Abstract

In time series analysis, the development of model framework in Banach space has facilitated. That allowed for the statistical inference in functional time series. One of the characteristics of functional time series is the periodicity and detecting it helps us understand the functional features better.
In this research, we propose a BIC-type information criterion for a trigonometric model in functional time series and the iterative algorithm to estimate the number of periodicities in functional time series.
Before introducing the proposed criterion, let us consider the parameter estimations in a trigonometric model with the known number of periodicities. To estimate the parameters, we convert the functional model to vectorized and matrix form, respectively, by utilizing the empirical functional principal component score. Under the above setting, the coefficient parameters are estimated by least squares method and the estimated frequency parameters are obtained by maximizing the value based on periodogram. We theoretically show that the estimated parameters have the consistency under some regularity condition.
Now, let us consider the estimation problem for the unknown number of periodicities. Note that we focus on the first principal component element of vectorized trigonometric model in functional time series. Here, we assume the spectral envelope of the true residuals is the same as the spectral density of h-order autoregressive model. We obtain the estimated prediction variance by fitting h-order autoregressive model to the estimated residual. The BIC type information criterion is defined as the summation of the logarithm of the estimated prediction variance and the penalty. Under the setting of the maximum values for the order of autoregressive model and the number of periodicities, we estimate the number of periodicities by obtaining it such as the proposed criterion becomes locally minimized. We proof theoretically the consistency of the estimated number of periodicities.
The penalty term of the proposed criterion has the parameter which is dependent on the number of observations in functional time series. In simulation, we illustrate the parameter is insensitive to select model as the number of observations increases.
To evaluate the effectiveness of the proposed criterion for the real data, we apply the 140-years sunspot data and 30-years daily average temperature data from Japan, Canada, Australia to the proposed criterion. Note that we set the number of the observations within each functional time series, called m. For the sunspot data, we obtain approximately 11 years cycle, by setting m to 15, 30, 91, and 182. That implies the solar activity. For the daily average temperature data, by setting m to 91, we obtain the periodicity of El Niño and La Niña phenomena, which are related to change in sea surface temperatures. It is well-known that those phenomena have a cycle of 3 to 7 years.