Study of long memory and periodicities in climate variables in different Meteorological Subdivisions of India

Abstract

Presence of long memory in climatic variables is frequently observed. The trend assessment becomes difficult in the presence of long-memory as the usual methods are not capable to take care of this property during trend estimation. In order to estimate the trend in presence of long memory, the non-parametric wavelet method has become popular in the recent time. The discrete wavelet transformation (DWT) re-expresses a time-series in terms of coefficients that are associated with a particular time and a particular scale. In the present study, DWT has been applied to estimate the monthly rainfall trend for the monsoon months: June-September in ten selected sub-divisions of India using “Haar” wavelet filter. The results from DWT were cross checked with the non-parametric Mann-Kendall (M-K) test. The investigation reveals that the monthly rainfall trend for the monsoon months of different subdivisions in India are significantly decreasing over the years. However, in some of the subdivisions, rainfall trend is increasing. DWT reveals significant trend in most of the subdivisions whereas M-K test reveals that most of the trends are not significant at 5% level. The variation in monthly rainfall for June-September for the studied zones were also investigated by wavelet using Haar filter. The local as well as global variation in rainfall is observed by DWT plot of the respective months in the respective locations. The variability in rainfall is evident in the recent decades in all the locations. The surface temperatures over a given region vary seasonally and annually depending upon latitude, altitude and location with respect to geographical features such as a water body (river, lake or sea), mountains, etc. Probably one of the most widely quoted aspects of climatic change is the significant increase in global mean air temperature during the past century. The modeling issue of maximum temperature series in India which possesses characteristic of long-term memory is addressed and TSF has proven to be a robust approach of capturing such long memory even in the presence of detected structural break. Time series modeling of temperature series assume prime importance both in the local and global levels. Wavelet transformation was applied to decompose the temperature series into time–frequency domain in order to study the local as well as global variation over different scale and time epochs. It is found that there is significant increase in maximum temperature over the years in India. The shift in maximum temperature in India occurred during mid of 1963 as detected by the statistical test. Temperature data series exhibiting long-range dependence property combined with structural break required to be modeled with a concrete and valid technique Summary 86 which can overcome the issue of loss of information, biased estimates and inaccurate forecast. In this regard, TSF approach has been found to serve the better results. For long memory processes with a change in mean level, the present study on maximum temperature data has established the outperformance of TSF methodology in terms of SSE, MSE and RMAPE criterions over the AR-Truncation approach. Time series analysis of weather data can be a very valuable tool to investigate its variability pattern and, maybe, even to predict short- and long-term changes in the time series. In this study, the long memory behaviour of monthly minimum and maximum temperature of India for the period 1901 to 2007 by means of fractional integration techniques has been investigated. The results show that the time series can be specified in terms of autoregressive fractionally integrated moving average (ARFIMA) process. Both the series were found to be integrated with orders of integration smaller than 0.5 ensuring the long memory stationarity. Wavelet methodology in frequency domain with Haar wavelet filter was applied in order to see the oscillation at different scale and at different time epochs of the series. Multiresolution analysis (MRA) was carried out to explore the local as well as global variations in both the temperature series over the years. The variability in minimum temperature is found to be more than maximum temperature. Though there is no clear significance trend in the temperature series in the long run, but there are pockets of change in the temperature pattern. The predictive ability of ARFIMA model was investigated in terms of relative mean absolute percentage error. Long memory time series have been analysed by using ARFIMA models. Model parameter d reflects the long memory in the maximum and minimum temperature series. It is found that in the both the series long memory parameter is significant. The study has revealed that the ARFIMA model could be used successfully for modelling the temperature series. The predictive ability of ARFIMA model was investigated in terms of relative mean absolute percentage error. The variability in minimum temperature is found to be more than maximum temperature. The study reveals that there are pockets of change in the temperature pattern (both in maximum as well as in minimum temperature) which may be clearly visible by vertical clustering of coefficients in MRA. It may happen that long memory and structural changes are easily confused and the time series is mistakenly detected as long memory process. However, most researchers choose to ignore the problem of structural break in testing for long memory. It is a known fact that short memory with structural break may exhibit the properties of long memory. To avoid the confusion test has to be performed to differentiate true long memory from spurious Summary 87 long memory. The main contribution of the paper is to detect if the DGP of monthly seasonal rainfall series of some zones across India is generated by a true long memory process. In this paper, we have employed exact local Whittle (ELW) estimator to estimate the long memory parameter. The results indicate that some of the series exhibit long memory pattern. Next, an empirical fluctuation process using the ordinary least square (OLS)-based Chow test is applied to detect the break date. Break dates are detected in two series of North-East and Central-North data sets in the year 1957 and 1965, respectively. The overconfidence and lack of reliability for regional rainfall forecasts is a common problem amongst the researchers. Moreover, the feature of rainfall in a location may not be always linear so that it can be modelled through the classical ARIMA model. To accommodate the pattern of nonlinearity and complexity, decomposition of the series under consideration is required. When the original series has much nonlinearity as its property, the MODWT has simplified it by breaking it into its sub-frequencies. Therefore, the ANN can now model the details and approximate components sufficiently so that the accuracy of the forecasting process is improved up to a marked extent. Therefore, the combination of wavelet approach along with classical time series model i.e. ARIMA model and promising machine learning technique i.e. ANN is applied for forecasting annual rainfall in 30 subdivisions of India. Superiority of Wavelet-ARIMA and Wavelet-ANN approach over traditional ARIMA model is demonstrated in terms of RMSE and MAPE. In Wavelet-ANN and Wavelet- ARIMA approach the minimum and maximum MAPE has been found in CHHAT and PUNJB sub-division respectively.