INCREASING ACCURACY IN ANALYSIS NDVI-PRECIPITATION RELATIONSHIP THROUGH SCALING DOWN FROM REGIONAL TO LOCAL MODEL

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1 INCREASING ACCURACY IN ANALYSIS NDVI-PRECIPITATION RELATIONSHIP THROUGH SCALING DOWN FROM REGIONAL TO LOCAL MODEL P. A. Propastn, M. Kappas Georg-August-Unversty, Department of Geography, Goldschmdtstr., 3, 37073; Goettngen, Germany, E-Mal KEY WORDS: NDVI-Ranfall Relatonshp, GWR Modellng, Uncertanty reducng, Kazakhstan ABSTRACT Spatal relatonshp between vegetaton and ranfall n Central Kazakhstan has been modelled usng Normalzed Dfference Vegetaton Index (NDVI) and ranfall data from weather statons. The modellng based on applcaton of two statstcal approaches: conventonal ordnary least squares (OLS) regresson, and geographcally weghted regresson (GWR). The results support the assumpton that the average mpresson provded by the OLS model may not accurately represent condtons locally. The OLS model appled to the whole study area was strong (R² = 0.63), however t gave no local descrpton of the relatonshp. Applcaton of the OLS at the scale of ndvdual land cover classes resulted n a better predcton power of the model (R² = 0.75) and revealed that the response of vegetaton to ranfall vares between the land cover classes. The GWR approach, dealng wth spatal non-statonarty, sgnfcantly ncreases the model s accuracy and predcton power (R² = 0.97), as well as hghlghts local condtons wthn every land cover class. In order to compare the OLS and the GWR models n terms of predcton uncertanty, we calculated Moran s I for ther resduals. Our results demonstrated that the GWR provdes a better soluton to the problem of spatally autocorrelated errors n spatal modellng compared to the OLS modellng. 1. INTRODUCTION Clmate s the most mportant factor affectng vegetaton condton. Great research effort has been made to derve models that predct spatal varatons n vegetaton by clmates whch are based on utlzng remotely sensng obtaned vegetaton ndces and clmatc data. The Normalzed Dfference Vegetaton Index (NDVI) s the most used mult-spectral vegetaton ndex whch s hghly correlated to green-leaf densty and can be consdered as a proxy for above-ground bomass (Tucker & Sellers, 1986). Several prevous studes already modelled relatonshps between spatal or temporal patterns of NDVI and that of clmatc factors. Partcularly strong relatonshps n the ard regons show NDVI and ranfall (Rchard & Poccard, 1998; Wang et al, 2001), NDVI and temperature or growng-degree days (Yang et al, 1998; L et al, 2002) as well as NDVI and evapotranspraton (J & Peters, 2004). Modellng the spatal NDVI-clmate relatonshp one should take nto account that one hase to deal wth a phenomenon of non-statonarty of ths relatonshp n space. However, the conventonal statstcal regresson method (global ordnary least squares regresson, OLS) s statonary n a spatal sense. Statonarty means that a sngle model s ftted to all data and s appled equally over the whole geographc space of nterest. Ths regresson model and ts coeffcents are constant across space assumng the relatonshp to be also spatally constant. That s usually not adequate for spatally dfferenced data, especally by quantfyng relatonshps at regonal or global scales. The dfferences between regresson models establshed at dfferent locatons can be large wth both the magntude and sgn of the model parameters varyng. The easest method to mprove the regresson model and to reduce these dfferences s the fttng of an ndvdual OLS model for each land-cover or vegetaton type. On ths way, the varance n regresson parameters between land-cover types can be hghlghted and the predcton power of the regresson model ncreases sgnfcantly (Wang et al, 2001; J & Peters, 2004, L et al, 2004). However, ths method does not make up the local non-statonarty n the relatonshp wthn the landcover type. An nterestng and effcent alternatve s to allow the parameters of the model to vary wth space. Such nonstatonary modellng shows a greater predcton precson because the model beng ftted locally s more attuned to local crcumstances. Local regresson technques, such as localzed OLS (movng wndow regresson) or geographcally weghted regresson (GWR) help to overcome the problem of non-statonarty and calculate the regresson model parameters varyng n space. Ths technques provde a more approprate and accurate bass for modellng relatonshp between varous spatal varables and sgnfcantly reduce uncertanty n model predcton. For example, the local regresson technques have been effectvely used to quantfy spatal relatonshps between dfferent varables at the feld of human and economcal geography (Fotherngham et al, 1996; Pavlov, 2000: Fotherngham et al, 2002; McMllen, 1996), n sol scence and clmatology (Murray & Backer, 1991; Brunsdon et al, 2001). In the feld of remote sensng there are only rare studes applyng local regresson technques for the analyss of spatal relatonshps between remotely sensng data and clmatc varables (Foody, 2003; Foody, 2005; Wang et al, 2005). In the submtted paper, we analyse spatal relatonshps between NDVI obtaned from the satelltes NOAA AVHRR and ranfall amounts n the southern margn of the Kazakh low hlls. The am of the study was to derve a regresson model wth strong focus on the accuracy of model predcton at a local scale. In order to fnd a model wth the best predc-

2 ton power, we tested three dfferent regresson technques, two OLS regresson models, and a local model based on geographcally weghted regresson (GWR). We demonstrated that scalng down from regonal to local relatonshps sgnfcantly ncreases the accuracy and the predcton power of a regresson model. 2. STUDY AREA The study area s located n the mddle part of Kazakhstan between 46 and 50 northern lattude and 72 and 75 eastern longtude. The clmate of the regon s dry, cold and hgh contnental. Average annual precptaton s above mm per year n the north of the study area, and below 150 mm n the south. Most part of the precptaton falls durng warm perod from March to October. The temperature ampltude s relatve hgh: average January temperature s below 12 C and average July temperature s about C. The south of the study regon s vegetated by sagebrush and perennal saltwort assocatons. Domnatng vegetaton speces here are Artemsa terrae-albae, Artemsa paucflora, Anabass salsa, Salsola orentals. The northern secton of the study regon s occuped by steppe vegetaton, where domnate short grassland speces such as Festuca sulcata, Stpa capllata and Stpa lessngana. The sem-desert vegetaton complex occupyng the md of the study area represents a complex combnaton of real steppe turf grasses and semshrubs wth halophytes. Dstrbuton of land cover types n the study area s shown n Fgure NDVI dataset 3. DATA USED IN THE STUDY We used a 10-day 1-km NDVI data set from the global AVHRR archve for every growng season (Aprl-October) durng the years 1992, 1993, and The data set s generated usng a maxmum value composte (MVC) procedure, whch selects the maxmum NDVI value wthn 10-day perod for every pxel (Holben, 1986). Ths procedure s used to reduce nose sgnal n NDVI data due to clouds or other atmospherc factors. In addton to that, we removed nosy pxels remaned n the NDVI maps characterzed by exceptonally hgh or low NDVI values relatvely to ther pxel neghbourhood. The method of the dentfcaton of nosy pxels used a wndow wth a sze of 3*3 pxels, whch was movng over NDVI scenes and calculated a mean value of the surroundng pxels for every pont. After subtractng the orgnal pxel value from the mean value of surroundng pxels, dfferences of NDVI more than 0.12 were consdered as nose. Then, pxels dentfed as nosy were replaced by the surroundng mean. From the AVHRR NDVI data set we computed a 3-year mean NDVI for every 10-day perod begnnng wth Aprl through October. At last, a NDVI data set accumulated over growng season,, was produced by summng up the 10-day mean values derved. Several studes used the as a measure of the magntude of greenness avalable through the growng season tme whch reflects the capacty of the land to support photosynthess and net prmary producton for a growng season. A close relatonshp between and precptaton, especally n ard and sem-ard regons has well been establshed n the lterature (L et al, 2004; Budde et al, 2005). Fgure 1. Map of land cover types. 3.2 Precptaton dataset The precptaton data n the study consst of 10-day ranfall data collected and calculated by the Natonal Hydro- Meteorologcal Centre of Kazakhstan for 9 clmate statons placed n the study area for the perod Aprl-October Usng these data, we calculated a total average precptaton over the growng season for each clmate staton durng the perod The preparaton of a grdded precptaton map was made by nterpolatng data between the statons. The nterpolaton method was krgng wth an external drft. Use of a secondary varable, elevaton, for the preparaton of the grdded map was mportant because of a strong nfluence of relef on spatal pattern n precptaton n the study area (Fgure 2, left). 4.1 Regresson models 4. METHODS Relatonshps between NDVI and precptaton were derved by usng conventonal ordnary least squares (OLS) and geographcally weghted regresson (GWR) analyss. The frst one was ftted both to the whole study regon (global OLS) and to each land-cover type (stratfed OLS). The second one uses the locaton nformaton for each observaton and allows the model s parameters to vary n space. As OLS and GWR have been well documented, we just brefly descrbe the theoretcal background, especally for the GWR method. A full descrpton of geographc weghted regresson and ts treatments s provded by (Fotherngham et al, 2002). The smple lnear model, usually ftted by ordnary least squares methods (OLS), s: y = α + β * x + ε (1) Where a s the ntercept of the lne on the y axs (where x = 0), β represents the slope coeffcent for ndependent varable x, and ε s the devaton of the pont from the regresson lne.

3 Fttng the best-ft regresson model ncorporates the problem to fnd a and β so that the total error ε 2 s mnmzed. Fgure 2. Growng season precptaton n mm (left) and ulated over growng season (rght). In ths model, the two varables to be related are y, the dependent varable (for ths study - NDVI), and x, the ndependent varable (ranfall). The regresson model s parameters a and β derved by the above approach are assumed to be statonary globally over the analyss space (the whole study regon or the geographcal space occuped by a land-cover type). In other words, applyng the conventonal global regresson model to studyng relatonshps between vegetaton dstrbuton and ts condtons and envronmental parameters, one bases hs calculaton on the assumpton, that at each pont of the study area ths model s absolutely representatve and the quantfed relatonshp s constant. Geographcally weghted regresson s a local regresson technque that deals wth the problem of non-statonarty through local dsaggregatng global statstcs and calculates the relatonshp between NDVI and ts explanatory varables for every pont. In geographcally weghted regresson, the regresson and ts parameters n each pont (pxel) of the study regon s quantfed separately and ndependently from o- ther ponts. The regresson model s calbrated on all data that le wthn the regon descrbed around a regresson pont and the process s repeated for all regresson ponts. The resultng local parameter estmates can then be mapped at the locatons of the regresson ponts to vew possble non-statonarty n the relatonshp beng examned. The sze of the movng wndow (kernel) s less than the regon sze and can be vared from one pont to another. GWR focused on dervng local parameters to be estmated. The above OLS model can be rewrtten as: y = α ( Θ) + β ( Θ) * x + ε (2) where Θ ndcates that the parameters are to be estmated at a locaton for whch the spatal coordnates are provded by the vector Θ. GWR beng a local technque s to be dstngushed from other local regressons, as t works n the way that each data pont s weghted by ts dstance from the regresson pont. Ths means that a data pont closer to the regresson pont s weghted more heavly n the local regresson than are data ponts farther away. For a gven regresson pont, the weght of a data pont s at maxmum when t has the same locaton as the regresson pont, and are more lghtly when t has a locaton at a range of the movng wndow. In GWR an observaton s weghted n accordance wth ts proxmty to locaton so that the weghtng of an observaton s no longer constant but vares wth. The matrx form of parameter estmaton for s expressed as: T 1 T ˆ α( θ ), ˆ( β θ) = ( X W ( θ ) X ) X W ( θ) y (3) where αˆ and βˆ are ntercept and slope parameter n locaton ; and W (θ) s weghtng matrx whose dagonal elements represent the geographcal weghtng assocated wth each ste at whch measurements were made for locaton of. Spatal weghtng functon can be calculated by several varous methods. For fxed kernel sze, the weght of each pont can be calculated by applyng Gaussan functon w 2 j = exp[ 1/ 2( dj / b)] (4) d j where s the dstance between regresson pont and data pont j, and b s referred to as a bandwdth. An alternatve way s the b-square functon w [ 1 ( d / b) 2 ] 2 j =, (5) j when d < b and w = 0 otherwse. j j

4 In the practce, for each varable from equaton (2) ts weghtng value can be calculated by applyng a weghtng matrx W(Θ). The weghtng matrx s an n by n matrx whose off-dagonal elements are zero and whose dagonal elements denote the geographcal weghtng of each of the n observed data for regresson pont. After that, movng a kernel over the space can derve a local regresson at each pont n the analyss area. Estmated parameters n geographcally weghted regresson depend on the weghtng functon of the kernel selected. As the bandwdth, b, becomes larger, the closer wll be the model soluton to that of global OLS. Conversely, as the bandwdth decreases, the parameter estmates wll ncreasngly depend on observatons n close proxmty to regresson pont and wll have ncreased varance. The problem s therefore how to select an approprate bandwdth n GWR. To establsh an approprate bandwdth, b, we used the cross-valdaton approach (CV) whch determnes b by mnmsaton of the sum of squared errors between predcted varables and those observed. Accordng to (11), the equaton for the crossvaldaton sum of squared errors CVSS s statstcally expressed as: n CVSS = [ y yˆ ( b)] = 1 2 where y s the observed value and s the ftted value yˆ ( b) of for bandwdth b. y As general rule, the lower the CVSS, the closer the approxmaton of the model to realty. The best model s the one wth the smallest CVSS. For our GWR model, the bandwdth of 9 pxels was decded to be the most approprate. 4.2 Accuracy analyss The results obtaned usng global OLS, stratfed OLS and GWR were compared by the amount of varance explaned by every regresson model. A general rule s that the hgher s R² the better s the understandng of the varables responsble for the varaton n values observed. Generally, a predcton power of a regresson model ncreases wth the ncrease of R². We plotted the observed NDVI accum values aganst the predcted NDVI accum values to compare the predcton power of each regresson model. The standard error, SE, was used as a gude to the accuracy of the predctons. Usng the standard error s based on central lmt theorem, whch says that for large sample sze n the condtonal dstrbuton of the error should be approxmately Gaussan (or normal). For a gven sample the standard error can be calculated by the equaton: where, s = (6) s SE = (7) n n = 1 ( z zˆ )² n 1 where, s s the standard devaton of the varable z and n s the number of data used. In the GWR, z exposes the mean (8) value of a gven kernel. The number of data depends on the kernel sze. Regresson resduals are devatons of the ponts from the regresson lne. They contan a very mportant nformaton about the predcton accuracy of a regresson model. We were nterested not only n values of resduals, but also n the spatal nformaton assocated wth error. The map of resduals mght hghlght areas of over-predcton (postve errors) and under-predcton (negatve errors). An ndependent dstrbuton of resduals over the analyss space s the sgn for a non-problematc regresson model. Spatal patterns of regresson resduals contanng postve autocorrelaton ndcate that a model created s problematc: the standard errors are underestmated and the correlaton coeffcent often ndcates a sgnfcant relatonshp between varables when n fact there s not (Clfford et al, 1989). For each regresson model, we calculated spatal autocorrelaton of the resduals. We were nterested n the comparson of the results from the global and the local models. In ths study, the Moran s I coeffcent was used as a measure of autocorrelaton. It s the most commonly used coeffcent n unvarate autocorrelaton analyss and s gven as: ( )( ) ( ) j y y y j y wj y y n I = (9) s where n s the number of samples, y and y are the data values n quadrats and j, y s the average of y and s an element of the spatal weghts matrx W. Under the null hypothess of no spatal autocorrelaton, Moran s I has an expected value near zero, wth postve and negatve values ndcatng postve and negatve autocorrelaton, respectvely. 5. RESULTS AND DISCUSSION j wj Spatal dstrbuton of roughly corresponds to that of ranfall (Fgure 2). Regresson analyss based on the applyng of conventonal global OLS regresson revealed that there was a strong relatonshp between spatal dstrbuton of the and precptaton. The global OLS regresson model between and precptaton ftted to all vegetated pxels explans about 64% of spatal varance n vegetaton dstrbuton and was expressed as: = * P (10) (R² = 0.64) where P s precptaton. The standard error used as a measure for predcton accuracy was 0.21 or about 5 % from the mean value. The relatvely low value of the standard error mght gve assumpton that the derved regresson model provdes an accurate descrpton of the relatonshp between varables. However, the two regresson varables, both and precptaton data contan postve autocorrelaton (graphs not shown). Ther Moran s I values up to a dstance of ca. 100 km are sgnfcantly larger than the values expected under the null hy-

5 pothess of no postve autocorrelaton. It s known that spatal autocorrelaton s problematc for statstcal analyss lke OLS regresson. When conventonal OLS regresson s appled to the analyss of data contanng postve autocorrelaton, there are two problems: (1) the standard error of the regresson coeffcent s underestmated, (2) the resdual mean square may serously underestmate the varance of the error term, hence the coeffcent of determnaton (R²) s overestmated (Clfford et al., 1989). Recent studes tred to overcome these problems by applyng spatal regresson technque that can adjust for spatal autocorrelaton nherent n the regresson model on the bass of a varogram functon (Ttelsdorf, 2000; J & Peters, 2004). The result of the global OLS analyss encouraged us for a further work that amed to ncrease the understandng of the relatonshp between varables. One of the ways to reduce the model uncertanty s to ntroduce other varables nto the model specfcaton. Another way may be an mprovng regresson model by dsaggregatng (stratfcaton) the global regresson model nto a separate model for each of land cover types. We tred to reduce the amount of unexplaned and the negatve nfluence of spatal autocorrelaton n that way that we performed the OLS regresson analyss separately to the four man vegetaton types represented n the study regon. Wth regard to vegetaton type, the results ndcate that the coeffcent of determnaton, R², ncreases from desert to sem-desert, to short grassland, and to steppe, wth value of 0.36, 0.44, 0.52, and 0.67 respectvely. The components of the regresson equaton vary n a wde range: there are notable dfferences n regresson slope and ntercept between the vegetaton types. The stratfcaton of the OLS model by land-cover types clearly llustrates presence of nonstatonarty n the general relatonshp between and precptaton, whch may now be wrtten as: = ( ) + ( ) * P (11) (R² = 0.75) Values for the range n both ntercept and slope parameters are gven n brackets. We concluded that the global OLS regresson can not possbly be consdered statonary. There s spatal varaton n ntercept and slope parameters as well as n the coeffcent of determnaton, R², between the landcover categores. These results assume a dfferent response of vegetaton to precptaton by varous land cover categores. That agrees wth the results of other research works about dry regons whch also deal wth the nfluence of NDVI-ranfall relatonshps by land cover type (Yang et al., 1998; L et al., 2002; Wang et al., 2001). Dsaggregatng of global OLS regresson model nto four stratfed OLS models has sgnfcantly mproved the modellng certanty and qualty. Although, the amount of varance n NDVI accum unexplaned remans relatvely hgh, but the accuracy of the predcton s ncreased: there s a sgnfcant decrease of standard error n comparson wth the global OLS model. The smallest SE was 0.10 for steppe grassland, whle the largest SE of predcton s equal to 0.17 and was calculated for short grassland. Fgure 3. Spatal varatons n regresson outputs from the GWR analyss of aganst precptaton: (a) model ntercept; (b) model slope parameter; (c) local coeffcent of determnaton, R; (d) standard error of the model predcton. The GWR method has also been appled for modellng the relatonshp between and precptaton. To obtan

6 localzed results, a 9 by 9 pxels wndow was placed over each pxel whch provdes 81 data ponts for the model calbraton at each pxel locaton. Ths was the wndow sze whch was determned by mnmsaton of the cross valdaton square sum, CVSS. The GWR model allows the regresson parameters to vary n space and establshes consderably stronger relatonshp between the two varables. The general regresson equaton may be gven as: = ( ) + ( ) * P (12) (R² = 0.97) In the brackets we have wrtten range values for regresson ntercept and slope parameters. Fgure 3 summarzes the results derved from the geographcally weghted regresson analyss between and ranfall. Panel a shows spatal dstrbuton of the ntercept that had a mean of 0.32 and a range of 4.98 to Large postve values are dstrbuted manly n the north of the regon where short grassland and steppe grassland domnate whle low values are manly n the md and n the south. Here domnate sem-desert and desert vegetaton. Panel b shows spatal varaton n the slope parameter. Ths parameter had a mean of wth a range of 2.36 to 1.97 and a standard devaton of Negatve values of the slope parameter ndcate that n some locatons decreases when precptaton ncreases. Negatve values are manly n the northern and western parts of the study regon where crop felds/grassland mosacs domnate. The valley bottoms n the northeast also exhbt negatve values of the slope parameter. Panel c dsplays the spatal varaton n the strength of the relatonshp. The goodness-of-ft, measured by the coeffcent of determnaton, R², vared n the space and ranged from to 0.99, wth R² > 0.75 for two-thrds of the study regon. Low values of R² are manly dstrbuted n the west and over a swath of land from the east to the northwest n upper part of the map. The entre model performance was sgnfcantly mproved both for standard error of predcton accuracy and for the predcton power. Panel d n Fgure 3 ndcates standard error term whch has been used as a gude to predcton accuracy. The standard error estmated for the GWR ranged from to Values of standard error are several tmes smaller than that estmated for the global OLS (SE = 0.21) and the stratfed OLS models (SE = ). The GWR model enables to map the standard error for every pxel. The spatal patterns n the standard error reveal the danger of usng the sngle estmate for SE derved from a global OLS locally, they vary n magntude from pxel to pxel. The spatal patterns of SE clearly correspond to that of land-cover categores. Ths suggests that the GWR model sgnfcantly mproved predcton of by ranfall over the OLS model. The spatal patterns n regresson resduals are mportant ndces to examne how accurate the regresson model reveals the real relatonshp. The valdty of the regresson statstcs depends on the dstrbuton of the resduals. There are three condtons whch have to be fulflled by the resduals: (a) the resduals must be normally dstrbuted; (b) the resduals must be homoscedastc; (c) the resduals must not be autocorrelated (Tefelsdorf, 2000). If the resduals exhbt some nonrandom patterns the model created s problematc. A dagnostc statstcs ndcatng problems n regresson modellng s the degree of spatal autocorrelaton exhbted by the resduals from the model. The standard errors are usually underestmated when postve autocorrelaton s present. Vsual nterpretatons of the resdual maps shown n Fgure 4 gve us a good mpresson that there s a clear separaton of the resduals from the global OLS n the space. In the northern part of the study area, the resduals tend to exhbt postve values, whle n the south

7 Fgure 4. Spatal patterns of regresson resduals (upper panels) and correspondng resduals hstograms (lower panels) for the global OLS model (a), the OLS model based on stratfcaton by land-cover types (b), the GWR model (c). demonstrated how the model accuracy and model predcton power ncrease through scalng down from regonal to local analyss. The analyss based on the use of two dfferent regresson technques: one s the global ordnary least squares regresson, OLS, whch assumes the relatonshp to be statonary n space, and the other s the geographcally weghted regresson, whch allows the regresson parameters to vary over space. The results of the GWR suggest that t provdes more accurate predctons than the OLS regresson model. the resduals values are manly negatve (Fgure 4, a). The global OLS model underestmates when s hgh and overestmates when NDVI s low. Patterns n the mapped resdual values appear to correspond clearly wth patterns n land-cover. The postve devatons are assocated wth the dry steppe vegetaton cover, whle the negatve devatons are manly observed n the desert zone. The spatal patterns of resduals from the OLS model stratfed by land-cover types are not so clear as those for the global OLS model, but the separaton n the space also remans (Fgure 4, b). Only the GWR model allowed destructng the spatal dependence of the regresson resduals (Fgure 4, c). The GWR resduals dsplay no clear spatal patterns and ther dstrbuton over the study area seems to be close to random. Spatal autocorrelaton measures the smlarty between samples for a gven varable as a functon of spatal dstance. For the global OLS model and the GWR model, we calculated the Moran s I of the resduals to examne the effect of calbratng the models locally by GWR rather than globally. It s proved that the local calbraton removes much of the problems of spatally autocorrelated error terms ncluded n tradtonal global OLS model (Fotherngham et al, 2002, pp ). We were nterested n the comparson of the results from the global and local models. Fgure 5 shows the spatal autocorrelograms for the global OLS model resduals and the resduals from the GWR model. As expected, the error terms are most strongly autocorrelated for the global OLS model. The OLS model resduals had sgnfcant spatal autocorrelaton up to crca 50 km. In comparson, no sgnfcant postve spatal autocorrelaton was found for the GWR model resduals. It suggests that the calbraton of local model reduces the problem of spatally autocorrelated error terms. The GWR model demonstrates the ablty to deal wth problems of spatal non-statonary. Fgure 5. Spatal autocorrelograms for OLS resduals and resduals from the GWR model. The autocorrelogram of the OLS resduals ndcates that Moran s I values up to a lag dstance of more than 40 km are sgnfcantly larger than the value expected under the null hypothess of no postve autocorrelaton. The autocorrelogram of the GWR resduals dsplays no sgnfcant postve autocorrelaton. 6. CONCLUSION In ths paper we modelled the spatal relatonshp between NDVI and ranfall n a sem-ard regon of Kazakhstan. We The study found a hgh spatal non-statonarty n the strength of relatonshp and regresson parameters both between the land-cover types and wthn each land-cover type tself. The ordnary least squares regresson model has been appled to the whole study area was superfcally strong (R² = 0.63), however t delvered no local descrpton of the relatonshp. Applyng the OLS at the scale of the separate land cover classes reduced sgnfcantly the amount of unexplaned varance n NDVI accum (for the whole model R² = 0.75) and revealed a dfferent response of varous vegetaton types to ranfall. The strength of the relatonshp between NDVI and ranfall ncreased from desert (R² = 0.36), to sem-desert (R² = 0.44), to short grassland (R² = 0.52), and to steppe grassland (R² = 0.67) respectvely. The coeffcent of determnaton, R², was hgher for the GWR model. The approach of geographcally weghted regresson provded consderably stronger relatonshps from the same data sets (R² value for the general regresson = 0.97), as well as hghlghted local varatons wthn the land cover classes. The amount of varance n NDVI unexplaned was not as large as had been antcpated from the OLS analyss. The standard error (SE) was used as a gude to accuracy of the predctons. For the global OLS modelng, SE was The SE calculated through the stratfed OLS model for the land-cover types were a few smaller than for the whole regon. Fttng the regresson model nto a pxel scale, what was acheved through applcaton of the GWR, sgnfcantly reduces error terms. As expected, the errors terms shown by the results of the GWR are several tmes lower rangng from to Applyng GWR method for dealng wth spatal relatonshp sgnfcantly reduces both the degree of autocorrelaton and absolute values of the regresson resduals. Fgure 4 (upper rght panel) dsplays that the resduals from the global OLS model clearly exhbt postve spatal autocorrelaton wth an area of postve resduals grouped together (n the north) and also an area of negatve resduals grouped together (n the south). The spatal autocorrelaton n the resduals from the equvalent GWR model, shown n Fgure 4 (lower rght panel), s no longer evdent. There are no obvous patterns to the resduals whch appear randomly over the regon. The results suggest that GWR provdes a better soluton to the problem of spatally autocorrelated error terms n spatal modellng compared to the global regresson modellng. The results also suggest that the calbraton of local rather than global models reduces the problem of spatally autocorrelated errors. The resduals from the global OLS model clearly exhbted postve spatal autocorrelaton up to approxmately 50 km. The resduals from the GWR model dsplayed no postve autocorrelaton, suggestng the ablty of GWR approach to deal wth spatal non-statonary problems. The GWR provdes a more drectly nterpretable soluton to the problem of spatally autocorrelated errors n spatal mod-

8 elng compared wth the global forms of spatal regresson modellng. In GWR, the spatal non-statonarty of the parameters s modelled drectly, rather than allowng the nonstatonarty to be reflected through the error terms n the global model. Ths agrees wth the results that have been dscussed by (Fotherngham et al, 2002; Wang et al, 2005). Our study proved the superorty of GWR over global OLS model n analyss the relatonshp between patterns n NDVI and precptaton. Ths superorty s manly due to the consderaton of the spatal varaton of the relatonshp over the study regon. Global regresson technques lkes OLS may gnore local nformaton and, therefore, ndcate ncorrectly that a large part of the varance n NDVI was unexplaned. The non-statonary modellng based on GWR approach has the potental for greater predcton precson because the model s more tuned to local crcumstances, although clearly a greater number of data s requred to allow relable local fttng. ACKNOWLEDGEMENTS We gratefully acknowledge Professor S. Fotherngham for provdng the software GWR 3.0. REFERENCES Brunsdon, C., McClatchey, J. and Unwn, D. J Spatal varatons n the average ranfall-alttude relatonshp n Great Brtan: An approach usng geographcally weghted regresson. Internatonal Journal of Clmatology, 21: Brunsdon, C. F., Fotherngham, A. S. and Charlton, M. E Geographcally weghted regresson: a method for explorng spatal non-statonarty. Geographcal Analyss, 28: Budde, M. E., Tappan, G., Rowland, J. Lews, J. and Teszen, L. L Assessng land cover performance n Senegal, West Afrca usng 1-km ntegrated NDVI and local varance analyss. Journal of Ard Envronments, 59: Clfford, P., Rchardson, S. and Hemon, D Assessng the sgnfcance of the correlaton between two spatal processes. Bometrcs, 45: Foody G. M Geographcal weghtng as a further refnement to regresson modellng: an example focused on the NDVI-ranfall relatonshp. Remote Sensng of Envronment, 88: Foody, G. M Spatal nonstatonary and scaledependancy n the relatonshp between speces rchness and envronmental determnants for the sub-saharan endemc avfauna. Global Ecology & Bogeography, 13: Fotherngham, A. S., Brunsdon, C. and Charlton, M Geographcally weghted regresson: the analyss of spatally varyng relatonshps. (Chchester, Wlley) 232 pp. Fotherngham, A. S., Charlton, M. E. and Brundson, C The geography of parameter space: and nvestgaton nto spatal non-statonarty. Internatonal Journal of GIS, 10: Holben, B. N Characterstcs of maxmum-value composte mages from temporal AVHRR data. Internatonal Journal of Remote Sensng, 7: J, L. and A. J. Peters A Spatal Regresson Procedure for Evaluatng the Relatonshp between AVHRR-NDVI and Clmate n the Nothern Great Plans. Internatonal Journal of Remote Sensng, 25: L, B., Tao, S. and Dawson, R. W Relaton between AVHRR NDVI and ecoclmatc parameters n Chna. Internatonal Journal of Remote Sensng, 23: L, J., Lews, J., Rowland, J., Tappan, G., Teszen, L Evaluaton of land performance n Senegal usng multtemporal NDVI and ranfall seres. Journal of Ard Envronments, 59: McMllen, D. P One hundred ffty years of land values n Chcago: a non-parametrc approach. Journal of Urban Economcs, 40: Murray, M. R. and Baker, D. E MWINDOW: an nteractve FORTRAN-77 program for calculatng movngwndow statstcs. Computers and Geoscence, 17: Pavlov, A. D Space-varyng regresson coeffcents: a sem-parametrc approach appled to real estate markets. Real Estate Economcs, 28: Rchard Y. and Poccard I A statstcal study of NDVI senstvty to seasonal and nter-annual ranfall varatons n southern Afrca. Internatonal Journal of Remote Sensng, 19: Tefelsdorf, M Modellng spatal processes: the dentfcaton and analyss of spatal relatonshps n regresson resduals by means of Moran s I. (Berln, Sprnger-Verlag) 345 pp. Tucker C. J. and P. J. Sellers Satellte remote sensng of prmary vegetaton. Internatonal Journal of Remote Sensng, 7: Wang, J., Prce, K. P. and P. M. Rch Spatal patterns of NDVI n response to precptaton and temperature n the central Great Plans. Internatonal Journal of Remote Sensng, 22: Wang, Q., N, J. and J. Tenhunen Applcaton of a geographcally weghted regresson analyss to estmate net prmary producton of Chnese forest ecosystem. Global Ecology & Bogeography, 14: Yang, L., Wyle, B., Teszen, L. L., Reed, B. C., An analyss of relatonshps among clmate forcng and tmentegrated NDVI of grasslands over the U.S. Northern and Central Great Plans. Remote Sensng of the Envronment, 65: