		More Information
	Back to Aabel 3 Overview Back to Aabel Statistics

Polynomial Trend Surface Analysis

Trend surface analysis is in most respects similar to normal regression analysis. The peculiarity with trend surface analysis is that the two independent variables represent two perpendicular spatial dimensions, and the dependent variable represents a regionalized variable e.g., elevation.

The trend surface analysis methods in Aabel can be used to e.g., derive a continuous smooth surface from irregular data or isolate regional trends from local variations.

Polynomial Trend Surface Analysis of XYZ Data

Polynomial Trend Surface Analysis of Matrix Data

Matrix Cellwise Operations

Polynomial Trend Surface Analysis of XYZ Data

The method requires three numeric variables representing X, Y, Z.

The analysis output includes the calculated trend grid, the XYZ estimates and residuals, and an ANOVA report for significance of regression of K^th-order polynomial trend surface.
The right-hand side image below is a matrix plot of 5^th-order trend generated from XYZ data representing subsurface structural elevations of top of Pennsylvanian Lansig-Kansas City Group (source of XYZ data: Davis, J. C., 2002). The left -hand side image is an XYZ contour diagram of the original data. The ANOVA report is shown for significance of regression of 5^th-order polynomial trend surface.

A Graphical Example of Applying Trend Surface Analysis to XYZ Data:
(a) Original XYZ Surface, and (b) the 5^th-Order Trend Generated Surface

The Corresponding Polynomial Trend Surface 5^th-Order Regression ANOVA

Polynomial Trend Surface Analysis of Matrix Data

The method requires two matrices each stored in a separate worksheet.

The analysis output includes the calculated trend and residual grids, the XYZ estimates and residuals, and an ANOVA report for significance of regression of K^th-order polynomial trend surface.
The example below shows a contour matrix map (left), the corresponding 6^th-order trend (middle), and the residual grid (right).

A Graphical Example of Applying Trend Surface Analysis to Matrix Data:
(a) Original Surface, (b) the 6^th-orderTrend Generated Surface, and (c) the Residual Matrix

The Corresponding Polynomial Trend Surface 6^th-Order Regression ANOVA

Matrix Cellwise Operations

These operations are performed cellwise (and must not be confused with matrix arithmetic). For example, cells of one matrix are added or subtracted from similar cells of another matrix.

If the dimensions of two matrices are dissimilar, the smallest overlapping dimension is used for the resulting matrix.