A Hierarchical Spatio-Temporal Dynamical Model for Predicting Precipitation Occurrence and Accumulation

Transcription

1 A Hierarchical Spatio-Temporal Dynamical Model for Predicting Precipitation Occurrence and Accumulation A Technical Report by Ali Arab Assistant Professor, Department of Mathematics and Statistics Georgetown University and Tolessa Deksissa Director Professional Science Master s Degree Program in Water Resources Management College of Agriculture, Urban Sustainability and Environmental Sciences University of the District of Columbia 30 August 2011

2 Table of Contents Project Information...2 Executive Summary Introduction and Project Summary Research Background and Literature Review Exploratory Data Analysis (EDA) Single Variable EDA Model- Based EDA Model Hierarchical Spatio- Temporal Model Predictive Statistical Model Statistical Bayesian Inference Results Discussion...10 Acknowledgements...10 References...11 Appendix A...12 Appendix B...20 Appendix C...37 Appendix D

3 PROJECT INFORMATION Title: A Hierarchical Spatio-Temporal Dynamical Model for Predicting Precipitation Occurrence and Accumulation Project Type: Research Focus Categories: FLOODS, METHODS Keywords: Precipitation Prediction, Probabilistic Flood Risk Assessment, Hierarchical Bayesian, Spatio-Temporal Analysis, Uncertainty, Climate Change. Start Date: March 1, 2010 End Date: August 30, 2011 (original end date: February 28, 2011) Principal Investigators: Ali Arab, Ph.D. (PI) Assistant Professor, Department of Mathematics and Statistics Georgetown University aa577@georgetown.edu Phone: (202) Tolessa Deksissa, Ph.D. (Co-PI) Research Associate and Lab Coordinator DC WRRI and Agricultural Experiment Station University of District of Columbia tdeksissa@udc.edu Phone: (202) Student researchers: Julie Menken (Georgetown University undergraduate student) Christianne Greer (Georgetown University graduate student) Conference Presentations JSM 2011 (July 30-August 5, 2011) An earlier version of this project was presented as an example of hierarchical modeling for environmental systems and ecological communities in the Joint Statistical Meetings (JSM) in Miami, Florida on August 2, (Contributed Oral Presentation) ISI 2011 (August 20-August 26) Final results of the project were presented at the International Statistics Institute (ISI) in Dublin, Ireland on August 22, (Contributed Oral Presentation) 2

4 Executive Summary The problem of predicting occurrence and accumulation of precipitation is of considerable interest in many disciplines such as atmospheric sciences, agriculture, and hydrology among others. The predictions based on climate models are often in a coarse resolution that can not provide accurate predictions for specific locations. Alternatively, statistical modeling of precipitation data can provide more reliable predictions at higher resolutions. There are several statistical models suggested in the literature, but most of these models ignore the spatial and/or temporal dependence of precipitation fields that results in lack of prediction accuracy. Our goal in this project was to develop a statistical method that yields predictive distributions for total monthly precipitation occurrence and accumulation while accounting for spatial and temporal correlation in the precipitation fields. The predictive distributions for precipitation accumulation can then be used to obtain exceedance probability of rainfall accumulation beyond a threshold in order to issue flash flood warnings, and optimize evacuation management in case of flooding events. In this project, we developed a hierarchical spatio-temporal dynamical model that conducts statistical learning on the parameters that govern the dynamics based on historic records of total monthly precipitation. The estimates of the parameters of the dynamical model then can be used to make predictions into the future. We conducted one-year ahead predictions and our results were successful in predicting total monthly precipitation values for a 12-month period into the future with a reasonable level of accuracy. We also used the same modeling approach to make predictions into the past for one of the weather stations that was relatively newly established (i.e., IAD Dulles airport stations established in 1963) based on data from two other stations in the Washington D.C. area (i.e,, DCA Reagan National Airport and BWI Baltimore Washington Airport) where data was able for both stations starting The obtained predictions for the past events were in line with observations from the other stations and we also constructed prediction intervals for each predicted value in order to account for uncertainties in the prediction procedure. 3

5 1. Introduction and Project Summary Historic records on total monthly precipitation values were obtained from the main weather stations in the DC area. These three stations are located at: Ronald Reagan Washington National Airport (DCA) Baltimore/Washington International Airport (BWI) Also known as: Thurgood Marshall Airport Washington Dulles International Airport (IAD) Student researchers have completed an extensive exploratory data analysis (EDA) of these data. The EDA results are essential for the statistical modeling. In the next stage of the project, a statistical model for predicting monthly rainfall values will be finalized based on work in progress. The first section describes a detailed exploratory data analysis of the data. Section 2 discusses the statistical modeling framework based on a hierarchical spatio-temporal modeling approach. Finally, results and discussion are presented. 2. Research Background and Literature Review Numerical weather prediction (NWP) models tend to provide forecasts that are often biased or over-predict precipitation accumulation (Berrocal et al., 2008). As a result, several statistical models for precipitation occurrence and accumulation have been proposed in the literature (Stidd, 1973; Bell, 1987, Bardossy and Plate, 1992; and Sanso and Guenni, 2004). Most of these methods make unrealistic assumptions about the distribution of precipitation data which includes many zero values (i.e., no precipitation), with a right-skewed distribution for precipitation accumulations greater than zero. Berrocal et al. (2008) proposes a spatial two-stage method that considers precipitation occurrence first, and then models nonzero precipitation accumulation under the condition that it has occurred. The nonzero precipitation accumulation is modeled using a continuous distribution such as exponential density, gamma density, or the mixture of several densities (Sloughter et al., 2007). We propose a spatio-temporal dynamical model that takes into account both spatial and temporal structure of data in a two stage model. The proposed two stage model consists of an occurrence model (stage 1) which predicts the precipitation occurrence at each location and time point and a precipitation accumulation model (stage 2) conditioned on the outcome of the occurrence model. There are several examples of related work on developing statistical models for precipitation occurrence and accumulation in the literature (Stidd, 1973; Bell, 1987, Bardossy and Plate, 1992; and Sanso and Guenni, 2004). Most of these methods make unrealistic assumptions about the distribution of precipitation data, which includes many zero values (i.e., no precipitation), with a right-skewed distribution for precipitation accumulations greater than zero. Stidd (1973) is a pioneer work on deriving climatic expectancies of flood or drought from the mean and variance of a precipitation record. The method is based on the cube root normal distribution of precipitation. This method does not account for spatial and temporal dependence and it provides an exploratory data analysis approach to analyzing precipitation data rather than a statistical modeling attempt. Bell (1987) discusses a model of the spatial and temporal distribution of precipitation that produces random spatial rainfall patterns defined on a grid with each grid point representing the average rain rate over the surrounding grid box. This method is 4

6 based on a correlated Gaussian random field that exceeds a threshold. The focus of the model is for use in evaluating sampling strategies for satellite remote-sensing of rainfall. Bardossy and Plate (1992) discuss a multidimensional stochastic model for the spatio-temporal distribution of daily precipitation. The rainfall is linked to the atmospheric circulation patterns using conditional distributions and conditional spatial covariance functions. The model is a transformed conditional multivariate autoregressive model, with parameters depending on the atmospheric circulation pattern. The model reproduces both the local rainfall occurrence probabilities and the distribution of the rainfall amounts at given locations. However, the methodology does not focus on obtaining predictive distributions of precipitation. Sanso and Guenni (2004) compare ground rainfall with purely deterministic Regional Climate Model (RCM) simulations within a Bayesian framework. The method considers spatial dependence and fits a truncated normal model to the observed ground data to represent spatial variability. The predictive posterior distribution of the spatially aggregated rainfall is obtained and compared to the RCM simulations. Sloughter et al. (2007) uses Bayesian model averaging (BMA) as a statistical way of postprocessing forecast ensembles to derive predictive probability density functions for weather quantities. Berrocal et al. (2008) proposes a spatial two-stage method which consider precipitation occurrence first, and then model nonzero precipitation accumulation conditioning on the occurrence. The nonzero precipitation accumulation is modeled using a continuous distribution. Our proposed method is an extension of the methodology discussed in Berrocal et al. (2008) to a spatio-temporal setting. 3. Exploratory Data Analysis (EDA) A thorough exploratory data analysis (EDA) was conducted in order to develop a better understanding of the data. We present the results of the EDA in two sections based on a single variable EDA that characterizes the properties of single variables, followed by a model-based EDA, which provides information on relationships between variables. 3.1 Single Variable EDA In this section, we discuss EDA for all three stations. The data for DCA and BWI stations are available for years 1871 through Note that IAD data is only available starting April 1963 (through 2010). In Appendix A, Table 1 shows a summary of historic maximum monthly values as well as historic maximum values for each station (or airport); Figures 1-12 show time series of historic monthly data for all three stations. 3.2 Model-Based EDA For brevity, we focus the exploratory data analysis on the data from the DCA weather station. It should be noted that large-scale patterns are very similar for the other two stations in the area and most conclusions and results for this station are valid for the other stations as well. In Appendix A, Figure 13 shows a scatterplot matrix of monthly observations against each other is given below. Here, our purpose is to investigate potential linear or nonlinear patterns that may exist between historic monthly rainfall data. We also conducted several simple linear regression analyses for the monthly data. For these simple linear regression models, we considered all the possible combinations of monthly data 5

7 (e.g., each monthly data was regressed on all the other 11 monthly data in a simple linear regression setting). Results for significant regression analyses are given in Table 2. Significant regressions at 0.05 level: January and March; June and November; October and November. 4. Model In this section, the statistical modeling approach we utilize is described. First we discuss a general spatio-temporal statistical modeling strategy that can be easily implemented in the context of a hierarchical Bayesian model. Finally, due to our emphasize on constructing a predictive model, we discuss our modeling approach for obtaining predictions for future precipitation monthly total values for the weather stations in the DC area. 4.1 Hierarchical Spatio-Temporal Model We consider a statistical model that will utilize data for all three stations (DCA, BWI, and IAD). Also, we will adopt a spatio-temporal modeling approach in a Bayesian framework. The spatial aspect of our modeling approach allows for taking into account similarities between values observed at weather stations that are located closer, ultimately allowing for borrowing strength across data for weather stations. We develop our model with the intention that it can be easily modified and used for cases where data from more than three stations are available. The temporal aspect of our model allows us to make realistic assumptions about the data (i.e., data are in form of a time series and thus, should not be considered as independent observations). Our speculation is that, by considering both spatial and temporal structures of the data, our model will be able to produce better prediction than most existing models which use faulty assumptions (such as independence over time and space). In particular, the data model is given by and the process model is Note this model requires data to be normally distributed. The normality assumption, if the data do not follow a normal distribution, can be achieved using a transformation of data such as the Box-Cox transformation (DeOliveira et al., 1997). See Cressie and Wikle (2011) for a complete discussion of spatio-temporal dynamical models. The proposed dynamical model structure is justified based on the fact that the joint spatiotemporal process can be factored into conditional models based on a Markovian assumption: where the notation [x] denotes the probability distribution of a random variable x, and the conditional distribution depends on a vector of parameters that govern the dynamics of the spatio-temporal process. In the dynamical model defined above, is a spatial error process, and is the propagator matrix which includes parameters that govern the dynamics of the process. 6

8 The propagator matrix can be modeled in a hierarchical fashion in order to obtain estimates of the parameters. The estimation of the hierarchical model will be done using Bayesian estimation where the posterior distribution of unknown parameters can be obtained using the sampling distribution of data and prior densities of the parameters. Once the estimates of parameters and models states are obtained a predictive distribution can be obtained for locations for which we do not have precipitation measurements (denoted by for ungauged locations ) described as In general, the estimation of the propagator matrix is often difficult due to its high dimensionality. We efficiently parameterize these matrices based on scientific and intuitive similarity structure between monthly rainfall data. The main assumption we will rely our modeling structure on is that consecutive months tend to have similar total rainfall values. This yields a sparse structure for the propagation matrix: The proposed structure described above requires estimation of two unknown parameters This sparse structure accounts for the effect of consecutive months (e.g., the rainfall values for January are only assumed to be affected by the values of December through parameter γ, and February through β, as well as, the effect of the values for January of the previous year through parameter α). We will also add extra parameters to this sparse structure based on the exploratory data analysis done on the data. In particular, we will add two extra parameters to account for potential correlation between monthly total rainfall values of January and March, and June and November. This parameterization is motivated by the exploratory data analysis discussed in the previous section. Another aspect of the proposed model is the ability to account for the spatial correlation between the rainfall values of the three weather stations. This assumption is accounted for in the covariance structure of the process model ( ). The spatial correlation we consider is based on an exponential covariogram model where the spatial correlation is based on the Euclidean distance (d) and a special range parameter, τ (which governs the strength of spatial correlation over spatial locations). Then, the covariance model can be written as Here, the symbol represents the Kronecker product of the two matrices. The rationale for accounting for spatial correlation is that there is spatial variability between the three weather stations. This spatial variability is shown in the figures shown in the previous section. Although, 7

9 in some cases, the amount of variability between the three locations is negligible, in certain years for certain months, this variability is significant and should be accounted for. For example, see the plots for January, May, July, and August. As part of our modeling strategy, we also account for different climate regimes based on the North Atlantic Oscillation (NAO) climate index. The NAO index is a climate index that is mostly associated with considerable variability on a wide range of time scales in the climate of the Atlantic sector (Visbeck 2001). This index is associated to well known effects on the weather conditions of the United States. 4.2 Predictive Statistical Model The main focus of our modeling is the ability to predict monthly total precipitation values and our proposed approach that can potentially strengthen the predictive power of the model is that we consider monthly time series for each month and parameterize a similarity structure for these data across month (e.g., rainfall values for January tend to get more affected by the rainfall values of the past few months). So our model assumes annual (within months variability) temporal effects (i.e., annual trends for each month are accounted for) as well as monthly temporal effects (between months variability). In this paper, we develop a statistical model for the data from all the weather stations in the DC area for the period1963 to The validation of the model is tested based on out-of-sample procedures as well as checking the precision of future predictions. Our goal is to predict monthly total rainfall values for year T+1 using data for years 1,...,T. This is a challenging prediction problem, however, if done with a reasonable measure of accuracy, these predictions can be very informative in making policy and management decision within the district and the immediate suburbs. For example, these predictions can be used in order to allocate funds as well as efficient planning of emergency strategy to deal with floods since we do obtain measures that indirectly indicate to risk of flooding for each month (based on predicted total rainfall value). 4.3 Statistical Bayesian Inference Due to the complex nature of our modeling approach, we considered a Bayesian estimation approach that in the context of a hierarchical modeling approach, allows for accounting for different scales of uncertainty (i.e, data, parameter and process uncertainties). The Bayesian estimation method comprises a heavily computational approach, the Markov Chain Monte Carlo (MCMC; Robert and Casella 2004) that allows for drawing samples from the conditional distribution of the unknown parameters (note that in the Bayesian approach all the unknown parameters are considered to be random variables and thus, have a probability distribution). In the Bayesian approach, a prior distribution is assigned to every unknown parameter. The prior density is then updated by information from the sampling distribution (i.e., the likelihood distribution), resulting in an updated version of the prior density, called the posterior distribution. The posterior distribution can be used to make inference about the parameter of interest. The prior distribution, if selected as a flat or non-informative density, does not affect the results (See Congdon 2006 and the references therein for a complete discussion of Bayesian methods). We wrote a set of code for the MCMC algorithm in MATLAB 7.1 R2010a. The algorithm, although very computational, take only a few minutes to run on a laptop computer (MacBook Pro 2.8 GHz Intel Core Duo Processor, 4 GB 1067 MHz DDR3 SDRAM). We ran the MCMC 8

10 algorithm for 10,000 iterations and after ignoring the first 1000 iterations for burn-in, we used the remaining MCMC realizations to make inference about the parameters. 5. Results We fitted several different models based on low, medium and high values of NAO. The notion of developing models for different levels of NAO values is motivated by previous studies that provide evidence of the relationship between NAO and environmental and ecological changes (e.g., Ottersen et al 2001). This was also supported by exploratory data analysis of the rainfall data for the DC area where an analysis of variance of data for the three categories of low, medium, high NAO years were significantly unequal at 5% (p-value=0.03). Thus, we divided the years into low, medium, and high NAO years based on the lowest 33.3% percentile (low) and highest 33.3% percentile (high) of the sorted NAO values (the middle category includes the middle 33.3% percentile of the sorted NAO values). The different models considered were based on data for years in the following categories: - Model 1 (M1): low NAO only, - Model 2 (M2): medium NAO only, - Model 3 (M3): high NAO only, - Model 4 (M4): low and high NAO combined, - Model 5 (M5): medium and high NAO combined. - Model 6 (M6): all years We then conducted a thorough model selection based on the predictive performance of the models. The details of the model selection are presented in Appendix B. We performed the predictive modeling for the last 5 years of the data (i.e., ) and in almost all cases Model 3, the model based on the years with high NAO values only, outperformed the other models. Figures show the one-year ahead predictions for the three stations DCA, IAD, and BWI. As described in tables 3-7, the range of successful predictive performance of the best model (Model 3) ranges between 58% and 84%. This is a reasonable accuracy given the challenging nature of the predictions. It should be noted that the total monthly precipitation values of some years are much harder to predict than others. For example, year 2006 is a difficult year to predict compared to years 2007, 2008, 2009, and Another advantage of the proposed hierarchical approach is that it allows us to handle missing values efficiently. It also allows for missing data imputation (predicting missing values). We used this aspect of our approach to reconstruct a historic record of monthly precipitation total values for 1871 through 1962 for the IAD station. Basically, our approach predicts the data for IAD for this period and the predicted values (within a level of uncertainty) provide us the observations that would have been recorded at this location if there was a weather station at this site pre The results for these predicted values are given in Appendix C. For each month, the posterior mean values as well as the lower and upper bounds for the 95% prediction interval are given. We also provide time series plots of these predicted values. The prediction intervals are wide due the high uncertainty in predicting many data points. However, the posterior means provide a reliable prediction of the actual observations that would have been observed at the location of IAD (if measurements were conducted). Figures show the reconstructed (predicted) total monthly values for the IAD station for the period The predicted values are given in Tables In Appendix D, Figures show the predicted total monthly precipitation values for the three stations for year The actual observations for the first half of 2011 are available and show 9

11 about 67% success rate for the predictions. However, the success of the remaining predictions cannot be measured until data are obtained for these future observations. The norm values (typical values) are plotted for reference. 6. Discussion In this project, we considered a hierarchical Bayesian spatio-temporal modeling approach with efficient parameterization of the dynamics to address a challenging problem in the field of water resources research and climate sciences. The proposed hierarchical dynamical modeling framework provides an easy to implement setting for making predictions into the future based on past data. We also implemented a separate but similar model that allowed us to predict missing values for the weather station located at IAD and thus, we were able to reconstruct a historic record of the data for this weather station for the period before the station was established (i.e., ). The main predictive model we developed provided very interesting result that the best predictions for one-year ahead monthly total precipitation values were obtained using the hierarchical dynamical model based on data from years with the highest third levels of NAO values. This result, aside from its scientific implications of the relationship between NAO levels and regional/local precipitation levels for the DC area, yields in a much computationally faster and more efficient prediction scheme since we only need to use a third of the historic data to make these predictions. Our proposed modeling approach is flexible and easy to implement (of course, conditioned on basic familiarity of the user with Bayesian methods and computational techniques). This model can be used in order to make management and policy related decisions based on local levels of precipitation for the DC area based on the one-year ahead predictions (with reasonable level of uncertainty). Also, since our model produces the predictive distribution of these predicted values, assessment of risk of flooding can be easily obtained and calculated based on the results. Of course, due to the resolution of the data used in this project (monthly rainfall accumulation) and the limited number of weather stations at hand, it will be more difficult to achieve realistic risk assessment of flash floods for the area. Our future efforts to develop a flooding risk assessment model should be focused on obtaining higher resolution data both in space and time (i.e., more stations, and weekly or daily levels of rainfall). Acknowledgements We would like to thank the DC Water Resources Research Institute of the University of the District of Columbia and the Department of Mathematics and Statistics of Georgetown University for providing support for the current research project. This research was also partially supported by Georgetown University through a Grant-in-Aid program and also a Junior Faculty Research Leave (Spring 2011). 10

12 References Bardossy, A. and Plate, E. J. (1992). Space-time model for daily rainfall using atmospheric circulation patterns. Water Resources Research, 28, Bell, T. L. (1987). A space-time stochastic model of rainfall for satellite remote-sensingstudies. J. Geophysical Research, 93, Berrocal, V. J., Raftery, A. E. and Gneiting, T. (2007). Probabilistic quantitative precipitation field forecasting using a two-stage spatial model. The Annals of Applied Statistics, Vol. 2, No. 4, Congdon, P. (2006). Bayesian Statistical Modeling, 2nd edition, Wiley, Chichester, West Sussex, U.K. Cressie, N., C. Wikle, (2011). Statistics for Spatio-Temporal Data. Wiley, NY. De Oliveira, V., Kedem, B. and Short, D. A. (1997). Bayesian prediction of transformed Gaussian random fields. J. Amer. Statist. Assoc. 92, Efron, Bradley (1987), "Better Bootstrap Confidence Intervals: Rejoinder", Journal of the American Statistical Association, Vol. 82, No pp Ottersen, G., Planque, B., Belgrano, A., Post, E., Reid, P.C., and Stenseth, N.C. (2001). Ecological effects of the North Atlantic Oscillation. Oecologia 128 (1): Reis, Jr., D.S. and Stedinger, J.R. (2005). Bayesian MCMC flood frequency analysis with historical information. Journal of Hydrology, 313, Robert CP, and Casella G Monte Carlo Statistical Methods, 2nd edition. Springer-Verlag, New York. Sanso, B. and Guenni, L. (2004). A Bayesian approach to compare observed rainfall data to deterministic simulations. Environmetrics, 15, Sloughter, J. M., Raftery, A. E., Gneiting, T. and Fraley, C. (2007). Probabilistic quantitative precipitation forecasting using Bayesian model averaging. Monthly Weather Review, 135, Stidd, C. K. (1973). Estimating the precipitation climate. Water Resources Research 9, Visbeck, M.H., Hurrell, J.W., Polvani, L., and Cullen H.M. (2001). The North Atlantic Oscillation: Past, present, and future, PNAS, 98 (23),

13 Appendix A Figures and tables for the exploratory data analysis are given below: Table 1. Maximum total precipitation (in inches) levels by month and airport Maximums by Month Month Maximum Value Year Airport Jan BWI Feb BWI Mar DCA Apr DCA May , 1889 DCA Jun IAD Jul DCA Aug BWI Sep DCA Oct DCA Nov IAD Dec BWI Maximums by Airport Maximum Value Year Month DCA September BWI August IAD June Table 2. Simple linear regression results for monthly data for DCA. Note that only the results for the models that are statistically significant are presented. Regression: January on March coefficient: p-value < 0.05 (0.029) Regression: June on November coefficient: p-value < 0.05 (0.031) Regression: October on November coefficient: p-value < 0.05 (0.022) Regression: March on January coefficient: p-value < 0.05 (0.029) Regression: November on June coefficient: p-value < 0.05 (0.031) Regression: November on October coefficient: p-value < 0.05 (0.022) Figures Time series for the monthly precipitation total values for all three stations ( ). Note that the data for IAD is only available after

14 January February 13

15 March April 14

16 May June 15

17 July August 16

18 September October 17

19 November December 18

20 Figure 13. Scatterplot matrix of monthly data for DCA. 19

21 Appendix B We conducted model selection between all the models we considered. In particular, we recorded the number of actual rainfall values (of the year we get predictions for) that do not fall within the 95% credible interval of the predicted values and used this as a measure of prediction performance. Thus, models with lower number of prediction intervals that do not contain the actual values were considered superior models. The model selection results are given below: Table 3 Model selection results for 2010 using data from The values in the table show the number of monthly total values not captured within the 95% credible interval of the predictions. The model with the lowest number of unsuccessful prediction is considered a superior model (highlighted in yellow). Model DCA IAD BWI M1: low NAO M2: medium NAO M3: high NAO M4: low & high NAO M5: medium & high NAO M6: all Table 4 Model selection results for 2009 using data from The values in the table show the number of monthly total values not captured within the 95% credible interval of the predictions. The model with the lowest number of unsuccessful prediction is considered a superior model (highlighted in yellow). Model DCA IAD BWI M1: low NAO M2: medium NAO M3: high NAO M4: low & high NAO M5: medium & high NAO M6: all Table 5 Model selection results for 2008 using data from The values in the table show the number of monthly total values not captured within the 95% credible interval of the predictions. The model with the lowest number of unsuccessful prediction is considered a superior model (highlighted in yellow). Model DCA IAD BWI M1: low NAO M2: medium NAO M3: high NAO M4: low & high NAO M5: medium & high NAO M6: all Table 6 Model selection results for 2007 using data from The values in the table show the number of monthly total values not captured within the 95% credible interval of the 20

22 predictions. The model with the lowest number of unsuccessful prediction is considered a superior model (highlighted in yellow). Model DCA IAD BWI M1: low NAO M2: medium NAO M3: high NAO M4: low & high NAO M5: medium & high NAO M6: all Table 7 Model selection results for 2006 using data from The values in the table show the number of monthly total values not captured within the 95% credible interval of the predictions. The model with the lowest number of unsuccessful prediction is considered a superior model (highlighted in yellow). Model DCA IAD BWI M1: low NAO M2: medium NAO M3: high NAO M4: low & high NAO M5: medium & high NAO M6: all The figures for the stations for each prediction year are given below. These results are based on the best predictive model (Model 3): 21

23 14 "BWI" Figure 14 Predictions of total monthly precipitation for 2010 using data (BWI). Posterior mean of predictions (red lines and circles), lower and upper bounds for the 95% credible interval (blue dashed lines), and actual observations (blue circles). 22

24 15 "IAD" Figure 15 Predictions of total monthly precipitation for 2010 using data (IAD). Posterior mean of predictions (red lines and circles), lower and upper bounds for the 95% credible interval (blue dashed lines), and actual observations (blue circles). 23

25 20 "DCA" Figure 16 Predictions of total monthly precipitation for 2010 using data (DCA). Posterior mean of predictions (red lines and circles), lower and upper bounds for the 95% credible interval (blue dashed lines), and actual observations (blue circles). 24

26 14 "BWI" Figure 17 Predictions of total monthly precipitation for 2009 using data (BWI). Posterior mean of predictions (red lines and circles), lower and upper bounds for the 95% credible interval (blue dashed lines), and actual observations (blue circles). 25

27 16 "IAD" Figure 18 Predictions of total monthly precipitation for 2009 using data (IAD). Posterior mean of predictions (red lines and circles), lower and upper bounds for the 95% credible interval (blue dashed lines), and actual observations (blue circles). 26

28 20 "DCA" Figure 19 Predictions of total monthly precipitation for 2009 using data (DCA). Posterior mean of predictions (red lines and circles), lower and upper bounds for the 95% credible interval (blue dashed lines), and actual observations (blue circles). 27

29 10 "BWI" Figure 20 Predictions of total monthly precipitation for 2008 using data (BWI). Posterior mean of predictions (red lines and circles), lower and upper bounds for the 95% credible interval (blue dashed lines), and actual observations (blue circles). 28

30 10 "IAD" Figure 21 Predictions of total monthly precipitation for 2008 using data (IAD). Posterior mean of predictions (red lines and circles), lower and upper bounds for the 95% credible interval (blue dashed lines), and actual observations (blue circles). 29

31 12 "DCA" Figure 22 Predictions of total monthly precipitation for 2008 using data (DCA). Posterior mean of predictions (red lines and circles), lower and upper bounds for the 95% credible interval (blue dashed lines), and actual observations (blue circles). 30

32 12 "BWI" Figure 23 Predictions of total monthly precipitation for 2007 using data (BWI). Posterior mean of predictions (red lines and circles), lower and upper bounds for the 95% credible interval (blue dashed lines), and actual observations (blue circles). 31

33 12 "IAD" Figure 24 Predictions of total monthly precipitation for 2007 using data (IAD). Posterior mean of predictions (red lines and circles), lower and upper bounds for the 95% credible interval (blue dashed lines), and actual observations (blue circles). 32

34 12 "DCA" Figure 25 Predictions of total monthly precipitation for 2007 using data (DCA). Posterior mean of predictions (red lines and circles), lower and upper bounds for the 95% credible interval (blue dashed lines), and actual observations (blue circles). 33

35 12 "BWI" Figure 26 Predictions of total monthly precipitation for 2006 using data (BWI). Posterior mean of predictions (red lines and circles), lower and upper bounds for the 95% credible interval (blue dashed lines), and actual observations (blue circles). 34

36 12 "IAD" Figure 27 Predictions of total monthly precipitation for 2006 using data (IAD). Posterior mean of predictions (red lines and circles), lower and upper bounds for the 95% credible interval (blue dashed lines), and actual observations (blue circles). 35

37 15 "DCA" Figure 28 Predictions of total monthly precipitation for 2006 using data (DCA). Posterior mean of predictions (red lines and circles), lower and upper bounds for the 95% credible interval (blue dashed lines), and actual observations (blue circles). 36

38 Appendix C Figure 29 Reconstructed historical records for IAD January total precipitation values. Posterior mean of predicted monthly total values (red), lower and upper bounds for the 95% credible interval for the predictions (dashed blue), DCA observations (magenta), BWI observations (green). 37

39 Figure 30 Reconstructed historical records for IAD February total precipitation values. Posterior mean of predicted monthly total values (red), lower and upper bounds for the 95% credible interval for the predictions (dashed blue), DCA observations (magenta), BWI observations (green). 38

40 Figure 31 Reconstructed historical records for IAD March total precipitation values. Posterior mean of predicted monthly total values (red), lower and upper bounds for the 95% credible interval for the predictions (dashed blue), DCA observations (magenta), BWI observations (green). 39

41 Figure 32 Reconstructed historical records for IAD April total precipitation values. Posterior mean of predicted monthly total values (red), lower and upper bounds for the 95% credible interval for the predictions (dashed blue), DCA observations (magenta), BWI observations (green). 40

42 Figure 33 Reconstructed historical records for IAD May total precipitation values. Posterior mean of predicted monthly total values (red), lower and upper bounds for the 95% credible interval for the predictions (dashed blue), DCA observations (magenta), BWI observations (green). 41

43 Figure 34 Reconstructed historical records for IAD June total precipitation values. Posterior mean of predicted monthly total values (red), lower and upper bounds for the 95% credible interval for the predictions (dashed blue), DCA observations (magenta), BWI observations (green). 42

44 Figure 35 Reconstructed historical records for IAD July total precipitation values. Posterior mean of predicted monthly total values (red), lower and upper bounds for the 95% credible interval for the predictions (dashed blue), DCA observations (magenta), BWI observations (green). 43

45 Figure 36 Reconstructed historical records for IAD August total precipitation values. Posterior mean of predicted monthly total values (red), lower and upper bounds for the 95% credible interval for the predictions (dashed blue), DCA observations (magenta), BWI observations (green). 44

46 Figure 37 Reconstructed historical records for IAD September total precipitation values. Posterior mean of predicted monthly total values (red), lower and upper bounds for the 95% credible interval for the predictions (dashed blue), DCA observations (magenta), BWI observations (green). 45

47 Figure 38 Reconstructed historical records for IAD October total precipitation values. Posterior mean of predicted monthly total values (red), lower and upper bounds for the 95% credible interval for the predictions (dashed blue), DCA observations (magenta), BWI observations (green). 46

48 Figure 39 Reconstructed historical records for IAD November total precipitation values. Posterior mean of predicted monthly total values (red), lower and upper bounds for the 95% credible interval for the predictions (dashed blue), DCA observations (magenta), BWI observations (green). 47

49 Figure 40 Reconstructed historical records for IAD December total precipitation values. Posterior mean of predicted monthly total values (red), lower and upper bounds for the 95% credible interval for the predictions (dashed blue), DCA observations (magenta), BWI observations (green). 48

50 Table 8 Posterior means and 95% credible intervals (2.5th and 97.5th percentiles of the posterior distribution) of predictions for years total monthly values for the month of January. Posterior Mean 2.50% 97.50%

51 * * The value for January total precipitation for 1963 is missing in the data. 50

52 Table 9 Posterior means and 95% credible intervals (2.5th and 97.5th percentiles of the posterior distribution) of predictions for years total monthly values for the month of February. Posterior Mean 2.50% 97.50%

53 * * The value for Februray total precipitation for 1963 is missing in the data. 52

54 Table 10 Posterior means and 95% credible intervals (2.5th and 97.5th percentiles of the posterior distribution) of predictions for years total monthly values for the month of March. Posterior Mean 2.50% 97.50%

55