A multi-objective fuzzy classification of large scale atmospheric circulation patterns for precipitation modeling

A?I~LIED, ~ATHI!~AT~C$ AND

CO~i PU'I'ATION ELSEVIER Applied Mathematics and Computation 91 (1998) 127-142

A multi-objective fuzzy classification of large scale atmospheric circulation patterns for precipitation modeling Ertunga C. Ozelkan a,,, Agnes Galambosi a, Lucien Duckstein a, Andr4s B4rdossy b ~t Department of Systems and Industrial Engineering, University of Arizona, Tucson, AZ 85721, USA b Institut.for Hydraulic Engineering, University of Stuttgart, Stuttgart, Germany

Abstract

A multi-objective fuzzy rule-based classification (MOFRBC) technique is applied in order to cluster and classify daily large scale atmospheric circulation patterns (CPs) and analyze the relationship between the CPs and local precipitation. The methodology is illustrated by means of an Arizona case study. For this purpose, three indices are calculated to measure the information content of the clustering method in terms of predicted precipitation. A thorough sensitivity analysis is provided to gain more understanding on the robustness of MOFRBC model. Furthermore, it is shown that extending the daily premises to two-day and three-day sequences of CPs improves the information content of the classification. The results are also compared with the original subjective clustering. For the Arizona case study MOFRBC seems to be a competitive technique with the advantage that the physical aspects can be better represented by fuzzy rules (which tend to mimic the human way of decision making) than by objective methods. © 1998 Elsevier Science Inc. All rights reserved. Keywords: Fuzzy rule-based classification; Multi-objective decision making; Subjective classification; Atmospheric circulation patterns; Arizona rainfall; Sequencing technique

*Corresponding author. 0096-3003/98/$19.00 © 1998 Elsevier Science Inc. All rights reserved. PII: S0096-3003(97) 10002-9

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1. Introduction and scope

The purpose of this paper 1s to develop a multi-objective fuzzy rule-based classification (MOFRBC) methodology to cluster (to define fuzzy types) and classify (construct the daily circulation pattern catalogue) macro-circulation patterns. The MOFRBC model is an extension of the work presented by B~rdossy et al. [1] who have shown that fuzzy rules can be used to classify circulation patterns (CPs) using a European case study. In this paper, we analyze the classification problem in a multi-objective optimization framework and propose a simple method to obtain the optimal model parameters. Three indices are calculated to measure the information content of the daily catalogue in terms of local precipitation. The fuzzy rules are based on the subjective classification of Bartholy and Duckstein [2] and used to analyze the relationship between the CPs and local precipitation events in Arizona. Furthermore, two-day and three-day CP sequences are analyzed as well as the daily CP sequence (catalogue) to improve the information content of the approach, in line with Comrie [3], who has shown that passing from one- to three-day CP window usually yields more information. It has been shown that there is a strong linkage between large-scale circulation and local climatic variables [4-11]. To model the linkage between CPs and a local climate variable taken herein as precipitation, one possible way is first to define the large scale CP types and then to find the connection between CPs and the local variable. Two basic ways to define CP types are manual (or subjective) and automated (or objective) clustering methods. The automated methods have the advantage of yielding reproducible categories with moderate time and effort. Examples are found in Refs. [12-15]. On the other hand, the physical aspects are better reflected by subjective methods, which however might not be duplicated and require much more time and effort [9,16 19]. MOFRBC can be considered as semi-subjective which is a combination of subjective and objective approaches. It is aimed at combining the advantages of both manual and automated clustering methods that is, using automation while preserving expert opinion. Forty-two years of daily observation of 500 hPa data have been obtained from the National Center for Atmospheric Research (NCAR), consisting of the National Meteorological Center (NMC) grid point analyses of the 500 hPa pressure field heights. Specifically, the data at 35 points on a diamond grid covering the sector 20°-50 ° N, 90°-130 ° W for the period January 1949-June 1989 are used (Fig. 1). Four seasons of equal length have been defined and are studied separately in Arizona, the first one being winter from 1 January to 31 March. The precipitation data are extracted from Earthinfo Climatedata. Eight Arizona precipitation stations have been selected for the analysis, namely, in

E.C. Ozelkan et al. /Appl. Math. Comput. 91 (1998) 127-142 DLa~tOND GRID

'*

26 * r

'''~];1 i=" T ']' tl,# $2 33 34

129

'i 35

Fig. 1. Pressure measurement grid and grid point numbers.

alphabetical order: Betatakin, Clifton, Fort Valley, Grand Canyon, Nogales, Roosevelt, Tucson, Yuma (Table 1, Fig. 2). The paper is organized as follows: the M O F R B C method is described next. An Arizona case study and sensitivity analysis are presented and the results are then analyzed. Section 4 consists of a discussion and a set of conclusions.

2. Methodology 2.1. Multi-objective f u z z y rule-based classification ( M O F R B C ) M O F R B C m o d e l s t r u c t u r e is c o m p o s e d o f t w o m a i n parts: f u z z y r u l e - b a s e d classification and multi-objective estimation of model parameters.

Table 1 Description of the Arizona precipitation stations Station

Latitude

Longitude

Elevation (ft)

Data reliability (%)

Betatakin Clifton Fort Valley Grand Canyon Nogales Roosevelt Tucson Yuma

36:41 33:03 35:16 36:03 31:21 33:40 32:15 32:37

110:32 109:17 111:44 112:08 110:55 111:09 110:57 114:39

7290 3460 7350 6890 3810 2210 2440 190

97 98 99 92 97 98 99 98

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I

/

Fig. 2. Weather stations under study.

2.1.1. F u z z y rule-based classification We have summarized below the fuzzy rule-based f r a m e w o r k used in this study. F o r some o f the basic definitions of fuzzy n u m b e r s the reader is referred to B~rdossy and Duckstein [20], B~irdossy et al. [1], or K a u f m a n and G u p t a [21], Z a d e h [22]. In a fuzzy rule-based classification, rules are given as " i f . . . t h e n . . . " but an observation vector usually belongs to m o r e then one rule. Formally, the rules are defined as follows. Ifa.jEBi.1~a.zEB~.2~...~a.vEBi.~.

then CP is i,

(1)

where V is the n u m b e r of explanatory variables or premises, and the premise a.,. = (al,,,, • • •, aN,,) T, V = 1 , . . . , V is a vector o f N selected m e a s u r e m e n t or grid points corresponding to a given fuzzy class Bi.~, o f CP type i, i = 1 , . . . , I, I being the total n u m b e r o f types. The logical o p e r a t o r "(~" here m a y be " A N D " or " O R " . Each fuzzy set B~.~, is associated with a m e m b e r s h i p function pB,.,(a,,) : Bi,~, ~ [0, 1], which shows basically how m u c h each point a.~, in the set belongs to the set Bi,, and performs the " E " function in Eq. (1). F o r any vector of premises (a i , . . . , a.v) the degree o f fulfillment ( D O F ) Dg of rule i, representing the extent of which a given rule is applicable, is defined as a function F of the m e m b e r s h i p function values #s,,,(a.~,)

E.C. Ozelkan et al. /Appl. Math. Comput. 91 (1998) 127-142

Di=F(g,,l(a.,),...,l~e,.~(av)).

13l

(2)

Function F in Eq. (2) may be an " O R " function Fo, an " A N D " function Fa according to the (~ operator chosen in Eq. (1), or a combination of Fo and Fa. Here Fo is taken as the " O R " function of fuzzy set theory [23]:

Fo(xl,x2) =xl + x 2 - x l x 2 , (x,,x2) E ~2. (3) For N variables, (xi,... ,XN) it is computationally convenient to define Fo recursively as:

Fo(x~,... ,XN) = Fo(Fo(xl,... ,XN ,),XN).

(4)

The " A N D " function Fa is defined as in Ref. [24] as: N

= Hx..

(s)

The main steps of the fuzzy rule-based approach are as follows: Step 1. (Partition of the input-output space into fuzzy regions): The pressure height data can be partitioned into a number of overlapping fuzzy sets or classes:/~ = {/~,; v = 1 , . . . , V}, where V is the number of classes and/~ is the fuzzy class v. An example is given in Fig. 3 for normalized values of pressure height. Bi.~, in Eq. (1) thus takes on fuzzy values from these classes: Bi,~ c/5. Let (,~., aT,~,) be the support of Bi.~,; the membership function/Je,,, here is chosen

i Very Low

>

i

, Medium Low

i Medium

, i Medium High

i Very High

0.8

0.6

E 0.4

0.2

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

Normalized 500 hPa Pressure Height

Fig. 3. Fuzzy partitioning of the 500 hPa height level.

0.9

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to be a fuzzy number (FN) defined as (7i.,-,~i.,. ~,,)Fn, where ~i., is the most credible value of a , with an assigned membership of 1. For example, in Fig. 3, the support of the triangular fuzzy number (TFN) "medium" is (0.25, 0.75); The TFN itself is denoted as (0.25, 0.50, 0.75)TVN, 0.50 being the most credible value. Step 2 (Assessment of fuzzy rules): First, a few representative grid points (a,., E B,,,) are selected such that each class from the subjectively defined CPs is described and then rules of the form of Eq. (1) are constructed. For example the selected grid points for each class for the winter CP type W3 (Fig. 4) give the following fuzzy rule: If

grid 25. 30, 33 are Very Low

~-

grid 20, 24, 35, are Medium Low

:+

grid 3, 5. 14, 28, 32 are Medium

~)

grid 6, 12, 16. 19, 31 are Medium High grid 17, 18, 26. 27 are Very High

then CP is WY

(6)

Step 3 (Selection of DOF function and calculation of rule fulfillments): To perform the combination of the fulfillment grades we use a combination of " A N D " and " O R " operations as in [1]. The partial DOF Di,, for (al,,,...,aN.,,) corresponding to class v is taken as a convex combination of " O R " and " A N D " fulfillment grades, by means of a weight ,,, 0 ~<7 ~< 1. Given 7, for each rule class, the value of D , is calculated as Di., = ~'Fo, ([~Bi, (al .i ), . . . , liB,, (a,\.., ) )

+ (1 - 7)F~, (/re, ' (a,.,) . . . . , t'z,, (av.,)).

-~

Winter/3.

se

13e I]| "--120 40 . ~

rio

(7)

too

to

[ 16~,8°An..40-160 1-240 -2S~ I

~

~

..

-2oo

/:

Fig. 4. 500 mb anomaly map tbr CP W3 of the semi-subjective clustering.

E.C Ozelkan et al. / Appl. Math. Comput. 91 (1998) 12~142

133

Then, the class fulfillment values D;.~,..., D;.t are combined into the DOF D; of rule i

D; = ~Yo,(D,.,,... ,D;.r) + (1 - B)F~i(D;.,,... ,D;j ).

(8)

Eq. (8) is different from the one used in [1], where only an " A N D " rule was used between classes, which corresponds to the special case [~ = 0. Step 4 (Selection of the index i with the highest D~ value): This approach selects the rule with the highest credibility level. This way each day is assigned to a CP type and, as a result, a daily catalogue of the CP types is obtained. One can also define a threshold for D~'s such as D T and identify an unclassified day whenever D, < DT.

2.2. Multi-objective estimation of 7 and [~ The parameter "/was proposed by Bfirdossy et al. [1] to be selected by the decision maker (DM) and/~ = 0 was chosen meaning that only " A N D " rules were used to combine the DOF values for each fuzzy class. Here, we propose a systematic approach for the selection of "best" 7 and/~ by maximizing predefined objective functions. Let I(X, Y, R; O) be a vector of functions showing the quality of classification in terms of input data X (e.g. CP data), desired output variable Y (e.g. rainfall), rule based system (R), and the parameter set O = {7,/~}. Given X, Y, and R the problem of estimating O becomes O ~ = {,,*,/~*} = argmax{I(O]X, Y,R): O ¢ [0, 1] x [0, 1]}.

(9)

This is a multi-objective mathematical programming (MOP) problem, where the aim is first to obtain the non-dominated solutions and then choose the best solution using DM's preferences on the objectives. In general, MOP problems are solved by converting the problem structure into many single-objective optimization problems which might require extensive amount of computations [25 27]. One should also keep in mind that in nonlinear systems such as physical systems, even a single optimization problem can be challenging to solve due to non-convexities [28]. Furthermore, techniques such as multi-objective genetic algorithms (MOGA) [29,30] and multi-objective complex evolution (MOCOM-UA) [27] do not always guarantee to find the non-dominated solution set. For more details on multi-objective decision making, the reader is referred to Refs. [27,31 36]. Because of the special structure of M O F R B C model, that is, two parameters to be calculated ( 6 / = {~,,fl}), both being bounded in [0,1], we propose the following heuristic approach to solve this problem. 1. Discretize the parameter space uniformly, e.g. 61 E [0.0,0.1,0.2,..., 1.0] x [0.0, 0.1,0.2 . . . . . 1.0].

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134

2. Compute the objective function I(.) for all parameter combinations, e.g. 11 × 11 = 121 simulations. 3. Obtain the non-dominated solution set. 4. Select best solution from the non-dominated solutions using DM's preferences on the objectives e.g. using compromise programming [31]. In the next section, three information measures selected as objective functions are briefly described.

2.3. Information measures Three information measures are calculated as defined in [1] to measure the quality of a classification for precipitation estimation and/or generation, namely:

I,=

-f ~t (pA, -- p)2

(10)

where T is the time horizon, PA, is the conditional probability of rainfall on day t, given that the CP is At and the unconditional probability of rainfall at a given site is p.

12=

F~t (mA,--m)2

(11)

where m is the unconditional mean daily rainfall amount at a given site and mA, the mean daily rainfall conditioned on the CP being of type A,. Finally, I3=1~

mA~'-l'm

(12)

High values of I~, 12, and/3 show strong precipitation conditioning. These three indices, whose ranges are calculated in Appendix A, appear to be in conflict, which indicates the necessity of using a multi-objective approach.

2.4. Sequencing As mentioned before, besides the daily CP types, sequences of CP types in two and three day windows are also used to see the influence on the information contents given in Eqs. (10)-(12). For the partial sequence of six consecutive days, let the CP types be for example {(1,3),(2,5),(3,5),(4,2), (5, 2), (6,4)}, where (t, At) denotes the CP type At occurring on day t. In such a case, the two-day sequences would be {(2, 35), (3, 55), (4, 52), (5, 22), (6, 24)}, and three-day sequences, {(3,355), (4, 552), (5,522), (6,224)}. Intuitively, the information content is expected to increase with sequence length because CP types usually tend to persist for more than one day over a given region.

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135

3. Application and results An Arizona case study has been selected to illustrate the fuzzy rule-based approach. The rules describing the types have been defined based on the original subjective classification of Bartholy and Duckstein [2] using the methodology given in Section 2.1.1. There are I = 6, 7, 7, 8 CP types in winter, spring, summer and fall, respectively. For comparison, the mean information content of eight precipitation stations in Arizona has been calculated for both the MOFRBC and the subjective classification of Bartholy and Duckstein [2]. In order to perform MOFRBC, several parameters should be selected such as the number of fuzzy classes (V), rule-based system, and the threshold parameter (Dr). To see the robustness of M O F R B C model, sensitivity results are obtained for V = 4, 5,DT = 0.00,0.10 and for four rule based systems (two for V = 4 and two for V = 5). The eight cases analyzed for sensitivity are shown in Table 2. T F N s are used as the membership functions for the rule classes. By following the steps of the approach given in section 2.1.1, the best parameters {7", fl*} are obtained. In step 4 of the algorithm, compromise programming [37] is used for the selection of the best solution among the nondominated solutions which minimizes the Lp n o r m defined as Lp(x) =

(13)

__ Wm -l£m~,,,*

where Wm are the objective function weights to be given by the decision maker and 1 ~

Rule base

# of classes (V)

Threshold (DT)

1 2

Ri Ri

4 4

0.0 0.1

3 4 5 6 7 8

R2 R2 R3 R3 R4 R4

5 5 4 4 5 5

0.0 O. 1 0.0 O. 1 0.0 0.l

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Table 3 Upper bounds for 11, 6, and 13 Information measure

Winter

Spring

Summer

Fall

I~

0.605 8.611 6.053

0.655 6.452 10.688

0.565 8.636 3.673

0.625 10.236 7.419

p = 1,2, and ~ ) , and the corresponding ranks are shown in Table 4. Compromise programming chooses the solution with the minimum Lp value. For winter, solution {7" = 1.0, fl* = 0.2} turns out to be superior. In spring, summer and fall, we ended up with 7, 6 and 5 non-dominated solutions, corresponding to the optimum values {7* = 0.6, fl* = 0.4},{7" = 0.6, fl* = 0.5} and {7* = 0.9, fl* = 0.1 }, respectively. Fig. 5 shows the sensitivity analysis results for winter, spring, summer and fall. Comparison of cases 1 to 2, 3 to 4, 5 to 6 and 7 to 8 show that changing the threshold level from D-r = 0.0 to DT = 0.1 does not substantially affect the results. In general, it seems that cases 5 and 6 provide superior results in winter, summer and fall (except for 13). In spring the index values do not vary much, (except for 13, cases 1 and 2 give slightly better values). M O F R B C yields robust index values, since the values did not vary much among the sensitivity cases. For further analysis, case 5 is chosen (R3, V = 4, DT = 0.0). Fig. 6 shows the information measures I1,12, and 13 for winter, spring, summer and fall, respectively. The indices show that M O F R B C appear to yield slightly better results (23 out of 36 cases) as compared to the subjective classification. Fig. 6 also shows the results using two and three-day CP sequences as premises. It turns out that two-day sequences improve the information content and three-day sequences have the highest information content from the viewpoint of precipitation for both M O F R B C and subjective techniques.

4. Conclusions Whereas Bfirdossy et al. [1] have applied fuzzy rule-based approach to classify CPs over western Europe which has a relatively simple climate and for which a well established subjective CP classification scheme exists, we have ap-

Table 4 Non-dominated solutions and compromise programming results for winter ?,

fl

1~

12

13

LI

L2

L:~

Rank

1.0 1.0

0.2 0.3

0.0746 0.0742

0.541 0.541

0.4540 0.4541

2.73868 2.73943

1.58182 1.58224

0.93703 0.93708

1 2

E.C. Ozelkan et al. / Appl. Math. Comput. 91 (1998) 127 142

137

:11 I 0.08

Case 1 Case 2 ~ = Case 3 ~ = , Case 4 Case 5 ~ = Case 6 Case 7 Case 8

0.06 0.04 0.02 0 Winter

Spring

Summer

Fall

0.8 0.6

I

E 0.4 ¢q 0.2 Winter

Spring

Summer

Fall

Winter

Spring

Summer

Fall

0.6 0.4 0.2

Fig. 5. Sensitivity analysis: Average information content 1L, h , and 13 in each season for eight sensitivity cases.

plied a modified version of the methodology to the complex case of Arizona climatic zones, using a fairly rudimentary subjective classification as a basis. The index values found are two to three times less than those computed by B~irdossy et al. [1] for the European case study. Again, this can be due first to the complexity of Arizona climate, and second, to the subjective types [12,13] upon which their model is based being presumably better defined. On the basis of the above developments, the following conclusions can be made: (l) A multi-objective fuzzy rule-based classification (MOFRBC) has been developed, where the model parameters (~,,fl) are estimated in a multi-objective fashion. A simple heuristic procedure is proposed for the solution. (2) The M O F R B C model works reasonably well in the Arizona case examined. It has slightly better information content than the subjective classification. Thus M O F R B C method is an effective way to construct a CP catalogue on the basis of an existing subjective cluster system.

E . C Ozelkan et al. /Appl. Math. Comput. 91 (1998) 127 142

138

0.2 0.15 ,0.1 0.05 0 Winter

Spring

Summer

Fall

1.5 3dy

1 0.5 0L Winter

Spring

Summer

Winter

Spring

Summer

Fall

1,2 / 1 0.8 (~ 0.6 0.4 0.2 0

-. m Subjective ~ FRB

J

Fall I

Fig. 6. Comparison of MOFRBC results with subjective classification: Average information content in each season for one-, two-, and three-day sequences.

(3) For M O F R B C we do not need precise, numerically defined types: it is sufficient to have a verbal description of the types, such as "Pacific region is high pressure", "Arizona is low pressure", etc. This confirms that fuzzy rules are similar to the human way of thinking. (4) Two-day and three-day sequences appear to improve the information content as measured by three indices, so that such a sequencing technique appears to be a useful method in the semi-arid case considered herein.

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139

Acknowledgements

The research in this paper has been partially supported by US National Science Foundation, the US Corps of Engineers and the US National Center for Global Change, Great Plains Center.

Appendix A. Upper and lower bounds on 11,12,13 Let I1,,/2,,/3, denote the lower bounds and I~', I~, I~ denote the upper bounds for the information indices introduced in Section 2.2. It is easy to verify that the lower bounds equal to zero for all indices. The following lemmas are provided to estimate the upper bounds.

Lemma 11. Assume an ideal case that there are only wet and dry types such that Prob(rainfalllwet type) = 1 and Prob(rainfallldry type) = O. Let us also assume that the number of wet and dry types are equal (l) and each type occurs the same total number of days (n). Let the unconditional probability of having rainfall be p, then it is to be shown that

I; = (p2 _ p + (1/2)),/2

(A.1)

Proof. 1

I,=

~

_ p)2

(PA,

1/2

=

-f

nj(pj

,

(A.2)

)

where pj is used to denote the probability of having rainfall given that CP is j. Note that A, E { C P I , . . . , CPj}. Since ni = n and p~ = 1 Vj E {wet types} and p~ = 0 Vj E {dry types} we can rewrite Eq. (A.2) as Ii=

0-p)Z+...+n(0-p)Z+n(1-p)Z+...+n(1-p 2nl =r

(A.3)

which gives Eq. (A.1). Lemma 12. Similar to the assumptions for 11, ideally we would want that the amount of rainfall produced by dry types is zero, and it would be desirable to have a distinct variability among the expected rainfall produced by each wet type. Let rmax be the maximum expected rainfall, m be the unconditional amount of rainfall and assuming that each wet type produces an expected rainfall amount m~ = (j/l)rmax, to provide the distinct variability, it can be shown that

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140

, (~ 12 =

1 ~/(.~ ) 2 ) ''~ + 71 r,..... - m

(A.4)

and assuming that an exponential distribution assumption holds f o r rainfall ?m~x ~ 3/" can be considered as a reasonable estimator Jbr rmax, where ~ is' the average rainJitll. Proof. The p r o o f o f Eq. (A.4) is trivial and similar to the p r o o f o f L e m m a Ii. Results for the estimation o f rmax can be obtained as follows: Let R be an exponential r a n d o m variable denoting daily rainfall Prob(R ~
'" do)

(A.5)

0

then for a chosen ~ risk level, rmax can be estimated by solving Prob(R~
(A.6)

which yields ?m~x = --ln(1 -- ~)r.

(A.7)

F o r ~ = 5%, ?max ~ 3r.

Lemma 13. Similarly to 12, it can be shown that I* =

3

1

1 ~~ "

]+57

"!"rmax

.

--1

(A.8)

Proof. The p r o o f o f Eq. (A.8) is similar to the p r o o f o f L e m m a 12.

References [1] A. Bfirdossy, L. Duckstein, 1. Bogardi, Fuzzy rule-based classification of atmospheric circulation patterns, Int. J. Climatol. 15 (1995) 1087 1097. [2] J. Bartholy, L. Dnckstein, A subjective macroclassification of atmospheric circulation in western United States, Annales Sciet, Budapest de Rol. Eotvos Nom., Sectio Geophysica et Meteorologica, Tomus, X, 1994. [3] A. Comrie, An enhanced synoptic climatology of ozone using a sequencing technique, Phys. Geog. 13 (1) (1992) 53-65. [4] H.H. Lamb, Climate, Present, Past and Future, Methuen, London, 1977, p. 835. [5] K.K. Hirschboeck, Catastrophic flooding and atmospheric circulation anomalies, in: L. Mayer, D.B. Nash (Eds.), Catastrophic Flooding, Allen & Unwin, Winchester, MA, 1987, pp. 23 56. [6] B. Yarnal, Synoptic climatology in Environmental Analysis, Studies in Climatology Series, Belhaven Press, London, 1993, pp. 195. [7] A. Bfirdossy, E. Plate, Space-time model for daily rainfall using atmospheric circulation patterns, Water Resources Res. 28 (1992) 1247-1259.

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