Calculation of the tilts of curved lines

From time to time, we encounter some dependence in a form of a curve. For the sake of comparison, the tilt of the curve could be taken as its characteristics. We present two methods for calculating a ratio of tilts of the curved lines. The first one consists in singular value decomposition of the data matrix. If the variability in the data is caused by single effect, i.e., the first singular value is about 99 times larger than the second one, then the first loadings vector reflects the ratio of tilts. As the useful variants of this method, the evolving and gliding calculation on the reduced data matrices were investigated; if they give the same (overlapping) results the method is considered as adequate. The other method comprises calculating suitable cosine and sine functions of amplitudes, respectively. In this cos–sine method each vector, and consequently each curve, is represented by two coordinates (xf, yf) of a point and the whole data matrix can be visualised as a cluster of points which can be circumscribed by confidence ellipses. If the shorter half-axis in such an ellipse is negligible, then the norms of the point vectors are in ratio of curve tilts.