fit parameters of an exponential smoothing model to a time series - Maple Help

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TimeSeriesAnalysis[Optimize] - fit parameters of an exponential smoothing model to a time series

Calling Sequence

Optimize(model, ts, init)

Parameters

model

-

Exponential smoothing model

ts

-

Time series consisting of a single data set

init

-

(optional) table of initial parameter values

Description

• 

The Optimize command will fit unassigned parameter values of model to maximize the likelihood of obtaining the time series ts.

• 

If a parameter was fixed when creating model, its value will not be subject to optimization. For example, if the calling sequence of ExponentialSmoothingModel includes the option alpha = 0.3, then calling Optimize on the resulting model keeps alpha fixed. (This is also true if Optimize is called automatically when ExponentialSmoothingModel gets a Time series as its first argument.)

• 

Optimize is only guaranteed to find a local optimum; it calls Optimization[NLPSolve]. It uses the nonlinear simplex method (also known as Nelder-Mead).

• 

The optimization process needs to be started with an initial point; this point is given by init. It uses the format returned by Initialize: it is a table with parameter names as indices and parameter values as values. If init is not given, Optimize calls Initialize and uses its output by default.

Examples

withTimeSeriesAnalysis:

Consider the following time series. It represents international tourist visitor nights in Australia.

ts:=TimeSeries41.7,24.0,32.3,37.3,46.2,29.3,36.5,43.0,48.9,31.2,37.7,40.4,51.2,31.9,41.0,43.8,55.6,33.9,42.1,45.6,59.8,35.2,44.3,47.9,startdate=2005,frequency=quarterly,header=Visitor nights

ts:=Time seriesVisitor nights24 rows of data:2005-Jan-01 - 2010-Oct-01

(1)

esm:=ExponentialSmoothingModelseasonal=A,M,constraints=admissible

esm:=< an ETS(*,*,*) model >

(2)

Transform to a collection of specialized models.

models:=Specializeesm&comma;ts

models:=< an ETS(A,A,A) model >&comma;< an ETS(A,Ad,A) model >&comma;< an ETS(A,N,A) model >&comma;< an ETS(M,A,A) model >&comma;< an ETS(M,A,M) model >&comma;< an ETS(M,Ad,A) model >&comma;< an ETS(M,Ad,M) model >&comma;< an ETS(M,M,M) model >&comma;< an ETS(M,Md,M) model >&comma;< an ETS(M,N,A) model >&comma;< an ETS(M,N,M) model >

(3)

Find initial points for optimization for all of these.

inits:=mapInitialize&comma;models&comma;ts&colon;

Optimize all of them.

foritonumelemsmodelsdoOptimizemodelsi&comma;ts&comma;initsiend do&colon;

Alternatively, we can let Optimize call Initialize for us.

models2:=Specializeesm&comma;ts

models2:=< an ETS(A,A,A) model >&comma;< an ETS(A,Ad,A) model >&comma;< an ETS(A,N,A) model >&comma;< an ETS(M,A,A) model >&comma;< an ETS(M,A,M) model >&comma;< an ETS(M,Ad,A) model >&comma;< an ETS(M,Ad,M) model >&comma;< an ETS(M,M,M) model >&comma;< an ETS(M,Md,M) model >&comma;< an ETS(M,N,A) model >&comma;< an ETS(M,N,M) model >

(4)

mapOptimize&comma;models2&comma;ts

43.87641045&comma;43.30741702&comma;47.25550594&comma;43.49539348&comma;42.02319299&comma;42.42526190&comma;40.68983138&comma;42.92489529&comma;40.42353080&comma;46.41831579&comma;46.75393519

(5)

Evaluate the Bayesian information criterion for each model.

mapmodel&rarr;printmodel&comma;BICmodel&comma;ts&comma;models&colon;

< an ETS(A,A,A) model >&comma;126.7819508

< an ETS(A,Ad,A) model >&comma;126.9258852

< an ETS(A,N,A) model >&comma;129.9242821

< an ETS(M,A,A) model >&comma;141.6667862

< an ETS(M,A,M) model >&comma;109.4702551

< an ETS(M,Ad,A) model >&comma;135.7502647

< an ETS(M,Ad,M) model >&comma;109.9821406

< an ETS(M,M,M) model >&comma;111.2692148

< an ETS(M,Md,M) model >&comma;109.4060877

< an ETS(M,N,A) model >&comma;140.6230023

< an ETS(M,N,M) model >&comma;112.5756460

(6)

Compare all models' fits.

colors:=ColorTools:-GradientNiagara Navy..Niagara Purple&comma;number&equals;numelemsmodels

colors:=RGB : 0 0.0549 0.471&comma;RGB : 0.0392 0.0503 0.469&comma;RGB : 0.0784 0.0458 0.467&comma;RGB : 0.118 0.0412 0.465&comma;RGB : 0.157 0.0366 0.463&comma;RGB : 0.196 0.032 0.461&comma;RGB : 0.235 0.0275 0.459&comma;RGB : 0.275 0.0229 0.457&comma;RGB : 0.314 0.0183 0.455&comma;RGB : 0.353 0.0137 0.453&comma;RGB : 0.392 0.00915 0.451&comma;RGB : 0.431 0.00458 0.449&comma;RGB : 0.471 0 0.447

(7)

TimeSeriesPlotseqOneStepForecastsmodelsi&comma;ts&comma;color&equals;ToPlotColorcolorsi&comma;legend&equals;modelsi&comma;i&equals;1..numelemsmodels&comma;ts&comma;color&equals;Niagara Green&comma;thickness&equals;3

See Also

Exponential smoothing model, Initialize, Loglikelihood, Specialize, TimeSeriesAnalysis

References

  

Hyndman, R.J. and Athanasopoulos, G. (2013) Forecasting: principles and practice. http://otexts.org/fpp/. Accessed on 2013-10-09.

  

Hyndman, R.J., Koehler, A.B., Ord, J.K., and Snyder, R.D. (2008) Forecasting with Exponential Smoothing: The State Space Approach. Springer Series in Statistics. Springer-Verlag Berlin Heidelberg.


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