There are different methods to estimate the trend in the time series analysis, such as the Ordinary least square (OLS), and the kernel estimation. The presence of correlated residuals will, however, render these methods inappropriate. In this paper we develop a new technique, based on the wavelet transform, to estimate a deterministic polynomial trend in the presence of correlated residuals.
The wavelet-based procedure for trends detection in time series use the Discrete Wavelet Transform (DWT) of the original time series, in order to decompose it into a low frequency (trend) and high frequency (noise) components. The smooth wavelet coefficients represent the long-term trends in the series, while the wavelet and smooth coefficients together represent the trends with cycles and local trends.
The DWT decompose, orthogonally, the global variance in term of scale and position. Hence, we use the analysis of variance to construct a new testing approach to discover possible trends in the time series. This test is based on the ratio between the variance of the smooth coefficients and the total variance of the time series. The limits and the properties of the test have been investigated under a variety of conditions using Monte Carlo simulations. Our results have shown that the test performs very well regarding both size and power properties. And, hence, we applied this technique to the Swedish wind speed data.
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