# Major and minor principal axes using kernel PCA on a 2D dataset

I have these two data sets:

x1 = [0.97955,0.7177,0.2374,-0.4374,-1.1126,-0.9457,-0.5329,0.2991,0.9846,1.2218];

y1 = [-0.1727,0.6582,0.7419,0.6433,0.3406,-0.0354,-1.0199,-0.9105,-0.6879,0.2234];


and

x2 = [1.7821,1.5386,0.4578,-0.7798,-1.5705,-1.8621,-1.2983,0.1988,1.3197,2.4700];

y2 = [-0.1231,1.4351,1.9311,1.9097,0.5310,-0.9644,-2.0165,-1.8719,-1.3210,-0.03921]


I used the Gaussian Kernel PCA algorithm with sigma = sqrt(10).

After finding my $K$ matrix, I then sorted the eigenvectors of $K$ and took the first two largest eigenvectors.

column1 [0.1528 0.1553 0.0666
-0.0741
-0.2206
-0.2197
-0.2078
-0.0366
0.1155
0.2222
0.2882
0.3119
0.1572
-0.050
-0.275
-0.4011
-0.3586
-0.1172
0.12466
0.36817]

and column2 [0.0930
-0.1122
-0.1716
-0.2018
-0.1777
-0.0861
0.15884
0.20136
0.20255
0.02489
0.13522
-0.1859
-0.3636
-0.4339
-0.2350
0.04714
0.26192
0.35898
0.33280
0.15149]


My k-PCA projection points were then calculated. I got this plot:

How do you go about finding the major and minor axes for the k-PCA projected points?

• +1 This is a sensible question. The answer is that kernel PCA maps your dataset from 2D into a space of some higher dimension and performs the regular PCA there. There are no "major and minor principal axes" in the original 2D space; those lie in the high-dimensional kernel space. In case of the Gaussian kernel this space has actually infinite dimensionality. So your question is ill-posed. Kernel PCA is nonlinear method and you should not be thinking in terms of linear projections (axes) as in regular PCA. – amoeba Oct 24 '16 at 22:26