Rule-Based Support Vector Machine Classifiers Applied to Tornado Prediction

Transcription

1 Rule-Based Support Vector Machne Classfers Appled to Tornado Predcton Theodore B. Trafals 1, Bud Santosa 1 and Mchael B. Rchman 2 1 School of Industral Engneerng, The Unversty of Oklahoma 202 W. Boyd, CEC 124, rman, OK {ttrafals, bsant}@ou.edu 2 School of Meteorology, The Unversty of Oklahoma 100 E. Boyd, SEC 1310, rman, OK mrchman@ou.edu Abstract. A rule-based Support Vector Machne (SVM) classfer s appled to tornado predcton. Twenty rules based on the Natonal Severe Storms Laboratory s mesoscale detecton algorthm are used along wth SVM to develop a hybrd forecast system for the dscrmnaton of tornadc from non-tornadc events. The use of the Weather Survellance Radar 1998 Doppler data, wth contnuous data streamng n every sx mnutes, presents a source for a dynamc data drven applcaton system. Scentfc nqures based on these data are useful for dynamc data drven applcaton systems (DDDAS). Senstvty analyss s performed by changng the threshold values of the rules. Numercal results show that the optmal hybrd model outperforms the drect applcaton of SVM by 12.7 percent. 1 Introducton Rule-based classfcaton methods have shown promse n physcal systems applcatons [1]. One bulds a rule-based model by ncorporatng pror nformaton. In the case of Support Vector Machnes (SVMs), pror knowledge s ncorporated nto the model as addtonal constrants n the form of polyhedral rule sets n the nput space of the gven data. These rule sets are supposed to belong to one of two categores nto whch all the data are dvded [2, 3]. Tornado forecastng s an actve area of research n the meteorologcal communty [4, 5]. State-of-the-scence weather radar scans volumes of the atmosphere, producng a large amount of nformaton that s updated every 5 to 6 mnutes. Scentfc nqures based on these data are useful for Dynamc Data Drven Applcatons Systems (DDDAS). Once the data are collected, they are quckly processed by algorthms that look for sgnatures of tornadoes n near-real tme, snce an extra mnute of lead-tme can save lves. The dynamc nature of DDDAS problems requres us to address the tme dependency or real

2 tme nature of the applcatons. Certan applcatons (e.g., tornado formaton) requre real tme response to observatons from data. Typcally, n the predcton of severe weather potental, data from observatons taken hours prevous to the formaton are used and these are not updated wth real data as they become avalable. Incorporatng new dynamcally njected data s a fundamental change n the desgn. The use of the Weather Survellance Radar 1998 Doppler (WSR-88D) data, wth contnuous data streamng n every sx mnutes, presents a source for data drven smulatons. One of the severe weather detecton algorthms, created by the Natonal Severe Storms Laboratory (NSSL) and n use at the WSR-88D, s the Mesocyclone Detecton Algorthm (MDA) [4]. Ths dynamc algorthm uses the data stream outputs of the WSR-88D and s desgned to detect storm crculatons assocated wth regons of rotaton n thunderstorms. The MDA s used by meteorologsts as one nput n ther decson to ssue tornado warnngs. Recent work by Trafals et al. [4, 5] has shown that SVMs appled to the MDA offer a promsng role n mproved tornado classfcaton. We present a novel approach by ncorporatng rules nto SVMs of the MDA attrbutes as they stream n just pror to tornado formaton. These rule based sets classfy the data nto one of three categores, tornado, non-tornado and unclassfed. Thus, the rules partton the nput space nto regons for whch we know, wth a hgh degree of certanty, the label of ponts located n those regons. Our approach s dfferent from [3] n the sense that the rules are combned wth SVM n a sequental approach. Ths paper s organzed as follows. In secton 2, the data descrpton s gven. In secton 3, we provde a descrpton of rule-based SVM classfers. Secton 4 descrbes the expermentaton procedure. Secton 5 provdes computatonal results and, n secton 6, analyss and conclusons are provded. 2. Data The MDA data set used for ths research s based on the outputs from the WSR-88D radar that s collected just pror to the formaton of a pre-tornadc crculaton. Any crculaton detected on a partcular volume scan of the radar can be assocated wth a report of a tornado. In the severe weather database, suppled by NSSL, there s a label for tornado ground truth that s based on temporal and spatal proxmty. If there s a tornado reported between the begnnng and endng of the volume scan, and the report s wthn a reasonable dstance of a crculaton detecton, then the ground truth value s flagged. If a crculaton detecton falls wthn the predcton "tme wndow" of -20 to +6 mnutes of the ground truth report duraton, then the ground truth value s also flagged. The key dea behnd these tmngs s to determne whether a crculaton wll produce a tornado wthn the next 20 mnutes, a sutable lead tme for advanced severe weather warnngs by the Natonal Weather Servce. Owng to the autocorrelaton n the MDA attrbutes, a samplng strategy s used to mnmze seral correlaton. These sampled data are dvded nto ndependent tranng and testng sets, wth 749 and 618 observatons, respectvely.

3 3 Rule-Based Support Vector Machne Classfers 3.1 Rule Generaton In ths work, we consder a rule-based approach of a decson tree type as shown n Fg. 1. des n the decson tree nvolve testng a partcular attrbute. The test at a node compares an attrbute value wth a constant threshold. Leaf nodes gve a classfcaton that apples to all nstances that reach the leaf. When a leaf s reached, the nstance s classfed accordng to the class assgned to the leaf. te that the output of the last node referrng to the unclassfed category becomes an nput to the SVM that provdes the fnal label to the unclassfed cases. There were 23 MDA attrbutes avalable for dscrmnatng tornadoes from nontornadoes [4]. For each attrbute, we consdered the correspondng probablty dstrbuton functon for tornado and non-tornado cases arsng from the tranng data. The selecton of the threshold for each rule was based on elmnatng msclassfcaton by nvestgatng f the mnmum for a non-tornado case had a value less than the mnmum for a tornado case for a specfc attrbute. If such a condton holds, then a regon unque to non-tornadoes s found. X1<90? X3<0? X2<617? X2>12219? T NT NT X3>13? :. T X23>28? unclassfed T SVM Fg. 1. Tree dagram of rule generaton and decsons. Ovals represent nodes, squares represents leaves

4 Smlarly, f the maxmum for a non-tornado case had a value less than the maxmum for a tornado case, for a specfc attrbute, a regon unque to tornado cases s found. Of the 23 attrbutes, only 20 were found to be useful for rule generaton. The thresholds used for tornado and non-tornado dscrmnaton are shown n Table 1. Table 1. Threshold values for each MDA attrbute. See [4] for descrpton of attrbutes n-tornado thresholds f x1 < 90, then non-tornado f x2 < 617, then non-tornado f x3 < 0, then non-tornado f x4 < 813, then non-tornado f x5 < 1091, then non-tornado f x6 < 124, then non-tornado f x7 < 6, then non-tornado f x8 < 10, then non-tornado f x9 < 122, then non-tornado f x10 <2, then non-tornado f x12 < 106, the non-tornado f x13 < 3, then non-tornado f x14 < 11, then non-tornado f x15 < 122, then non-tornado f x16 < 106, then non-tornado f x17 < 617, then non-tornado Tornado thresholds f x2 > 12219, then tornado f x3 > 13, then tornado f x10 >77, then tornado f x11 >83, then tornado f x18 >113, then tornado f x22 > 26, then tornado f x23 > 28, then tornado 3.2 Support Vector Machnes (SVMs) n Gven a set of data ponts { }, the SVM fnds a ( x, y ), 1,..., l wth x R and y = ± 1 = classfer that separates the two classes of ponts wth maxmum margn separaton (Fg. 2). The SVM formulaton can be wrtten as follows [6], mn w, b, η C l = 1 η w 2 (1) st y (wx + b) + η 1 0 = 1,... l η

5 where C s a parameter to be chosen by the user that controls msclassfcatons, w s referrng to the vector perpendcular to the separatng hyperplane, η refers to the msclassfcaton error varables and b s the bas of the separatng hyperplane. A larger C corresponds to assgnng a larger penalty to errors. Introducng postve Lagrange multplers α, to the nequalty constrants n model (1) we obtan the followng dual formulaton: mn α 1 2 l l y y j α α j x x = 1j = 1 l j l α = 1 st α = 0, (2) = 1 y 0 α C = 1,...l The soluton of the prmal problem s then gven by w = Σ α I y x where w s the vector that s perpendcular to the separatng hyperplane. The free coeffcent b can be found from the relaton α (y (w x + b) - 1) = 0, for any such that α s not zero. The use of a kernel functon allows the SVM to operate effcently n nonlnear hgh-dmensonal feature space [7]. x 2 wx + b = 1 wx + b = 0 wx + b = 1 Data pont Margn = 2 w x 1 Fg 2. The geometrc llustraton of SVM

6 4 Experments In our expermentaton, the data are splt nto tranng and testng sets. The testng set s sampled ndependently fve tmes. The frst set of experments s performed by usng SVM only on the fve testng samples. The total msclassfcaton error s computed as the average of the msclassfcaton error of each sample. The second set of experments s performed by extractng the rules from the tranng data and applyng those rules n the testng phase. Based on the rules, each testng sample s dvded nto three dfferent sets: non-tornado, unclassfed, and tornado. In the testng phase, those observatons not classfed by the rules are used as nputs to SVM. The SVM s traned on the tranng set then tested on fve dfferent unclassfed samples. For each testng set, the msclassfcaton error for the non-tornado rules set, SVM and tornado rules set are computed. The OSU SVM Classfer Matlab Toolbox [8] was used to run experments of SVM. 5 Computatonal Results The results of the experments are presented n Fg. 3 and Table 2. The values n the table are msclassfcaton error rates for non-tornado and tornado and SVM components of the total hybrd system. After ntal expermentaton, t was noted that the rules components of the system had a lower error rate than the SVM component of the system. Accordngly, alterng the rules to admt addtonal cases was consdered by creatng a multpler for the threshold values n Table 1. Ths multpler controls the level of threshold values (e.g., n Table 1, for attrbute 1, the orgnal threshold, 90, corresponds to multpler 1 and 1.05 tmes 90 equals 94.5 and ths value admts addtonal observatons nto the non-tornado category). Table 2 and Fg. 3 llustrate the senstvty of msclassfcaton error wth respect to the threshold. Table 3 shows the msclassfcaton error for SVM for each testng sample and the average of the fve samples. 6 Analyss and Conclusons Tables 2 and 3, show that the best msclassfcaton error for the hybrd model (0.1449) s 12.7% lower than the one for the model based solely on SVM (0.1706). The reason for the total system mprovement can be seen n Fgure 3a,c and Table 2, where the nontornado rules, based on the threshold gven n Table 1, have a mean error rate of Smlarly, the tornado rules have a mean error rate of at the same multpler. In contrast, the SVM component has an error rate of The behavor of the rules, as seen n Fg. 3 a, c s nterestng as the msclassfcaton rate s remarkably low (approxmately 5 percent) for threshold multplers of 0.90 to The trade-off s that

7 fewer observatons are classfed as tornadoes or non-tornadoes. As the threshold multplers ncrease to 1.05 and beyond, the msclassfcaton error ncreases consderably to approxmately 15 percent ndcatng a poorer dscrmnaton between tornadoes, and non-tornadoes. In contrast, the SVM, based on unclassfed data (Fg. 3c), s nsenstve to the threshold multpler. Msclassfcaton rate Msclassfcaton rate Threshold multpler (a) Threshold multpler (c) Msclassfcaton rate Msclassfcaton rate svm Threshold multpler (c) Threshold multpler (d) Fg. 3. Boxplots of msclassfcaton error due to (a) non-tornado rules set, (b) SVM, (c) tornado rules set and (d) total hybrd system. Threshold multplers are shown on X-axs and the numbers of cases classfed are shown above the boxplots n (a), (b) and (c)

8 Table 2. Msclassfcaton error of the hybrd system components and total system Multpler n-tornado rules Tornado rules SVM Total system Table 3. Msclassfcaton error for SVM Sample 1 Sample 2 Sample 3 Sample 4 Sample 5 Average SVM Therefore, gven the lower rates n the non-tornado and tornado rules, t s logcal to create a hybrd system to captalze on the dsparty n error rate. By admttng addtonal observatons nto the leaves of the decson tree pror to sendng the remanng observatons to the SVM, the optmal system s found. Ths occurs at a multpler of 1.05 tmes the threshold values n Table 1. Experments are planned for njecton of nformaton from successve volume scans to assess addtonal predctve capablty n a constantly updatng form of DDDAS. Acknowledgements. Ths work has been supported by NSF grant EIA References 1. Mtchell, T.M., Machne Learnng, McGraw-Hll, New York, Fung, G.M., Mangasaran, O.L., Shavlk, J.W.: Knowledge-based Support Vector Machnes Classfers, Data Mnng Insttute. Techncal Report Computer Scences Department, Unversty of Wsconsn (2001) 3. Fung, G.M., Mangasaran, O.L., Shavlk, J.W.: Knowledge-based nlnear Kernel Classfers. Data Mnng Insttute Techncal Report Computer Scences Department, Unversty of Wsconsn (2003) 4. Trafals, T.B., Santosa B., Rchman, M.B.: Tornado Detecton wth Kernel-based Methods. In: Dagl, C.H., Buczak, A.L., Ghosh, J., Embrechts, M., Ersoy, O. (eds.): Intellgent Engneerng Systems Through Artfcal Neural Networks. ASME Press, Vol. 13 (2003) Trafals, T.B., Santosa B., Rchman, M.B.: Tornado Detecton wth Kernel-Based Classfers From WSR-D88 Radar. Submtted to: Darema, F. (ed.) Dynamc Data Drven Applcaton Systems, Kluwer (2004) 6. Haykn, S.: Neural Networks: A Comprehensve foundaton, 2 nd edton, Prentce-Hall, Upper Saddle Rver New Jersey (1999) 7. Schölkopf, B., Smola, A.: Learnng wth Kernels. MIT Press, Cambrdge Massachusetts (2002) 8. Junshu, M., Zhao, Y. Ahalt, S.: OSU SVM Classfer Matlab Toolbox. Avalable at