Information criteria are functions used to evaluate goodness of fit for a model representing a time series.

The functions take into account both the goodness of fit itself and the number of parameters of the model: a model is better if it fits more closely and if it has fewer parameters.

Akaike's information criterion is defined by

where $k$ is the number of parameters and $l$ the log likelihood of obtaining the given time series from the given model.

Akaike's information criterion gives very good results if used to evaluate goodness of fit against a large sample size (i.e., a long time series), but for smaller sample sizes a correction is needed. This criterion is obtained as follows:

where $k$ and $l$ are as before and $n$ is the size of the sample.

Finally, the Bayesian information criterion is given by

where $k$, $l$, and $n$ are as above.

The number of parameters of the model is always computed by the information criterion procedure, as is the sample size. The log likelihood can also be computed, but if the log likelihood is known beforehand (e.g. because of running the Optimize command), then it can be passed in using the loglikelihood option. This prevents recomputing the log likelihood and thereby increases efficiency very slightly.

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

$\mathrm{ts}≔\mathrm{TimeSeries}\left(\left[1.8\,3.4\,2.1\,2.9\,2.4\,2.9\,2.5\,3.1\right]\,\mathrm{period}=2\right)$

$\mathrm{ts}≔\left[\begin{array}{c}\mathrm{Time\; series}\\ \mathrm{data\; set}\\ \mathrm{8\; rows\; of\; data:}\\ \mathrm{2016\; -\; 2023}\end{array}\right]$

$\mathrm{models}≔\mathrm{Specialize}\left(\mathrm{ExponentialSmoothingModel}\left(\right)\,\mathrm{ts}\right)$

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

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

$\left[\mathrm{-0.014065563}\&comma;\mathrm{-5.416379083}\&comma;0.031961103\&comma;\mathrm{-5.145765873}\&comma;0.002827297\&comma;\mathrm{-5.804974813}\&comma;\mathrm{-0.603105141}\&comma;\mathrm{-1.517769899}\&comma;\mathrm{-6.694364663}\&comma;\mathrm{-0.487294335}\&comma;\mathrm{-1.376178599}\&comma;\mathrm{-6.769140123}\&comma;\mathrm{-0.975463079}\&comma;\mathrm{-6.912940013}\&comma;\mathrm{-1.134387223}\&comma;\mathrm{-6.715408243}\&comma;\mathrm{-1.772582973}\&comma;\mathrm{-2.135892023}\&comma;\mathrm{-6.808173573}\right]$

$$\begin{gathered} \mathbf{for}m\mathbf{in}\mathrm{models}\mathbf{do} \\ \mathrm{print}\left(m\&comma;\mathrm{AIC}\left(m\&comma;\mathrm{ts}\right)\right) \\ \mathbf{end}\mathbf{do} \end{gathered}$$

$\mathrm{<\; an\; ETS(A,A,A)\; model\; >},12.01066840$

$\mathrm{<\; an\; ETS(A,A,N)\; model\; >},18.83275817$

$\mathrm{<\; an\; ETS(A,Ad,A)\; model\; >},13.89411477$

$\mathrm{<\; an\; ETS(A,Ad,N)\; model\; >},20.29153175$

$\mathrm{<\; an\; ETS(A,N,A)\; model\; >},7.994336806$

$\mathrm{<\; an\; ETS(A,N,N)\; model\; >},15.60994963$

$\mathrm{<\; an\; ETS(M,A,A)\; model\; >},13.26051935$

$\mathrm{<\; an\; ETS(M,A,M)\; model\; >},15.27182966$

$\mathrm{<\; an\; ETS(M,A,N)\; model\; >},21.38872933$

$\mathrm{<\; an\; ETS(M,Ad,A)\; model\; >},15.51428223$

$\mathrm{<\; an\; ETS(M,Ad,M)\; model\; >},16.93433454$

$\mathrm{<\; an\; ETS(M,Ad,N)\; model\; >},23.53828025$

$\mathrm{<\; an\; ETS(M,M,M)\; model\; >},13.94926348$

$\mathrm{<\; an\; ETS(M,M,N)\; model\; >},21.82588003$

$\mathrm{<\; an\; ETS(M,Md,M)\; model\; >},16.21600049$

$\mathrm{<\; an\; ETS(M,Md,N)\; model\; >},23.43081649$

$\mathrm{<\; an\; ETS(M,N,A)\; model\; >},11.54039483$

$\mathrm{<\; an\; ETS(M,N,M)\; model\; >},12.29990591$

$\mathrm{<\; an\; ETS(M,N,N)\; model\; >},17.61634715$

$$\begin{gathered} \mathbf{for}m\mathbf{in}\mathrm{models}\mathbf{do} \\ \mathrm{print}\left(m\&comma;\mathrm{AICc}\left(m\&comma;\mathrm{ts}\right)\right) \\ \mathbf{end}\mathbf{do} \end{gathered}$$

$\mathrm{<\; an\; ETS(A,A,A)\; model\; >},96.01066840$

$\mathrm{<\; an\; ETS(A,A,N)\; model\; >},32.16609150$

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

$\mathrm{<\; an\; ETS(A,Ad,N)\; model\; >},50.29153175$

$\mathrm{<\; an\; ETS(A,N,A)\; model\; >},21.32767014$

$\mathrm{<\; an\; ETS(A,N,N)\; model\; >},18.00994963$

$\mathrm{<\; an\; ETS(M,A,A)\; model\; >},97.26051935$

$\mathrm{<\; an\; ETS(M,A,M)\; model\; >},99.27182966$

$\mathrm{<\; an\; ETS(M,A,N)\; model\; >},34.72206266$

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

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

$\mathrm{<\; an\; ETS(M,Ad,N)\; model\; >},53.53828025$

$\mathrm{<\; an\; ETS(M,M,M)\; model\; >},97.94926348$

$\mathrm{<\; an\; ETS(M,M,N)\; model\; >},35.15921336$

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

$\mathrm{<\; an\; ETS(M,Md,N)\; model\; >},53.43081649$

$\mathrm{<\; an\; ETS(M,N,A)\; model\; >},24.87372816$

$\mathrm{<\; an\; ETS(M,N,M)\; model\; >},25.63323924$

$\mathrm{<\; an\; ETS(M,N,N)\; model\; >},20.01634715$

$$\begin{gathered} \mathbf{for}m\mathbf{in}\mathrm{models}\mathbf{do} \\ \mathrm{print}\left(m\&comma;\mathrm{BIC}\left(m\&comma;\mathrm{ts}\right)\right) \\ \mathbf{end}\mathbf{do} \end{gathered}$$

$\mathrm{<\; an\; ETS(A,A,A)\; model\; >},12.48731765$

$\mathrm{<\; an\; ETS(A,A,N)\; model\; >},19.15052434$

$\mathrm{<\; an\; ETS(A,Ad,A)\; model\; >},14.45020556$

$\mathrm{<\; an\; ETS(A,Ad,N)\; model\; >},20.68873946$

$\mathrm{<\; an\; ETS(A,N,A)\; model\; >},8.312102973$

$\mathrm{<\; an\; ETS(A,N,N)\; model\; >},15.76883271$

$\mathrm{<\; an\; ETS(M,A,A)\; model\; >},13.73716860$

$\mathrm{<\; an\; ETS(M,A,M)\; model\; >},15.74847891$

$\mathrm{<\; an\; ETS(M,A,N)\; model\; >},21.70649550$

$\mathrm{<\; an\; ETS(M,Ad,A)\; model\; >},16.07037302$

$\mathrm{<\; an\; ETS(M,Ad,M)\; model\; >},17.49042533$

$\mathrm{<\; an\; ETS(M,Ad,N)\; model\; >},23.93548796$

$\mathrm{<\; an\; ETS(M,M,M)\; model\; >},14.42591273$

$\mathrm{<\; an\; ETS(M,M,N)\; model\; >},22.14364620$

$\mathrm{<\; an\; ETS(M,Md,M)\; model\; >},16.77209128$

$\mathrm{<\; an\; ETS(M,Md,N)\; model\; >},23.82802420$

$\mathrm{<\; an\; ETS(M,N,A)\; model\; >},11.85816099$

$\mathrm{<\; an\; ETS(M,N,M)\; model\; >},12.61767207$

$\mathrm{<\; an\; ETS(M,N,N)\; model\; >},17.77523023$

The TimeSeriesAnalysis[AIC], TimeSeriesAnalysis[AICc] and TimeSeriesAnalysis[BIC] commands were introduced in Maple 18.

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