Examples of Autocorrelation Function (ACF)

Explore diverse and practical examples of autocorrelation function (ACF) in time series analysis.
By Jamie

Introduction to Autocorrelation Function (ACF)

The autocorrelation function (ACF) is a crucial tool in time series analysis, helping to identify patterns and correlations within a dataset over time. It measures the relationship between a time series and its past values, allowing researchers and analysts to understand the inherent structure of the data. ACF is particularly useful in determining the seasonality and cyclic behavior of time series data, which can be invaluable in fields like finance, meteorology, and economics.

Example 1: Monthly Average Temperature Analysis

Context

In meteorology, analyzing temperature trends over time can reveal seasonal patterns and help in climate modeling. Understanding how past temperatures influence future readings is essential for predicting weather patterns.

To illustrate this, we will analyze the monthly average temperatures of a specific city over the past five years.

When we compute the ACF for this temperature data, we expect to see strong correlations at lags corresponding to the seasonal cycle (e.g., lag 12 for yearly seasonality).

Example

  • Data: Monthly average temperatures (°C) for City XYZ from January 2018 to December 2022.
Month Temperature (°C)
Jan 5
Feb 6
Mar 10
Apr 15
May 20
Jun 25
Jul 30
Aug 29
Sep 23
Oct 15
Nov 10
Dec 6
  • ACF Results: ACF shows significant spikes at lags 1 (0.8), 12 (0.9) indicating that past temperatures, especially from the same month in the previous year, strongly influence current temperatures.

Notes

  • This example highlights how ACF can reveal seasonal trends. Variations in the dataset (like extreme temperatures) may affect the correlation values.

Example 2: Stock Prices Analysis

Context

In finance, understanding the behavior of stock prices over time can help investors make informed decisions. ACF is used to analyze the dependency of a stock’s current price on its past prices.

We will examine the daily closing prices of a stock over a 30-day period to visualize how past prices correlate with current prices.

Example

  • Data: Daily closing prices of Stock ABC for 30 days.
Day Closing Price (USD)
1 50
2 51
3 52
4 49
5 50
6 48
7 51
8 53
9 55
10 54
11 52
12 51
13 50
14 49
15 52
16 54
17 56
18 58
19 57
20 59
21 60
22 61
23 62
24 64
25 65
26 66
27 67
28 68
29 69
30 70
  • ACF Results: The ACF reveals strong correlations at lag 1 (0.75) and decreasing correlations as the lag increases, indicating that the most recent price is a good predictor of the next day’s price.

Notes

  • ACF can help identify momentum in stock prices. However, external factors such as market news can disrupt these correlations.

Example 3: Daily Website Traffic Analysis

Context

For digital marketers and web analysts, understanding traffic patterns on a website is crucial for optimizing content and improving user engagement. ACF can help in identifying trends in daily visits.

We will look at the number of daily visits to a website over a month to see how past traffic influences future traffic.

Example

  • Data: Daily visits to Website DEF for 30 days.
Day Daily Visits
1 200
2 220
3 180
4 210
5 240
6 230
7 250
8 300
9 310
10 290
11 280
12 320
13 330
14 340
15 360
16 370
17 400
18 410
19 450
20 460
21 480
22 500
23 520
24 530
25 550
26 560
27 570
28 590
29 600
30 620
  • ACF Results: The ACF shows strong correlations at lag 1 (0.85) and lag 2 (0.65), indicating that traffic from the previous days significantly influences the current day’s traffic.

Notes

  • Variations in traffic, such as promotional events or external campaigns, can alter the expected autocorrelation patterns. Monitoring these changes is essential for effective traffic analysis.