Comparative Analysis of Single and Mixed Spatial Interpolation Methods for Variability Prediction of Temperature Prediction

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1 Vol. 6, No. 1, Jauary, 2013 Comparatve Aalyss of Sgle ad Mxed Spatal Iterpolato Methods for Varablty Predcto of Temperature Predcto Yuhua Gu, Lgme Su ad J Wag Jagsu Egeerg Ceter of Network Motorg, School of Computer & Software, Najg Uversty of Iformato Scece & Techology, Najg, , Cha, yhgu6655@163.com, slm.happy@163.com, wagj@ust.edu.c Abstract Ths paper aalyzes the valdty of temperature maps obtaed by meas of sgle ad mxed terpolato methods. I ths cotext, several terpolato methods are used for the temperature mappg: verse dstace square (IDS), ordary krgg (OK) ad co-krgg (CK) ad mxed methods (combed global, local ad geostatstcal methods). Ad the valdty of the maps s checked through the cross-valdato error statstcs preseted terms of Mea absolute error (MAE) ad root-mea-square error (RMSIE). The results show that the spatal temperature dstrbuto formed by IDS, OK ad CK show smlarty geeral whle the CK method better reflects the temperature dstrbuto. Moreover, the best result for temperature mappg s obtaed usg the mxed method, whch suggests that the correcto of regresso models usg resduals mproves the terpolato accuracy. Relable calculato for agrcultural maagemet ad mprovemet of clmatc models at local scales ca be obtaed wth creased effcecy. Keywords: spatal terpolato, sgle terpolato method, mxed terpolato method, ArcGIS, exploratory data aalyss 1. Itroducto Spatal terpolato s a essetal tool processg the data of atural ad socal scece. It has bee wdely used especally the dscple of hydrologcal, meteorologcal clmate, ecology, evromet, geology etc. [1, 2]. The essece of spatal terpolato s to estmate the values of uobserved pots based o kow sample data. Due to o-avalablty of abudat measuremet pots, relable estmato of temperature dstrbuto poses a great challege. At preset, may domestc ad foreg scholars ad sttutos study the spatal terpolato of meteorologcal elemets. The spatal terpolato predcto techques (lke sple, verse dstace weghtg ad krgg) provde better estmato of temperature tha covetoal methods [3]. There s o sgle preferred method for data terpolato whch ca meet wth the selecto crtera of requred level of accuracy, the tme ad/or computer resources etc. The commo approach to select the optmal spatal terpolato method has become the focus. Domestc studes ted to compare the dfferet effects of sgle terpolato methods, few studes are volved the comparatve aalyss of sgle ad mxed spatal terpolato methods [4]. The ma purpose of ths study s to provde the bass for the selecto of a approprate terpolato method ad map the spatal varablty. Take the spatal terpolato of Hea provce as a example, ths paper maly focuses o the followg specfc objectves: (1) to make the preprocessg data aalyss; (2) to utlze the sgle terpolato methods (verse dstace square, ordary krgg, co-krgg); (3) to coduct the multple regresso aalyss ad utlze the mxed method; (4) to determe the valdty of terpolated 67

2 Vol. 6, No. 1, Jauary, 2013 temperature maps through statstcal crtera ad subjectve commets. Accordg to crossvaldato error statstcs, the spatal temperature dstrbuto formed by the three terpolato methods show smlarty geeral whle the CK method better reflects the temperature dstrbuto. Mxed terpolato method shows better accuracy measuremet tha other three methods, the correcto of regresso models usg resduals mproves the terpolato accuracy. The result dcates that the proposed method ad procedures are hghly effectve to predct temperature. 2. Exploratory Data Aalyss The data we get ofte cotas lots of ose or cosstet data due to a varety of errors from huma or system. To meet actual terpolato eeds, data preprocessg s always dspesable. I ths paper, the orgal data of 32 clmatc statos do t follow ormal dstrbuto accordg to the Kolmogorov Smrov test. To meet eeds of ormal dstrbuto, logarthmc trasformato ad Box-Cox trasformato are carred out for the log-term mea mothly temperature. Ad the skewess, kurtoss ad P after logarthmc trasformato best meet the ormal dstrbuto. The statstcal fttg parameters ad the varogram model fttg are accomplshed through use of ArcGIS ad GS+ software. The aalyss ths paper dcates that all ugget/sll rato values of three semvarograms are the rage of 25%-75% [5, 6]. They all have moderate ugget/sll rato. Fgure 1 shows the sphercal, expoetal ad Gaussa model fttg curve for aul temperature. Fgure 1. Semvarogram Models for Temperature: (a) Sephercal Model; (b) Expoetal Model; (c) Gaussa Model The small rectagle the fgure dcates the separato dstace for the average dstace lmted by the step value of all pots. The vertcal axs dcates sem-varace of the pots. Smaller the overall devato from the fttg curve, better the curve fts. I ths study, the Gaussa model ft better tha the other two models. 3. Sgle Iterpolato Methods The sgle determstc fucto methods (global terpolators, local terpolators ad geostatstcal methods) are called as sgle terpolato methods ths artcle. The sgle terpolato methods assessed ths study clude verse dstace square (IDS), ordary krgg (OK) ad co-krgg (CK) Iverse Dstace Square (IDS) The IDS method s based o the frst law of geography, whch dcates earby values cotrbute more to the terpolated values tha dstat observatos. Whe the weght value s gve by the verse of the square of the dstace, we call t the verse dstace square. Ths 68

3 Vol. 6, No. 1, Jauary, 2013 supposes that predctos are obtaed from the earest samplg pots. The formula s show as (1). Z Z 2 1 d 1 Where Z refers to the predcted value; Z s the observed value of a eghborg weather stato ; d s the dstace betwee Z ad Z ; ad refers to the umber of sample pots. The terpolato accuracy of the method usually depeds o the o-uformty of the sample pots. IDS s sestve to outlers, whle uevely dstrbuted data clusters results troduced errors Ordary Krgg (OK) Krgg s a statstcally-based terpolato method. Based o spatal autocorrelato, t uses the structural propertes of the raw data ad the varogram. It s a terpolato method that gets the partal valuato of regoalzed varables of ukow sample pots [5]. OK assumes that samplg pots do ot have the potetally global treds, ad that t ca well estmate the ukow values by usg oly local factors. The clmatc value at pot Z(x 0 ) s expressed by formula (2). Z 1 d 2 x z x (1) 0 (2) Where λ refers to the weght of the regoalzed varable z(x ), whch s used to represet the cotrbuto of the observato z(x ). z(x ) s the estmated value of z(x 0 ). The weght coeffcet λ s obtaed by formula (3) x, x x, x j Where γ (x, xj) refers to the sem-varace of Z betwee the sample pots x ad x j ; γ (x, x 0 ) refers to the sem-varace of Z betwee the sample pot x ad estmates pot x 0 ; φ s the Lagrage multpler the mmal treatmet. These varables are obtaed through the varogram ad ts formula s show as (4). 0 j (3) 1 2 ( h) Z ( x h) (4) 2 1 As formula (4), h refers to the step; γ (h) refers to sem-varace of Z whose step s h; refers to the umber of expermetal data splt by h. The most commoly used varogram model OK are rg model, expoetal model, sphercal model, gaussa model, etc [6] Co-Krgg (CK) There are dfferet types of krgg, ad several papers descrbe them detal [7] (Goovaerts 1997). Krgg method to estmate the spatal varables wth the spatal correlato ca be classfed as CK method. Wth ths method, you ca make spatal estmato of oe varable or multple varables usg the correlato betwee several space varables, so that 69

4 Vol. 6, No. 1, Jauary, 2013 the accuracy ad reasoableess of the estmato ca be mproved. The key to use CK s calculatg the covarato fucto betwee varables. If v 1 (x), v 2 (x)... v m (x) deote the values of the varables v 1, v 2,..., v m the posto x. If x 1, x 2,..., x s the posto of the sample pots ad the measured values are Vx 1, Vx 2... Vx, the the Co-krgg method ca be expressed as formula (5). x V V x (5) 1 As formula (5), Γ refers the weght vector gve by the covarato fucto, whch s calculated as formula (6). h E v x h v x v x h v x Y (6) CK depeds o spatal ad statstcal relatoshps to calculate the surface. There are some advatages of ths method, such as the corporato of varable terdepedece, the avalable error surface output, etc. A dsadvatage s that t requres substatally more computg ad modelg tme [8]. 4. Mxed Methods 4.1. Overvew of Mxed Methods Global methods are exact terpolators, sce there s a kow error betwee the fal predcted data ad the orgal observed data (resdual). Mxed terpolato method uses a correcto method (terpolato of resduals) to further mprove the terpolato accuracy. It s a effectve terpolato meas, whch combes the advatages of global terpolato ad local terpolato methods [9]. resdual = observed data - predcted data (7) The, terpolato of resduals must be coducted to obta correcto maps. May terpolato methods ca be used to terpolate the resduals of clmatc maps, such as sples, verse dstace weghtg, krgg etc. The sum of resdual terpolato ad predcted regresso maps modfes the tal results. The real values ca be obtaed as: observed data = predcted data + resdual terpolato (8) 4.2. Uversal Processes Mxed terpolato method s a combato of multple regresso ad geostatstcal methods. To establsh the regresso equato betwee the depedet varables ad depedet varables, the depedet varables must be cofrmed advace. After a thoroughly aalyss of correlato betwee temperature ad geographc features (lattude ad logtude alttude ad terra, etc), features wth strog correlato are take as depedet varables. The multple regresso equato ca be obtaed as formula (9). T 3 a0 a1 X a2y a Z (9) Where T refers to temperature, the depedet varables X, Y, Z are the geographc features such as lattude, logtude, elevato, slope, etc. Ad a0 ~ a3 are the costat ad coeffcet of the depedet varables. The the resduals betwee observed data ad the 70

5 Vol. 6, No. 1, Jauary, 2013 predcted data ca be calculated, ad the terpolato for resduals ca be carred o. A dversty of methods ca be used for resdual terpolato, clude sples, verse dstace weghtg, seres of krgg etc. After terpolato for resduals, the corrected temperature map ca be obtaed by addg resdual terpolato results ad the predcted regresso maps A Istace The mxed terpolato method ths paper s a combato of multple regresso ad krgg terpolato method. It s based o the assumpto that the resdual retas the heret spatal structure of the orgal data. Sce the collearty statstc has a great mpact o the result of stepwse regresso aalyss, collearty aalyss s essetal. Ad the tolerace ad varace flato factor are the ma parameters of collearty aalyss. If the tolerace s less tha 0.2, t meas the exstece of collearty amog factors. Ad the tolerace less tha 0.1 dcate a obvous collearty. The logtude, lattude ad elevato are take as varables of stepwse regresso aalyss. Table 1 shows that 3 tolerace values are larger tha 0.2, so the lattude, logtude, ad elevato ca be take as depedet varables of stepwse regresso. Table 1. Multcollearty Dagoss Idexes for Varables Idepedet Varable Tolera ce varace factor logtude lattude elevato flato The stepwse multple regresso aalyss s performed wth temperature as depedet varables ad geographc varables as predctors. So, the predcted temperature (T) ca be created by the regresso equato. It s show as formula (10). T X 0.378Y Z (10) Where X refers to the logtude, Y s the lattude ad Z represets the elevato. To obta correcto maps geerated by resduals terpolato, CK s used to terpolate the resduals of the temperature map created by meas of the regresso-based method. The sum of resdual terpolato ad predcted regresso maps comes to the fal temperature map. 5. Iterpolato Effect Assessmet Spatal terpolato s wdely used for creatg cotuous data from data collected at dscrete locatos. Whe a terpolated surface s used as part of larger research project [8] both the method ad accuracy of the terpolato techque are mportat. To create a surface best represet the emprcal realty, the method selected must be assessed for accuracy. To test the accuracy of the terpolato results of meteorologcal statos the clmate dcator, actual valdato ad cross-valdato are commoly used [10].For the purpose of maxmze the use of observatos, cross-valdato s chose to comparatvely aalyze the terpolato results. Temperature maps must be assessed by statstcs that dcate the degree of cocordace betwee models ad realty. Mea absolute error (MAE) ad root-measquare error (RMSIE) are amog the best overall measures of model performace, as they 71

6 Vol. 6, No. 1, Jauary, 2013 summarze the mea dfferece the uts of observed ad predcted values. The MAE provdes a measure of bas; the RMSIE provdes a measure of accuracy [11]. Measuremets of MAE ad RMSIE are show as formula (11), (12), Where T o s the measured value of Z at locato, T e s the predcted value at the same locato. 1 MAE 1 ABS ( T o T e ) (11) RMSIE 1 T o T e 1 2 (12) Nevertheless, ot oly statstcal crtera are used to determe the valdty of terpolated temperature maps. Subjectve commets, based o the kowledge of spatal dstrbuto of temperature, are also added. 6. Expermet ad Verfcato As a essetal meteorologcal elemet of bologcal survval, temperature has a very mportat mpact o Earth. Despte abudat temperature data have bee probed through log-term observato, but these data oly represet the meteorologcal statos wth the lmted cofes of the locato of the temperature codtos durg the regoal / global ecologcal system Data The locato of the study area s dcated Fgure 2. The total area of the study area s 167,000 km2 wth oly thrty-two meteorologcal statos to measure temperature all over the Hea provce. It s located betwee lattude ~36.22, east logtude ~ Fgure 2. Mothly Mea Temperature Dstrbuto of Hea Provce The clmate of Hea provce s warm temperate-subtropcal mosoo wth mea aual temperatures ragg betwee 12 C ad 16 C. Based o avalable hstorc records, a umber of studes have bee carred out to support crop producto systems plag, meteorologcal dsaster warg ad so o. I ths cotext, t s essetal to select a approprate terpolato method to map the spatal varablty for the whole study area. 72

7 Vol. 6, No. 1, Jauary, Statstcal Results The results of the accuracy measuremets the dfferet temperature models are show Table 2. It dcates that the mothly average MAE values of terpolato IDS, OK, CK ad mxed method are 0.41, 0.43, 0.41 ad Respectvely, the average RMSIE values are 0.61, 0.63, 0.58 ad OK ca be cosdered the worst model statstcal terms, as t gves the hgher values of MAE ad RMSIE. Table 2. Accuracy Measuremets for Temperature Models: IDS, OK, CK ad Mxed Methods are Expressed Moth IDS OK CK Mxed MAE RMSIE MAE RMSIE MAE RMSIE MAE RMSIE Aual The, we try to evaluate the performace of IDS, OK, CK ad the mxed methods by comparg the statstcal errors geerated by cross-valdato. The MAE ad RMSIE of each terpolato method are respectvely show Fgure 3(a) ad Fgure 3(b). (a) Fgure 3. MAE ad RMSIE of IDS, OK, CK ad Mxed Methods: (a) MAE; (b) RMSIE It ca be foud that the MAE ad RMSIE of IDS, OK ad CK terpolato methods show smlarty geeral, wth good geeral results all cases. Nevertheless, CK produces lower error values tha the other two sgle terpolato methods. I addto, the mxed methods have lowest MAE ad RMSIE. It s proved to be a effectve approach for the detfcato of best spatal terpolato techques for further aalyss. (b) 73

Vol. 6, No. 1, Jauary, 2013 6.3. Temperature Maps Nevertheless, ot oly statstcal crtera are used to determe the valdty of the terpolated clmatc maps.

8 Vol. 6, No. 1, Jauary, Temperature Maps Nevertheless, ot oly statstcal crtera are used to determe the valdty of the terpolated clmatc maps. Drect emprcal clmatc kowledge ca help determe the models that reflect realty best, as log as ther statstcal values are reasoable. Therefore, subjectve commets, based o the kowledge of spatal dstrbuto of temperature Hea provce, are also added ths paper. The aual mea temperature maps are show Fgure 4. Fgure 4. Results of Iterpolato Methods for Aual Mea Temperature ( ): (a) IDS Iterpolato; (b) OK Iterpolato; (c) CK Iterpolato; (d) Mxed Iterpolato It ca be foud that orth-south gradet s detected by the four terpolato methods, whle the geeral spatal patters have substatal dffereces specfc areas. Large spatal dffereces betwee the dverse terpolated maps are observed. The sgle terpolato methods show smlar results, although CK produces the better map tha IDS ad OK. The model obtaed by IDS method show has cocetrc crcles. Mxed terpolato method shows better accuracy measuremet tha other three methods, the correcto of regresso models usg resduals mproves the terpolato accuracy. 7. Dscusso Most of the terpolato models just take the factor of temperature to accout. Ad the terpolato results wthout cosderg the mpact of topography ad other factors do have hgher errors predcto [12]. How to take the factors of elevato, slope ad so o to terpolato models has bee a major problem of spatal terpolato. I ths paper, the use of dfferet topographc ad geographc factors that determe the spatal dstrbuto of clmatc varables makes t possble to mprove the terpolato accuracy. Moreover, mxed method s used to map the clmatc varables, whch geerates a adequate temperature map. At preset, researches from home ad abroad show that data of oe stato always represet for data of earby rego. Less dstrbuto of statos meas larger area the observed value of oe sgle stato wll represet for. Usually, characterstcs of a sgle meteorologcal stato ca t accurately represet the regoal clmate, especally the rego wth complex clmate or large regoal dfferece. Sce there s o approprate method to 74

9 Vol. 6, No. 1, Jauary, 2013 drectly terpolate for temperature of a large scale rego, the temperature formato of the large rego ca oly be obtaed from the aalyss of the very lmted space. Dowscalg terpolato s to reduce a large rego to several small regos, meas a large rego s dvded to a few small regos. To mprove the terpolato accuracy, the the operato of terpolato for each small rego separately s made o the bass of rego dvso. The aual temperature maps developed by dowscalg ca be very useful for hydrologcal ad agrcultural maagemet the rego. These maps must be the frst step the creato of a more complete spatal clmatc database, whch cludes the dversty offered by seasoal or mothly temporal scales. The latter, whch are very useful for plag, expad our uderstadg of spatal clmatc complexty ad ts varato over tme the study area. 8. Coclusos Creato of dgtal grd maps makes t possble to obta clmatc formato at ay pot, whether there s a weather stato or ot. Multple factors codto the dffculty of map creato, such as the locato of the ste samples, spatal desty, spatal varablty etc. The use of dfferet topographc ad geographc factors that determe the spatal dstrbuto of clmatc varables makes t possble to map more local clmatc features. Sce the same aalyss ca gve the dfferet results dfferet areas, pror tests eed to be establshed to determe the most adequate method. Before geographc spatal terpolato, may procedures must be take to accout such as data preprocessg, aalyss of spatal varablty ad correlato, the varato of key parameters ad theoretcal model. Based the characterstcs ad er laws, the parameter settgs of terpolato methods ad the choce of varogram model are dscussed ths study. Ad the mxed terpolato method s very useful the mappg of clmatc varables, because t adapts to almost ay space ad usually geerates adequate maps. The proposed method ad procedures ths paper ca ot oly reduce the workload of the spatal terpolato model comparg ad screeg, but also mprove the accuracy of spatal terpolato. It s proved to be a effectve approach for the detfcato of best spatal terpolato techques for further aalyss. The ma cocluso to be draw from ths paper s that, although the topography of Hea provce s ot very complex, spatal clmate dversty s sgfcat. The aual temperature map developed ca be useful for agrcultural maagemet the rego. Relable calculato for agrcultural maagemet ad mprovemet of clmatc models at local scales ca be obtaed wth creased effcecy. Ackowledgemets The work has bee supported by the publc servce sectors (WMO) research project, project NO.GYHY Ths work was also supported by the Natural Scece Foudato of Jagsu Provce (No. BK ). Refereces [1] X. L, Y. Y. Xao, W. C. Lu ad J. Y. Tag, Meteorologcal Scece ad Techology, vol. 1, (2002), pp. 41. [2] D. Brkes ad Y. Dodge, Alteratve Methods of Regresso, Wley Ole Lbrary, Hoboke, (2009). [3] S. B. Lao ad Z. H. L, Meteorologcal Scece ad Techology, vol. 5, (2004), pp [4] A. Sarag, C. A. Cox ad C. A. Madramootoo, Geostatstcal methods for predcto of spatal varablty of rafall a moutaous rego, Trasactos of the ASAE, vol. 48, o. 3, (2005), pp [5] C. A. Cambardella, T. B. Moorma, J. M. Novak, T. B. Park, D. L. Karle, R. F. Turco ad A. E. Koopka, Feld-Scale Varablty of Sol Propertes Cetral IOWA Sols, Sol Scece Socety of Amerca, vol. 58, (1994), pp

Vol. 6, No. 1, Jauary, 2013 [6] H. V. Jase, N. R. Tas ad J. W. Bereschot, Ecyclopeda of Naoscece ad Naotechology, Edted H. S. Nalwa, Amerca Scetfc Publshers, Los Ageles, vol. 5, (2004), pp. 163-275.

Cabral, Statstcal Evaluato of Spatal Iterpolato Methods for Small-Sampled Rego: A Case Study of Temperature Chage Pheomeo Bagladesh, B. Murgate, O. Gervas, A. Iglesas, D. Taar & B. Apduha (Eds.

Wllmott, Some commets o the evaluato of model performace, Bullet of the Amerca Meteorologcal Socety, vol. 63, (1982), pp. 1309 1313. [10] H. Ya, Geography ad Geo-Iformato Scece, vol. 5, (2003), pp.

10 Vol. 6, No. 1, Jauary, 2013 [6] H. V. Jase, N. R. Tas ad J. W. Bereschot, Ecyclopeda of Naoscece ad Naotechology, Edted H. S. Nalwa, Amerca Scetfc Publshers, Los Ageles, vol. 5, (2004), pp [7] P. Goovaerts, Geostatcs for Natural Resources Evaluato, Oxford Uversty Press Publshers, New York (1997). [8] A. Bhowmk ad P. Cabral, Statstcal Evaluato of Spatal Iterpolato Methods for Small-Sampled Rego: A Case Study of Temperature Chage Pheomeo Bagladesh, B. Murgate, O. Gervas, A. Iglesas, D. Taar & B. Apduha (Eds.), Computatoal Scece ad Its Applcatos - ICCSA 2011, Sprger Berl Publshers, Hedelberg, vol. 6782, (2011), pp [9] C. J. Wllmott, Some commets o the evaluato of model performace, Bullet of the Amerca Meteorologcal Socety, vol. 63, (1982), pp [10] H. Ya, Geography ad Geo-Iformato Scece, vol. 5, (2003), pp. 27. [11] H.D. Zeg, Geo-formato Scece, vol. 7, (2005), pp. 82. [12] D. T. Prce, D. W. Mckeey, L. A. Nalder, M. F. Hutchso ad J. L. Kesteve, A Comparso of Two Statstcal Methods for Spatal Iterpolato of Caada Mothly Clmate Data, Agrcultural ad Forest Meteorology, vol. 101, (2000), pp Authors Yuhua Gu Professor Yuhua Gu was bor She receved the B.E. degree the Computer Scece ad Techology from Shagha Jaotog Uversty 1987, ad M.E. degree the Computer Scece ad Techology from Southeast Uversty Now, she s a professor the Computer ad Software Isttute, Najg Uversty of Iformato Scece ad techology. Her ma research terests are computer etwork, database, GIS systems. She s a member of ACM ad a seor member of CCF. Lgme Su Lgme Su was bor She receved the B.E. degree the Computer Scece ad Techology from Najg Uversty of Iformato Scece & Techology (NUIST). Now, she s a master degree caddate of NUIST,major computer applcato techology. Her ma research areas are database, GIS systems. J Wag Dr. J Wag receved the B.S. ad M.S. degree the Electrocal Egeerg from Najg Uversty of Posts & Telecomm., Cha 2002 ad 2005, respectvely. He receved Ph.D. degree Ubqutous Computg laboratory from the Computer Egeerg Departmet of Kyug Hee Uversty Korea Now, he s a professor the Computer ad Software Isttute, Najg Uversty of Iformato Scece ad techology. Hs research terests maly clude routg protocol ad algorthm desg, performace evaluato ad optmzato for wreless ad hoc ad sesor etworks. He s a member of the IEEE ad ACM. 76

Simple Linear Regression

Simple Linear Regression Statstcal Methods I (EST 75) Page 139 Smple Lear Regresso Smple regresso applcatos are used to ft a model descrbg a lear relatoshp betwee two varables. The aspects of least squares regresso ad correlato