I currently have a problem whereby I need to combine the magnitude and direction of a vector (in my case a length in mm, and angle in degrees, although can be converted to radians) into one single figure, that will not repeat itself for all lengths up to 5 mm within 360 degrees. This is for use in Statistical analysis using correlation coefficients to test against one of my output variables.
I have tried a couple of methods, however am now unsure as to whether this is mathematically possible? Additionally I have failed to find any literature that has attempted this.
I've tried to keep this message as simple as possible, however if you need any more information as to methods I've tried or more information on my study please let me know.
Looks like your vector has two components. This means you want to transform 2D objects into real numbers (1D space). Consequently, you will loose some information, and the inverse transform will not be possible (assuming you want the continuous transform). How about
Required Number = Lenght * sin(Angle/4) ?
Thank you Marek for your quick response. Do you mind be asking why the angle is divided by 4 in the equation? This might be something quite simple but I am not from a Mathematics background.
Thank you
Ashley
Marek
I have just tried that formula and like the other trigonomic functions and formulas I have tried, creates duplicates for a number of angle/length combinations and therefore would not be suitable for my analysis.
I could equally use X, Y coordinates instead of the length and angle, but have fell short myself of finding a method to combine these coordinates into one number for correlation analysis.
Ashley
Ashley
Well, when Angle changes while Length does not, then there is no repetition (and vice versa). This is why the Angle is divided by 4. But if you want the required expression to take unique value for arbitrary combination of Angle/Length then I can't see any 'nice' solution. This is because the surface f(Angle,Length) will have contour lines, not isolated points, whenever f is 'nice' (i.e. its surface is smooth or at least continuous).
There is another solution, but I'm afraid you will not like it. Namely: write Angle and Length in (finite) decimal notation, both with the same precision (number of decimal digits). Then, starting at decimal point, write the consecutive digits of Length at every odd position and digits of Angle at every even position. Repeat the procedure in front of decimal point as well. This way you will have a single number, with digits belonging to Angle, Length, Angle, Length, ... with decimal point somewhere in this sequence. This transformation is ugly, I admit, but it is reversible! More nice solution, but still in this spirit may be constructed using Peano curve or Cantor set - both even more complicated and most likely equally inapropriate for further statistical computations, I think.
One possibility is what Marek just mentioned of using x and y deciding precision for both (truncating the rest of decimals), and creating an integer out of it of the form:
Value=10^(2*decimals)*x+10^decimals*y
another possibility is creating a sufficiently dense spiral and calculating the total length along the spiral form the origin to each of your points.
both are reversible
but probably the easiest is taking x and y separately and perform multilinear regression analysis
I think that a 2 variable statistical analysis would be better.
As I understood your question, you would like to get a formula which combines 2 input variables into 1 output value. Samuel Arba Mosquera's method is good but for small change in x, the "Value" changes a lot, therefore for close points in your original 2D domain this "Value" can be very different. It may be a bad feature for statistical analysis.
An another method for combining x and y into one variables should be mixing the bits of x and y. If x and y is represented with B bit integers with digits (x1;x2;x3;...;xB) and (y1;y2;...;yB), then the "output" value should be a 2B-digit binary number: (x1;y1;x2;y2;...;xB;yB). It is a bijection between two sets, and for small change in x or y the 2B bit output changes slightly in most of the cases.
But again: I think that a 2D statistical method can be better for you.
Thank you all for your answers. I am going to trial some of the methods you suggested and will inform you as to the decision I make.
Ashley
It is possible to run a correlation with two dependent variables (length and angle which I assume are dependent) in the model. This might provide the least 'biased' answer as any transformation of inherently 2D data into a 1D variable could produce mathematical anomalies. If there is a possibility that angle and length are not independent, you might be able to use principle component analysis on your length/angle data to transform it into a single variable encompassing the largest combined variance in both features.
Hope that helps.
How about the following: r=2.52 mm, angle=1 degree => 1.252 (if you do only use integer degrees). that is, take the degree as integer part and the radius as decimal places. Of course, this has to fit the accuracy of your distances and angles. Alternatively, you could take x + i*y = z as a single complex number :)
I can't but expanding Herbert's example: The moment I write this is 2013.0708184729137 where I combine even 6 numbers ( year, month, day, hour, minute, second with 3 decimals :).
decimals are great, aren't they? :) You can even map a square to a real interval ... Of course, there are limits to this search for a 1-dim representation of a multidim-space. For storage purposes, this can be helpful, but for the purpose of the question, I am not so sure.
Thank you all for your insightful responses. For the moment I have completed multiple regression as the statistical procedure, which I deemed appropriate in the end as I had minimal issues with normality and no issues with collinearity. However I may pursue the methods to combine a vector at a later date, however the PhD has a deadline! So I have to move on.
Ashley