Adam OlsenRegression Analysis Fall 201812/12/17Final Paper Analysis: SpatialAutocorrelation from Least Squares Regression In this paper we will analyze an open access paper aboutthe topic of Spatial Autocorrelation and how it can be used to test theresiduals from a Least Squares Regression equation. The scholastic paper waswritten by Yanguang Chen from Peking University in China. The editor of thepaper was Guy J-P. Schumann who was a professor at the University ofCalifornia, Los Angeles. We are going to be examining the implications ofSpatial Autocorrelation techniques and how it may be able to provide a moreaccurate answer than the Durbin-Watson test which was covered in our textbook.The overall paper has interesting connections for examining cross-sectionaldata stemming from random spatial sampling. The beginning of this paper first describes theimplementations of least squares regression and how it can be used to describereal world systems.
In this part of the paper Chen states, “By means ofregression modeling, we can explain the causes of effects or predict theeffects with causes” (Chen 1). I believe this is a respectable statement on hisbehalf on what regression modeling can produce. Chen then goes into discussinghow a good regression model would look like and how the residuals of the modelwould be a random series and not have autocorrelation. Autocorrelation is theidea that there is a correlation between your predictors variables or elementswhen the same regression is used over a specific period.
The way scientists goaround this problem however to analyze the residuals and test forautocorrelation is the Durbin-Watson test. However, this test is only usefulwhen the regression analysis is only in a one-dimensional field. This is do tothe fact that the Durbin-Watson test only contains a one-dimensionalautocorrelation coefficient. The paper states, “the aim of this study is todevelop simple methods to test residuals of regression analysis based onspatial data from a new point of view” (Chen 2). Chen will be looking at otheralternatives for analyzing autocorrelation of residuals when it comes tomultidimensional regression analysis. The next part of Chen’s paper goesinto discussing the models and methods of the regression equation and showingthe deficiency of the Durbin-Watson test. He begins by describing the multivariablelinear equation as follows: (1)Itis the same equations as a linear regression equation we learned in class, howeverthe residuals are slightly changes and must satisfy a set of conditions. TheDurbin-Watson test is described as follows: (2)When the Durbin-Watsontest statistic is close to two, then the residuals can be considerednon-auto-correlating.
The Durbin-Watson test however will only providemeaningful results when the data being analyzed from the regression equation istime series or an ordered spatial series. When you perform a regressionanalysis on data that is “cross-sectional from spatial random samples, theresiduals will form a space series” (Chen 4) and make the Durbin-Watson testnull. The next section of Chen’s paper discusses methods to be able to testrandom serial correlation. From the perspective of Chen, thebest way to approach this problem is to implement Moran’s index.
Moran’s indexis “a measure of spatial autocorrelation that is characterized by a correlationin a signal among nearby locations and heavily used in geography” (Chen 5).Chen first uses a series of residuals from predicted values and standardizesthem with the following equation: (3)Youthen will be able be able to create a special weights matrix once you have therandom sampling points. Then the spatial autocorrelation coefficient can becalculated by the following formula: (4)From this point we candevelop mathematical alternative forms that may be easier to compute. Astatistical measurement will contain two segments of data one that is based onthe population and the other that is based on samples. For determining aprinciple component analysis theory set we mustbe able to base that theory on sample data size. A way to construct the spatialcontiguity matrix is: (5)Thiscan then be used to develop of a new set of indices where we are able to testserial correlation which is as follows: (6)The approximate residualcorrelation index can then be calculated: (7)When you compare theDurbin-Watson test statistic to the approximate residual correlation index, youcan see some of the similarities between them.
The Durbin-Watson test containsa one order time lag, whereas the approximate residual correlation indexcontains a spatial weight function. The next part of the scholastic paperwritten by Yanguang Chen describes a case study of testing for spatialauto-correlation in regression. In his case study he applies the autocorrelation analysisto the relationship of urbanization and economic development. The paperprefaces that a nonlinear function can be used to model the relationship; but alinear equation will be able to produce an approximation. The case study willbe analyzing 31 provinces in China and will contain two variables in theregression. The first variable is known as the level of urbanization and thesecond variable is gross regional product which is similar to GDP. The level ofurbanization can be calculated as the proportion of urban population vs thetotal population in a region.
The first step in Chen’s case study is to developthe regression equation. The equation will result in the residual andstandardized residuals. The next step in solving for autocorrelation is computingthe Moran’s index.
To calculate the residual correlation index however, thereare a few steps needed. The first step in this process is to standardize theresidual vector and then calculate the spatial weighted matrix that willreplace the normal one-dimensional regression equation. From there Chencomputes the spatial autocorrelation index and finally from there calculatedthe residual correlation index. The next test Chen provides is the ‘Test forserial correlation for linear regression analyses” (). Chen in his second section states”The correlation between the level of urbanization and level of economicdevelopment is currently a hot topic in China” (Chen 12-13).
In this section hecreates a linear regression model that fits his data set with the following equation: (8)This is easily done in Rwhich will be provided in a separate file. From his model he assessed hisr-squared value which provided a high “goodness of fit”. From there he went onto use the Durbin-Watson statistic to be able to obtain the spatialautocorrelation index and the residual correlation index. One item that isnoteworthy is that if the elements are rearranged the Durbin-Watson value ischanged as well but the residual correlation index remains the same. This is doto the fact that the weight function generates a residual correlation indexthat is unique in nature.
Chen also discusses in his paper on the third sectionthat you can transform the regression equation such that it is a log linearrelation and the spatial autocorrelation analysis can be applied to thosemodels as well. One of the last sections of this paper discusses thebasic framework of his methods and how to apply them to different situationsother than the economic development of China. This methodology can beillustrated best by Figure 4 (Chen 15) in the scholastic paper. The first stepis to analyze a spatial data set using regression analysis. The next step is tostandardize the residuals and create the spatial weight matrix. From this pointyou can calculate Moran’s index and find the index for the residual correlation.
In the last step you calculate the spatial Durbin-Watson statistic and then usethe test of residuals of serial correlation. However, one must also realizethere are deficiencies when it comes to any model or test and we must look atwhat those might be. The first deficiency in this method must deal with theweight functions and how they are used in the regression. There are only fourmain types of weight functions in geographical analysis and the problem lieswith determining which function to use. It is very difficult to pick a weightfunction because one must know the physical meanings behind the differentfunctions and the statistician will need to have a lot of background knowledgeof the subject. Another huge challenge as with any statistical modeling is thequality of the data provided. These techniques are not known to work as wellwith big data methodologies. Overall, this paper has provided some new insightsinto knowledge about autocorrelation and how it is used in different areasother than pure statistics, a more applied arena.
The conclusion of this paperends with three main points that were produced by Chen. The first is thatspatial autocorrelation can be used to test serial correlation of least squaresregression residuals. The next point is that the testing of residualcorrelation when it pertains to a spatial random series can be constructed inmore than one way.
The last point Chen makes is that the Durbin-Watson tables canbe adopted for testing the autocorrelation of spatial serial data. References1.