TripleExpSmooth Routines

Summary

Triple exponential smoothing.

Declaration

Function TripleExpSmooth(Y: TVec; S, B, L: TVec; var Alpha, Beta, Gamma: TSample; const Period: Integer): TSample;

Parameter	Description
Y	Time series data set.
S	Smoothed values (see above equation). Size and complex properties of S are set automatically.
b	Trend values (see above equation). Size and complex properties of b are set automatically.
L	Seasonal indices (see above equation). Size and complex properties of L are set automatically.
Alpha	Defines initial estimate for Alpha, returns Alpha which minimizes MSE.
Beta	Defines initial estimate for Beta, returns Beta which minimizes MSE.
Gamma	Defines initial estimate for Gamma, returns Gamma which minimizes MSE.
Period	Period length. An exception is raised if Y.Length mod Period is not 0.

Result

MSE, evaluated at minimum.

Description

Performs double exponential smoothing (also known as Holt-Winters smoothing) using the following equations:

where Y are the observations, S are the smoothed observations, b trend factors, L the seasonal indices and P is the period length. To initialize triple exponential smoothing method we need at least one complete season's data to determine initial estimates of the seasonal indices L[0]..L[P-1]. Again, there are several ways to initialize L values. The algorithm uses approach, described at NIST pages. For initial estimate for S and b, the following equations are being used:

Please note that there are no S[0]..S[P-2]; the smoothed series starts with the smoothed version of the Y[P] observation. Also note that the internal algorithm automatically accounts for this by resizing S,b vector to Y.Length-Period.

See Also
TripleExpForecast

Example 1

Generate 24 random values representing 4 quarters x 6 years = 24, perform smoothing and read Alpha,Beta,Gamma + MSE.

  Uses MtxExpr,StatTimeSerAnalysis, Math387;
  procedure Example;
  var Data,S,b,L: Vector;
      Alpha,Beta,Gamma,MSE: TSample;
  begin
    Data.Size(24,false);
    Data.RandGauss;
    // smooth data, initial alpha = 0.1, beta=0.1, gamma = 0.3
    Alpha := 0.1;
    Beta := 0.1;
    Gamma := 0.3;
    // Period = 4
    MSE := TripleExpSmooth(Data,S,Alpha,BetamGamma,4);
    // results: MSE and MLE estimate for Alpha,Beta,Gamma
  end;

  #include "MtxVecCpp.h"
  #include "Math387.hpp"
  #include "StatTimeSerAnalysis.hpp"
  void __fastcall Example();
  {
    Vector Data,S,b,L;
    Data->Size(24,false);
    Data->RandGauss();
    // smooth data, initial alpha = 0.1, beta=0.1, gamma = 0.3
    double alpha = 0.1;
    double beta = 0.1;
    double gamma = 0.3;
    // Period = 4
    double MSE = TripleExpSmooth(Data,S,alpha,beta,gamma,4);
    // results: MSE and MLE estimate for Alpha,Beta,Gamma
  }

  using Dew.Math;
  using Dew.Stats;
  using Dew.Stats.Units;
  namespace Dew.Examples
  {
    private void Example()
    {
      Vector Data = new Vector(0);
      Vector S = new Vector(0);
      Vector b = new Vector(0);
      Vector L = new Vector(0);
      Data.Size(24,false);
      Data.RandGauss();
      // smooth data, initial alpha = 0.1, beta=0.1, gamma = 0.3
      double alpha = 0.1;
      double beta = 0.1;
      double gamma = 0.3;
      // Period = 4
      double MSE = TripleExpSmooth(Data,S,ref alpha,ref beta, ref gamma,4);
      // results: MSE and MLE estimate for Alpha,Beta,Gamma
    }
  }

Declaration

Procedure TripleExpSmooth(Y: TVec; S, B, L: TVec; const Alpha, Beta, Gamma: TSample; out MSE: TSample; const Period: Integer);

Parameter	Description
MSE	Returns MSE, evaluated for constant Alpha, Beta and Gamma.

Description

In this case a fixed smoothing constants Alpha, Beta and Gamma are used in smoothing equations (no minimization is performed).