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Simple Exponential Smoothing

Simple Exponential Smoothing (SES) is a forecasting method for univariate time series data that does not exhibit a clear trend or seasonal pattern. It's a straightforward yet effective technique that computes future values based on a weighted average of past observations, with the weights decaying exponentially over time. This means that more recent observations are given more weight than older ones.

How Simple Exponential Smoothing Works

The core of SES is the smoothing equation, which updates the smoothed level at each time step:

\(S\_t = \\alpha y\_t + (1 - \\alpha) S\_{t-1}\)

Where:

  • \(S\_t\) is the smoothed value (the level) at time \(t\).
  • \(y\_t\) is the actual observation at time \(t\).
  • \(S\_{t-1}\) is the smoothed value from the previous time step.
  • \(\\alpha\) is the smoothing factor, where \(0 \< \\alpha \\le 1\).

A higher value of alpha makes the model react more quickly to recent changes, while a smaller alpha results in a smoother forecast as it gives more weight to past observations.

Once the model has processed the entire signal, the forecast for all future time points is the last calculated level (\(S\_n\), where \(n\) is the length of the signal).

Configuration (SimpleExponentialSmoothingOptions)

The behavior of the SES algorithm is controlled by the SimpleExponentialSmoothingOptions record:

  • Alpha (double): The smoothing factor for the level. This is a required parameter and must be between 0 and 1, inclusive. Higher values give more weight to recent observations.
  • InitialLevel (double?): An optional initial value for the smoothed level. If you don't provide one, the first data point of the signal is used as the initial level.

Usage Examples

Here are some practical examples of how to use the SimpleExponentialSmoothingExtrapolator in C#.

Example 1: Basic Extrapolation

This example demonstrates the basic usage of the SimpleExponentialSmoothingExtrapolator with a given Alpha.

using SignalSharp.Extrapolation.ExponentialSmoothing;
using System;

// Sample signal with no clear trend or seasonality
double[] signal = { 10.0, 12.0, 15.0, 11.0, 13.0 };

// 1. Configure SES with an Alpha value
var options = new SimpleExponentialSmoothingOptions { Alpha = 0.3 };

// 2. Create the extrapolator instance
var extrapolator = new SimpleExponentialSmoothingExtrapolator<double>(options);

// 3. Fit the model to the historical data
extrapolator.Fit(signal);

// 4. Extrapolate to predict the next 3 data points
int horizon = 3;
double[] forecast = extrapolator.Extrapolate(horizon);

Console.WriteLine("Signal: " + string.Join(", ", signal));
Console.WriteLine("Forecast: " + string.Join(", ", forecast));
// The forecast will be an array of 3 elements, all having the value of the last calculated level.

Example 2: Using an Initial Level

You can also provide a specific starting level for the smoothing process.

using SignalSharp.Extrapolation.ExponentialSmoothing;
using System;

double[] signal = { 10.0, 12.0, 15.0 };

// 1. Configure SES with an Alpha and a specific initial level
var options = new SimpleExponentialSmoothingOptions { Alpha = 0.2, InitialLevel = 8.0 };

// 2. Create the extrapolator and use the FitAndExtrapolate convenience method
var extrapolator = new SimpleExponentialSmoothingExtrapolator<double>(options);
int horizon = 2;
double[] forecast = extrapolator.FitAndExtrapolate(signal, horizon);

Console.WriteLine("Signal: " + string.Join(", ", signal));
Console.WriteLine("Forecast with Initial Level: " + string.Join(", ", forecast));
// L_initial = 8.0
// L1 = 0.2*10.0 + 0.8*8.0 = 2.0 + 6.4 = 8.4
// L2 = 0.2*12.0 + 0.8*8.4 = 2.4 + 6.72 = 9.12
// L3 = 0.2*15.0 + 0.8*9.12 = 3.0 + 7.296 = 10.296
// Expected forecast: [10.296, 10.296]

Example 3: Impact of Alpha

This example shows how different Alpha values affect the forecast. An Alpha of 1 means the forecast will be the last value of the signal.

using SignalSharp.Extrapolation.ExponentialSmoothing;
using System;

double[] signal = { 10.0, 12.0, 15.0 };

// High Alpha: The model gives most weight to the last observation.
var highAlphaOptions = new SimpleExponentialSmoothingOptions { Alpha = 1.0 };
var highAlphaExtrapolator = new SimpleExponentialSmoothingExtrapolator<double>(highAlphaOptions);
double[] highAlphaForecast = highAlphaExtrapolator.FitAndExtrapolate(signal, 1);

Console.WriteLine("Forecast with Alpha=1.0: " + string.Join(", ", highAlphaForecast));
// Expected: [15.0]

// Low Alpha: The model is smoothed more, relying more on past values.
var lowAlphaOptions = new SimpleExponentialSmoothingOptions { Alpha = 0.1 };
var lowAlphaExtrapolator = new SimpleExponentialSmoothingExtrapolator<double>(lowAlphaOptions);
double[] lowAlphaForecast = lowAlphaExtrapolator.FitAndExtrapolate(signal, 1);

Console.WriteLine("Forecast with Alpha=0.1: " + string.Join(", ", lowAlphaForecast));
// L_initial = 10.0
// L1 = 0.1*10 + 0.9*10 = 10
// L2 = 0.1*12 + 0.9*10 = 1.2 + 9 = 10.2
// L3 = 0.1*15 + 0.9*10.2 = 1.5 + 9.18 = 10.68
// Expected: [10.68]

API References

  • Extrapolator: SignalSharp.Extrapolation.ExponentialSmoothing.SimpleExponentialSmoothingExtrapolator<T>
  • Options: SignalSharp.Extrapolation.ExponentialSmoothing.SimpleExponentialSmoothingOptions
  • Interface: SignalSharp.Extrapolation.IExtrapolator<T>
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