
A technique for fitting a generalized surface to a set of geographical point data. The method also encompasses evaluating the degree of fit, representing the surface with isolines, and mapping the residuals, that is, the vertical deviations between the observed data values and the bestfit surface. Because identification and removal of an obvious trend can yield a more meaningful map, the residuals are often of greater interest than the trend surface itself (Chorley and Haggett, 1965).
Trend surface analysis is usually a form of multiple regression in which dependent variable Z is a function of orthogonal geographical coordinates X and Y (Swan and Sandilands, 1995, pp. 290300). The simplest configuration is the linear trend surface, a plane represented by the equation:Â Z = bo + b1X + b2Y
in which bo is the Z intercept and slope coefficients b1 and b2 describe the plane\'s inclination along the X and Y axes. Trend surfaces are easily mapped because the polynomial can be evaluated readily at any point. If the contour interval is constant, a linear trend surface yields a map of parallel, uniformly spaced straightline contours. Multiple regression measures the degree of fit with the coefficient of determination R2, which ranges from 0.0 for data with no discernible trend to 1.0 for a surface passing through every data point. Addition of quadratic, cubic or higherorder terms affords a more flexible surface and a better fit (see figure). The quadratic polynomialÂ Z = bo + b1X + b2Y + b3X2 + b4XY + b5Y2allows a single warp, which can represent a single peak, pit, ridge or valley, whereas the cubic polynomialÂ Z = bo + b1X + b2Y + b3X2 + b4XY + b5Y2 + b6X3 + b7X2Y + b8XY2 + b9Y3,provides a second warping, which can accommodate a peak and a pit. Still higher orders of trend surface are possible, at least in theory, as long as the number of terms in the polynomial does not exceed the number of data points. In practice, rounding error and other computational difficulties obviate the exact fit of higherorder trend surfaces to large data sets (Unwin, 1975).
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trend surface analysis four orders of trend surface, based on the date of arrivalof each country\'s first settler, describe the generalized advance of the settlement frontier in New York State
Trend surface analysis supports hypotheses testing as well as exploratory pattern analysis. Whereas a researcher using trend surface analysis as a confirmatory tool might specify the order or number of warpings â€” for example, a quadratic surface to model a phenomenon declining uniformly with distance from a single peak â€” the analyst looking for interesting patterns (see exploratory data analysis) might examine the relationship between flexibility and fit for a range of trend surfaces. As a rule of thumb, a surface merits visual examination if the number of data points is at least three times greater than the number of terms in the polynomial and the fit represents a marked improvement over the next lowerorder surface. Researchers willing to ignore underlying assumptions of inferential statistics might use the analysis of variance to assess a surface\'s statistical significance (see significance test). Geographers who base trend surfaces on area data can use Monte Carlo simulation to explore the effects of representing areal units by arbitrarily assigned data points.Â (MM)
References Chorley, R.J. and Haggett, P. 1965: Trend surface mapping in geographical research. Transactions, Institute of British Geographers 37: 4767.Â Swan, A.H.R. and Sandilands, M. 1995: Introduction to geological data analysis. Oxford: Blackwell.Â Unwin, D.J. 1975: An introduction to trend surface analysis, Concepts and techniques in modern geography, no. 5. Norwich: Geo Abstracts Ltd. 
