation analysis in ML Python

Introduction

Autocorrelation analysis helps us find patterns in data by checking if values repeat over time or space.

To see if daily temperatures follow a pattern over weeks.

To check if stock prices depend on their past values.

To find repeating signals in sensor data like heartbeats.

To detect seasonality in sales data over months.

To understand if errors in a model are related over time.

Syntax

ML Python

import numpy as np

def autocorrelation(x, lag=1):
    n = len(x)
    x_mean = np.mean(x)
    numerator = np.sum((x[:n-lag] - x_mean) * (x[lag:] - x_mean))
    denominator = np.sum((x - x_mean) ** 2)
    if denominator == 0:
        return float('nan')
    return numerator / denominator

The function calculates autocorrelation for a given lag (how far apart values are).

Lag 1 means comparing each value with the next one.

Examples

Calculates autocorrelation between each number and the next one.

ML Python

autocorrelation([1, 2, 3, 4, 5], lag=1)

Calculates autocorrelation between each number and the one two steps ahead.

ML Python

autocorrelation([1, 2, 3, 4, 5], lag=2)

Returns NaN because all values are the same, so no variation.

ML Python

autocorrelation([5, 5, 5, 5, 5], lag=1)

Sample Model

This program calculates how daily sales relate to the previous day (lag 1) and two days before (lag 2).

ML Python

import numpy as np

def autocorrelation(x, lag=1):
    n = len(x)
    x_mean = np.mean(x)
    numerator = np.sum((x[:n-lag] - x_mean) * (x[lag:] - x_mean))
    denominator = np.sum((x - x_mean) ** 2)
    if denominator == 0:
        return float('nan')
    return numerator / denominator

# Example data: daily sales over 7 days
sales = np.array([10, 12, 13, 12, 11, 13, 14])

# Calculate autocorrelation for lag 1 and lag 2
ac_lag1 = autocorrelation(sales, lag=1)
ac_lag2 = autocorrelation(sales, lag=2)

print(f"Autocorrelation at lag 1: {ac_lag1:.3f}")
print(f"Autocorrelation at lag 2: {ac_lag2:.3f}")

Important Notes

Autocorrelation values range from -1 to 1, where 1 means perfect positive correlation.

A value near 0 means no correlation at that lag.

High autocorrelation at certain lags can show repeating patterns or cycles.

Summary

Autocorrelation checks if data points relate to past points at different distances.

It helps find patterns like seasonality or trends in time series data.

You can calculate it easily with a simple formula comparing shifted data.