Simulate ARMA time series

Simulate an ARMA(2,1) time series with 100 observations.

Mdl0 = arima('持续的',0.2,'AR',{0.75,-0.4},...'MA'，0.7，'Variance',0.1); rng(5) Y = simulate(Mdl0,100); figure plot(Y) xlim([0,100]) title('Simulated ARMA(2,1) Series')

Plot Sample ACF and PACF

Plot the sample autocorrelation function (ACF) and partial autocorrelation function (PACF) for the simulated data.

figure subplot(2,1,1) autocorr(Y) subplot(2,1,2) parcorr(Y)

Both the sample ACF and PACF decay relatively slowly. This is consistent with an ARMA model. The ARMA lags cannot be selected solely by looking at the ACF and PACF, but it seems no more than four AR or MA terms are needed.

Fit ARMA(p,q) Models to Data

要识别最佳滞后，请拟合具有不同滞后选择的几种型号。在这里，适合所有组合p= 1，...，4和q= 1,……4(总共16模型)。存储loglikelihood objective function and number of coefficients for each fitted model.

LogL = zeros(4,4);% Initializepq =零（4,4）;forp = 1:4forq = 1：4 mdl = arima（p，0，q）;[estmdl，〜，logl（p，q）] =估计（mdl，y，'Display','off'); PQ(p,q) = p + q;endend

Calculate BIC

Calculate the BIC for each fitted model. The number of parameters in a model isp+q+ 1 (for the AR and MA coefficients, and constant term). The number of observations in the data set is 100.

logL = LogL(:); pq = PQ(:); [~,bic] = aicbic(logL,pq+1,100); BIC = reshape(bic,4,4)

BIC =4×4102.4215 96.2339 100.8005 100.3440 89.1130 93.4895 97.1530 94.0615 93.6770 93.2838 100.2190 103.4779 98.2820 97.8331 100.3024 107.5245

In the output BIC matrix, the rows correspond to the AR degree (p) and the columns correspond to the MA degree (q). The smallest value is best.

minBIC = min(BIC,[],'all')

minBIC = 89.1130

[minP,minQ] = find(minBIC == BIC)

minP = 2

minQ = 1

The smallest BIC value is89.1130in the (2,1) position. This corresponds to an ARMA(2,1) model, which matches the model that generated the data.