A NOTE CONCERNING THE PROPER CHOICE FOR MARKOV MODEL ORDER FOR DAILY PRECIPITATION IN THE HUMID TROPICS: A CASE STUDY IN COSTA RICA

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1 INTERNATIONAL JOURNAL OF CLIMATOLOGY Int. J. Climatol. 20: (2000) SHORT COMMUNICATION A NOTE CONCERNING THE PROPER CHOICE FOR MARKOV MODEL ORDER FOR DAILY PRECIPITATION IN THE HUMID TROPICS: A CASE STUDY IN COSTA RICA MICHAEL HARRISON a, * and PETER WAYLEN b a Department of Geography and Planning, The Uni ersity of Southern Mississippi, Box 5051, Hattiesburg, MS , USA b Department of Geography, The Uni ersity of Florida, 3141 Turlington Hall, Gaines ille, FL 32611, USA Recei ed 6 October 1999 Re ised 2 May 2000 Accepted 5 May 2000 ABSTRACT The use of chain-dependent hydroclimatological models (sometimes referred to as combined models or two-part models ) in analysing daily precipitation requires that rainfall be modelled using both occurrence and intensity statistics. Markov processes in the context of precipitation climatology have been studied in such regions as monsoonal Asia, sub-saharan Africa and South America. Many studies have indicated that the use of a first-order Markov model is often adequate when describing daily precipitation occurrences, particularly when working in temperate regions, but relatively little work has been done in the humid tropics regarding proper Markov model order, particularly in the western hemisphere. This research examines the occurrence characteristics of Costa Rican daily precipitation by comparing the Akaike and Bayesian information criteria (AIC and BIC) for three long-term meteorological stations. It is found that the most parsimonious models generally are those of first order (winter) or zero order (summer). Overall, the BIC yields less ambiguous results than the AIC, and thus, a higher level of model confidence is achieved when using the BIC as the model-order selection criteria. Copyright 2000 Royal Meteorological Society. KEY WORDS: Costa Rica; Central America; chain-dependent processes; Markov models; information criteria; daily precipitation 1. INTRODUCTION The creation of statistical models representing daily precipitation has formed one of the most useful ways of examining precipitation climatologies. One of the most common statistical models in current use is the chain-dependent model, (sometimes referred to as combined models or two-part models ), which incorporates precipitation frequencies and magnitude models into a single unit. Inherent in the creation of the frequency portion of the chain-dependent process is modelling the occurrence of rainfall events at discrete intervals, often on a daily basis. The daily rainfall occurrences are generally modelled as Markov processes. Many studies that have examined rainfall occurrence have employed a two-state first-order Markov model. While the rationale for choosing a two-state occurrence Markov model is justified because the magnitude information is contained in a separate part of the larger chain-dependent model, choosing an order for the Markov process is more problematic, and has important implications from the standpoint of model effectiveness and parsimony. Many studies, particularly those which have focused on temperate latitude processes, have found that a first-order Markov model was often sufficient to adequately describe the occurrence characteristics of most types of daily data. However, few studies have examined the properties of rainfall occurrence in the humid tropics of the Americas. * Correspondence to: Department of Geography and Planning, The University of Southern Mississippi, Box 5051, Hattiesburg, MS , USA; mike.harrison@usm.edu Copyright 2000 Royal Meteorological Society

2 1862 M. HARRISON AND P. WAYLEN Besides being interesting from a purely climatological perspective, understanding the occurrence processes governing rainfall production is of considerable importance to a wide variety of human activities. Effective use of agricultural and hydraulic resources becomes critical when such systems become part of the infrastructure of developing countries. For example, the nation of Costa Rica receives approximately 60% of its electrical power from hydroelectric sources, making it necessary to understand as completely as possible the causes of precipitation variability, and the effects that such variability will have on forecast models which are used for decision-making by administrators (Gleick, 1993). The purpose of this research is to examine the Markov model of precipitation occurrence using daily rainfall data from the Central Valley and Pacific Slope of Costa Rica. Model orders up to third order will be compared for efficiency and parsimony, and seasonal changes in preferred model order will be examined in light of the governing climate mechanisms that dictate the rainfall patterns. 2. STUDY AREA Costa Rica lies at the confluence of several important atmospheric processes, which give rise to the seasonal precipitation patterns seen in Figures 1 and 2. The region is dominated by the northeast trade winds, which keep the eastern lowland well-watered, but result in a rain shadow to the west. The trade winds reach their greatest strength during the boreal summer months, when the pressure gradient between the Atlantic subtropical high and the inter-tropical convergence zone (ITCZ) is maximized. This low-level flow results in the advection of a large amount of precipitable water from the Caribbean, which is condensed when the moist air is forced over the cordillera. The ITCZ experiences a north south migration in the eastern equatorial Pacific, from approximately 3 N in the boreal winter to 10 N in the summer. The ITCZ promotes considerable instability in the region, bringing heavy rains to the western areas of central America. In addition, cross-equatorial westerlies arise south of the ITCZ, bringing frequent precipitation to the extreme south of the country as the ITCZ reaches its maximum northerly extent (Hastenrath, 1976). It is also during the northward transit of the ITCZ that a strengthening of the trade wind flow occurs, resulting in the Veranillos de San Juan (or simply eranillos, literally, little summer ), during which the rainfall abates somewhat; this generally occurs during July, and ends when the ITCZ comes to its most northerly position in late summer (Waylen et al., 1994). A third seasonal influence is the incursion of polar air from the north during the winter months, following cold front penetration into the region; these are referred to as the nortes. This results in periods of prolonged precipitation in the northeastern part of the country, a consequence of stress differential induced convergence by the northerly winds (Bryson and Khun, 1961; Schultz et al., 1997, 1998). While less effective at creating precipitation over long periods than the summer climate components, the nortes are nevertheless important rainfall components, as the rains they produce come at a time when the country is experiencing considerably drier conditions than in the summer. The norte-driven rainfalls are often locally quite intense, resulting in the unusual situation where the annual maximum rainfalls for the area will occur during the winter months (Waylen et al., 1994, 1996); this situation is very specific to the central American isthmus and northern South America, and is an important feature in understanding the specific climatology of the region. Three stations were chosen for detailed analyses. The stations were chosen to be representative of the central valley, Pacific slope and the transitional zone between them, because of their representative geographic locations and relatively long data records. Sanatorio Duran (station ID number ) was selected to represent the central valley (length of record: 49 years, , 98% complete), Villa Mills (073033) was chosen for the transition zone (length of record: 50 years, , 94% complete), and Nagatac (080005) was chosen to represent the Pacific coastal zone (length of record: 39 years, , 97% complete). These stations are shown on Figure 3.

3 MARKOV MODELS AND DAILY PRECIPITATION 1863 Figure 1. Average Costa Rican precipitation, by monthly triad. Top: December January February; bottom: March April May

4 1864 M. HARRISON AND P. WAYLEN Figure 2. Average Costa Rican precipitation, by monthly triad. Top: June July August; bottom: September October November

5 MARKOV MODELS AND DAILY PRECIPITATION METHODOLOGY Markov chain models are created by conditioning the probability of the occurrence of rainfall on 1 day, upon whether measurable rainfall was observed on the previous days. These are characterized by transition probabilities : Pr{X t+1 X t, X t-1, X t 2,...,X 1 }=Pr{X t+1 X t, X t 1,...,X t m } (1) in which the order of the model is given by m, the number of previous days to be examined for rainfall. In this study, a precipitation threshold of 0.25 mm (0.01 inches) was used to indicate whether measurable precipitation occurred on a given day. Rainfall events spanning different days were considered to be separate events in order to simplify the modelling process. Conceivably, the number of states to be used in a Markov rainfall model is unlimited, but more generally a two-state model is used: X t =1 wet day (2) X t =0 dry day (3) Using this notation, the transition probabilities for a two-state Markov precipitation model of order m is given by: P i,j,k,...,m =Pr{X m X m 1,X m 2, X m 3,...,X j, X i } (4) Figure 3. Costa Rican topography, with rainfall stations indicated

6 1866 M. HARRISON AND P. WAYLEN where i, j, k,...,m 1 refer to the previous days state (i.e. wet or dry). The maximum likelihood estimators for the transition probabilities is given by: p h,i,j,...,m = n h,i,j,...,m h, i, j,...,m=0, 1 (5) n h for a multi-order, two-state Markov model, where n represents the number of transitions which have been counted for a given time period, and i, j, k,... represent the state of previous days, upon which the transition probability is conditioned. To decide the order to use in modelling the rainfall occurrence process, it is necessary to determine how much information is being included in a model of given order, and compare this with other possible model orders. Of concern is the actual amount of information about the rainfall occurrence process, as well as the number of parameters to be generated to account for this information. Ideally, we desire the maximum amount of information and the fewest parameters (i.e. greatest degree of parsimony); in other words, a balance must be struck between having too many, and too few, parameters generated. Information criteria are employed to make this determination. The two most common are the Akaike information criteria (AIC) (Akaike, 1974) and the Bayesian information criteria (BIC) (Schwartz, 1978). Both methods rely on generating the log-likelihood functions for the estimated transition probabilities of the fitted Markov model. Given a two-state Markov model, the log-likelihoods for orders 0, 1, 2 and 3 are given by Wilks (1995, p. 301) as: 1 L 0 = n j ln(p j ) (6a) j=0 1 L 1 = 1 L 2 = 1 L 3 = 1 i=0 j=0 n ij ln(p ij ) 1 1 h=0 i=0 j=0 n hij ln(p hij ) n ghij g=0 h=0 i=0 j=0 These log-likelihood formulae are then used to calculate the AIC and BIC scores; the order giving the lowest score is the most parsimonious (i.e. the most efficient ) model. The criteria are calculated using the following equations for an m-order, s-state model, using a data stream of n-length: AIC(m)= 2L m +2s m (s 1) BIC(m)= 2L m +s m (ln(n)) Given that this study will use two-state (i.e. rain/no rain ) models, these equations simplify to: AIC(m)= 2L m +2 m+1 BIC(m)= 2L m +2 m (ln(n)) The BIC is generally considered the more conservative of the two criteria, and will often give more definitive results than the AIC; however, the BIC employs the data length, and as pointed out in Katz (1981) and Wilks (1995), the BIC may only be preferable over sufficiently long periods, ranging from n=100 to over n=1000, depending upon the degree of serial correlation inherent in the data. For this study, both the AIC and BIC are calculated and compared, in order to have the most information available for determining the best order to be used for the Costa Rica data. Calculation of the transition probabilities is best illustrated using the first-order case. In this instance, the probability of rainfall occurring on any given day is conditioned upon whether measurable rainfall occurred on the immediately preceding day. This allows the occurrence process to be represented by four parameters, of which two must be calculated: (6b) (6c) (6d) (7a) (7b) (8a) (8b)

7 MARKOV MODELS AND DAILY PRECIPITATION 1867 p ij =Pr{X j =n X i =m} m, n={0, 1} (9) where X n = 1 if precipitation occurred 0 otherwise (10) and p 00 +p 01 =1 (11a) p 10 +p 11 =1 (11b) The procedure can be generalized to any higher order, as shown in Haan (1977). For this study, each 3-month calendar sequence is passed through a computer routine which tallies the sequences of wet and dry days; in this way, each month of data actually represents monthly triads of data centred on the month listed, such that n 90 in every case. These totals are used to create the transition probabilities, for orders or 0, 1, 2 and 3. The AIC and BIC statistics are computed, and the resulting monthly time series of probabilities and information criteria are segregated by station. In addition, a tally is kept of all years for all stations, to be used as a baseline for comparison, indicating the number of times a particular order is chosen according to the AIC/BIC criteria. An evaluation of the order results is made, and the order with the best (i.e. lowest) overall AIC/BIC is used. In the case of mixed results, the higher order is chosen so as not to inadvertently hamper future analysis with loss of information. 4. RESULTS As a general rule, the BIC seems to give much more definitive guidance concerning the proper choice of model order than the AIC, as shown in the total station tallies (Figure 4). During the summer months, as well as during the late winter months of February and March, the zero-order model is found to be superior more often than other model order choices. During early and mid-winter (November through January), the first-order model emerges as the preferred choice a majority of the time. There is no month in which the second- and third-order models are deemed appropriate according to the BIC. This result is borne out when looking at the average values of the AIC versus BIC for each station (Figures 5 7). The BIC averages show a reasonably consistent pattern, with the second- and third-order model values noticeably higher than the zero- and first-order values, indicating a decided preference towards the lower order models. However, comparing the tallies (Figure 4) with the average values shows that while the zero-order model is chosen most often over the first-order model, the choice is often made by very narrow margins, indicating that the choice of a zero-order model over the first-order is not nearly as clear-cut as the tallies would make it seem. The AIC is much more ambiguous. While the zero- and first-order models are generally chosen most often (as with the BIC), the higher order models have a much stronger showing; in December, for instance, the third-order model ranks just behind the first-order model as being chosen most often. The problem with using the AIC seems to be two-fold. First, the AIC values nearly always tend to be more tightly grouped, making the values subject to more variation as a result of the time frame from which the station data was chosen. Second, the AIC seems to be more sensitive to rapid changes in the inter-monthly rainfall climatology than the BIC, as evidenced in the behaviour of the average values for Nagatac (Figure 7). Nagatac is more subject to the eranillos than the other stations, resulting in a rapid switchover from very wet conditions early in the summer, to a somewhat drier eranillos period in July, and black to near-daily rainfall during the latter part of the summer season; during the drier periods, the average value of the AIC for each of the model orders become very similar, and the choice of proper model order is difficult to assess. This sensitivity to the eranillos manifests itself in the AIC tallies shown in Figure 4. While the zero-order model is clearly chosen most often during the summer months, the

8 1868 M. HARRISON AND P. WAYLEN Figure 4. Number of times each order is chosen as best using AIC and BIC criteria. Top: AIC; bottom: BIC choice becomes somewhat less clear during the eranillos, in which the rainfall abates somewhat over the region. The use of the BIC statistic tends to diminish the eranillos effect in the statistics, making the summer month order choices more consistent, even with a clear change in rainfall patterns over the region during this time. The winter season tends to maintain the overall average AIC and BIC patterns, with one important exception: during the early- to mid-winter (November February), the BIC shows that the first-order model is preferred over all others, while the AIC gives no clear-cut answer. In addition, the overall values of the AIC and BIC rise markedly during the winter months, indicating a sensitivity of the statistics to changes in seasonal precipitation regimes. It is important to note that, without exception, the winter AIC and BIC measures are higher than the summer measures (although the degree to which the measures are higher is less noticeable at Nagatac, which generally experiences extremely dry conditions during the winter months).

9 MARKOV MODELS AND DAILY PRECIPITATION DISCUSSION It is important to understand at the outset that minimizing the AIC and BIC statistics to achieve the lowest possible Markov model order, without regard to other statistical considerations, will not always yield an ideal model for the original data stream. As reported by Katz and Parlange (1998), a common result of employing too low an order for daily precipitation measurements is a model which underestimates the observed inter-annual and inter-seasonal variance in total precipitation; this is referred to as the overdispersion phenomenon. Blind acceptance of the AIC/BIC results is therefore not advised, but detailed testing of the various model orders is definitely suggested. Such testing might involve, for example, comparison of distributions of wet and dry spells of varying lengths estimated from models with the historical data sets. A further difficulty in using the AIC/BIC statistic in assessing precipitation climatology is the nature of the statistical calculations. From the maximum likelihood formulae 6a to 6d, an important feature must be noted: the transition probability (p... x... ) is modified using a logarithmic function, which would tend to inordinately increase the absolute value of the log-likelihood when the calculated transition probabilities are closer to zero, thereby increasing the ultimate value of the AIC or BIC parameter. In other words, Figure 5. Average AIC and BIC values for Sanatorio Duran (073011). Top: AIC; bottom: BIC

10 1870 M. HARRISON AND P. WAYLEN Figure 6. Average AIC and BIC values for Villa Mills (073033). Top: AIC; bottom: BIC the more certain you are that rainfall will not occur on a given day, the less useful the Markov model will be in describing the pattern of rainfall occurrence. This adds weight to the idea that precipitation sequences that are nearly entirely dominated by continuous dry (or wet) spells are not easily modelled using Markov methods and the AIC/BIC criteria. This having been said, it is possible to examine the results of the AIC/BIC computations for the three stations used in this study, and relate the results back to the underlying climate mechanisms in place over the region. The first most striking feature is the prevalence of the zero-order model preference during the summer months. It is important to note that the summer months represent the primary rainy season across central America, the period during which the trade winds and the ITCZ dominate the climate scene. As such, it is normal to expect there will be a high probability of precipitation on any given day in most parts of the region. From this standpoint, it stands to reason that conditioning rainfall probabilities on previous days occurrence is unlikely to add much useful information to the occurrence model under most circumstances, and a commensurate bias towards lower model orders is to be expected. There is a noted sensitivity in the average AIC/BIC measures at Nagatac during the July/August timeframe, more so than in the other stations. This clearly corresponds to the occurrence of the eranillos, during which the frequency of wet days is reduced. The eranillos has been noted in previous studies to be much more pronounced on the Pacific slope region of Costa Rica (Waylen et al., 1994). In fact, both

11 MARKOV MODELS AND DAILY PRECIPITATION 1871 the AIC and BIC show themselves to be particularly sensitive to the changes in the number of wet days across the summer season, picking up the bi-modal rainfall distribution inherent in the western part of the country. The effects of the eranillos are somewhat diminished as you move towards the eastern part of the country, and for this reason the AIC/BIC values do not show the same degree of variation across the summer months. Changes in winter climatology are also clearly visible in the AIC/BIC statistics, which can be related to the frequency with which the nortes change during the November March timeframe. Early winter (November December January triad) calculations have much higher values than late winter (March) values, showing that the winter rains tend to be less concentrated from the standpoint of occurrence; rainfall in the early winter is generally a consequence of cold-front intrusion from the North American landmass, and the occurrence of rainfall is much less certain than in the summer. During the winter spring transition period, however, tropical waves begin moving onshore from the Caribbean Sea, allowing more frequent rainfalls to occur in conjunction with the nortes, some of which can be quite heavy. For this reason, the variation between early winter to late winter values of the AIC/BIC are much more pronounced in the eastern stations in the Central Valley (Sanatorio Duran) than in the western parts of the country (Nagatac), where nortes and tropical waves have less effect on the local climates. Figure 7. Average AIC and BIC values for Nagatac (080005). Top: AIC; bottom: BIC

12 1872 M. HARRISON AND P. WAYLEN Ultimately, the choice of model order to be employed to describe the occurrence of daily rainfall in Costa Rica must be made by considering both the AIC/BIC statistics, as well as the sensitivity of the model to the effects of overdispersion. The results of this study would indicate that the decision to use a high-order model ( 2) is difficult to justify. However, given the overdispersive effects of an extremely low-order model (zero-order), it would make more sense to use a model of at least first-order, provided the AIC and BIC do not in turn rise inordinately as a result. It would appear reasonable, therefore, to use a first-order model to characterize rainfall occurrence in the absence of extreme underestimation of inter-annual or inter-seasonal variance. 6. CONCLUSION AND FUTURE DIRECTIONS From the earlier work of Chin (1977), Katz (1977, 1981), Coe and Stern (1982), Stern and Coe (1983), and Katz and Parlange (1995, 1998), the choice for model order when creating Markov models to understand daily rainfall occurrence is a matter of considerable concern. The blind acceptance of the results of an AIC/BIC-type analysis can lead to significant underestimation of variance of the original data, and thereby lead to erroneous conclusions regarding the nature of precipitation occurrence in the study area. However, even with these limitations in effect for using these analysis techniques, they are still useful as indicators of local climate variability. The AIC and BIC indicators seem to be quite sensitive to variations in the duration of consecutive wet-day and dry-day periods, which likely accounts for the variability of AIC and BIC values between winter and summer months, as well as variations arising from the geographic distribution of the stations. Clearly, more research needs to be done on the nature of daily rainfall occurrence in the humid tropics. This study represents a first attempt at performing such an analysis, and before further conclusions can be reached it will be necessary to perform a significant amount of simulation using the data and the results of these analyses. In addition, future work will include the incorporation of the occurrence results with an analysis of magnitudes across the region, as well as understanding how inter-annual climate variations, such as El Niño Southern Oscillation (ENSO), affect the occurrence parameters. This represents an important step in making the models useful outside the arena of theoretical climatology, allowing professionals in other fields to apply the results into such practical areas as flood management, agriculture, and the management of hydroelectric resources. REFERENCES Akaike H A new look at the statistical model identification. IEEE Transactions on Automatic Control AC-19: Bryson RA, Khun PM Stress-differential induced divergence with application to littoral precipitation. Erkunde 15: Chin EH Modeling daily precipitation occurrence process with Markov chain. Water Resources Research 13: Coe R, Stern RD Fitting models to daily rainfall data. Journal of Applied Meteorology 21: Gleick PH Water in Crisis: A Guide to the World s Fresh Water Resources. Oxford University Press: New York. Haan CT Statistical Methods in Hydrology. The Iowa State University Press: Ames, IA. Hastenrath SL Variations in low latitude circulation and extreme climatic events in the tropical Americas. Journal of the Atmospheric Sciences 33: Katz RW Precipitation as a chain-dependent process. Journal of Applied Meteorology 16: Katz RW On some criteria for estimating the order of a Markov chain. Technometrics 23: Katz RW, Parlange MB Generalizations of chain-dependent processes: Applications to hourly precipitation. Water Resources Research 31: Katz RW, Parlange MB Overdispersion phenomenon in stochastic modeling of precipitation. Journal of Climate 11: Schultz DM, Bracken WE, Bosart LF, Hakim GJ, Bedrick MA, Dickenson MJ, Tyle KR The 1993 superstorm cold surge: frontal structure, gap flow, and extratropical tropical interaction. Monthly Weather Re iew 125: Schultz DM, Bracken WE, Bosart LF Planetary- and synoptic-scale signatures associated with central American cold surges. Monthly Weather Re iew 126: Schwartz G Estimating the dimension of a model. The Annals of Statistics 6: Stern RD, Coe R A model fitting analysis of daily rainfall data. Journal of the Royal Statistical Society 147: Waylen PR, Caviedes CN, Quesada ME Interannual variability of monthly precipitation in Costa Rica. Journal of Climate 9: Waylen PR, Quesada ME, Caviedes CN The effects of El Niño Southern Oscillation in San José, Costa Rica. International Journal of Climatology 14: Wilks DS Statistical Methods in the Atmospheric Sciences. Academic Press: New York.