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.

$\mathrm{with}\left(\mathrm{TimeSeriesAnalysis}\right)\:$

$\mathrm{ts}≔\mathrm{TimeSeries}\left(⟨41.7\,24.0\,32.3\,37.3\,46.2\,29.3\,36.5\,43.0\,48.9\,31.2\,37.7\,40.4\,51.2\,31.9\,41.0\,43.8\,55.6\,33.9\,42.1\,45.6\,59.8\,35.2\,44.3\,47.9⟩\,\mathrm{startdate}=''2005''\,\mathrm{frequency}=\mathrm{quarterly}\,\mathrm{header}=''Visitor\; nights''\right)$

$\mathrm{ts}≔\left[\begin{array}{c}\mathrm{Time\; series}\\ \mathrm{Visitor\; nights}\\ \mathrm{24\; rows\; of\; data:}\\ \mathrm{2005-Jan-01\; -\; 2010-Oct-01}\end{array}\right]$

$\mathrm{esm}≔\mathrm{ExponentialSmoothingModel}\left(\mathrm{seasonal}=\left\{A\,M\right\}\,\mathrm{constraints}=\mathrm{admissible}\right)$

$\mathrm{esm}≔\mathrm{<\; an\; ETS(*,*,*)\; model\; >}$

$\mathrm{models}≔\mathrm{Specialize}\left(\mathrm{esm}\&comma;\mathrm{ts}\right)$

$\mathrm{models}≔\left[\mathrm{<\; an\; ETS(A,A,A)\; model\; >}\&comma;\mathrm{<\; an\; ETS(A,Ad,A)\; model\; >}\&comma;\mathrm{<\; an\; ETS(A,N,A)\; model\; >}\&comma;\mathrm{<\; an\; ETS(M,A,A)\; model\; >}\&comma;\mathrm{<\; an\; ETS(M,A,M)\; model\; >}\&comma;\mathrm{<\; an\; ETS(M,Ad,A)\; model\; >}\&comma;\mathrm{<\; an\; ETS(M,Ad,M)\; model\; >}\&comma;\mathrm{<\; an\; ETS(M,M,M)\; model\; >}\&comma;\mathrm{<\; an\; ETS(M,Md,M)\; model\; >}\&comma;\mathrm{<\; an\; ETS(M,N,A)\; model\; >}\&comma;\mathrm{<\; an\; ETS(M,N,M)\; model\; >}\right]$

$\mathrm{inits}≔\mathrm{map}\left(\mathrm{Initialize}\&comma;\mathrm{models}\&comma;\mathrm{ts}\right)\&colon;$

$$\begin{gathered} \mathbf{for}i\mathbf{to}\mathrm{numelems}\left(\mathrm{models}\right)\mathbf{do} \\ \mathrm{Optimize}\left(\mathrm{models}\left[i\right]\&comma;\mathrm{ts}\&comma;\mathrm{inits}\left[i\right]\right) \\ \mathbf{end}\mathbf{do}\&colon; \end{gathered}$$

$\mathrm{models2}≔\mathrm{Specialize}\left(\mathrm{esm}\&comma;\mathrm{ts}\right)$

$\mathrm{models2}≔\left[\mathrm{<\; an\; ETS(A,A,A)\; model\; >}\&comma;\mathrm{<\; an\; ETS(A,Ad,A)\; model\; >}\&comma;\mathrm{<\; an\; ETS(A,N,A)\; model\; >}\&comma;\mathrm{<\; an\; ETS(M,A,A)\; model\; >}\&comma;\mathrm{<\; an\; ETS(M,A,M)\; model\; >}\&comma;\mathrm{<\; an\; ETS(M,Ad,A)\; model\; >}\&comma;\mathrm{<\; an\; ETS(M,Ad,M)\; model\; >}\&comma;\mathrm{<\; an\; ETS(M,M,M)\; model\; >}\&comma;\mathrm{<\; an\; ETS(M,Md,M)\; model\; >}\&comma;\mathrm{<\; an\; ETS(M,N,A)\; model\; >}\&comma;\mathrm{<\; an\; ETS(M,N,M)\; model\; >}\right]$

$\mathrm{map}\left(\mathrm{Optimize}\&comma;\mathrm{models2}\&comma;\mathrm{ts}\right)$

$\left[\mathrm{-43.87641045}\&comma;\mathrm{-43.30741702}\&comma;\mathrm{-47.25550594}\&comma;\mathrm{-43.49539348}\&comma;\mathrm{-42.02319299}\&comma;\mathrm{-42.42526190}\&comma;\mathrm{-40.68983138}\&comma;\mathrm{-42.92489529}\&comma;\mathrm{-40.42353080}\&comma;\mathrm{-46.41831579}\&comma;\mathrm{-46.75393519}\right]$

$\mathrm{map}\left(\mathrm{model}\mapsto \mathrm{print}\left(\mathrm{model}\&comma;\mathrm{BIC}\left(\mathrm{model}\&comma;\mathrm{ts}\right)\right)\&comma;\mathrm{models}\right)\&colon;$

$\mathrm{<\; an\; ETS(A,A,A)\; model\; >},126.7819508$

$\mathrm{<\; an\; ETS(A,Ad,A)\; model\; >},126.9258852$

$\mathrm{<\; an\; ETS(A,N,A)\; model\; >},129.9242821$

$\mathrm{<\; an\; ETS(M,A,A)\; model\; >},141.6667862$

$\mathrm{<\; an\; ETS(M,A,M)\; model\; >},109.4702551$

$\mathrm{<\; an\; ETS(M,Ad,A)\; model\; >},135.7502647$

$\mathrm{<\; an\; ETS(M,Ad,M)\; model\; >},109.9821406$

$\mathrm{<\; an\; ETS(M,M,M)\; model\; >},111.2692148$

$\mathrm{<\; an\; ETS(M,Md,M)\; model\; >},109.4060877$

$\mathrm{<\; an\; ETS(M,N,A)\; model\; >},140.6230023$

$\mathrm{<\; an\; ETS(M,N,M)\; model\; >},112.5756460$

$\mathrm{colors}≔\mathrm{ColorTools}:-\mathrm{Gradient}\left(''Niagara\; Navy''..''Niagara\; Purple''\&comma;\mathrm{number}=\mathrm{numelems}\left(\mathrm{models}\right)\right)$

$\mathrm{colors}≔\left[⟨RGB\; :\; 0\; 0.0549\; 0.471⟩\&comma;⟨RGB\; :\; 0.0392\; 0.0503\; 0.469⟩\&comma;⟨RGB\; :\; 0.0784\; 0.0458\; 0.467⟩\&comma;⟨RGB\; :\; 0.118\; 0.0412\; 0.465⟩\&comma;⟨RGB\; :\; 0.157\; 0.0366\; 0.463⟩\&comma;⟨RGB\; :\; 0.196\; 0.032\; 0.461⟩\&comma;⟨RGB\; :\; 0.235\; 0.0275\; 0.459⟩\&comma;⟨RGB\; :\; 0.275\; 0.0229\; 0.457⟩\&comma;⟨RGB\; :\; 0.314\; 0.0183\; 0.455⟩\&comma;⟨RGB\; :\; 0.353\; 0.0137\; 0.453⟩\&comma;⟨RGB\; :\; 0.392\; 0.00915\; 0.451⟩\&comma;⟨RGB\; :\; 0.431\; 0.00458\; 0.449⟩\&comma;⟨RGB\; :\; 0.471\; 0\; 0.447⟩\right]$

$\mathrm{TimeSeriesPlot}\left(\mathrm{seq}\left(\left[\mathrm{OneStepForecasts}\left(\mathrm{models}\left[i\right]\&comma;\mathrm{ts}\right)\&comma;\mathrm{color}=\mathrm{ToPlotColor}\left(\mathrm{colors}\left[i\right]\right)\&comma;\mathrm{legend}=\mathrm{models}\left[i\right]\right]\&comma;i=1..\mathrm{numelems}\left(\mathrm{models}\right)\right)\&comma;\left[\mathrm{ts}\&comma;\mathrm{color}=''Niagara\; Green''\&comma;\mathrm{thickness}=3\right]\right)$

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.

The TimeSeriesAnalysis[Optimize] command was introduced in Maple 18.

For more information on Maple 18 changes, see Updates in Maple 18.