Consequences of Continuous Zero Values and Constant Values in Time Series Modeling: Understanding through Chaotic Approach

This study is aimed at understanding the behavior of a rainfall time series having a large number of continuous zero values. Forty-nine years of daily rainfall data pertaining to the Koyna Reservoir catchment in India is employed in the study. The majority of rainfall happens during the monsoon period from June to September; the rainfall during the non-monsoon period (October to May) is almost negligible. This phenomenon has been observed every year. Hence, 64% of the time series contains zero values. Six sets of rainfall time series along with the observed series are analyzed: (1)daily observed average rainfall data; (2)daily transformed average rainfall data; (3)daily wet-period average rainfall data; (4)phase-randomized average rainfall data; (5)daily average rainfall anomaly data; and (6)standardized daily average rainfall anomaly data. To understand the consequence of a greater length of zero values and constant values, daily observed average rainfall data results are compared with daily wet-period average rainfall data and daily transformed average rainfall data. The phase-randomized and the anomaly data of the rainfall series were used to cross verify the behavior. The correlation dimension method (CDM) based on the Grassberger-Procaccia algorithm was used in this study for behavioral analysis as well as to find the embedding dimension of the time series. The results reveal that the CDM underestimated the correlation dimension as one owing to a higher percentage of continuous zeros/constant values in the case of full-year rainfall data and transformed daily rainfall data, respectively, whereas the correlation dimension of the wet-period rainfall data is five. On the other hand, the optimum embedding dimension in all cases of full-year and wet-period rainfall data is estimated by the CDM turned out to be five. It is found that the correlation dimension method underestimates the correlation dimension if the series has a large number of zeros or constant values.