Learning a probabalistic model of rainfall using graphical models Byoungkoo Lee Computational Biology Carnegie Mellon University Pittsburgh PA 15213 byounko andrew cmu edu Jacob Joseph Computational Biology Carnegie Mellon University Pittsburgh PA 15213 jmjoseph andrew cmu edu Abstract We present an analysis of historical precipitation data for the United States Pacific Northwest measured for the years 1949 1994 on a grid of approximately 50km resolution We have implemented a Bayesian network with nodes representing individual geographic grid regions Directed weighted edges represent dependence relationships between regions Using a modified K 2 learning algorithm we build a heuristically optimal Bayesian network We examine degree of dependence between regions the predictive capacity of a minimal set of measurements and evaluate the utility of additional strategically selected measurements in enhancing local predictions 1 Introduction Although weather prediction is essential to many of our social and economic processes accurate prediction remains an open field of research On the most simplistic level weather derives from a variety of interdependent physical factors including wind speed air pressure temperature ocean currents and local topology Meteorologists typically rely upon numerical atmospheric circulation models ACMs to predict local and global weather at short and long time scales These models are most effective at low resolution predicting large scale events 1 An orthogonal approach to weather prediction involves statistical models constructed from local historical data Such models are typically designed to represent local effects Many machine learning techniques such as Markov chains auto regressive models and neural networks have been used with limited success In particular these models fail to represent spatial and temporal dependencies between neighboring locales 1 In this study we examine the use of Bayesian networks to better capture regional dependencies in the limited context of precipitation prediction We are particularly interested in determining a minimal set of measurement sites sufficient to quantitatively predict local rainfall Central to these goals we exploit the interdependence between geographically disparate measurements to evaluate the utility of each existing measurement site and potential new sites From the given historical data we construct a high resolution 50km grid probabilistic Figure 1 Histogram of Rainfall mm 5 16 x 10 14 12 Count 10 8 6 4 2 0 0 1 1 5 5 15 15 40 mm day 40 100 100 model of rainfall throughout the Pacific Northwest We are particularly interested determining a minimal set of measurement sites sufficient to quantitatively predict local rainfall Central to these goals we exploit the interdependence between measurements at distinct stations and geographic regions to evaluate the utility of each data source 2 Data We have been provided precipitation data derived from a number of measurement sites throughout the United States Pacific Northwest 2 This data is formatted to a grid of 17 discrete latitudes and 16 discrete longitudes The actual measurement stations within each grid cell have been consolidated Several cells have no measurement sites For each geographical area a daily measurement of rainfall is provided for the years 1949 1994 totaling 16801 daily measurements Due to the nature of the data collection some locations do not include daily measurements over the period considered The few grid points or nodes with incomplete measurements over the full time series have been omitted for simplicity All analyses have been performed using the 167 nodes with complete data series Data pre processing consisted of conversion from the provided netCDF format to a native three dimensional Matlab array more amenable to analysis without additional Matlab interfaces Several such Matlab netCDF interfaces are available though none proved usable with the particular Matlab environment available to us An indirect approach was accomplished by first transforming the netCDF format to ASCII using native libraries and finally reconstructing a multidimensional Matlab array 2 1 Discretization Daily rainfall measurements are supplied as continuous values of millimeters per day To facilitate construction of a discrete Bayesian network we opted to to discretize rainfall to six categories corresponding to 0 1 no rain 1 5 5 15 15 40 40 100 and 100 mm day respectively This approach as been used previously to represent light medium and heavy rain 1 The histogram within Figure 1 illustrates the number of measurements observed within each category We sought to minimize data skew by empirically selecting thresholds to represent equal sized populations within each category A roughly exponen Figure 2 Modified K 2 Algorithm Input Quantized data of n nodes an ordering of n nodes an ordering of neighbors for each node max parents Output Adjacency matrix representing all directed edges in the network For i 1 to n parent i Initial condition no parent node for any node P old f i parent i Probability of data i node given parent i Gonext true While Gonext size parent i max num P new f i parent i another parent i choose from neighbors If P new P old P old P new parent i parent i another parent i else Gonext false end Save parent nodes for node i in adjacent matrix end return adjacency matrix tial decrease from 0mm day in the number of measurements is observed resulting in increasing category bin widths Note that all values within the 0 1 category are exactly 0 and are thus insensitive to threshold selection Note that Euclidean distances and correlation were calculated with the original continuous data series 3 Methods 3 1 Bayes Network Construction As each node in a Bayes network may conditioned upon any other node in the network exhaustively learning an optimal network structure for all but the smallest networks is computationally intractable Indeed this problem is NP hard As such a number of heuristics are commonly used to approximate a globally optimal DAG structure These include the Metropolis Hastings Markov Chain Monte Carlo MCMC method to sample the DAG space hill climbing methods to explore node neighbors incrementally active structure learning 3 and structural EM 6 We utilized the K 2 algorithm 7 due to its ease of implementation and suitability for subsequent modification An ideal network structure maximizes the probability of the network given the observed data
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