Holt-Winters Forecasting and Exponential Smoothing Simplified - Orange Matter (2024)

Holt-Winters forecasting is a way to model and predict the behavior of a sequence of values over time—a time series. Holt-Winters is one of the most popular forecasting techniques for time series. It’s decades old, but it’s still ubiquitous in many applications, including monitoring, where it’s used for purposes such as anomaly detection and capacity planning.

Unfortunately, Holt-Winters forecasting is confusing, so it’s often poorly understood. We want to fix that, so we wrote this post: a visual introduction to Holt-Winters.

What Is the Holt-Winters Method?

Holt-Winters is a model of time series behavior. Forecasting always requires a model, and Holt-Winters is a way to model three aspects of the time series: a typical value (average), a slope (trend) over time, and a cyclical repeating pattern (seasonality).

Time series anomaly detection is a complicated problem with plenty of practical methods. It’s easy to get lost in all of the topics it encompasses. Learning them is certainly an issue, but implementing them is often more complicated. A key element of anomaly detection is forecasting—taking what you know about a time series, either based on a model or its history, and making decisions about values that arrive later.

You know how to do this already. Imagine someone asked you to forecast the prices for a certain stock, or the local temperature over the next few days. You could draw out your prediction, and chances are it’s a pretty good one. Your brain works amazingly well for problems like this, and our challenge is to try to get computers to do the same.

If you take an introductory course on time series, you’ll learn how to forecast by fitting a model to some sample data, and then using the model to predict future values. In practice, especially when monitoring systems, this approach doesn’t work well, if at all! Real systems rarely fit mathematical models. There’s an alternative. You can do something a lot simpler with exponential smoothing.

Holt-Winters Triple Exponential Smoothing Formula Explained

The Holt-Winters method uses exponential smoothing to encode lots of values from the past and use them to predict “typical” values for the present and future. Exponential smoothing refers to the use of an exponentially weighted moving average (EWMA) to “smooth” a time series. If you have some time seriesx_t, you can define a new time seriess_tthat’s a smoothed version ofx_t.

s_t=αx_t+(1−α)s_t−1

In the late 1950s, Charles Holt recognized the issue with the simple EWMA model with time series with trend. He modified the simple exponential smoothing model to account for a linear trend. This is known as Holt’s exponential smoothing. This model is a little more complicated. It consists of two EWMAs: one for the smoothed values ofxt, and another for itsslope. The termslevelandtrendare also used.

s_t=αx_t+(1−α)(s_t−1+b_t−1)

b_t=β(s_t−s_t−1)+(1−β)b_t−1

Notice how the smoothed values are much better at following the original time series with double exponential smoothing. This means you’ll get much better forecasts.

To forecast with this model, you have to make a slight adjustment. Because there’s another term for the slope, you’ll have to consider that in the forecast. Suppose you’re trying to forecast the value inmtime steps in the future. The formula for them-step-ahead forecast,F_t+m, is:

F_t+m=s_t+mb_t

Notice how it’s essentially the formula for a line. What if your time series doesn’t have a linear trend, but rather some sort of seasonality? For this, you’ll need yet another EWMA.

Holt’s student, Peter Winters, extended his teacher’s model by introducing an additional term to factor in seasonality. Notice how there’s another variableL, which depends on the period of the seasonality and has to be known in advance.The three aspects of the time series behavior—value, trend, and seasonality—are expressed as three types of exponential smoothing, so Holt-Winters is called triple exponential smoothing. The model predicts a current or future value by computing the combined effects of these three influences. The model requires several parameters: one for each smoothing (ɑ, β, γ), the length of a season, and the number of periods in a season.

Holt-Winters Forecasting Example

The hardest parts of Holt-Winters forecasting are understanding how the model works and choosing good parameters. To tackle the first, we’ll walk through an example model by doing Holt-Winters “by hand.”

The usual way to explain Holt-Winters is by showing a bunch of complicated equations with Greek letters and subscripts. We’ll skip the math and show how it works, which is a lot simpler. We’re going to be working with this time series: