Evaluation of Various Linear Regression Methods for Downscaling of Mean Monthly Precipitation in Arid Pichola Watershed

In this paper, downscaling models are developed using various linear regression approaches namely direct, forward, backward and stepwise regression for downscaling of GCM output to predict mean monthly precipitation under IPCC SRES scenarios to watershed-basin scale in an arid region in India. The effectiveness of these regression approaches is evaluated through application to downscale the predictand for the Pichola lake region in Rajasthan state in India, which is considered to be a climatically sensitive region. The predictor variables are extracted from (1) the National Centers for Environmental Prediction (NCEP) reanalysis dataset for the period 1948–2000, and (2) the simulations from the third-generation Canadian Coupled Global Climate Model (CGCM3) for emission scenarios A1B, A2, B1 and COMMIT for the period 2001–2100. The selection of important predictor variables becomes a crucial issue for developing downscaling models since reanalysis data are based on wide range of meteorological measurements and observations. Direct regression was found to yield better performance among all other regression techniques explored in the present study. The results of downscaling models using both approaches show that precipitation is likely to increase in future for A1B, A2 and B1 scenarios, whereas no trend is discerned with the COMMIT.

section_content: Global circulation models (GCMs) are important tool in assessment of climate change. These are numerical models that have been designed to simulate the past, present, and future climate [1]. These models remain relatively coarse in resolution and are unable to resolve significant subgrid scale features. In most climate change impact studies, such as hydrological impacts of climate change, impact models are usually required to simulate sub-grid scale phenomenon and therefore require input data at similar sub-grid scale. The methods used to convert GCM outputs into local meteorological variables required for reliable hydrological modeling are usually referred to as "downscaling" techniques [2, 3]. Precipitation is an important parameter for climate change impact studies. A proper assessment of probable future precipitation and its variability is to be made for various water resources planning and hydro-climatology scenarios. 

A number of papers have previously reviewed down-scaling concepts, including 1) low-frequency rainfall events [4] 2) daily precipitation [5] 3)seasonal precipitation [6] 4) daily and monthly precipitation [7] 5) monthly precipitation [8] 6) monthly precipitation [9] 7) monthly precipitation [10] 8) monthly precipitation [11] 9) annual precipitation [3]. 

In this paper, we explore four linear regression approaches; namely, (a) direct regression, (b) forward regression, (c) backward regression and (d) stepwise regression as a downscaling methodology to study climate change impact over Pichola lake basin in an arid region. Apparently, in the literature, there appears no evidence of any study dealing with simultaneous evaluation of various regression approaches. In the light of this, the objective of this study is to 1) to rank various regression approaches 2) to downscale mean monthly precipitation using best available regression approach from simulations of CGCM3
section_name: Introduction

section_content: The area of the this study is the Pichola lake catchment in Rajasthan in India that is situated from 72. 5°E to 77. The Pichola watershed, located in Udaipur district, Rajasthan is one of the major sources for water supply for this arid region. During the past several decades, the streamflow regime in the catchment has changed considerably, which resulted in water scarcity, low agriculture yield and degradation of the ecosystem in the study area [12]. Regions with arid and semi-arid climates could be sensitive even to insignificant changes in climatic characteristics [13]. Temperature affects the evapotranspiration [14], evaporation and desertification processes and is also considered as an indicator of environmental degradation and climate change. Understanding the relationships among the hydrologic regime, climate factors, and anthropogenic effects is important for the sustainable management of water resources in the entire catchment hence this study area was chosen because of aforementioned reasons. The location map of the study region is shown in Figure 1.
section_name: Study Region

section_content: The monthly mean atmospheric variables were derived from the National Center for Environmental Prediction (NCEP/NCAR) (hereafter called NCEP) reanalysis data set [15] for a period of January 1948 to December 2000. The data have a horizontal resolution of 2. 5° latitude X longitude and seventeen constant pressure levels in vertical. The atmospheric variables are extracted for nine grid points whose latitude ranges from 22. 5 to 27. 5 °N, and longitude ranges from 72. 5 to 77. 5 °E at a spatial resolution of 2. 5°. The precipitation are used at monthly time scale from records available for Pichola Lake which is located in Udaipur at 24° 34'N latitude and 73°40'E longitude. The data is available for the period January 1990 to December 2000 [12]. The Canadian Center for Climate Modeling and Analysis (CCCma) provides GCM data for a number of surface and atmospheric variables for the CGCM3 T47 version which has a horizontal resolution of roughly 3. 75° latitude by 3. 75° longitude and a vertical resolution of 31 levels. The data comprise of present-day (20C3M) and future simulations forced by four emission scenarios, namely A1B, A2, B1 and COMMIT. The nine grid points surrounding the study region are selected as the spatial domain of the predictors to adequately cover the various circulation domains of the predictors considered in this study. The GCM data is re-gridded to a common 2. 5° using inverse Pichola lake [16]. The utility of this interpolation algorithm was examined in previous downscaling studies [17, 18].
section_name: Data Extraction

section_content: In statistical methods, the order in which the predictor variables are entered into (or taken out of) the model is determined according to the strength of their correlation with the criterion variable. 

In direct regression, all available predictor variables are put into the equation at once and they are assessed on the basis of proportion of variances in the criterion variable (Y) they uniquely account for. 

In Forward selection, the variables are entered into the model one at a time in an order determined by the strength of their correlation with the criterion variable. The effect of adding each is assessed as it is entered, and variables that do not significantly add to the success of the model are excluded [19]. 

In Backward selection, all the predictor variables are entered into the model. The weakest predictor variable is then removed and the regression re-calculated. If this significantly weakens the model then the predictor variable is re-entered-otherwise it is deleted. This procedure is then repeated until only useful predictor variables remain in the model [20, 21]. 

Stepwise is the most sophisticated of these statistical methods. Each variable is entered in sequence and its value assessed. If adding the variable contributes to the model then it is retained, but all other variables in the model are then re-tested to see if they are still contributing to the success of the model. If they no longer contribute significantly they are removed. Thus, this method should ensure that one end up with the smallest possible set of predictor variables included in one's model [22].
section_name: Regression Approaches

section_content: For downscaling predictand, the selection of appropriate predictors is one of the most important steps in a downscaling exercise. Various authors have used large-scale atmospheric variables, namely air temperature (at 925, 500 and 200mb pressure levels), geopotential height (at 500 and 200mb pressure levels), zonal (u) and meridional (v) wind velocities (at 925 and 200mb pressure levels), as the predictors for downscaling GCM output to mean monthly precipitation over a catchment [8, 10, 23]. 

Predictors have to be selected based both on their relevance to the downscaled predictands and their ability to be accurately represented by the GCMs. Cross-correlations are in use to select predictors to understand the presence of nonlinearity/linearity trend in dependence structure [23, 24]. These cross-correlations between each of the predictor variables in NCEP and GCM datasets are useful to verify if the predictor variables are realistically simulated by the GCM. Cross-correlations are computed between the predictor variables in NCEP and GCM datasets (Table 1 ). The cross correlations are estimated using three measures of dependence namely, product moment correlation, Spearman's rank correlation and Kendall's tau Scatter plots and cross-correlations between each of the predictor variables in NCEP and GCM datasets are useful to verify if the predictor variables are realistically simulated by the GCM. Cross-correlations between each of the predictor variables in NCEP and GCM datasets are useful to verify if the predictor variables are realistically simulated by the GCM.
section_name: Selections of Predictors

section_content: For downscaling precipitation, the probable predictor variables that have been selected to develop the models are considered at each of the nine grid points surrounding and within the study region. In this study, various linear regression approaches are used to downscale mean monthly precipitation in this study. The data of potential predictors is first standardized. Standardization is widely used prior to statistical downscaling to reduce bias (if any) in the mean and the variance of GCM predictors with respect to that of NCEP-reanalysis data [24]. Standardization is done for a baseline period of 1948 to 2000 because it is of sufficient duration to establish a reliable climatology, yet not too long, nor too contemporary to include a strong global change signal [24]. 

A feature vector (standardized predictor) is formed for each month of the record using the data of standardized NCEP predictor variables. The feature vector is the input to the linear regression models, and the contemporaneous
section_name: Development of Downscaling Models

section_content: Downscaling models were developed following the methodology as discussed in preceding section. The results and discussion are presented in this section.
section_name: Results and Discussions

section_content: The most relevant probable predictor variables necessary for developing the downscaling models are identified by using the three measures of dependence following the procedure. The cross-correlations enable verifying the reliability of the simulations of the predictor variables by the GCM, are shown in Table 1. In general, the most of predictor variables are realistically simulated by the GCM. It is noted that air temperature at 925mb (Ta 925) is the most realistically simulated variable with a CC greater than 0. 8, while meridional wind at 200mb (Va200) is the least correlated variable between NCEP and GCM datasets (CC = -0. 17). It is clear from Table 1 that air temperature at 925mb (Ta 925), air temperature at 500 mb (Ta500), air temperature at 200 mb (Ta200), meridional wind at 925mb (Va 925), zonal wind at 925mb (Ua925), zeo-potential height at 200mb (Zg200) and zeo-potential height at 500mb (Zg500) are better correlated than meridional wind at 200mb (Va200) and zonal wind at 200mb (Ua200).
section_name: Potential Predictor Selection

section_content: Seven predictor variables namely air temperature at 925 mb, 500 mb and 200 mb, zonal wind (925 mb); meridoinal wind (925 mb); zeo-potential height 500 mb and 200 mb at 9 NCEP grid points with a dimensionality of 63, are used as the standardized data of potential predictors. These feature vectors are provided as input to the various regressions downscaling model. Results of the different regression models (viz. M1 to M4) as discussed in previous section are tabulated in Table 2. Some of the precipitation values using this technique resulted in negative precipitation. However, this is physically not possible to have negative precipitation on a basin. Hence, these negative values are considered zero to compute various errors. 

For predictand precipitation, coefficient of correlation From the box plots of downscaled predictand (Figure 3 ), it can be observed that precipitation are projected to increase in future for A1B, A2 and B1 scenarios. The projected increase of precipitation is high for A1B and A2 scenarios whereas it is least for B1 scenario. This is because among the scenarios considered, the scenario A1B and A2 have the highest concentration of atmospheric carbon dioxide (CO2) equal to 720 ppm and 850 ppm, while the same for B1 and COMMIT scenarios are 550 ppm and ≈ 370 ppm respectively. Rise in concentration of CO2 in the atmosphere causes the earth's average temperature to increase, which in turn causes increase in evaporation especially at lower latitudes. The evaporated water would eventually precipitate [10, 25]. In the COMMIT scenario, where the emissions are held the same as in the year 2000, no significant trend in the pattern of projected future precipitation could be discerned. The overall results show that the projections obtained for precipitation are indeed robust.
section_name: Downscaling and performance of GCM Models

section_content: This paper investigates the applicability of the various linear regression approaches such as direct, forward, backward and stepwise to downscale precipitation from GCM output to local scale. The effectiveness of this model is demonstrated through the application of lake catchment in arid region in India. The predictand is downscaled from simulations of CGCM3 for four IPCC scenarios namely SRES A1B, A2, B1 and COMMIT. Four regression models are developed and the performance of the models is evaluated using the statistical measures CC, SSE, MSE, RMSE, NMSE, η and MAE. 

The overall conclusions of this evaluation study are as follows:

1) Overall direct regression performed best followed by backward regression method. Backward regression was followed by forward regression and stepwise regression which yielded the similar results. 

2) Direct regression yielded better results for training data set while forward regression performed better for validation data set. 

3) The results of downscaling models show that precipitation is projected to increase in future for A2 and A1B scenarios, whereas it is least for B1 and COMMIT scenarios using predictors.
section_name: Conclusions