Local regression models for spatial interpolation of urban heat island—an example from Wrocław, SW Poland
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Abstract
Geographically weighted regression algorithm (GWR) has been applied to derive the spatial structure of urban heat island (UHI) in the city of Wrocław, SW Poland. Seven UHI cases, measured during various meteorological conditions and characteristic of different seasons, were selected for analysis. GWR results were compared with global regression models (MLR), using various statistical procedures including corrected Akaike Information Criterion, determination coefficient, analysis of variance, and Moran’s I index. It was found that GWR is better suited for spatial modeling of UHI than MLR models, as it takes into account nonstationarity of the spatial process. However, Monte Carlo and F3 tests for spatial stationarity of the independent variables suggest that for several spatial predictors a mixed GWR–MLR approach is recommended. Both local and global models were extended by the interpolation of regression residuals and used for spatial interpolation of the UHI structure. The interpolation results were evaluated with the crossvalidation approach. It was found that the incorporation of the spatially interpolated residuals leads to significant improvement of the interpolation results for both GWR and MLR approaches. Because GWR is better justified in terms of statistical specification, the combined GWR + interpolated regression residuals (GWR residual kriging; GWRK) approach is recommended for spatial modeling of UHI, instead of widely applied MLR models.
Keywords
Multiple Linear Regression Normalize Difference Vegetation Index Land Surface Temperature Urban Heat Island Multiple Linear Regression Model1 Introduction
Urban heat island (UHI; see Appendix 1 for abbreviations) is probably the most significant phenomenon of urban climate, with further strong impact on various aspects of urban environment. Depending on climate regime and season, socioeconomic and health impacts of UHI are either positive or negative. For example, the UHI influence on human comfort, mortality, and energy usage in cold and moderate climate is positive in winter and negative in summer, while in hot climate, it is negative regardless the season. Besides that, UHI influences air pollution dispersion in the cities, water usage, bioclimatic conditions, and others (Unger 2004). The most important features describing UHI are its magnitude and spatiotemporal structure. All the information is expected by townplanners, municipal services, and are essential input for various modeling studies (e.g., air pollutants dispersion).
Providing spatially continuous information on weather and climate at any time is a crucial but difficult task because meteorological observations and measurements are usually discretely distributed. Such a procedure requires data transformation (from discrete to continuous in space) that can be performed by various spatial interpolation (or spatialization) methods. Spatialization algorithms can be divided into several groups: deterministic and stochastic (or their combinations), exact and inexact, global and local, and one and multidimensional. Numerous algorithms have been successfully introduced into meteorology and climatology (Dobesch et al. 2007; Tveito et al. 2008).
The general spatial structure of UHI is characterized by the occurrence of three distinct zones, named cliff, plateau, and peak (Oke 1976).This general structure of UHI can be strongly modified depending on landuse types and urban structures. During calm and clearsky meteorological conditions, which are favorable for UHI development, and especially during nighttime, UHI takes a multicellular, irregular shape (e.g. Park 1986; Kłysik and Fortuniak 1999) and is sometimes called an urban heat archipelago (Unger 2004). The first attempts to analyze the UHI structure were based on manually interpolated isotherm maps (Duckworth and Sandberg 1954). More sophisticated interpolation algorithms became popular with the increasing access to effective computers and development of geographic information system (GIS) (Svensson et al. 2002; Bottyán and Unger 2003; VicenteSerrano et al. 2005; Alcoforado and Andrade 2006). Most of the recent studies on spatial characteristic of UHI are based on multidimensional interpolation algorithms, with the multiple linear regression (MLR) being the most often applied (Unger et al. 2010). This is because of the strong correlation of UHI with urban environment characteristics, which can be described and analyzed quantitatively in space with GIS tools. Good performance of multidimensional interpolation techniques (both MLR and its extension—residual kriging, RK), especially in the case where observations are sparse, unevenly distributed, and do not cover the entire city area, was confirmed by earlier studies (Szymanowski and Kryza 2009). Despite providing better results of interpolation of UHI than univariate geostatistical techniques, MLR could lead to distorted results when the spatial process is nonstationary, e.g., due to wind influence. Spatial nonstationarity is common for meteorological data; therefore, applicability of the given interpolation algorithm can be strongly limited if the method is not able to deal with it. This is the main problem when applying multidimensional algorithms like MLR.
The main goal of this paper is the application and evaluation of GWR for determination of the spatial structure of seven selected UHI cases measured in Wroclaw (SW Poland). In the following sections, study area and measurement data are briefly described. The next sections introduce the new set of potential spatial predictors which were used for interpolation with both MLR and GWRbased algorithms. The set of spatial predictors was significantly extended in comparison with previous study of Szymanowski and Kryza (2009), with the aim to verify if there is a significant gain in terms of interpolation results when utilizing more complex approaches for predictor calculation. Next, the global and local regression models are introduced. An indepth statistical analysis is performed to verify if there is methodological (statistical) justification for a more complex approach with local models. Finally, local and global regression models, both raw and extended by interpolation of the regression residuals (RK and GWR residual kriging (GWRK) for geographically weighted regression with residual kriging) are used for spatial interpolation of UHI. Interpolation results are evaluated with crossvalidation (CV) approach to quantify if there is a gain in terms of smaller interpolation error when approaching the spatial structure of urban heat island with GWR and GWRK algorithms vs. global models.
2 Study area
Average characteristics of Wroclaw climate for the period 1971–2000 and UHI magnitude: April 1997–March 2000 (Szymanowski 2004)
Season  UHI magnitude [K]  Air temperature [°C]  Wind speed [m s^{−1}]  Relative humidity [%]  Cloudiness [0–10]  Rainfall [mm] 

Winter  0.9  0.2  2.6  82.4  7.8  97.7 
Spring  1.2  8.8  2.4  71.0  7.3  126.2 
Summer  1.1  17.6  1.9  72.6  7.0  226.6 
Autumn  0.9  8.6  2.1  82.0  7.5  125.0 
Year  1.0  8.8  2.3  77.0  7.4  575.5 
The magnitude of the UHI was calculated as the air temperature difference dT = T _{U} − T _{R} measured at the same time on stations U and R (Fig. 1, Table 1). Also, the occurrence of UHI in the city center was calculated as the frequency of dT stated above. Detailed average, extreme, and frequency of UHI values in Wrocław in the period April 1997–March 2000 were introduced by Szymanowski (2004, 2005) and Szymanowski and Kryza (2009). UHI phenomena in the city center rises the annual mean temperature by 1.0 K. Thermal excess is weaker in large housing estates (0.7 K) and in residual areas (0.3 K). Similarly to other cities of this size, the average magnitude of UHI in the night is two to three times higher than the average value for daytime. The maximum difference between the city center and suburban areas may exceed 9 K (Szymanowski and Kryza 2009). Positive values of UHI in the central parts of the city are observed during >96% of night hours and >80% of daytime, but strong UHI effect (>5.0 K) are measured in 3.8% of night hours and only randomly during daytime. The annual cycle of the UHI magnitude is dependent on meteorological conditions and the release of artificial heat. The most favorable conditions for UHI occur in warm season, but due to increasing convective cloudiness in the midsummer, the highest values are observed in May and August. Secondary maximum of UHI intensity is observed in January (heating season), and the minima are observed in October and February. More detailed analysis of UHI in Wroclaw is provided by Szymanowski (2004, 2005) and Szymanowski and Kryza (2009).
3 Meteorological data
Meteorological conditions in the city outskirts and UHI magnitude in the city center during measurements
Date  UTC  Wind speed [m s^{−1}]  Prevailing wind direction  Cloudiness [0–8]  UHI magnitude [K] 

22 May 2001  00.00  0–1  WWSW  0  6.0 
26 June 2001  00.00  0–1  WSSW  0–3  4.9 
30 July 2001  22.30  1–2  SWWSW  1–4  3.8 
13 October 2001  21.30  1–2  NNW  0  3.4 
03 January 2002  02.00  0–1  NNNE  0–2  6.2 
15 January 2002  01.00  3–4  ESESE  0  0.7 
15 February 2002  00.30  1–3  NEENE  0  1.9 
4 Methods
 1.
Preparation of a spatially continuous set of potential UHI predictors required for multidimensional interpolation algorithms
 2.
Specification and evaluation of the MLR models
 3.
Specification and evaluation of the GWR models and selection of the kernel type and size, testing for spatial nonstationarity of parameter estimates
 4.
Comparison of regression models using ANOVA
 5.
Extension of the regression models by interpolation of residuals
 6.
Evaluation of the spatial interpolation results calculated with four models: MLR, GWR, RK, and GWRK.
The set of potential UHI predictors was prepared with GIS tools, provided with GIS GRASS (GRASS Development Team 2010) and ArcGIS systems. The statistical analysis (points 2–6) was performed with R statistical package (R Development Core Team 2010) and GWR3 software for Geographically Weighted Regression (Charlton et al. 2010).
4.1 Potential UHI predictors
Highrise development, introduction of new surface materials (mostly waterproof, opaque, and airtight), emission of artificial heat, moisture, and pollutants are among the leading factors responsible for aerodynamic, radiative, thermal, and moisture modifications of the local climate in cities and responsible for UHI phenomena (Oke 1987). Most of the features describing size, geometry, thermal properties, and “metabolism” of the cities may be derived from maps, digital databases, and satellite imagery by GIS techniques. All the spatially continuous information can be used as additional explanatory variables in the UHI spatialization process with multidimensional methods (Bottyán and Unger 2003; Alcoforado and Andrade 2006; Szymanowski and Kryza 2009).
In the previous study, the authors of this paper compared various interpolation algorithms for the UHI spatial interpolation and used a set of six spatial predictors for multidimensional spatialization, which were derived mostly from the landuse map of Wrocław and the buildings database available only for the selected areas of the city (Szymanowski and Kryza 2009). The regression analysis showed that for some UHI cases, over 30% of temperature variance remained unexplained. Therefore, the question appeared if calculation (for example of roughness length) of relatively simple landuse map derived predictors is detailed enough for spatial interpolation procedure, or the interpolation results can be improved by providing other spatial predictors or derived with more complex approaches. Here, the state of the art LIDARoriginated database, together with 3D trees database and digital elevation model (DEM), were used to expand a set of potential predictors and develop the new ones. These were supported by the extensive set of Landsat ETM+derived information. All potential predictors are described in the following sections, including short introduction of the previously applied independent variables (Section 4.1.1) and newly derived (Sections 4.1.2–4.1.5).
4.1.1 Landuse map derivatives
 1.
Roughness length (z _{0}; meters), which is one of the most important parameters describing properties of the urban boundary layer, was calculated using the modified formula proposed by Lettau (1969). Due to limited information on buildings geometry, some simplifications were assumed: the lot area was held equal to the area of the given landuse class and wind direction was not incorporated in the silhouette parameter. The high values of the predictor are related with the areas of decreased wind speed and turbulent fluxes. The >0 regression coefficient is expected for this predictor.
 2.
Percentage of artificial surfaces (AS, percent) in a given landuse class took into consideration both horizontal (e.g., roofs, roads) and vertical surfaces (walls). Artificial surfaces were added together and linked to the lot area. Buildings were represented by boxes and roof structures were not considered during calculations. The predictor describes jointly the areas of altered energy balance leading to positive thermal anomaly, as described by Oke (1987). The >0 regression coefficient is expected for this predictor.
 3.
Percentage of seminatural surfaces (NS, percent) in a given landuse class. Calculations for this parameter were similar to AS, but only horizontal surfaces were considered. The expected regression coefficient is <0.
 4.
Thermal admittance (μ, Joules per squared meter per root second per Kelvin), estimated as a weighted value of the ratio of vegetated surfaces to artificial surfaces. Thermal admittance for concrete (builtup classes) and moderately moist (40%) clay soil covered by grass (nonbuiltup classes excluding water) were used as starting values, after Boeker and van Grondelle (1995). The predictor describes the areas of increased sensible heat storage and is expected to be positively correlated with air temperature.
4.1.2 Landsat ETM+ derivatives
Days of air temperature measurements and corresponding Landsat ETM+ data
UHI cases  Landsat ETM+ data 

22 May 2001, 26 June 2001, 30 July 2001  24 May 2001 
13 October 2001  15 October 2001 
03 January 2002, 15 January 2002, 15 February 2002  03 January 2002 
 1.Albedo (a, unitless [0, 1]), considered as reflectance for panchromatic band 8 (Landsat ETM+ band 8):where L _{PAN} is the spectral radiance for panchromatic band [Watts per square meter per steradian per micrometer], d the Earth–Sun distance in astronomical units, ESUN_{PAN} the mean solar exoatmospheric irradiances (Landsat 7 Science Data Users Handbook 2010), and θ _{s} the Solar zenith angle in degrees. Negative correlation with UHI is expected.$$ a = \frac{{\pi \cdot L_{{{\text{PAN}}}} \cdot d^{2} }}{{{\text{ESUN}}_{{{\text{PAN}}}} \cdot \cos \theta _{{\text{s}}}}} $$(1)
 2.Vegetation indices:
 Normalized Difference Vegetation Index (NDVI) is modulation ratio of reflectance (ρ) for nearinfrared (NIR) and red bands (RED) as it indicates vegetation (Tucker 1979):$$ {\text{NDVI}} = \frac{{{\rho_{\text{NIR}}}  {\rho_{\text{RED}}}}}{{{\rho_{\text{NIR}}} + {\rho_{\text{RED}}}}} $$(2)
 Soil Adjusted Vegetation Index (SAVI) is a superior vegetation index for lowcover environments (Heute 1988):where L is an empirically determined constant to minimize the vegetation index sensitivity to soil background reflectance variation (Schowengerdt 2007). In this case, L is set to 0.5.$$ {\text{SAVI}} = \left( {\frac{{{\rho_{\text{NIR}}}  {\rho_{\text{RED}}}}}{{{\rho_{\text{NIR}}} + {\rho_{\text{RED}}} + L}}} \right)(1 + L) $$(3)
 Normalized Difference Moisture Index (NDMI) that contrasts the NIR, sensitive to the reflectance of leaf chlorophyll content to the midinfrared band (MIR), sensitive to the absorbance of leaf moisture (Wilson and Sader 2002):$$ {\text{NDMI}} = \frac{{{\rho_{\text{NIR}}}  {\rho_{\text{MIR}}}}}{{{\rho_{\text{NIR}}} + {\rho_{\text{MIR}}}}} $$(4)
Vegetation indices are unitless and its range is [−1, +1].
In Wrocław, deciduous trees dominates over coniferous, so the vegetation indices mentioned above, based on chlorophyll content, are for winter at the same level as for the wooded and builtup areas. Negative correlation with UHI is expected (Szymanowski and Kryza 2011).

 3.Land surface temperature (T _{ls}, kelvin) was calculated using emissivity (ε) and atsatellite temperature T _{as} [kelvin] with the singlechannel algorithm (JiménezMuñoz et al. 2009). Atmospheric parameters were estimated using Atmospheric Correction Parameter Calculator (2010). T _{as} is converted from spectral thermal infrared radiance and is considered as effective atsatellite temperature of the viewed Earth–atmosphere system under the assumption of unity emissivity. The conversion formula is:where L _{TIR} is the spectral radiance for TIR band [W m^{−2} sr^{−1} μm^{−1}] and calibration constants K _{1} and K _{2} are equal to 666.09 W m^{−2} sr^{−1} μm^{−1} and 1,282.71 K, respectively (Landsat 7 Science Data Users Handbook 2010).$$ {T_{\text{as}}} = \frac{{{K_2}}}{{\ln (\frac{{{K_1}}}{{{L_{\text{TIR}}}}} + 1)}} $$(5)
Uncorrected T _{ls} is equal to T _{as} on the assumption that ε = 1, so all emitting materials are ideal blackbodies with 100% radiative efficiency. If emissivity of thermal region is known, surface temperature can be calculated more precisely.
The correlation between the land surface and air temperature changes seasonally (Szymanowski and Kryza 2011). For winter, when artificial build up areas are the warmest, high negative correlation coefficients were calculated (R = −0.90). If the snow cover is observed outside the city center, which is the case of January, air temperature is strongly correlated with land surface temperature. In summer and particularly in autumn, after harvest, when fields are bare, day and night thermal condition of land surface differs significantly in the city outskirts, while builtup areas are warm irrespectively to diurnal cycle.
 4.Emissivity (ε, unitless [0–1]) is defined as the ratio of the spectral radiant exitance of a graybody to that emitted by a blackbody at the same temperature (Schowengerdt 2007). In urban areas, the emissivity of typical manmade materials in TIR band of Landsat ETM+ ranges from 0.40 to 0.98 (Stathopoulou et al. 2007). There are numerous techniques to retrieve emissivity from satellite multispectral imagery (Sobrino and Raissouni 2000; Sobrino et al. 2008; Stathopoulou et al. 2007). The method depends on reclassification of the study area due to NDVI values and then separately for three NDVI classes:
 a.for bare soil, rocks and artificial materials in urban environment (NDVI < 0.2):where ρ _{RED} is reflectance for RED band;$$ \varepsilon = 1  {\rho_{\text{RED}}} $$(6)
 b.
for vegetated areas (NDVI > 0.5), ε is assumed to be constant and equal to 0.98
 c.
 a.
Further corrections are applied for the emissivity layer based on landuse map. The emissivity for water areas is often too low (0.90–0.93), therefore all are reclassified to 0.99. Similarly, areas covered by snow in the winter case (a > 0.5) are set to 0.99 (Arnfield 1982).
The seasonal change of sign of correlation coefficients can be observed when analyzing emissivity, with negative correlation in warm season and positive in cold season, if snow cover is present (Szymanowski and Kryza 2011).
4.1.3 LIDAR scan derivatives
 1.Roughness length (z _{0}; meters). Various procedures for estimation of z _{0} are available (e.g., Grimmond et al. 1998; Grimmond and Oke 1999), and here, the formula proposed by Bottema (1997) and Bottema and Mestayer (1998) was used, with the simplification proposed by Gal and Unger (2009):where h is averaged building height, λ _{P} is plan area ratio, and λ _{F} is frontal area ratio (calculated for eight main wind directions). For the purpose of this study, the algorithm proposed by Gal and Unger (2009) was modified to provide spatially continuous information on z _{0}. The maximum distance from a building (or a group of buildings) to the border of its lot area is assumed to be maximum 10h, while if not limited, it led to overestimation of lot areas for sparse, low development. The gaps between lot areas and nonbuiltup areas in the city boundaries are filled with the same values as used in the previous paper (Szymanowski and Kryza 2009) for a given landuse class. The >0 regression coefficient is expected for this predictor (see Section 4.1.1. above for details).$$ {z_0} = h(1  \lambda_{\text{P}}^{{0.6}})\exp \left( {  \sqrt {{\frac{{0.4}}{{{\lambda_{\text{F}}}}}}} } \right) $$(8)
 2.Porosity (P, unitless [0, 1]) is a measure of how penetrable the area is for the airflow and could be defined as the ratio of the volume of the open air and the volume of the urban canopy layer referring to the same area. In this case, all calculations were performed for squared lot areas equal to 1 ha (A _{T} = 10,000 m^{2}) with 1m resolution buildings, trees, and shrubs raster datasets. The formula designed for the porosity of buildings proposed by Gal and Unger (2009) was modified due to influence of trees and shrubs:where P _{b} (P _{ts}) is the buildings porosity (or trees and shrubs), h _{b} (h _{ts}) is the mean height buildings (trees and shrubs) in the lot area, V _{b} (V _{ts}) is the sum of volumes of the buildings (trees and shrubs), and p is the porosity index of trees. The value of p is equal to 0.2 when deciduous trees are in leafs and it is set to 0.6 for the leafless period (Heisler and DeWalle 1988). The predictor works in the opposite way to roughness length, and negative correlation with UHI is expected.$$ P = {P_{\text{b}}} + {P_{\text{ts}}} = \frac{{{A_{\text{T}}}{h_{\text{b}}}  {V_{\text{b}}}}}{{{A_{\text{T}}}{h_{\text{b}}}}} + (1  p)\frac{{{A_{\text{T}}}{h_{\text{ts}}}  {V_{\text{ts}}}}}{{{A_{\text{T}}}{h_{\text{ts}}}}} $$(9)
 3.
Sky View Factor (SVF, unitless [0, 1]), defined as the hemispherical fraction of unobstructed sky visible from a given location. Here, the computationally efficient approach based on hillshading algorithm proposed by Corripio (2003) was used to derive spatial information on SVF for the Wrocław area. The solar azimuth and elevation steps were set to 2° for computational efficiency, with 1 m spatial resolution of the digital elevation model. The predictor is related with the geometry of buildings and street canyons, and high values are related with decreased longwave radiation loss (positive correlation with UHI is expected).
 4.
Daily sums of solar irradiation (DSI—excluding walls, DSI_{w}—including walls; Watthours per square meter), calculated using r.sun model implemented in GIS GRASS system (Šuri and Hofierka 2004; Hofierka and Kaňuk 2009). Sums of daily total solar irradiation for the day preceding nighttime UHI were calculated. The shadowing effects of the nearby buildings were included. Because the r.sun works only with 2D raster elevation layers, the model can be applied specifically to the selected building surfaces—roofs and to the interbuilding areas. The solar energy reaching building walls was also approximated here by setting specific values of aspect, slope, and height to 1m resolution raster elements representing walls. The aspect was set according to the real orientation of the wall calculated from the vector model, and the slope was set to 90°. The relative height of the wall was set to the half of the real height to account for shadowing effect of the wall due to the surrounding buildings. The shadowing effect of trees was not included. DSI is negatively correlated with air temperature (Szymanowski and Kryza 2011) and can be explained by the strong shadowing effect of the compact development in densely builtup parts of the city. This causes relatively low sums of energy incoming to the areas between the buildings, where the measurements were performed. The idea of DSI_{w} incorporates façade surfaces that can surpass the role of relatively flat terrain and roofs, especially in winter when the sun position is low and is expected to be positively correlated with UHI.
4.1.4 Artificial heat emission
Anthropogenic heat release (Q _{A}, watts per square meter) was earlier estimated by Chudzia and Dubicka (1998) for the Wrocław area based on detailed inventory of energy (electricity and fuel) consumption in the late 1990s. Q _{A} was estimated for nonheating (April to October) and heating (November to March) seasons in various parts of the city and in various landuse classes. Positive correlation with UHI is expected for Q _{A}, regardless of the season.
4.1.5 Spatial predictors’ derivatives
Spatially continuous variables described above were considered as potential predictors of the spatial structure of the UHI. However, spatial gradients of air temperature are smoothed due to air flow and turbulence, and therefore less pronounced, than “sharp” transitions typical of highresolution satellite imagery and LIDAR data. Moreover, the air temperature in a given location is influenced by thermal conditions of the surrounding areas (source region), with the effective radius of ∼0.5 km (screen level ruleofthumb; Oke 2004; Szymanowski and Kryza 2009), and depends mainly on building density. To incorporate the source region effect in the interpolation procedure, a set of raster layers for each parameter described above was calculated with the focal mean filter tool. For each raster element, the filter calculates the average of the values within a specified neighborhood of the input raster map. The averaging reduces isolated high values and smoothes sharp gradients in the original highresolution data. The averaging matrices applied here are circular in shape with radii varying from 25 to 1,000 m.
4.2 Global linear regression model
Independent variables were selected for each UHI case from the set of potential predictors described in Section 4.1. Due to proper specification of the regression model from 206 measurement points, the number of independent variables was assumed to be equal or less than five. Selection of the predictors was performed stepwise, taking into account their statistical significance and the lack of colinearity with other independent variables, by analyzing the variance inflation factor (VIF). Also, the direction of dependence (sign of the β coefficient) between air temperature and independent variable was checked to ensure that the final equation can be explained in terms of known physical processes that influence UHI formation. Expected level of confidence was 95% (p value < 0.05), and VIF should not exceed the value of 10, unless the considered variable significantly improve overall regression model. Usually the same sign of β is expected for a given parameter throughout a whole year. However, in some cases, it can change respectively to the meteorological conditions, for example due to snow cover as it was observed for T _{ls} in winter cases (Szymanowski and Kryza 2011).
4.3 Geographically weighted regression model
 1.
AICc—corrected Akaike Information Criterion (Hurvich et al. 1998), which is a measure of model performance and is helpful for comparing different regression models. Taking into account model complexity, the model with a lower AICc value provides a better fit to the observed data;
 2.
Estimated standard deviation for the residuals (σ). The models with smaller values of this statistic are preferable.
 3.
Global and local minimum and maximum determination coefficients (R ^{2}) as measures of goodness of fit.
Last question that should be answered in the process of local regression model specification is whether each set of parameter estimates exhibits significant spatial variation over the study area. The reason is that if localized parameter estimates do not meet statistically significant differences, the GWR model can be considered as equivalent to the global regression model, although the local parameter estimates show spatial variation. Moreover, if any independent variable shows spatial stationarity by the test, a mixed GWR model may be more appropriate (Fotheringham et al. 2002; Yu and Wu 2004; Yu 2006). In this paper, two tests were employed to address the issue: Monte Carlo (Charlton et al. 2010) and F3 (Leung et al. 2000). Both tests are applied for verification weather the local model (GWR) offers an improvement over the global model (MLR). For Monte Carlo approach, the significance of variability of individual coefficients is tested—for a given number of times, the geographical coordinates of the observations are randomly permuted against the variables, resulting in n values of variance of the coefficient of interest, which are used as an experimental distribution. The actual value of the variance is compared against this list to calculate an experimental significance level. Analytical F3 method is less computationally intensive than Monte Carlo algorithm, and it tests the variability of the variance under a null hypothesis of a stationary coefficient. The detailed equations are provided by Leung et al. (2000).
4.4 Comparison of regression models
Both models, MLR and GWR, for all analyzed UHI cases were evaluated and compared using a set of statistics to check a goodnessoffit of the models to the observations. Additionally, spatial autocorrelation of regression residuals (Moran’s I statistics) was analyzed to detect possible problems with proper specification of the model in the nonstationary conditions of the spatial processes.
To test whether the GWR model offers an improvement over the MLR model, an analysis of variance (ANOVA) was used (Fotheringham et al. 2002). The analysis of variance is used here to compare MLR and GWR models by providing a statistical test of whether the means of residuals of both models differ significantly or not. The ANOVA tests the null hypothesis that the GWR model represents no improvement over a MLR model, using the F test. As pointed out by Leung et al. (2000) and Fotheringham et al. (2002), the GWR model certainly fits a given dataset better than a global MLR model. However, in practice, a simpler model is usually preferred over a more complex one if there is no significant improvement from the latter, and this was addressed with ANOVA.
4.5 Extension of the local and global models by interpolation of the residuals—the residual kriging approach
While GWR can be used for spatial interpolation of climate elements (Lloyd 2007), the novelty of this paper is the introduction of geostatistical (kriging, OK) interpolation of GWR residuals that has not been implemented before for climatological purposes. The composition of GWR and OK of residuals is named here as geographically weighted regression (residual) kriging (GWRK).
4.6 Evaluation of the interpolation results
Interpolation results derived with spatialization techniques applied in this study (MLR, GWR, RK, and GWRK) were evaluated and compared using CV approach in which a single observation is removed from the original sample dataset (“leaveoneout” method) to be used as the validation data and the remaining observations are used for interpolation. The procedure is repeated consecutively for all measuring sites and the interpolation errors are calculated as the difference between the modeled and the observed values. The CV errors were used to calculate diagnostic measures, including mean bias (BIAS), the root mean square error (RMSE), the mean absolute error (MAE), and maximum and minimum errors (Willmott and Matsuura 2006). Statistical validation of the interpolation algorithms was complemented by visual analysis of the UHI spatial patterns and spatial distribution of the crossvalidation errors.

The magnitude and sign of CV errors, which has been previously standardized and classified to be comparable between cases. This is symbolized by the proper symbol size and shape.

The tendency for clustering of high or low CV errors was described with the Local Moran’s index (Anselin 1995) and marked on maps. This is done because the quality of the spatial model can be evaluated also in terms of clustering tendency of errors (Fotheringham et al. 2002). Significant spatial tendency for clustering of very high or low errors (outliers) suggests the model misspecification in the region of clustered errors. The spatial relations among the features in Moran’s index calculation were conceptualized by the inverse distance method with threshold set to the maximum distance of the first neighbor (∼1,160 m). Due to the irregular distribution of sites, the row standardization was used to generate spatial weights matrix. The Moran’s index is used here as quantitative measure of local spatial autocorrelation of CV errors. The white or black filling of the symbol distinguishes between a statistically significant (0.05 level) cluster of high values (HH) or cluster of low values (LL). The gray color indicates either not significant cluster/outliers process or significant outlier in which a high value is surrounded primarily by a low (HL) value or a low value is surrounded primarily by a high (LH) value.
5 Results
5.1 Regression models
Regression analysis and spatial autocorrelation of regression residuals for the MLR model (for all UHI cases n = 206)
Case  Regression analysis  R ^{2} adjusted from Szymanowski and Kryza (2009)  Spatial autocorrelation of residuals  

Predictor  p value  R ^{2} adjusted  AICc  F  Standard error  Moran’s I E(I) = −0.0049  p value  
22 May 2001  Q _{ A }  0.0000  0.79  562.3  150.9  0.94  0.77  0.2710  0.0000 
P  0.0003  
SAVI  0.0000  
DSI  0.0083  
z _{0}  0.0001  
26 June 2001  a  0.0000  0.81  476.8  175.6  0.76  0.81  0.1610  0.0000 
NDVI  0.0000  
P  0.0000  
z _{0}  0.0000  
DSI  0.0634  
30 July 2001  a  0.0005  0.71  569.5  167.9  0.96  0.69  0.2460  0.0000 
NDVI  0.0000  
z _{0}  0.0000  
13 October 2001  μ  0.0000  0.74  482.2  291.3  0.78  0.72  0.0841  0.0000 
z _{0}  0.0000  
03 January 2002  Q _{A}  0.0000  0.85  479.1  373.8  0.77  0.66  0.3142  0.0004 
a  0.0395  
T _{ls}  0.0000  
15 January 2002  NDVI  0.0002  0.75  218.3  153.9  0.40  0.74  0.1237  0.0000 
T _{as}  0.0000  
z _{0}  0.0374  
SVF  0.0392  
15 February 2002  ε  0.0582  0.76  420.3  160.6  0.67  0.69  0.2262  0.0000 
NDMI  0.0000  
NDVI  0.0000  
T _{as}  0.0000 
Regression analysis and spatial autocorrelation of regression residuals for the GWR model
Case  GWR analysis  Spatial autocorrelation of residuals  

Optimal adaptive kernel (number of objects)  Global R ^{2} adjusted  Min local R ^{2} adjusted  Max local R ^{2} adjusted  AICc  Moran’s I E(I) = −0.0049  p value  
22 May 2001  49  0.84  0.80  0.88  523.8  −0.0589  0.0077 
26 June 2001  22  0.88  0.76  0.91  436.3  −0.0519  0.0203 
30 July 2001  37  0.80  0.64  0.83  513.7  −0.0126  0.7057 
13 October 2001  27  0.80  0.71  0.86  447.6  −0.0189  0.4900 
03 January 2002  60  0.88  0.82  0.91  437.7  −0.0237  0.3540 
15 January 2002  62  0.79  0.69  0.81  194.9  −0.0178  0.5253 
15 February 2002  72  0.79  0.78  0.82  405.4  0.0408  0.0244 

Not smaller then assumed minimum of points required for proper local model calibration (20 points in case of this study);

The smallest possible for which the β values do not change sign over the study area and therefore can be physically explained.
The resulting kernel sizes that meet these conditions are summarized in Table 5.
The analysis of Moran’s I index of GWR residuals showed significantly better specification of GWR model in comparison to MLR. For the MLR model, a spatial autocorrelation of regression residuals was detected for all UHI cases, suggesting the misspecification of the model due to the nonstationarity of the spatial process (Table 4). After the implementation of GWR, statistically significant tendency for clustering of similar residuals was observed only for 15 February 2002 (Table 5).
Comparison between MLR and GWR model—ANOVA test
Case  Source  SS  DF  F 

22 May 2001  MLR Residuals  174.7  6.00  
GWR Improvement  120.6  61.24  
GWR Residuals  54.1  136.76  4.98  
26 June 2001  MLR Residuals  114.9  6.00  
GWR Improvement  86.5  95.85  
GWR Residuals  28.4  102.15  3.24  
30 July 2001  MLR Residuals  184.8  4.00  
GWR Improvement  120.9  42.05  
GWR Residuals  63.9  157.95  7.11  
13 October 2001  MLR Residuals  121.7  3.00  
GWR Improvement  75.4  41.85  
GWR Residuals  46.3  159.15  6.18  
03 January 2002  MLR Residuals  118.6  4.00  
GWR Improvement  68.5  25.2  
GWR Residuals  50.2  174.8  9.47  
15 January 2002  MLR Residuals  32.7  5.00  
GWR Improvement  16.5  27.95  
GWR Residuals  16.2  171.05  6.25  
15 February 2002  MLR Residuals  88.2  5.00  
GWR Improvement  38.4  26.16  
GWR Residuals  49.8  172.84  5.10 
Monte Carlo and F3 tests for local parameters estimates nonstationarity
Case  Parameter  Monte Carlo  F3 

22 May 2001  Intercept  0.110  0.004* 
Q _{A}  0.100  0.152  
P  0.030*  0.000*  
SAVI  0.330  0.985  
DSI  0.380  0.000*  
z _{0}  0.000*  0.001*  
26 June 2001  Intercept  0.020*  0.558 
a  0.700  0.001*  
NDVI  0.050*  0.000*  
P  0.000*  0.001*  
z _{0}  0.000*  0.000*  
DSI  0.890  1.000  
30 July 2001  Intercept  0.930  0.399 
a  0.930  1.000  
NDVI  0.010*  0.000*  
z _{0}  0.000*  0.000*  
13 October 2001  Intercept  0.000*  0.000* 
μ  0.000*  0.000*  
z _{0}  0.000*  0.000*  
03 January 2002  Intercept  0.030*  0.000* 
Q _{A}  0.480  0.000*  
a  0.000*  0.115  
T _{ls}  0.020*  0.000*  
15 January 2002  Intercept  0.150  0.021* 
NDVI  0.000*  0.282  
T _{as}  0.000*  0.000*  
z _{0}  0.140  0.993  
SVF  0.150  0.000*  
15 February 2002  Intercept  0.000*  0.000* 
ε  0.000*  0.000*  
NDMI  0.000*  0.000*  
NDVI  0.000*  0.002*  
T _{as}  0.010*  0.000* 
The results presented so far suggest that for all analyzed UHI cases, local geographically weighted regression models are able to significantly better describe the UHI structure than global, ordinary leastsquares models, and there are strong statistical basis that supports the application of local over global regression models for UHI spatialization.
5.2 Spatial interpolation
The GWR and MLR models, together with their extension by kriging of residuals, have been applied for interpolation of UHI. The interpolation results are evaluated with the crossvalidation procedure to quantify the possible interpolation and extrapolation error for each approach.
Crossvalidation results for the selected interpolation methods
Case  Method  BIAS  MAE  RMSE  Min  Max  Range 

22 May 2001  MLR  0.00  0.75  0.95  −3.10  2.55  5.65 
RK  0.00  0.52  0.66  −1.65  1.76  3.41  
GWR  0.08  0.68  0.86  −2.60  2.20  4.80  
GWRK  0.00  0.51  0.66  −1.78  1.72  3.50  
26 June 2001  MLR  0.00  0.60  0.77  −2.14  2.55  4.69 
RK  0.00  0.48  0.62  −1.31  2.04  3.35  
GWR  0.04  0.55  0.71  −1.85  2.21  4.06  
GWRK  0.00  0.52  0.65  −1.47  2.04  3.51  
30 July 2001  MLR  0.00  0.78  0.97  −2.83  2.62  5.45 
RK  0.00  0.60  0.75  −2.17  2.28  4.45  
GWR  0.02  0.68  0.84  −2.31  2.51  4.82  
GWRK  0.00  0.58  0.74  −2.06  2.31  4.37  
13 October 2001  MLR  0.00  0.60  0.78  −2.37  2.12  4.49 
RK  0.00  0.40  0.58  −1.95  2.06  4.01  
GWR  0.02  0.54  0.72  −2.31  1.97  4.28  
GWRK  0.00  0.44  0.58  −1.92  1.93  3.85  
03 January 2002  MLR  0.00  0.63  0.78  −1.64  2.27  3.91 
RK  0.00  0.44  0.58  −1.72  1.91  3.63  
GWR  0.00  0.56  0.70  −1.48  2.19  3.67  
GWRK  0.00  0.42  0.55  −1.91  1.91  3.82  
15 January 2002  MLR  0.00  0.34  0.41  −0.93  0.97  1.90 
RK  0.00  0.23  0.30  −0.77  0.87  1.64  
GWR  0.01  0.32  0.39  −0.91  0.89  1.80  
GWRK  0.00  0.21  0.28  −0.74  0.89  1.63  
15 February 2002  MLR  0.00  0.55  0.68  −1.62  1.80  3.42 
RK  0.00  0.40  0.51  −1.23  1.57  2.80  
GWR  0.03  0.53  0.65  −1.47  1.82  3.29  
GWRK  −0.01  0.41  0.51  −1.20  1.57  2.77 
For 22 May 2001 (Fig. 5), the application of the MLR model resulted in underestimations over the NE part of the city and in the northern part of the “peak” zone of the UHI. In contrary, MLR resulted in overestimations in SSW part of the city. This tendency is decreased while using GWR but is not entirely removed. In the case of 3 January 2002, MLR is not able to detect exterior factor, i.e., the regional NEENE wind, which can be recognized by the analysis of spatial distribution of CV errors and results in the underestimations in southern parts and overestimations in northern part of the city (Fig. 6). The GWR model works better (crossvalidation errors are smaller than for MLR) in this case and improves the quality of the interpolation, but still does not fully recognize the role of wind on shift of the UHI structure.
It should to be stressed that the incorporation of stochastic part in the process of spatialization (RK, GWRK) improves the crossvalidation results both from the quantitative and visual point of view. Statistically significant tendency to cluster similar CV errors is actually eliminated, and the zones of over or underestimation do not longer exist (Figs. 5 and 6).
6 Summary and conclusions
For the purpose of this study, a set of new potential predictors of the UHI was derived from satellite imagery (Landsat ETM+) and 3D LIDARoriginated database. Computationally intensive derivatives of these variables, including: daily sums of solar irradiation, roughness length, porosity, sky view factor, or land surface temperature did not improve the global regression model, compared to the results published earlier by Szymanowski and Kryza (2009). This leads to the conclusion that in the lack of 3D database and remotely sensed data, landuse map and its derivates are sufficient for spatial interpolation of UHI. The gain from applying more complex independent variables was significant here only in one UHI case out of seven.
With the given set of spatially continuous UHI predictors, it was not possible to propose one general regression model, build on universal subset of independent variables. This is because for various mesoscale meteorological conditions or seasons, various factors may be responsible for the spatial pattern of UHI.
The analysis of spatial autocorrelation of regression residuals with the Moran’s I index showed a statistically significant tendency for clustering of MLR residuals that meant the model was misspecified due to the nonstationarity of the spatial process for all UHI cases. For the GWR model, such misspecification was observed only in one case. That is the reason why, dedicated to nonstationary processes, local regression techniques should be used for the analysis of meteorological phenomena like UHI. This conclusion is also supported with other tests used to compare the MLR and GWR models, including AICc, R ^{2}, and ANOVA. However, the Monte Carlo and F3 tests for the significance of spatial variance of local β estimates pointed out that for two cases analyzed, a mixed GWR–MLR approach could be justified.
Locally weighted regression model (GWR) was built using the same independent variables that were used for the global MLR model, and the basic assumption was to retain, in all subareas of the city, the possibility of physical interpretation of the model (the deterministic regression model). Due to irregular distribution of the sampling points in space, the adaptive kernel (Gaussian shape) instead of the fixed one was used. The bandwidth size was selected with the iterative procedure including the analysis of corrected Akaike Information Criterion, standard deviation for the residuals, global and local determination coefficients, and local parameter estimates. The objective when choosing optimum bandwidth size was to keep it as small as possible and assure that for the entire study area, the final GWR model is physically explainable for all independent variables. The physical correctness of the regression equation is crucial if the model is applied to derive air temperature over the areas not covered with measurements, i.e., used for extrapolation.
Comparing the spatialization results achieved by the MLR and GWR techniques, one should stress that despite the maps looks similar, the latter has a strong advantage in better recognition of nonstationarity characteristics of spatial process, which was proved with various statistics above. Generally, MLR assumes constant relationships with landuse and remotely sensed derivatives, while GWR is dedicated to perform locally and the combination of local models gives better fit to observed data when an external, nonstationary process is noticeable. The goodnessoffit of the GWR model is the function of the kernel size: the smaller the kernel, the better fit is expected. There are two main reasons limiting decreasing the kernel size. First is statistical: too many independent variables for too few observations leads to the misspecification of the model. Secondly, the physical interpretation of the local model is often lost for a given predictor if too small kernel size is selected. The main assumption of our model was to assure proper deterministic relations over the study area. It was also shown, by comparison of the current and previously published results, that the incorporation of the more advanced spatial predictors does not necessarily lead to the improvement of the interpolation results, expressed in terms of crossvalidation errors. The GWR and MLR results can be significantly improved by adding the stochastic part of the process, i.e., interpolation of the regression residuals RK and GWRK procedures. The results of those procedures are similar while comparing the CV errors statistical characteristics and spatial distribution. The main reason that is decisive in recognizing GWRK as the most proper method is its statistical correctness due to unexplained (by independent variables) and nonstationary phenomena.
Notes
Acknowledgments
The authors thank Ryszard Kryza for his help in improving the manuscript and Joanna Brzuchowska, the Bureau of City Development, City Hall, Wrocław for providing 3D database for the project. The study was supported by the Polish Ministry of Science and Higher Education (science financial resources), project no. N N306 155038.
Open Access
This article is distributed under the terms of the Creative Commons Attribution Noncommercial License which permits any noncommercial use, distribution, and reproduction in any medium, provided the original author(s) and source are credited.
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