Use of normalized radial basis function in hydrology

In hydrology we work a lot with time series of river runoff, temperature and precipitation. Those time series are nonlinear and stohastic. They are ussualy contaminated with noise. Despite their complexity we built models from them. Models can be deterministic or stohastic. We use those models for prediction of future values, for interpolation of missing values and for extrapolation to scenarios that we do not have similar data for. In our case we used a normalized radial basis function for interpolation of missing Reka river mean monthly runoff. We have mean monthly temperature and monthly precipitation time series from 1851 to 2006 from station in Trieste, Italy. We have Reka river runoff time series from 1952 to 2006 form station Cerkvenikov mlin in Ilirska Bistrica, Slovenia. Those two stations are roughly 40 kilometers apart. From those data alone we tried to estimate Reka river runoff from 1851 to 1951. Data from 1952 to 1990 were used for learning of model, data from 1991 to 2006 were used for verification of model. For radial basis function w we chose multidimensional normal distribution, where Σ is covariance matrix: w(x,Σ) = -----1---- (2π)D/2|Σ|1/2exp[ ] - 1(x - x)T Σ-1 (x - x) 2(1) The prediction or in other words interpolating process is given by: ŷi(xi) = ∑ n=1NynCn(xi)(2) Weighting coefficient C, which is also a function, is: