Function mapping using a polynomial polynomial

Question

Consider that we have a data matrix data points , and we are interested in mapping these data points to higher dimensional spatial space. We can do this using d-degree polynomials. So for a sequence of data points new data matrix

I studied a relevant script (Andrew Ng. English course) that makes such a conversion for two-dimensional point data a higher object space. However, I could not understand the way of generalization in arbitrary samples of a higher size, . Here is the code:

d = 6; m = size(D,1); new = ones(m); for k = 1:d for l = 0:k new(:, end+1) = (x1.^(kl)).*(x2.^l); end end 

Can we vectorize this code? Also, given the data matrix , can you suggest a way how we can convert data points of arbitrary dimension to a higher one using the d-dimensional polynomial?

PS: A generalization of d-dimensional data points would be very useful.

Answer 1

This solution can process the variables k and generate all members of a polynomial of degree d , where k and d are non-negative integers. Most of the code length is associated with the combinatorial complexity of generating all members of a polynomial of degree d in variables k .

n_obs required per k data matrix X , where n_obs is the number of observations and k is the number of variables.

Helper function

This function generates all possible strings, so each record is a non-negative integer, and the string is summed with a positive integer:

 the row [0, 1, 3, 0, 1] corresponds to (x1^0)*(x1^1)*(x2^3)*(x4^0)*(x5^1) 

Function (which can almost certainly be written more efficiently):

 function result = mg_sums(n_numbers, d) if(n_numbers<=1) result = d; else result = zeros(0, n_numbers); for(i = d:-1:0) rc = mg_sums(n_numbers - 1, d - i); result = [result; i * ones(size(rc,1), 1), rc]; end end 

Initialization code

 n_obs = 1000; % number observations n_vars = 3; % number of variables max_degree = 4; % order of polynomial X = rand(n_obs, n_vars); % generate random, strictly positive data stacked = zeros(0, n_vars); %this will collect all the coefficients... for(d = 1:max_degree) % for degree 1 polynomial to degree 'order' stacked = [stacked; mg_sums(n_vars, d)]; end 

Final Step: Method 1

 newX = zeros(size(X,1), size(stacked,1)); for(i = 1:size(stacked,1)) accumulator = ones(n_obs, 1); for(j = 1:n_vars) accumulator = accumulator .* X(:,j).^stacked(i,j); end newX(:,i) = accumulator; end 

Use either method 1 or method 2.

Final step: Method 2 (requires that all data in the X data matrix be strictly positive (the problem is that if you have 0 elements, -inf does not propagate properly when you call matrix algebra routines.)

 newX = real(exp(log(X) * stacked')); % multiplying log of data matrix by the % matrix of all possible exponent combinations % effectively raises terms to powers and multiplies them! 

Run example

 X = [2, 3, 5]; max_degree = 3; 

The folded matrix and its polynomial term:

  1 0 0 x1 2 0 1 0 x2 3 0 0 1 x3 5 2 0 0 x1.^2 4 1 1 0 x1.*x2 6 1 0 1 x1.*x3 10 0 2 0 x2.^2 9 0 1 1 x2.*x3 15 0 0 2 x3.^2 25 3 0 0 x1.^3 8 2 1 0 x1.^2.*x2 12 2 0 1 x1.^2.*x3 20 1 2 0 x1.*x2.^2 18 1 1 1 x1.*x2.*x3 30 1 0 2 x1.*x3.^2 50 0 3 0 x2.^3 27 0 2 1 x2.^2.*x3 45 0 1 2 x2.*x3.^2 75 0 0 3 x3.^3 125 

If the data matrix X is [2, 3, 5] , this correctly generates:

 newX = [2, 3, 5, 4, 6, 10, 9, 15, 25, 8, 12, 20, 18, 30, 50, 27, 45, 75, 125]; 

Where the first column is x1 , the second is x2 , the third is x3 , 4th is x1.^2 , 5th is x1.*x2 , etc.