Evaluating Regression Models Fitted on some Features of High and Widespread Extreme Precipitation in the Caspian Region

Due to the socio-economic consequences, the analysis of the climatic extreme has been studied by many climatologists, environmental scientists and even scholars of humanity and social sciences. Extreme events are said to be rare events far from normal conditions (Bartolini et al., 2008). Time changes in frequency and intensity of high extreme precipitation prevent the floods and itsconsequent risks. High extreme variability of precipitation has widespread effects and  severe consequences on human societies, natural ecosystems and physical structures as these structures are consistent with normal climatic conditions and their adaptation to the extreme conditions is hardly possible. Due to the environmental and human impacts and the importance of high perception, the trends in these events have been widely studied, detected and modeled worldwide. But there are few studies focusing on high and widespread precipitation in the Caspian region; most of these studies have adapted a synoptic approach to do the research. From the perspective of statistical analysis, less attention has been paid to this part. Therefore, concerning the importance of the behavior of precipitation and its changes, the present paper studies the frequency and  the average intensity of high precipitation in the Caspian region  because the increasing and decreasing fluctuations of the extreme precipitation can affect the ecosystem of the area and make management and planning face serious issues.

To determine the frequency and the average of the intensity of widespread and high extreme precipitation events resulted by percentile threshold of 90-95, 95-99 (heavy precipitation) and 99 and more (very heavy precipitation), the daily precipitation data related to synoptic stations, the climatology and the rain gauge in Meteorological Organization and also the Ministry of Energy of Caspian Sea Region have been used from 1966 to 2016 (51 years). Due to different data records regarding the length of station and their non-uniform distribution and the changing of the station data into network data, the Kriging method is used as the optimal method in the interpolation of the observation. The result of the interpolation of daily precipitation is a matrix with 6479  18628 (the rows are cells and the columns are the days of precipitation). Thus, the spatial resolution of the resulted maps from interpolation is 3  To study the extreme precipitation in climatology, various definitions and absolute or relative indexes (the index of curve area of particular precipitation and the index of percentile threshold) are presented (Mofidi, 2007). Accordingly, in the present study the percentile threshold of 90-95 and the percentile threshold of 95-99 were considered as the index of heavy precipitation, and the percentile of 99 and more were regarded as an extreme precipitation index (see Equation 1). Thus, the percentile threshold of precipitation for each cell in each day of the year was calculated in the study area using the MATLAB software. Next, the frequency and the average of the intensity of extreme precipitation were calculated for one up to five days persistence. The frequency of extreme precipitation events refers to the number of the days along with high percentile precipitation in different months. The sum of these frequencies in each month were calculated for the whole statistical period. The average of the intensity of precipitation refers to the amount of precipitation in time unit, mm per day (see Equation 2). Finally, for the purpose of modeling the long-term behavior of the extreme precipitation, linear and nonlinear regression models (35 models) were fitted data using the curve expert software. Regression analysis is a statistical technique for analyzing and modeling the relationship between variables (Bazargan Lary, 2006). Under normal and very simple conditions, the best line is determined by the coordinates of the points obtained from two variables x and y on one page. This kind of regression is known as linear regression. In most cases, the relationship between climate variables (here, extreme precipitation and time indexes) cannot be represented by a line. Therefore, nonlinear regression (logarithmic, exponential, and parabolic) is used to model relations. Non-linear regression is a method for finding a non-linear model to find the relation between dependent and independent variables. After fitting the pattern on the data, the significance of the coefficients of the regression equation was checked. For this purpose, the statistical value of t-student was investigated for the significance of regression coefficients. A model based on coefficients with a significant error of 0.05 was regarded as a fitted pattern. Then, by using the indexes of coefficients of determination ( ), the root mean square error ( ) was considered appropriate among the other fitted models. The coefficient of determination is dimensionless. The sum of squared deviations of the observations around the mean equals the total changes observed in the observations. Also, if we  consider equation 3 as the  squared error or unexplained changes by regression line (or curve), the ratio of these two represents the ratio of the unexplained changes to the regression line  in proportion to the total variation (Asakereh, 2011) In addition, the best value of the coefficient of determination is equal to one. The mean of the squared error represents the error rate of the model, the best value of which is zero and is calculated by Equation (4) (Karamoz. 2006) In the above relations, o and p are respectively the observed and predicted values, and n is the number of data. The threshold of the percentiles examined is 90-95 and 95-99 percentiles for heavy precipitation and  the  percentile  99  and more is for extreme precipitation that have been widespread, the frequency and the average precipitation intensity is considered as independent variable, and time is considered as a dependent variable. At 95% confidence level, the optimal models for each of the high extreme precipitation indexes (i.e. frequency and average intensity) were fitted and evaluated.

This study examines the persistence of frequency and the average intensity of precipitation of extreme precipitation in the Caspian region from 1966 to 2016 based on the thresholds 90-95, 95-99, 99 and more through the linear and non-linear regression model.  The low correlation between dependent and time variables, and consequently the low coefficient of determination among variables, the low variations in these two variables and the high errors of the model based on   indexes show that regression model does not fit the data.

Few studies have been carried out on extreme precipitation in the Caspian Region, these studies have mostly applied synoptic approaches (e.g. Halabian et al., 2011, 2016; Mofidi et al., 2007; Sleigheh, 2016). Here, the frequency and the average of the intensity of extreme precipitation in the Caspian Region have been studied from 1966 to 2016 based on the thresholds 90-95, 95-99, 99 and more. Due to the low correlation between the dependent variables and time, the very there was a low coefficient of determination found between the variables, and the variation of these two variables was found very low.  Despite the significant trend in the observations which has been empirically deducible and tangible within the recent years, the regression patterns have not been able to justify these changes. Therefore, it seems that these patterns do not have the ability to justify the trend of observations in this field or in many others. Thus, two types of evaluation techniques including the analysis of observations based on probability knowledge and random process, as well as the study of observations through artificial intelligence techniques (e.g. artificial neural networks) can be recommended for such observations. Therefore, the probabilistic analysis of observations and the use of artificial neural networks are suggested as methods for evaluating trends in observations.