Regression analysis Control of built engineering objects, comparing to the plan Surveying observations – position of points Linear regression Regression plan Regression curve Least squares method Deformation analysis

Linear regression I. Correlation coefficient, tightness of relation of x and y coordinates 1. Only the y coordinates are supposed to have error x y

Linear regression I. cont.

Linear regression II. x y  v i – distance from the line

Linear regression II. cont. Moving the origin of the coordinate system to the weight point:

Regression plan

Regression polynomial Badly conditioned equation system

Distance calculation Point-line distance Point-plane distance x y t t x z y

Coordinate transformation Helmert (orthogonal)

Solution using least squares method unknowns: y A, x A, r, m Matrix form: Weight point coordinates:

Transformation using three parameters Only rotation and offset (k = 1) Unknowns: , y A, x A Correction equation is not linear, Series development

Affine transformation Different scale along the coordinate axis Two independent equation system of three unknown Weight point coordinates simplify the equation system

Polynomial transformation Used for large areas 3rd power polynomials 20 unknowns, min. 10 common points 4th power polynomials 30 unknowns, min. 15 common points 5th power polynomials 42 unknowns, min. 21 common points Weight point coordinates reduce the effect of rounding errors

Interpolation Linear interpolation Lagrange interpolation (polynomial) Spline interpolation Low order polynomials between the points Cubic spline 1 2 … n Continuous curves 1st and 2nd derivates are the same at the points (n-1) * 4 unknowns (n-1) * 2 equation (through the points) (n-2) equation for the 1st derivates (n-2) equation for the 2nd derivates +2 boundary condition