Learning a probabalistic model of rainfall usinggraphical modelsByoungkoo LeeComputational BiologyCarnegie Mellon UniversityPittsburgh, PA [email protected] JosephComputational BiologyCarnegie Mellon UniversityPittsburgh, PA [email protected] present an analysis of historical precipitation data for the UnitedStates’ Pacific Northwest, measured for the years 1949-1994 on a gridof approximately 50km resolution. We have implemented a Bayesiannetwork with nodes representing individual geographic grid regions. Di-rected, weighted edges represent dependence relationships between re-gions. Using a modified K-2 learning algorithm, we build a heuristicallyoptimal Bayesian network. We examine degree of dependence betweenregions, the predictive capacity of a minimal set of measurements, andevaluate the utility of additional strategically selected measurements inenhancing local predictions.1 IntroductionAlthough weather prediction is essential to many of our social and economic processes, ac-curate prediction remains an open field of research. On the most simplistic level, weatherderives from a variety of interdependent physical factors, including wind speed, air pres-sure, temperature, ocean currents, and local topology. Meteorologists typically rely uponnumerical atmospheric circulation models (ACMs) to predict local and global weather atshort and long time scales. These models are most effective at low resolution, predictinglarge-scale events [1].An orthogonal approach to weather prediction involves statistical models constructed fromlocal historical data. Such models are typically designed to represent local effects. Manymachine learning techniques such as Markov chains, auto-regressive models, and neuralnetworks have been used with limited success. In particular, these models fail to representspatial and temporal dependencies between neighboring locales [1].In this study, we examine the use of Bayesian networks to better capture regional depen-dencies in the limited context of precipitation prediction. We are particularly interestedin determining a minimal set of measurement sites sufficient to quantitatively predict localrainfall. Central to these goals, we exploit the interdependence between geographically dis-parate measurements to evaluate the utility of each existing measurement site and potentialnew sites.From the given historical data, we construct a high-resolution (<=50km grid) probabilisticFigure 1: Histogram of Rainfall (mm)0 − 1 1 − 5 5 − 15 15 − 40 40 − 100 >1000246810121416x 105mm/dayCountmodel of rainfall throughout the Pacific Northwest. We are particularly interested deter-mining a minimal set of measurement sites sufficient to quantitatively predict local rainfall.Central to these goals, we exploit the interdependence between measurements at distinctstations and geographic regions to evaluate the utility of each data source.2 DataWe have been provided precipitation data derived from a number of measurement sitesthroughout the United States’ Pacific Northwest [2]. This data is formatted to a grid of 17discrete latitudes and 16 discrete longitudes. The actual measurement stations within eachgrid cell have been consolidated. Several cells have no measurement sites. For each geo-graphical area, a daily measurement of rainfall is provided for the years 1949-1994,totaling16801 daily measurements. Due to the nature of the data collection, some locations do notinclude 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 nativethree-dimensional Matlab array more amenable to analysis without additional Matlab in-terfaces. Several such Matlab-netCDF interfaces are available, though none proved usablewith the particular Matlab environment available to us. An indirect approach was accom-plished by first transforming the netCDF format to ASCII using native libraries, and finallyreconstructing a multidimensional Matlab array.2.1 DiscretizationDaily rainfall measurements are supplied as continuous values of millimeters per day. Tofacilitate construction of a discrete Bayesian network, we opted to to discretize rainfallto six categories, corresponding to 0-1 (’no rain’), 1-5, 5-15, 15-40, 40-100, and >100mm/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 measurementsobserved within each category. We sought to minimize data skew by empirically selectingthresholds to represent equal-sized populations within each category. A roughly exponen-Figure 2: Modified K-2 AlgorithmInput: Quantized data of n nodes, an ordering of n nodes,an ordering of neighbors for each node, max_parentsOutput:Adjacency matrix representing all directed edges in thenetworkFor i = 1 to nparent_i = []; #Initial condition: no parent node for any nodeP_old = f(i, parent_i); #Probability of data (i node) given parent_iGonext = true;While Gonext & size(parent_i) < max_numP_new = f(i, parent_i, another parent_i); # choose from neighborsIf P_new > P_oldP_old = P_new;parent_i = parent_i + another parent_i;else Gonext = false;endSave parent nodes for node i; # in adjacent matrixendreturn adjacency matrixtial decrease from 0mm/day in the number of measurements is observed, resulting in in-creasing category bin widths. Note that all values within the 0-1 category are exactly 0 andare thus insensitive to threshold selection. Note that Euclidean distances and correlationwere calculated with the original continuous data series.3 Methods3.1 Bayes Network ConstructionAs 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 com-putationally intractable. Indeed, this problem is NP-hard. As such, a number of heuristicsare commonly used to approximate a globally optimal DAG structure. These include theMetropolis-Hastings Markov Chain Monte Carlo (MCMC) method to sample the DAGspace, hill climbing methods to explore node neighbors incrementally, active structurelearning[3], and structural EM [6]. We utilized the K-2 algorithm[7] due to its ease ofimplementation and suitability for subsequent
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