Theor Appl Climatol DOI 10.1007/s00704-014-1172-5 ORIGINAL PAPER Prediction of periodically correlated processes by wavelet transform and multivariate methods with applications to climatological data Mitra Ghanbarzadeh · Mina Aminghafari Received: 24 December 2013 / Accepted: 28 April 2014 © Springer-Verlag Wien 2014 Abstract This article studies the prediction of periodically correlated process using wavelet transform and multivariate methods with applications to climatological data. Period- ically correlated processes can be reformulated as mul- tivariate stationary processes. Considering this fact, two new prediction methods are proposed. In the first method, we use stepwise regression between the principal compo- nents of the multivariate stationary process and past wavelet coefficients of the process to get a prediction. In the sec- ond method, we propose its multivariate version without principal component analysis a priori. Also, we study a generalization of the prediction methods dealing with a deterministic trend using exponential smoothing. Finally, we illustrate the performance of the proposed methods on simulated and real climatological data (ozone amounts, flows of a river, solar radiation, and sea levels) compared with the multivariate autoregressive model. The proposed methods give good results as we expected. 1 Introduction Stochastic prediction methods mostly depend on the stationarity assumption. But this assumption does not hold in many cases of interest such as climatic series. Peri- odically correlated (PC) processes offer an alternative for analysis of this type of data. Climatological data as flows of a river, solar radiation, sea levels, and ozone amounts M. Ghanbarzadeh · M. Aminghafari () Department of Statistics, Faculty of Mathematics and Computer Science, Amirkabir University of Technology, Tehran, Iran e-mail: Aminghafari@aut.ac.ir M. Ghanbarzadeh e-mail: Ghanbarzadeh@aut.ac.ir are examples of these processes which are studied in this article. Gladyshev (1961) presented these processes for the first time and studied the structure of their autocovariance function and spectral representation. There exists a periodic rhythm in their mean and autocovariance function. They are nonstationary but exhibit many properties of stationary series because they can be written as multivariate stationary series (Gladyshev 1961). Wavelets play an important role in the time series analy- sis which can form an orthonormal basis for L 2 (R) (Mallat (1998), Chapter 7). Wavelet transform decomposes sig- nal into a low-frequency component and the sum of the details (high frequency). Sufficiently regular wavelets sep- arate automatically deterministic trend. Since the wavelets are local, they can play an important part to manage the nonstationary series for analyzing its local aspect. Similarly, the complex structure of the process can often be simplified using this transform. For these reasons, we use the wavelet transform in this paper. Cambanis et al. (1995) show that PC processes are characterized by the property that their contin- uous wavelet transform (CWT) is PC (with the same period) at all scales under some conditions. Because of computing the cost of CWT, Averkamp and Houdr´ e(2000) show that the time series is PC if and only if the corresponding discrete wavelet coefficients are PC at each scale. Let us now give a history on the prediction of PC processes. Marseguerra et al. (1992) study the predic- tion of autoregressive moving average (ARMA) and sea- sonal ARMA by artificial neural networks. Makagon et al. (1994) give a theorem for Wold decomposition of infi- nite dimensional stationary processes for continuous time PC processes. Bittanti et al. (1988) focus on the predic- tion problem of the discrete time periodic Riccati equation which is associated with periodic autoregressive processes. Li and Hinich (2000) propose a method for the prediction