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Fit curve in space.
This is very tricky topic in data analysis. Not to say that it's special difficult but demand very clear understanding...
What is the formula of the curve in the
space? For specificity sake, what is the formula of the line in space? The
answer is surprising in the way. There is not one. There are at least
two equations needed. The line in space is the result of interception of two
planes. So two equations of a planes is enough. There are several styles of
presentations of a line in space. Most obvious approach is based on the notion
that a live is defined by coordinates of two points in space. So you need six
numbers of coordinates of two points:
The example of visual presentation of of two lines in three dimensional space are shown on the picture:
If we set only one parameter as an argument we can at once get fitting formulas for projections of these two lines in space on the planes y-x and z-x:
Instantly by going in "Result" tab and clicking on the "Copy formulas on screen" we get them in clipboard memory and then past here:
y1 = +7.4 +3.5*x
There are four coefficients defined for line. Looks like not enough... Where could we get two more? Let's calculate coordinates of two points with x=0 and x=1.
For line #1 P0(0, 7.4, -10.8) and P1(1, 10.9, -6.5); for line #2 P0(0, 56, 2.71) and P1(1, 51.6, -0.43). That is it. Had happened that projections of the line in space on two coordinates planes are quite enough. We just have to remember that x domain of lines has no limits and we select any values for x coordinates of base points.
The approximation of the nonlinear curves in space is much more difficult of course. The ultimately helpful feature of LeoStatistic is a visual presentation of correlation between parameters. This pictures can be considered as a projections of the data on the all possible combination of two parameters. If all projections of data are curves the figure in multidimensional space can be considered as a curve too.
As a example of the such task is presented on the next picture of vortex like curve:
If to set statistical scheme to "Correlation" it's very clear how this vortex like picture is built:
The actual reconstruction of the parametric equations x(t), y(t), z(t) not
difficult too much if to know that parametric equations for spiral are like
Screenshots of the LeoStatistic software:
click on picture to enlarge
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