Geographically Weighted Regression Technique for Spatial Data ...

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Geographically Weighted Regression Technique for Spatial Data Analysis Chang-Lin Mei School of Science, Xi’an Jiaotong University (E-mail: [email protected])

1. Introduction In many practical fields such as geography, economics, environmental science and epidemiology, the data are generally related with the geographical locations where they are observed. This type of data is called spatial data. How to analyze spatial dada has long been one of very important problems in statistics. In today’s speech, I will first introduce a recently developed spatial data analysis method called geographically weighted regression (GWR) technique which is originally proposed by Brunsdon et al (1996; 1998). Then some work of ours on this technique are summarized. Furthermore, sseveral problems that need to be further studied will be discussed. 2. Geographically weighted regression model and its fitting method 2.1 The geographically weighted regression model

Motivated by the idea of nonparametrical regression methods, Brunsdon et al (1996; 1998) have proposed a so-called geographically weighted regression (GWR) technique for exploring spatial non-stationarity of a regression relationship for spatial data by locally fitting a spatially varying coefficient regression model of the form yi =

p X

βj (ui , vi )xij + εi , i = 1, 2, · · · , n,

(1)

j=1

where (yi ; xi1 , · · · , xip ) are observations of the response y and explanatory variables x1 , x2 , · · · , xp at location (ui , vi ) in the studied geographical region, βj (ui , vi )(j = 1, 2, · · · , p) are p unknown functions of geographical locations and εi (i = 1, 2, · · · , n) are error terms with mean zero and common variance σ 2 . βj (ui , vi )(j = 1, 2, · · · , p) 1

are locally estimated at each location (ui , vi ) by the weighted least-squares procedure in which some distance-decay weights are used. Each set of the estimated coefficients at n locations can produce a map of variation which may give useful information on non-stationarity of the regression relationship. The GWR technique has a great appeal in analysis of spatial data and has been successfully applied to many practical problems. The main results on this topic are summarized in Fotheringham et al (2002). 2.2 The fitting method—geographically weighted regression technique

The parameters in the GWR model are locally estimated by the weighted least squares approach. The weights at each location (ui , vi ) are taken as a function of the distance from (ui , vi ) to other locations where the observations are collected. Suppose that the weights at location (ui , vi ) are wj (ui , vi ), j = 1, 2, · · · , n. Then the parameters at location (ui , vi ) is estimated by minimizing n X

wj (ui , vi )[yj − β1 (ui , vi )xj1 − β2 (ui , vi )xj2 − · · · − βp (ui , vi )xjp ]2

j=1

Let

   X=  

x11 x12 x21 x22 .. .. . . xn1 xn2

· · · x1p · · · x2p . . . .. . · · · xnp





    , Y =     

y1 y2 .. .

   .  

yn

and W(ui , vi ) = Diag[w1 (ui , vi ), w2 (ui , vi ), · · · , wn (ui , vi )]. Then according to the theory of the weighted least squares, the estimated parameters at (ui , vi ) are ˆ i , vi ) = [XT W(ui , vi )X]−1 XT W(ui , vi )Y β(u Let xT i = (xi1 , xi2 , · · · , xip ) be the ith row of X. Then the fitted value of y at (ui , vi ) is obtained by T T −1 T ˆ yˆi = xT i β(ui , vi ) = xi [X W(ui , vi )X] X W(ui , vi )Y.

ˆ = (ˆ Denote respectively by Y y1 , yˆ2 , · · · , yˆn )T and εˆ = (ˆ ε1 , εˆ2 , · · · , εˆn )T the vector of fitted values of y and the vector of residuals at n locations (ui , vi ), i = 1, 2, · · · , n. 2

Then

(

where



ˆ = LY; Y ˆ = (I − L)Y, εˆ = Y − Y

(2)

T −1 T xT 1 [X W(u1 , v1 )X] X W(u1 , v1 )

    xT [XT W(u , v )X]−1 XT W(u , v ) 2 2 2 2 L= 2  ..  . 

T −1 T xT n [X W(un , vn )X] X W(un , vn )

        

is an n × n matrix and I is an identity matrix of order n. 2.3 Choices of the weights

The function of the weights is to place different emphases on different observations in generating the estimated parameters. In spatial analysis, observations close to a location (ui , vi ) generally exert more influence on the parameter estimates at location (ui , vi ) than those farther away. When the parameters at location (ui , vi ) are estimated, more emphases should be placed on the observations which are close to location (ui , vi ). Like the weights in nonparametric regression, one obvious choice is wj (ui , vi ) = exp[−(dij /h)2 ], j = 1, 2, · · · , n, where dij is the distance from the location (ui , vi ) to (uj , vj ) and h is called bandwidth. Another choice of the weights is as follows. (

wj (ui , vi ) =

2

[1 − (dij /h)2 ] , if dij ≤ h , 0, if dij > h

j = 1, 2, · · · , n.

The bandwidth h can be determined the cross-validation procedure. Let ∆(h) =

n X

(yi − yˆ(i) (h))2 ,

i=1

where yˆ(i) (h) is the fitted value of yi with the observation at location (ui , vi ) omitted from the fitting process. Choose h0 as a desirable value of the bandwidth such that ∆(h0 ) = min∆(h). 3. Testing for global linear regression based on the geographically weighted regression technique (Leung, Mei and Zhang, 2000a) 3

The geographically weighted regression technique provides a feasible way for testing a global linear regression relationship for spatial data. This amounts to test the following hypotheses: H0 : βj (ui , vi ) = βj , j = 1, 2, · · · , p versus H1 : At least one of the βj (ui , vi )’s is varying with the locations. 3.1 Construction of the test statistic

We construct the test statistic based on the residuals sum of squares respectively obtained under H0 and H1 . Under the H0 , we fit the corresponding linear regression model by ordinary least squares approach and obtain the residuals sum of squares as RSS(H0 ) = YT (I − H)Y,

(3)

where H = X(XT X)−1 XT . Under the H1 , the spatially varying coefficient regression model (1) is fitted by the geographically weighted regression technique and we obtain the residuals sum of squares as RSS(H1 ) = εˆT εˆ = YT (I − L)T (I − L)Y.

(4)

The test statistic is then constructed as F =

RSS(H0 ) − RSS(H1 ) YT (M0 − M1 )Y = , RSS(H1 ) Y T M1 Y

(5)

where M0 = I − H and M1 = (I − L)T (I − L). 3.2 Calculation of p-value of the test

Since larger value of T tends to support H1 , the p-value of the test is p0 = PH0 (F > f ),

(6)

where f is the observed value of the test statistic F . It is observed that LX = X and E(Y) = Xβ when H0 is true, where β = (β0 , β1 , · · · , βp )T . We then obtain that RSS(H1 ) = εT M1 ε and that RSS(H0 ) = εT M0 ε. Therefore, under the null hypothesis H0 , F can be expressed as F =

εT (M0 − M1 )ε , εT M1 ε 4

and p0 can be calculated by Ã

εT (M0 − M1 )ε p0 = P >f εT M1 ε

!

= P{εT [M0 − (1 + f )M1 ]ε > 0}.

(7)

That is, the p-value has been expressed as the probability that a ratio of quadratic forms takes positive value. If we assume the error vector ε ∼ N (0, σ 2 I), we can obtain both the exact and the approximate formulae for calculating p0 . 3.2.1 The exact formula Theorem 3.1 Suppose that the error terms ε1 , ε2 , · · · , εn are independent and identically distributed random variables with common distribution N (0, σ 2 ). Then 1 1 Z ∞ sin[θ(t)] p0 = PH0 (F > f ) = + dt, 2 π 0 tρ(t) where

   θ(t)  

=

1 2

(8)

Pm

−1 k=1 [hk tan (λk t)],

2 2 hk /4 ρ(t) = Πm , k=1 (1 + λk t )

λ1 , λ2 , · · · , λm are the distinct nonzero eigenvalues of the matrix M0 − (1 + f )M1 and h1 , h2 , · · · , hm are their respective orders of multiplicity. 3.2.2 Three-moment χ2 approximation Computing numerically the eigenvalues of an n × n matrix and an integral on an infinite interval is in fact computationally expensive. Some approximate methods are available in this case. Here, we introduce a so-called three-moment χ2 approximation method to compute the p-value of the test. This approximate method can significantly reduce the computational overhead. The main idea of this approximation is to approximate the distribution of a quadratic form in normal variables by that of a linear function of a χ2 variable with appropriate degrees of freedom, say a + bχ2d . The coefficients a and b of the linear function and the degrees of freedom d are chosen in such a way that the first three moments of a + bχ2d are made to match those of the quadratic form. Theorem 3.2 Suppose that the error terms ε1 , ε2 , · · · , εn are independently and identically distributed as N (0, σ 2 ). If three-moment χ2 approximation is used to approximate the p-value p0 , then we have p0 ≈

 2  P(χ  µ d > d − h),  

Φ √tr[M0 −(1+f )M1 ]

 2tr[M0 −(1+f )M1 ]2    P(χ2 < d − h), d

if tr[M0 − (1 + f )M1 ]3 > 0;



, if tr[M0 − (1 + f )M1 ]3 = 0; 3

if tr[M0 − (1 + f )M1 ] < 0. 5

(9)

where

    d=    h=

{tr[M0 −(1+f )M1 ]2 }3 , {tr[M0 −(1+f )M1 ]3 }2 tr[M0 −(1+f )M1 ]2 tr[M0 −(1+r)M1 ] . tr[M0 −(1+f )M1 ]3

4. Testing for spatial autocorrelation among the residuals of the geographically weighted regression (Leung, Mei and Zhang, 2000b) Similar to the case in the ordinary linear regression, spatial autocorrelation in error terms can invalidate the standard assumption of homoscedasticity of the error terms and mislead the results of statistical inference. Therefore, developing some statistical methods to test for spatial autocorrelation in the error terms of model (1) is a very important issue. Here, the well known two statistics, Moran’s I and Geary’ C in the regional science are used to explore spatial autocorrelation among the residuals of the geographically weighted regression. let εˆ = (ˆ ε1 , εˆ2 , · · · , εˆn )T = (I − L)Y be the residual vector obtained by fitting the spatially varying coefficient model ¯ = (w¯ij )n×n be a specific spatial weight matrix with the GWR technique and W which is defined by the underlying spatial structure such as the spatial contiguity or adjacency between the geographical units where observations are observed. After neglecting a constant coefficient, the Moran’s I and Geary’s C of the residuals with ¯ = (w respect to W ¯ij )n×n are respectively Pn

i=1

I= and

Pn

C=

i=1

Pn

¯ij εˆi εˆj j=1 w Pn ˆ2i i=1 ε

Pn

εi j=1 wij (ˆ Pn 2 ˆi i=1 ε

− εˆj )2

εˆT Wεˆ = T , εˆ εˆ

=

¯ εˆ εˆT (D − 2W) , εˆT εˆ

(10)

(11)

where D = Diag(w¯1· + w¯·1 , w¯2· + w¯·2 , · · · , w¯n· + w¯·n ) and w¯i· =

Pn

j=1

w¯ij , w ¯·i =

Pn

j=1

w¯ji .

The p-values for testing spatial autocorrelation are respectively pI = PH0 (I > I0 ) and pC = PH0 (C > C0 ), 6

(12)

where I0 and C0 are respectively the observed values of I and C. When the bias of the fitted value of y at each location is negligible, the Moran’s I and Geary’s C can be respectively expressed as I=

¯ ¯ εT NT WNε εT NT (D − 2W)Nε , C = , εT NT Nε εT NT Nε

(13)

where N = I − L. If we assume that the error vector ε ∼ N (0, σ 2 I), we can calculate the p-values with the same methods introduced in Sections 3.2.1 and 3.2.2. Along with the same derivation as above, Leung, Mei and Zhang (2003) have proposed a approach for testing local patterns of spatial association based on the recently proposed local statistics of local Moran’s Ii , local Geary’s Ci and Anselin’s LISA. 5. Mixed geographically weighted regression model In consideration of the situations where certain explanatory variables influencing the response may be global in nature, whist others are local, Brunsdon et al (1999) have proposed a mixed geographically weighted regression (MGWR) model in which some coefficients in the model (1) are assumed to be constant and the others are allowed to vary across the studied region. After re-ordering the explanatory variables, a MGWR model is specified as yi =

q X

βj xij +

j=1

p X

βj (ui , vi )xij + εi , i = 1, 2, · · · , n.

(14)

j=q+1

By taking xi1 = 1 or xi,q+1 = 1 for all i, the model can involve a constant or a spatially varying intercept. 5.1 Identification of constant coefficients in a MGWR model (Mei, He and Fang, 2004)

When a MGWR model is applied to analyze a real-world data set, one should first determine which coefficients can be kept fixed and which ones cannot. For a given k(1 ≤ k ≤ p), to test whether or not the coefficient βk (ui , vi ) of the kth explanatory variable xk is constant across the geographical region amounts to test the following hypotheses (

H0 : βk (u1 , v1 ) = βk (u2 , v2 ) = · · · = βk (un , vn ), H1 : not all βk (ui , vi ) (1 ≤ i ≤ n) are equal, 7

Firstly, fit the data to the spatially varying coefficient model (1) and let ˆ i , vi ) = (βˆ0 (ui , vi ), βˆ1 (ui , vi ), · · · , βˆp (ui , vi ))T = [XT W(ui , vi )X]−1 XT W(ui , vi )Y β(u be the estimated coefficient vector at location (ui , vi ). The n estimated values of the kth coefficient βj (ui , vi ) at the n locations where the data are observed are βˆk (uj , vj ) = ek [XT W(ui , vi )X]−1 XT W(ui , vi )Y, j = 1, 2, · · · , n,

(15)

where ek is a column vector of p dimensions with unity for the kth element and zero for others. Let βˆk = (βˆk (u1 , v1 ), βˆk (u2 , v2 ), · · · , βˆk (un , vn ))T .

(16)

When neglecting the constant 1/n, the sample variance of βˆk (uj , vj ), j = 1, 2, · · · , n can be expressed as ¶

µ

µ



1 1 V (k) = βˆkT I − J βˆk = YT BT I − J BY, n n where



T −1 T eT k [X W(u1 , v1 )X] X W(u1 , v1 )

    eT [XT W(u , v )X]−1 XT W(u , v ) 2 2 2 2 B= k  ..  . 

T −1 T eT k [X W(un , vn )X] X W(un , vn )

(17)

     ,   

and J is an n × n matrix with unity for each of its elements. The test statistic is constructed as F (k) =

YT BT (I − n1 J)BY βˆkT (I − n1 J)βˆk = . εˆT εˆ YT (I − L)T (I − L)Y

(18)

The p-value of F (k) is p(k) = PH0 [F (k) > f (k)],

(19)

where f (k) is the observed value of F (k). Under the null hypothesis and some conditions, we have F (k) = where

εT BT (I − n1 J)Bε εT M1 ε = , εT (I − L)T (I − L)ε εT M2 ε

   M1  

= BT (I − n1 J)B

M2 = (I − L)T (I − L). 8

Therefore, the p-value can be calculated by the same methods introduced in sections 3.2.1 and 3.2.2. 5.2 Estimation and inference on the MGWR model (Mei, Wang and Zhang, 2004) 5.2.1 Estimation of the MGWR model

After the constant coefficients in a MGWR model are identified, we can estimate both the constant coefficients and spatially varying coefficients which are important to reflect the spatial nonstationarity of the regression relationship. Brunsdon et al (1999) have proposed an iterative estimation method based on the back-fitting procedure. However, this method is computationally expensive. Motivated by the approach in Speckman (1988) for fitting a partially linear model, we propose the following estimation method which can significantly reduce the computational overhead. Let    Xc =   

and

x11 x12 x21 x22 .. .. . . xn1 xn2    βc =   

· · · x1q · · · x2q . .. . .. · · · xnq β1 β2 .. .





     , Xv =     





     , βv (ui , vi ) =     

x1,q+1 x1,q+2 x2,q+1 x2,q+2 .. .. . . xn,q+1 xn,q+2 βq+1 (ui , vi ) βq+2 (ui , vi ) .. .

βq

· · · x1p · · · x2p . .. . .. · · · xnp





    , Y =     

y1 y2 .. .

   ,  

yn

    , i = 1, 2, · · · , n.  

βp (ui , vi )

Firstly, we rewrite the MGWR model (4) as y˜i = yi −

q X j=1

βj xij =

p X

βj (ui , vi )xij + εi , i = 1, 2, · · · , n.

j=q+1

Using the GWR technique, we obtain the estimated spatially varying coefficients at location (ui , vi ) as βˆv (ui , vi ) = (βˆq+1 (ui , vi ), βˆq+2 (ui , vi ), · · · , βˆp (ui , vi ))T −1 T ˜ = (XT v W(ui , vi )Xv ) Xv W(ui , vi )Y,

where ˜ = (˜ Y y1 , y˜2 , · · · , y˜n )T = Y − Xc βc . 9

(20)

Then, substituting the elements of βˆv (ui , vi ) into the original MGWR model (4) and rewrite it as yi −

p X

βˆj (ui , vi )xij =

j=q+1

q X

βj xij + εi , i = 1, 2, · · · , n.

(21)

j=1

Because  Pp j=q+1

βˆj (u1 , v1 )x1j

j=q+1

βj (un , vn )xnj

   Pp  fv =  j=q+1 βˆj (u2 , v2 )x2j  ..  .  Pp ˆ





ˆ xT v1 βv (u1 , v1 )

        T ˆ  =  xv2 βv (u2 , v2 )   ..   .  

ˆ xT vn βv (un , vn )

     ˜ = Sv (Y − Xc βc ),  = Sv Y   

(22)

equation (21) can be expressed with the matrix notation as Y − Sv (Y − Xc βc ) = Xc βc + ε or (I − Sv )Y = (I − Sv )Xc βc + ε. According to the ordinary leat-squares method, we obtain the estimates of the constant coefficients as T −1 T T βˆc = (βˆ1 , βˆ2 , · · · , βˆq )T = [XT c (I − Sv ) (I − Sv )Xc ] Xc (I − Sv ) (I − Sv )Y. (23)

Substituting βˆc into (20), we finally obtain the estimated spatially varying coefficients at location (ui , vi ) as −1 T ˆ βˆv (ui , vi ) = [XT v W(ui , vi )Xv ] Xv W(ui , vi )(Y − Xc βc ), i = 1, 2, · · · , n.

(24)

Then according to (22), the fitted values of the spatially varying coefficient part at n locations are fv = Sv (Y − Xc βˆc ).

(25)

Therefore, the fitted values of the response at n locations are ˆ = (ˆ Y y1 , yˆ2 , · · · , yˆn )T = fv + Xc βˆc = Sv (Y − Xc βˆc ) + Xc βˆc = Sv Y + (I − Sv )Xc βˆc = SY,

10

(26)

where T T −1 T S = Sv + (I − Sv )Xc (XT c (I − Sv ) (I − Sv )Xc ) Xc (I − Sv ) (I − Sv ).

(27)

Here, we suggest a generalized cross-validation method for selecting the value of the bandwidth which can reduce the computational overhead considerably. In order to clearly show the dependence between the fitted values of the response and the bandwidth h, we write (26) as ˆ Y(h) = (ˆ y1 (h), yˆ2 (h), · · · , yˆn (h))T = S(h)Y, where S(h) is shown in (27) for the back-fitting method or in equation (18) for the two-step method. Let GCV (h) =

n X

Ã

i=1

yi − yˆi (h) 1 − sii (h)

!2

,

where sii (h) is the ith diagonal element of S(h) and yˆi (h) is the ith fitted value of y. Select h0 as a desirable value of h such that GCV (h0 ) = minh>0 GCV (h).

(28)

5.2.2 An inference framework of the MGWR model

We shall describe in this section a framework of statistical inference on the MGWR model. Considering that, like a partially linear model, the constant coefficients in a MGWR model are frequently of primary interest because of their explanatory power, we henceforth mainly focus the inference on the constant coefficients to illustrate the inference framework. One of the important inference problems is whether or not certain variable in the constant coefficient part is statistically significant. This amounts to test the following hypotheses H0 : βk = 0 vs H1 : βk 6= 0, for some 1 ≤ k ≤ q. We first fit the full MGWR model (4) (that is, under H1 ) by the method proposed before and denote by S1 the hat matrix in (27). Then the residual sum of squares under H1 is RSS(H1 ) = (Y − S1 Y)T (Y − S1 Y) = YT (I − S1 )T (I − S1 )Y = YT R1 Y, (29) 11

where R1 = (I − S1 )T (I − S1 ). We then fit the reduced MGWR model under H0 (that is, let βk = 0 in the model (4)) with the same method and the same value of bandwidth as those under H1 . Denote by S0 the resulted hat matrix. Then the residual sum of squares under H0 is RSS(H0 ) = (Y − S0 Y)T (Y − S0 Y) = YT (I − S0 )T (I − S0 )Y = YT R0 Y, (30) where R0 = (I − S0 )T (I − S0 ). If H0 is indeed true, there should not be significant difference between RSS(H0 ) and RSS(H1 ). Otherwise, RSS(H0 ) − RSS(H1 ) will tend to be larger. Therefore it is natural to propose the test statistic as T =

RSS(H0 ) − RSS(H1 ) YT (R0 − R1 )Y = . RSS(H1 ) Y T R1 Y

(31)

Here,we introduce a bootstrap procedure to derive the p-value of the test as follows. Step 1. Fix the bandwidth at some properly given value, say h∗ , and respectively fit the MGWR model under H0 and H1 by the aforementioned estimation method. Then calculate the residual sums of squares RSS(H1 ) and RSS(H0 ) in (29) and (30) as well as the observed value t of statistic T in (31). Furthermore, obtain the residual vector εˆ = (ˆ ε1 , εˆ2 , · · · , εˆn )T = Y − S1 Y under H1 and compute εˆic = εˆi −

1 n

Pn

j=1

εˆj for i = 1, 2, · · · , n, to form the centered residual vector

εˆc = (ˆ ε1c , εˆ2c , · · · , εˆnc )T . Step 2. Draw a bootstrap sample ε∗ = (ε∗1c , ε∗2c , · · · , ε∗nc )T with replacement from εˆc = (ˆ ε1c , εˆ2c , · · · , εˆnc )T . Let Y∗ = S0 Y + ε∗ and T ∗ = (Y∗ )T (R0 − R1 )Y∗ /(Y∗ )T R1 Y∗ .

(32)

Step 3. Repeat step 2 for B times and obtain a bootstrap sample of the statistic T as T1∗ , T2∗ , · · · , TB∗ . The bootstrap p-value of the test is p∗ = #{Ti∗ ; Ti∗ ≥ t}/B, 12

(33)

where t is the observed value of T obtained from step 1 and #A represents the number of elements in the set A. Extensive simulations demonstrate that the proposed test method with the bootstrap procedure for deriving the p-value of the test are quite accurate and powerful. 6. Some problems in future research 1. Up till now, the studies on the GWR technique all assume that the error terms in the model (1) are independent and identically distributed random variables. Generally, the error terms are spatial correlated. The influence of the spatially correlated error terms on the GWR technique needs to be further investigated. Furthermore, when the error terms follow some specific forms of spatial correlation such as spatial ARMA process, how to apply the GWR technique to explore spatial non-stationarity of the data remains to be studied. 2. Spatial-temporal data analysis is more useful in practice, because most of data sets in, for example, economics, environmental science and epidemiology are related to not only the geographical locations but also the time. It is an interesting topic to apply the GWR technique to analyze this kind of data sets. One of the possible way for the study is to assume the coefficients in the model (1) are functions of both spatial location and time. But how to efficiently deal with the problem of ”curse of dimensionality” may be an important issue in the study.

References Brunsdon C, Fotheringham A S, Charlton M, 1996, “Geographically weighted regression: a method for exploring spatial nonstationarity” Geographical Analysis 28 281–298 Brunsdon C, Fotheringham A S, Charlton M, 1998, “Geographically weighted regression — modelling spatial nonstationarity” The Statistician 47 431–443 Brunsdon C, Fotheringham A S, Charlton M, 1999, “Some notes on parametric significance tests for geographically weighted regression” Journal of Regional Science 39 497–524 13

Fotheringham A S, Brunsdon C, Charlton M, 2002 Geographically Weighted Regression— the Analysis of Spatially Varying Relationships, Wiley, Chichester Leung Y, Mei C L, Zhang W X, 2000a, “Statistical tests for spatial nonstationarity based on the geographically weighted regression model” Environment and Planning A 32 9–32 Leung Y, Mei C L, Zhang W X, 2000b, “Testing for spatial autocorrelation among the residuals of the geographically weighted regression” Environment and Planning A 32 871–890 Leung Y, Mei C L, Zhang W X, 2003, “Statistical test for local patterns of spatial association” Environment and Planning A 35 725–744 Mei C L, He S Y, Fang K T, 2004, “A note on the mixed geographically weighted regression model” Journal of Regional Science 44 143–157 Mei C L, Wang N, Zhang W X, 2004, “Estimation and inference on mixed geographically weighted regression model”, to appear in Environment and Planning A. Speckman P, 1988, ”Kernel smoothing in partial linear model” Journal of the Royal Statistical Society, Series B 50 413–436

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