Akaike or Bayesian information criteria - MATLAB aicbic
aicbic
Akaike or Bayesian information criteria
Syntax
aic = aicbic(logL,numParam)
[aic,bic] = aicbic(logL,numParam,numObs)
Description
aic = aicbic(logL,numParam) returns Akaike information criteria (AIC) corresponding to optimized
loglikelihood function values (logL), as returned by estimate, and the model parameters, numParam.
[aic,bic] = aicbic(logL,numParam,numObs) additionally returns Bayesian information criteria (BIC)
corresponding to logL, numParam, and the sample sizes associated with each logL value.
Examples
example
example
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Compare AIC Statistics
Calculate and interpret the AIC for four models.
The loglikelihood function values (logL) and the number of model
parameters (numParam) from four multivariate time series analyses
are:
logL1
logL2
logL3
logL4
=
=
=
=
Try This Example
View MATLAB Command
-681.4724;
-632.3158;
-663.4615;
-605.9439;
numParam1
numParam2
numParam3
numParam4
=
=
=
=
12;
27;
18;
45;
Calculate the AIC.
aic = aicbic([logL1,logL2,logL3,logL4], ...
[numParam1,numParam2,numParam3,numParam4])
aic = 1×4
103 ×
1.3869
1.3186
1.3629
1.3019
The model with the lowest AIC has the best fit. Therefore, the fourth model fits best.
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Akaike or Bayesian information criteria - MATLAB aicbic
Information Criteria Statistics for Simulated Data
Compare information criteria statistics for several model fits.
Specify the model
Try This Example
View MATLAB Command
where
is Gaussian with mean 0 and variance 2. Simulate data
from this model.
rng(1); % For random data reproducibility
T = 100; % Sample size
DGP = arima('Constant',-4,'AR',[0.2, 0.5], ...
'Variance',2);
y = simulate(DGP,T);
Define three competing models to fit to the data.
EstMdl1 = arima('ARLags',1);
EstMdl2 = arima('ARLags',1:2);
EstMdl3 = arima('ARLags',1:3);
Fit the models to the data.
logL = zeros(3,1); % Preallocate loglikelihood vector
[~,~,logL(1)] = estimate(EstMdl1,y,'Display','off');
[~,~,logL(2)] = estimate(EstMdl2,y,'Display','off');
[~,~,logL(3)] = estimate(EstMdl3,y,'Display','off');
Compute the AIC and BIC for each model.
[aic,bic] = aicbic(logL, [3; 4; 5], T*ones(3,1))
aic = 3×1
381.7732
358.2422
358.8479
bic = 3×1
389.5887
368.6629
371.8737
The model containing two autoregressive lag parameters fits best since it yields the lowest information criteria. The
structure of the best fitting model matches the model structure that simulated the data.
Input Arguments
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Akaike or Bayesian information criteria - MATLAB aicbic
logL — Optimized loglikelihood values
scalar | vector
Optimized loglikelihood objective function values associated with various model fits, specified as a scalar or vector.
Obtain an optimized loglikelihood value using estimate, infer, estimate, or an Optimization Toolbox™ function
such as fmincon or fminunc.
Data Types: double | single
numParam — Number of estimated parameters
scalar | vector
Number of estimated parameters associated with each corresponding fitted model in logL, specified as a positive
integer, or a vector of positive integers having the same length as logL.
If numParam is a scalar, then aicbic applies it to all logL values.
For univariate time series models, use length(info.X) to obtain numParam from a fitted model returned by
estimate.
For VAR models, obtain numParam using summarize from an estimated varm model object.
Data Types: double | single
numObs — Sample sizes
scalar | vector
Sample sizes of the observed series associated with each corresponding fitted model in logL, specified as a
positive integer, or a vector of positive integers having the same length as logL.
aicbic requires numObs to compute the BIC.
If numObs is a scalar, then aicbic applies it to all logL values.
Data Types: double | single
Output Arguments
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aic — AIC statistics
scalar | vector
AIC statistics associated with each corresponding fitted model in logL, returned as a vector with the same length
as logL.
bic — BIC statistics
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Akaike or Bayesian information criteria - MATLAB aicbic
scalar | vector
BIC statistics associated with each corresponding fitted model in logL, returned as a vector with the same length
as logL.
More About
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Akaike Information Criterion
A model fit statistic considers goodness-of-fit and parsimony. Select models that minimize AIC.
When comparing multiple model fits, additional model parameters often yield larger, optimized loglikelihood values.
Unlike the optimized loglikelihood value, AIC penalizes for more complex models, i.e., models with additional
parameters.
The formula for AIC, which provides insight into its relationship to the optimized loglikelihood and its penalty for
complexity, is:
Bayesian Information Criterion
A model fit statistic considers goodness-of-fit and parsimony. Select models that minimize BIC.
Like AIC, BIC uses the optimal loglikelihood function value and penalizes for more complex models, i.e., models
with additional parameters. The penalty of BIC is a function of the sample size, and so is typically more severe than
that of AIC.
The formula for BIC is:
References
[1] Box, G. E. P., G. M. Jenkins, and G. C. Reinsel. Time Series Analysis: Forecasting and Control. 3rd ed. Englewood
Cliffs, NJ: Prentice Hall, 1994.
See Also
Objects
arima | garch | regARIMA | varm
Functions
estimate | estimate | estimate | estimate | fmincon | fminunc | infer | lmtest | waldtest
Topics
Information Criteria
Time Series Regression V: Predictor Selection
Determine Minimal Number of Lags Using Information Criterion
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Akaike or Bayesian information criteria - MATLAB aicbic
Choose ARMA Lags Using BIC
Compare Conditional Variance Models Using Information Criteria
VAR Model Case Study
Introduced before R2006a
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