Daily relative humidity projections in an Indian river basin for IPCC SRES scenarios

Abstract

A two-stage methodology is developed to obtain future projections of daily relative humidity in a river basin for climate change scenarios. In the first stage, Support Vector Machine (SVM) models are developed to downscale nine sets of predictor variables (large-scale atmospheric variables) for Intergovernmental Panel on Climate Change Special Report on Emissions Scenarios (SRES) (A1B, A2, B1, and COMMIT) to R _H in a river basin at monthly scale. Uncertainty in the future projections of R _H is studied for combinations of SRES scenarios, and predictors selected. Subsequently, in the second stage, the monthly sequences of R _H are disaggregated to daily scale using k-nearest neighbor method. The effectiveness of the developed methodology is demonstrated through application to the catchment of Malaprabha reservoir in India. For downscaling, the probable predictor variables are extracted from the (1) National Centers for Environmental Prediction reanalysis data set for the period 1978–2000 and (2) simulations of the third-generation Canadian Coupled Global Climate Model for the period 1978–2100. The performance of the downscaling and disaggregation models is evaluated by split sample validation. Results show that among the SVM models, the model developed using predictors pertaining to only land location performed better. The R _H is projected to increase in the future for A1B and A2 scenarios, while no trend is discerned for B1 and COMMIT.

Appendix 1

The downscaled scenarios are constructed by translating GCM-simulated information from coarser scale to finer watershed scale using spatial downscaling models, based on the assumption that regional climate is conditioned by the climate on a relatively larger scale (e.g., continental). The spatial downscaling techniques can be broadly classified into dynamic downscaling and statistical downscaling. In the dynamic downscaling approach, a Regional Climate Model (RCM) is embedded into GCM. The RCM is essentially a numerical model in which GCMs are used to fix boundary conditions. The major drawback of RCM, which restricts its use in climate impact studies, is its complicated design and high computational cost. Whereas, the statistical downscaling involves deriving empirical relationships that transform large-scale features of the GCM (LF) to regional-scale variables (RSV)

$$ {\text{RSV}} = g\left( {\text{LF}} \right) $$

where RSV represents predictands, LF refers to predictors, and g is a downscaling function which could be deterministic or stochastic.

The classical statistical downscaling techniques include weather classification methods, weather generators, and transfer functions. The simple and commonly used statistical downscaling approaches are based on transfer functions, which model relationships between predictors and predictand using methods such as linear and nonlinear regression, artificial neural networks, canonical correlation, principal component analysis, and SVM. In this paper, the transfer function-based statistical downscaling method is chosen for determining plausible future scenarios of relative humidity.

1.1 Least-Square Support Vector Machine

The Least-Square Support Vector Machine (LS-SVM) has been used in this study for downscaling. Details of the same can be found in Suykens (2001) and Tripathi et al. (2006). This subsection presents the underlying principle of the LS-SVM.

Consider a finite training sample of N patterns $\left\{ {\left( {{{\text{x}}_i},{y_i}} \right),i = 1,...,N} \right\}$, where x _i is the ith pattern in n-dimensional space (i.e., ${x_i} = \left[ {{x_{{1i}}}, \ldots, {x_{{ni}}}} \right] \in {\Re^n}$), and it constitutes input to LS-SVM, whereas Y _i ∈ ℜ is the corresponding value of the desired model output. Further, let the learning machine be defined by a set of possible mappings $x \mapsto f\left( {x{,}w} \right)$, where f _(·) is a deterministic function which, for a given input pattern x and adjustable parameters w ($w \in {\Re^n}$), always gives the same output. The training phase of the learning machine involves adjusting the parameter w. These parameters are estimated by minimizing the cost function Ψ _L (w,e).

$$ {\psi_{\text{L}}}\left( {w{,}e} \right) = \frac{1}{2}{w^{\text{T}}}w{ + }\frac{1}{2}C\sum\limits_{{i = 1}}^N {e_i^2} $$

(5)

subject to the equality constraint

$$ {y_i} - {\hat{y}_i} = {e_i}\quad i = 1,...,N $$

(6)

where C is a positive real constant, and ${\hat{y}_i}$ is the actual model output. The first term of the cost function represents weight decay or model complexity–penalty function. It is used to regularize the weight sizes and to penalize the large weights. This helps in improving the generalization performance (Hush and Horne 1993). The second term of the cost function represents penalty function.

The solution of the optimization problem is obtained by considering the Lagrangian as

$$ L\left( {w,b,e{,}a} \right) = \frac{1}{2}{w^{\text{T}}}w{ + }\frac{1}{2}C\sum\limits_{{i = 1}}^N {e_i^2} -\sum\limits_{{i = 1}}^N {{\alpha_i}} \left\{ {{{\hat{y}}_i} + {e_i} - {y_i}} \right\} $$

(7)

where α _i are Lagrange multipliers, and b is the bias term. The conditions for optimality are given by

$$ \left\{ {\begin{array}{*{20}{c}} {\frac{{\partial L}}{{\partial w}} = w - \sum\limits_{{i = 1}}^N {{\alpha_i}\phi \left( {{x_i}} \right) = 0\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,} } \\ {\frac{{\partial L}}{{\partial b}} = \sum\limits_{{i = 1}}^N {{\alpha_i} = 0\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,} } \\ {\frac{{\partial L}}{{\partial {e_i}}} = {\alpha_i} - C{e_i} = 0\quad i = 1,...,N\,\,\,\,\,\,} \\ {\frac{{\partial L}}{{\partial {\alpha_i}}} = {{\hat{y}}_i} + {e_i} - {y_i} = 0\quad i = 1,...,N} \\ \end{array} } \right. $$

(8)

The above conditions of optimality can be expressed as the solution to the following set of linear equations after elimination of w and e _i .

$$ \left[ {\begin{array}{*{20}{c}} {0\quad \quad {{\overrightarrow 1 }^{\text{T}}}} \\ {\overrightarrow 1 \quad \Omega + {C^{{{ - 1}}}}I} \\ \end{array} } \right]\left[ {\begin{array}{*{20}{c}} b \\ {{\alpha }} \\ \end{array} } \right] = \left[ {\begin{array}{*{20}{c}} 0 \\ y \\ \end{array} } \right] $$

(9)

where

$$ y = {\left[ {\begin{array}{*{20}c} {{y_{1} }} \\ {{y_{2} }} \\ { \vdots } \\ {{y_{N} }} \\ \end{array} } \right]};\,{\overrightarrow{1}} = {\left[ {\begin{array}{*{20}c} {1} \\ {1} \\ { \vdots } \\ {1} \\ \end{array} } \right]}_{{N \times 1}} ;\,\alpha = {\left[ {\begin{array}{*{20}c} {{\alpha _{1} }} \\ {{\alpha _{2} }} \\ { \vdots } \\ {{\alpha _{N} }} \\ \end{array} } \right]};\,I = {\left[ {\begin{array}{*{20}c} {1} & {0} & { \ldots } & {0} \\ {0} & {1} & { \ldots } & {0} \\ { \vdots } & { \vdots } & { \vdots } & { \vdots } \\ {0} & {0} & { \ldots } & {1} \\ \end{array} } \right]}_{{N \times N}} $$

(10)

In Eq. (9), Ω is obtained from the application of Mercer s theorem.

$$ {\Omega_{{i,j}}} = K\left( {{x_i},{x_j}} \right) = \phi {\left( {{x_i}} \right)^{\text{T}}}\phi \left( {{x_j}} \right)\quad \forall i,j $$

(11)

where ϕ(·) represents nonlinear transformation function defined to convert a nonlinear problem in the original lower dimensional input space to linear problem in a higher dimensional feature space.

The resulting LS-SVM model for function estimation is:

$$ f(x) = \sum {\alpha_i^{ * }} K\left( {{x_i},x} \right) + {b^{*}} $$

(12)

where $\alpha_i^{ * }$and b ^* are the solutions to Eq. (7), K(x _i , x) is the inner product kernel function defined in accordance with Mercer s theorem (Courant and Hilbert 1970; Mercer 1909), and b ^* is the bias. There are several choices of kernel functions, including linear, polynomial, sigmoid, splines, and radial basis function (RBF). The linear kernel is a special case of RBF (Keerthi and Lin 2003). Further, the sigmoid kernel behaves like RBF for certain parameters (Lin and Lin 2003). In this study, RBF is chosen to map the input data into higher dimensional feature space, which is given by:

$$ K\left( {{x_i},{x_j}} \right) = \exp \left( { - \frac{{{{\left\| {{x_i},{x_j}} \right\|}^2}}}{\sigma }} \right) $$

(13)

where, σ is the width of the RBF kernel, which can be adjusted to control the expressivity of RBF. The RBF kernels have localized and finite responses across the entire range of predictors.

The advantage of RBF kernel is that it maps the training data non-linearly into a possibly infinite-dimensional space, and thus, it can effectively handle the situations when the relationship between predictors and predictand is nonlinear. Moreover, the RBF is computationally simpler than polynomial kernel, which requires more parameters. It is worth mentioning that developing LS-SVM with RBF kernel involves a judicious selection of RBF kernel width σ and parameter C.

The software used in this study is the “LS-SVMlab: a MATLAB toolbox for Least Squares Support Vector Machines.” Details of the software can be found at http://www.esat.kuleuven.ac.be/sista/lssvmlab/tutorial/lssvmlab_paper0.pdf. The running time for the worst case (the maximum of the running times over all nine cases considered in the study) was 15 min, while the same for the best case was 5 min (on a Pentium PC). The average run time over all the cases was 9 min.