StatTimeSerAnalysis.DoubleExpSmooth Method

Overload List

#	Signature	Description
1	procedure DoubleExpSmooth(const Y: TVec; const S: TVec; const B: TVec; const Alpha: Double; const Gamma: Double; out MSE: Double; const InitMethod: Integer);	In this case a fixed smoothing constants Alpha, Gamma are used in smoothing equations (no minimization is performed).
2	function DoubleExpSmooth(const Y: TVec; const S: TVec; const B: TVec; var Alpha: Double; var Gamma: Double; const InitMethod: Integer): Double;	Double exponential smoothing.

Overload 1: procedure DoubleExpSmooth(const Y: TVec; const S: TVec; const B: TVec; const Alpha: Double; const Gamma: Double; out MSE: Double; const InitMethod: Integer);

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

#	Name	Description
1	MSE	Returns MSE, evaluated for constant Alpha and Gamma.
2	Y	Time series data set.
3	S	Smoothed values (see above equation). Size and complex properties of S are set automatically.
4	B	Trend values (see above equation). Size and complex properties of b are set automatically.
5	Alpha	Defines initial estimate for Alpha, returns Alpha which minimizes MSE.
6	Gamma	Defines initial estimate for Gamma, returns Gamma which minimizes MSE.
7	InitMethod	Defines how the initial values for b[0] are calculated.

Result: stored in self (calling object)

Examples

Uses MtxExpr, StatTimeSerAnalysis, Math387;
procedure Example;
var Data,S,b: Vector;
MSE: double;
begin
    Data.LoadFromFile('aerosol_particles.vec');
    // smooth data with Alpha=0.2, Gamma=0.18
    DoubleExpSmooth(Data,S,b,0.2,0.18,MSE,1);
    // results: MSE
end;

Overload 2: function DoubleExpSmooth(const Y: TVec; const S: TVec; const B: TVec; var Alpha: Double; var Gamma: Double; const InitMethod: Integer): Double;

Double exponential smoothing.

#	Name	Description
1	Y	Time series data set.
2	S	Smoothed values (see above equation). Size and complex properties of S are set automatically.
3	B	Trend values (see above equation). Size and complex properties of b are set automatically.
4	Alpha	Defines initial estimate for Alpha, returns Alpha which minimizes MSE.
5	Gamma	Defines initial estimate for Gamma, returns Gamma which minimizes MSE.
6	InitMethod	Defines how the initial values for b[0] are calculated.

Returns: Double - MSE,evaluated at minimum.

Remarks:

Performs double exponential smoothing using the following equations:

\begin{aligned} S[i] &= \alpha \cdot Y[i] + (1-\alpha)(S[i-1]+b[i-1]) \quad , \quad 0\leq \alpha \leq 1 \\ b[i] &= \gamma (S[i]-S[i-1])+(1-\gamma)b[i-1] \quad, \quad 0\leq \gamma \leq 1 \end{aligned}

Smoothing scheme begins by setting S[0] to Y[0] and b[0] to pne of the following choices:

\begin{aligned} b[0] &= Y[1]-Y[0] \\ b[0] &= \cfrac{1}{3}(Y[3]-Y[0]) \\ b[0] &= \cfrac{1}{n-1}\left(Y[n-1]-Y[0]\right) \end{aligned}

Different initialization methods are controlled by the InitMethod parameter. Default value (0) uses first equation, setting it to (1) means the second equation will be used and setting it to (2) means the third equation will be used to initialize b[0].

The first smoothing equation adjusts S[i] directly for the trend of the previous period, b[i-1], by adding it to the last smoothed value, S[i-1]. This helps to eliminate the lag and brings S[i] to the appropriate base of the current value. The second smoothing equation then updates the trend, which is expressed as the difference between the last two values. The equation is similar to the basic form of single smoothing, but here applied to the updating of the trend.

Examples

Uses MtxExpr, StatTimeSerAnalysis, Math387;
procedure Example;
var Data,S,b: Vector
Alpha,Gamma,MSE: double;
begin
    Data.LoadFromFile('aerosol_particles.vec');
    // smooth data, initial alpha = 0.1, gamma = 0.3
    Alpha := 0.1;
    Gamma := 0.3;
    MSE := DoubleExpSmooth(Data,S,b,Alpha,Gamma,1);
    // results: MSE and MLE estimate for Alpha,Gamma
end;

See Also: StatTimeSerAnalysis.DoubleExpForecast