Multivariate second order polynomial regression python
I am dealing with multivariate regression problems. My dataset is something like X = (nsample, nx) and Y = (nsample, ny). nx and ny may vary based on different dataset of different case to study, so they should be general in the code.
I would like to determine the coefficients for the multivariate polynomial regression minimizing the root mean square error. I thought to split the problem in ny different regressions, so for each of them my dataset is X = (nsample, nx) and Y = (nsample, 1). So, for each depended variable (Uj) the second order polynomial has the following form:
I coded the function in python as:
def func(x,nx,pars0,pars1,pars2):
y = pars0 #pars0 = bias
for i in range(nx):
y = y + pars1[i]*x[i] #pars1 linear coeff (beta_i in the equation)
for j in range(nx):
if (j < i ):
continue
y = y + pars2[i,j]*x[i]*x[j]
#diag pars2 = coeff of x^2 (beta_ii in the equation)
#upper triangle pars2 = coeff of x_i*x_k (beta_ik in the equation)
return y
and the root mean square error as:
def resid(nsample,nx,pars0,pars1,pars2,x,y):
res=0.0
for i in range(nsample):
y_pred = func(nx,pars0,pars1,pars2,x[i])
res=res+((y_pred - y[i]) ** 2)
res=res/nsample
res=res**0.5
return res
To determine the coefficients I thought to use scipy.optmize.minimize but it does not work example_1 example_2. Any ideas or advices? Should I use sklearn?
-> EDIT: Toy test data nx =3, ny =1
0.20 -0.02 0.20 1.0229781
0.20 -0.02 0.40 1.0218807
0.20 -0.02 0.60 1.0220439
0.20 -0.02 0.80 1.0227083
0.20 -0.02 1.00 1.0237960
0.20 -0.02 1.20 1.0255770
0.20 -0.02 1.40 1.0284888
0.20 -0.06 0.20 1.0123552
0.24 -0.02 1.40 1.0295350
0.24 -0.06 0.20 1.0125935
0.24 -0.06 0.40 1.0195798
0.24 -0.06 0.60 1.0124632
0.24 -0.06 0.80 1.0131748
0.24 -0.06 1.00 1.0141751
0.24 -0.06 1.20 1.0153533
0.24 -0.06 1.40 1.0170036
0.24 -0.10 0.20 1.0026915
0.24 -0.10 0.40 1.0058125
0.24 -0.10 0.60 1.0055921
0.24 -0.10 0.80 1.0057868
0.24 -0.10 1.00 1.0014004
0.24 -0.10 1.20 1.0026257
0.24 -0.10 1.40 1.0024578
0.30 -0.18 0.60 0.9748765
0.30 -0.18 0.80 0.9753220
0.30 -0.18 1.00 0.9740970
0.30 -0.18 1.20 0.9727272
0.30 -0.18 1.40 0.9732258
0.30 -0.20 0.20 0.9722360
0.30 -0.20 0.40 0.9687567
0.30 -0.20 0.60 0.9676569
0.30 -0.20 0.80 0.9672319
0.30 -0.20 1.00 0.9682354
0.30 -0.20 1.20 0.9674461
0.30 -0.20 1.40 0.9673747
0.36 -0.02 0.20 1.0272033
0.36 -0.02 0.40 1.0265790
0.36 -0.02 0.60 1.0271688
0.36 -0.02 0.80 1.0277286
0.36 -0.02 1.00 1.0285388
0.36 -0.02 1.20 1.0295619
0.36 -0.02 1.40 1.0310734
0.36 -0.06 0.20 1.0159603
0.36 -0.06 0.40 1.0159753
0.36 -0.06 0.60 1.0161890
0.36 -0.06 0.80 1.0153346
0.36 -0.06 1.00 1.0159790
0.36 -0.06 1.20 1.0167520
0.36 -0.06 1.40 1.0176916
0.36 -0.10 0.20 1.0048287
0.36 -0.10 0.40 1.0034699
0.36 -0.10 0.60 1.0032798
0.36 -0.10 0.80 1.0037224
0.36 -0.10 1.00 1.0059301
0.36 -0.10 1.20 1.0047114
0.36 -0.10 1.40 1.0041287
0.36 -0.14 0.20 0.9926268
0.40 -0.08 0.80 1.0089013
0.40 -0.08 1.20 1.0096265
0.40 -0.08 1.40 1.0103305
0.40 -0.10 0.20 1.0045464
0.40 -0.10 0.40 1.0041031
0.40 -0.10 0.60 1.0035650
0.40 -0.10 0.80 1.0034553
0.40 -0.10 1.00 1.0034699
0.40 -0.10 1.20 1.0030276
0.40 -0.10 1.40 1.0035284
0.40 -0.10 1.60 1.0042166
0.40 -0.14 0.20 0.9924336
0.40 -0.14 0.40 0.9914971
0.40 -0.14 0.60 0.9910082
0.40 -0.14 0.80 0.9903772
0.40 -0.14 1.00 0.9900816
After some trial and error, I finally came up with a solution. The problem can be seen as linear using a change of variables. I used scikit-learn to build the model. After some tests on real cases it works really well