StatTimeSerAnalysis.TripleExpSmooth Method

Overload List

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

Overload 1: procedure TripleExpSmooth(const Y: TVec; const S: TVec; const B: TVec; const L: TVec; const Alpha: Double; const Beta: Double; const Gamma: Double; out MSE: Double; const Period: Integer);

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

#	Name	Description
1	MSE	Returns MSE, evaluated for constant Alpha, Beta 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	L	Seasonal indices (see above equation). Size and complex properties of L are set automatically.
6	Alpha	Defines initial estimate for Alpha, returns Alpha which minimizes MSE.
7	Beta	Defines initial estimate for Beta, returns Beta which minimizes MSE.
8	Gamma	Defines initial estimate for Gamma, returns Gamma which minimizes MSE.
9	Period	Period length. An exception is raised if Y.Length mod Period is not 0.

Result: stored in self (calling object)

Overload 2: function TripleExpSmooth(const Y: TVec; const S: TVec; const B: TVec; const L: TVec; var Alpha: Double; var Beta: Double; var Gamma: Double; const Period: Integer): Double;

Triple 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	L	Seasonal indices (see above equation). Size and complex properties of L are set automatically.
5	Alpha	Defines initial estimate for Alpha, returns Alpha which minimizes MSE.
6	Beta	Defines initial estimate for Beta, returns Beta which minimizes MSE.
7	Gamma	Defines initial estimate for Gamma, returns Gamma which minimizes MSE.
8	Period	Period length. An exception is raised if Y.Length mod Period is not 0.

Returns: Double - MSE, evaluated at minimum.

Remarks:

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

\begin{aligned} S[i] &= \alpha \cfrac{Y[i]}{L[i-P]} + (1-\alpha)\left(S[i-1]+b[i-1]\right) \quad , \quad \text {overral smoothing} \\ b[i] &= \gamma \left( S[i]-S[i-1]\right) + (1-\gamma)b[i-1] \quad , \quad \text{trend smoothing} \\ L[i] &= \beta \cfrac{Y[i]}{S[i]} + (1-\beta)L[i-P] \quad , \quad \quad \text{seasonal smoothing} \end{aligned}

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 L0]..L[P-1]. Again, there are several ways to initialize L values. The algorithm uses approach, described at [www.itl.nist.gov/div898/handbook/pmc/section4/pmc435.htm page. For initial estimate for S and b, the following equations are being used:

\begin{aligned} S[P-1] &= \cfrac{1}{P}\sum _{i=0} ^{P-1} Y[i] \quad , \\ b[P-1] &= \cfrac{1}{P^2} \left((Y[P]-Y[0]) + (Y[P+1]-Y[1]) + \cdots + (Y[2P-1]-Y[P-1])\right) \quad . \end{aligned}

Note
There are no S[0]..S[P-2] values; 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.

Examples

Uses MtxExpr,StatTimeSerAnalysis, Math387;
procedure Example;
var Data,S,b,L: Vector;
Alpha,Beta,Gamma,MSE: double;
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,b,L,Alpha,Beta,Gamma,4);
    // results: MSE and MLE estimate for Alpha,Beta,Gamma
end;

See Also: StatTimeSerAnalysis.TripleExpForecast