Correlation, Method of Least Squares, Linear Regression and the General Linear Model - Geomathematics: Theoretical Foundations, Applications and Future Developments - page 106

Keywords Correlation • Functional relationship • Least squares • Linear regression
• 95 %-confidence belts • Multiple regression • Mineral potential estimation
• Abitibi copper hindsight study • Pareto distribution • Abitibi copper deposits
4.1 Correlation and Functional Relationship
It is good to keep in mind that the linear relationship resulting from linear regression
only provides an estimate for linear functional relationship between two variables if
one of the variables is not a random variable; i.e., free of measurement error. This is
illustrated in Fig. 4.1 (based on original data from Krige 1962 ). Along the X -axis, it
shows the average value of amount of gold in inch-dwt values for panel faces in a
Witwatersrand gold mine. Along the Y-axis, similar values are plotted for panels which
are located 30 ft. (9 m) ahead of the panels whose values are plotted along the X -axis.
Each value for a panel is the average of ten single values for narrow channel samples
cut across the gold reef. The ten channel samples cover a distance of 150 ft. across the
panel along which mining proceeded. Figure 4.1 can be considered as a scattergram.
Individual values are not shown; their frequency was counted for small blocks in this
diagram. Since panel values are approximately lognormally distributed, a ratio scale is
used for the coordinate axes rather than an arithmetic scale. The correlation coefficient
amounts to 0.59. The elliptical contour shown in the diagram contains most data points
and this suggests that the bivariate frequency distribution is bivariate normal.
Fig. 4.1 Contour ellipse contains most of gold values (from Krige 1962 ); frequencies shown for
blocks. Krige's regression line ( A ) is used for prediction; line B represents solution of Kummell's
equation for linear functional relationship between two random variables ( cf . Deming 1943 ).
Kummell's method represents a special case of MLFR. The resulting line coincides approximately
with the axis of the contour ellipse because, on average, y is about equal to x (Source: Agterberg