24 Trend Surface Analysis: A Technique for land-Surface Interpretation of Geographical Attributes

Prof Surendra Singh

 1.0  Introduction:

Trend Surface Analysis (TSA) originally introduced by the Earth scientists and Geo-mathematician during the 1960s and early 1970s when Georges Matheron, a French Geo-statistician,authored book ‘ Les variables regionalises et leur estimation’ in 1965 introducing stochastic process to minimize variance of surface in relation to input data. Later on, Watson (1971) developed new dimension of TSA. A set of residuals computed through the regression technique for analyzing land surface of particular attribute was used to consider its surface position. In course of time, a number of applications of TSA added many subjects and now it has become the subject matter of goe-statistics to understand the spatial trend of a particular geographical attribute. In due course of time, methods of TSA were modified from handling of simple surface data to compute complex multi-variate analysis of the surface of geographic attributes. Mapping the surface of geographical attribute mathematically is the prime aim to understand the relatively large scale systematic change(Fig-1).Through mapping, the essential non-systematic small scale variations occurring due to local effects can be depicted. Development of Geographical Information Systems (GIS) tool and computation of geographical data of land surfaces through the use of coordinates and grid systems have provided new dimensions for TSA to interpret the results of complex surface data.

Fig.-1: Three- Dimensional Land surface Feature and its Intepolated contours for the Trend Surface Analysis

A number of points need to compute the local coverage of surface featurein a better way to interpolate a weighted attribute than a single function of distance used through the coordinates of points (point geometry) and its error (variation) term. Such aspects are associated with uncerteinity and interpolated values of attributes. The related issues were addressed by Georges Matheron and D.G. Krige (a South African mining engineer) to develop suitable method of interpolation. It was developed later a geo-statistical interpolation known as ‘Kriging’ method as modules of GIS. Irregular (random) variation of a variable over land surface was systemetised through Kriging technique by obtaining residual error term.

We should describe details about the methods of TSA in the coming part of discussion.Our aim here is two-fold: we would describe the techniques of TSA and its associated methods which are to support for development of its procedure and,secondly, the example with exercise for deeps and clear understanding of the uses of TSA in geographical studies.

2.0 How Do WeUse Trend Surface Analysis?

In fact, TSA now-a- days has become tool to understand geo-statistics of residuals (error) in the distribution of a particular attribute. It is used in various ways in the surface analysis. Some of the following methodological aspects are associated with its use in different subjects.

2.1 TSA as an Approach:

Of course, it follows a set procedure to analysis the residuals of an attribute. For consideration of interacted regressed dependent variable (Y) with explanatory ones (X1X2, X3, …, Xn), we have to estimate dependent variable (Ŷ) which deviates from observed one (Y). One tries by using ‘least square’ method through the data of Y and Xn thereby minimizing the ‘sum of squares’ for ∑(Y- Ŷ) (details on regression analysis are given in the other module entitled‘Multiple regression analysis through matrix algebra’). In TSA, we use coordinate of n number of points ( Xi, Yi), where the data of a surface attribute of interest, Z, for n points are collected as independent variable and attribute Z is dependent on these point coordinates in order to understand the trend of Z subject to its land surfacepoint coordinates. It is to note here that Z may be the linear function of X and Y as Z = f( x,y). But in many cases, Z may notbe the linear function of its point coordinates and varies in more complicated manner. It may followquadratic or higher degree polynomials; the nature and characteristics of such polynomials are given somewhere else in ‘non-linear functions and curve fitting in module entitled ‘Non-Linear Geographical Distribution: A Curve-Fitting Technique’. In two-dimensional feature of a polynomial function derived from multiple-linear regression model, a simplest case of curvilinear surface analysisis chosen. Its coordinate form expressed as :

f( x,y) = ∑(b, x,y).                                …                    …… (1)

The forms of the different polynomial functions are as the following

Flat	: b0
Linear	: b0	+ b1X + b2Y, and
Quadratic	: b0	+ b1X + b2Y + b3X2 + b4 XY + b5 Y2.

In these forms of polynomial functions of an attribute Z, the coefficient are normally chosen to minimize as ∑(Z-Zc) where Zc is predicted value from function. In this process of regression analysis of an attribute Z subject to its point-surface coordinates (x,y), the TSA deals with the spatial pattern of Z attribute in order to make a logical classification of the residual values of surface points. Positive residuals above and negative residuals below the predicted surface must be the same in space or volume. However, it provides a procedure for separating the relatively large scale systematic changes in coordinate based mapped data from essentially irregular small-scale variation due to local effects. TSA approaches to show the variation of local phenomena occurring over land surface in its deeper understanding systematically.

The use of specific degree of polynomial function is used as per the surface variation of phenomena ( or considered attribute). R2 that is ‘test of fit’ is used to understand the degree of surface variation and to determine the form of polynomial function. For instance, if variation of an attribute over space is less; it means surface is almost flat, a linear function of polynomial form may be used. But, it is not true for complex surface features. The trend surfaces of plain region follow linear functions while hilly tract (if Z is elevation attribute) and X,Y are coordinate of Z point) is shown by quadratic equation of second degree or so.

2.1.1 The Significance of Trend Surface:As trend surface is essentially based on residual analysis,the surface features of an attribute are mapped. It is dependent on its significance test. As order of polynomial function is increased to show the attribute pattern over surface, R2 increases in consequence of increasingits degree and the surface-pattern becomes more clear. The degree of freedom, d.o.f., for the derivation from the regression is [(n-1)-m] where m = degree of polynomial. The analysis of variance must provide the variation test (called variance ratio, F-test) that is dependent on sum of squares, d.o.f. and mean squares for linear regression is given below (Table-1).

Table-1: Analysis of Variance for Linear Regression

     Source: Burrough and Mcdonell(1998).

2.2 TSA as a Method:

It includes the interpolation method for interpretation of surface features of a geographical attribute and also a method of residual-identification for the analysis of phenomenal variation. There are many methods of interpolating the data of given prints. For example, classification of entities in distribution and specification of equidistant classes for contour mapping of a given attribute are dimensions used in TSA. ‘Thiessen Polygon’ is most accurate method which is being used now-a-days for interpolation of surface trends through the use of GIS in digital cartography. Recently, geo-statistical method using the method of spatial autocorrelation for understanding spatial clusters of a variable is used through compute aid ( Burrough and McDonell 1998). This method is called ‘Kriging’ which requires understanding of spatial autocorrelation as described in detail in the module entitled ‘Spatial Autocorrelation –Measurements with Examples’.

On the whole, interpolation through classification required to show attributes variability over space is the main aim of this method. This method is used here in the last part of discussion to analyze the trend surface of West Bengal state for the variation occurring in agricultural productivity in the State.

Secondly, TSA is method to map ‘error’ or residual term data of a distribution of geographical attribute. Though it is the subject matter of ‘regression analysis’ and given somewhere else to find error term given in regression equation, however, it is important to highlight here that mapping (either by using isopleth or choropleth method) of residuals distribution is the essential part of TSA. One should be aware that isopleths mapping is based on interpolation method, while choropleth is dependent on classified sheds/ tints for showing trend surfaces. Interpolation is used in two ways as: first, to consider the whole area is view and interpolate to draw iso-lines with in the area. Secondly, to consider the local area interpolation and expanding iso-lines out side the local area to fit them maximum possible.

3.0  Procedure of Solving Multi-dimensional Linear Equation for Trend Surface Analysis:

Of course, least square method for finding the coefficient of multi-dimensional linear equation is applicable in two ways:

3.1  The Usual Matrix Method:

Since multi- (or minimum three-) dimensional land surface features based on linear equations are more complicated and have three parts of equations, the solution can be obtained by using matrix algebra. For example, we have linear equations with a, b, c as coefficients and d as constants:

a1X + b1 Y + c1 Z = d1

a2X + b2 Y + c2 Z = d2

a3X + b3 Y + c3 Z = d3

In order to proceed for solution of these given equation, it has three matrices, namely, (i) the [A] which is called coefficient matrix( 3 x3 matrix in the present case )that includes a1,a2,a3,b1,b2,b3 and c1,c2,c3 coefficients of equations in the same sequence; (ii) the [B] of column Vector ( 3×1 in the present case ) of all constants; d1,d2 andd3 and (iii) the outputvariable vector[X] of obtaining the result (the value of x,y and z). The given simultaneous equationscan be written in matrix form as:

[A].[X] = [B] and the solution vector will be;            ………….. (2)

[X] = A-1. B.                                                                ……………….(3)

In this equation A-1 = A inverse matrix, that is computed through applying a standard matrix procedure, given in module on Matrix. A-1 is an square matrix ( 3 x3 in the present case ) and B is a column vector. The product will also give a column vector giving the solution for [X] i.e. first value of it will give the value of x, second value will give the value of y and the third will give the value of z.

  For solving the above simultaneous equations we can also apply Cramer’s Rule ( given in standard books on matrix algebra).

This procedure is used here to derive Trend Surface Analysis of agricultural productivity pattern of the plains of West Bengal State considering three dimensional features of land surface.

4.0 Example and Exercise:

A data set of the coordinates (x,y) of thirteen points as district Headquarters of the Bengal plains was generated and the agricultural productivity attribute is considered as third dimension of these points (z) to show the trend surfaces of agricultural productivity. The statistics of x,y and z variables are given in Table-2. The numerical procedure of TSA is given below.

Table-2 : District-wise Agricultural Productivity of the Plains of West Bengal State (2006-07, at constant price of 2000-01)

4.1 Calculation ofCoefficients and Constants in the Form of Equations and Generation of Concerned Matrices, [A] and [B] from the Given Data:

A set of normal (called simultaneous) equations was used to make ‘best fit solution’ through the use of statistical procedure. The statistical form of these equations is:

an+b∑X + c∑Y = ∑Z

a∑X+ b∑X2 + c∑XY = ∑XZ

a∑Y+ b∑XY + c∑Y2 = ∑YZ, where n= number of observations.

These equations are written in their numerical forms by adopting a statistical procedure by generating additional columns as given inTable-3.

Table-3: Statistical Procedure for Generation of Simultaneous Equations for Trend Surface Analysis

From the Table-3, the numerical form of the above equations are arranged in the following manner by using x,y and z as variables and a,b,c as coefficients

The solution can also be found by using the Cramer’rule which is given below:

Simultaneous Fraction Procedure (Application of Cramer’s Rule):

It is a short cut method of obtaining the elements of [X] matrix through calculating determinants of the matrices generated through simultaneous equations. Suppose, we have matrices [A], [B] and [X]. The elements of coefficient matrix [A] are written in matrix form replacing column of this matrix with the elements of column matrix [B]. Then the procedure of solving it is now simplier than the earlier one. The steps of this procedure are given below

     Step-III : Calculate determinant of Matrices Ax, Ay and Az.

Step-IV: Find solution dividing the determinants of replaced matrices by the determinant of x = ( Ax/Ad), y= ( Ay/Ad) and z= (Az/Ad). [A] matrix as

This procedure is used here to derive Trend Surface Analysis of agricultural productivity pattern of the plains of West Bengal State considering three dimensional features of land surface.

x= Ax/Ad = 1082,796/597,048 = 1.813583

y = Ay/Ad = 115,104/ 597,048 = 0.192789

z = Az/Ad = 18,542/597,048 = 0.031056. Both the methods give identical results and are same.

Thus, the multiple-linear equation for the TSA of agricultural productivity is derived as

Zc = 1.8136 + 0.19279 X + 0.031056 Y .                        …                            … (4)

The computed values of agricultural productivity Zc of each point (district headquarters) are arranged with its observed values Z and the residuals (Z-Zc) were calculated (Table-4).

Table-4: To Derive the Residuals and the Index of Goodness-of-fit of the Trend Surface

N.B.: Z* denotes the mean

Interpretation:

On order to analyse the surface trends of agricultural productivity in the plains of West Bengal, we have to interpret the trend surface equation which after calculating its parameters in the preceding part, is derived as

Zc = 1.8136 + 0.19279 X + 0.031056 Y             ,            ……(5)

with R2 = 3.95569/8.31344 = 0.47582 (i.e, 47.58% appro). It means the X-coordinate (East-West) has higher gradient (0.1928) of the spatial trend of agricultural productivity than the Y-coordinate (North-South direction). So the linear trend of productivity distribution follows a direction from West to East as shown through trend surface line (Figs-2 and 3). It is to note here that the degree of determination (R2) is weak despite the negligible differences of 0.361 thousand Rs/ha in this case (Fig-3). So, instead of agricultural productivity of linear trend, a polynomial of higher degree may be used to show more correct depiction of trend lines.

Fig.- 2: District Headquarters of the Plains of West Wengal State at their Coordinate System

Fig.-3: Derived Trend Surfaces of Agricultural productivity using Multiple Linear Regression (values in 10 thousand Rs per hectare)

Further, the trend of productivity residuals (Z-Zc) is also shown by mapping it through using ‘interpolation’ method. The distribution shows the spatial deviations of the predicted trend surface values (Zc). Since general trend of productivity distribution is from West to East direction in the study area, the residual variations also follow the same trend. The Eastern and Northern parts of West Bengal have two areas (Maldah in the North and Nadia area of the central East) of very high variability and high values of productivity residuals ( more than 6 thousnd Rs/ha). On the other hand, the mid-western parts ( Bakura and Midnipur) of the state have negative values of productivity residuals (Fig-4).

Fig.-4: Surface Trends of Agricultural Productivity Residuals (Z- Zc) (values in 10 thousand Rs per Hectare)

    Summary:

Trend Surface Analysis provides a strong base of mapping the systematic variation of geographic phenomena. Although it is an extension of regression residuals (error-term) analysis, it is viewed in context with analyzing the land surface variation by geo-physists and geomorphologists. Geo-statisticians developed the complex parametric relationship to make this tool more accurate and effective. Interpolation and extrapolation are methods used for mapping the trend surfaces of a particular attribute. However, now-a-days, social scientists especially economists used it as multi-variate technique to replace landsurface geometric coordinates (x,y) with the multi-dimensional independent variables to analyse variability of complex spatial features of geographic phenomena in order to explain ‘cause-effect’ based paradigm of land surface features. The ‘Matlab’ is software developed by mathematicians to map surface features of multi-dimensional variables. It is now being used by the scientists of different disciplines for trend surface analysis. In present module, a simple single-dimensional trend surface features of agricultural productivity emerging in the plains of West Bengal state are analysed by adopting multiple-linear regression technique in context with using geometric coordinates showing land surface (x,y) and agricultural productivity (z) as third dimension of land surfaces.

References

• Pal, S.K. (1998): Statistics for GeoScientists, Concept Pub. Co. , New Delhi
• Burrough P.A. and Mc Donald RA (1998): Principles of Geographical Information Systems, Oxford University Press, U.K.
• Higucji Y. (1995): Lectures on Multivariate Analysis for Human Settlements Planning, Deptt. Of Mechanical and Environmental Informatics, Graduate School of Information Science and Engineering Tokyo Institute of Technology, Tokya.
• WWW.jstor.org/discover/10.2307/20000596?sid. 
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• Singh, Manjeet (1997): Mathematics, Dhanpat Rai & Co., Delhi

• Mahmood, Aslam (1997): Statistical Method in Geographical Studies, Rajesh Pub., New Delhi.