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# Python - Coeff (scores) [[-2.809 0.097 0.244 0.050] [-1.834 0.286 0.010 -0.135] [-0.809 0.963 -0.341 0.078] [-0.155 -1.129 0.548 0.026] [0.707 -0.723 -0.736 -0.024] [1.830 -0.290 -0.157 0.030] [3.070 0.796 0.431 -0.026]] # m1e <- empca(x=B1, w=B1wt, ncomp=4) # Un-sweep the eigenvalues to compare to python results # R round( sweep( m1e$scores, 2, m1e$eig, "*"), 3) PC1 PC2 PC3 PC4 G1 -2.809 0.097 -0.244 0.050 G2 -1.834 0.286 -0.010 -0.135 G3 -0.809 0.963 0.341 0.078 G4 -0.155 -1.129 -0.548 0.026 G5 0.707 -0.723 0.736 -0.024 G6 1.830 -0.290 0.157 0.030 G7 3.070 0.796 -0.431 -0.026 # Matlab - P (scores) 0.5590 0.0517 0.2210 0.2910 0.3650 0.1520 0.0095 -0.7840 0.1610 0.5120 -0.3080 0.4530 0.0309 -0.6010 0.4950 0.1510 -0.1410 -0.3850 -0.6640 -0.1380 -0.3650 -0.1540 -0.1420 0.1760 -0.6110 0.4230 0.3890 -0.1490 # R round(m1e$scores, 3) PC1 PC2 PC3 PC4 G1 -0.559 -0.052 0.221 -0.291 G2 -0.365 -0.152 0.009 0.784 G3 -0.161 -0.512 -0.308 -0.453 G4 -0.031 0.601 0.495 -0.151 G5 0.141 0.385 -0.664 0.138 G6 0.365 0.154 -0.142 -0.176 G7 0.611 -0.423 0.389 0.149

# Python with initial Identity matrix [[2.791 0.125 0.325 -0.035] [1.528 -0.989 -0.211 0.172] [0.990 -0.651 -0.117 -0.186] [0.159 1.463 0.530 0.020] [-0.628 0.862 -0.730 -0.032] [-1.738 0.406 -0.139 -0.071] [-2.917 -0.712 0.520 -0.047]] Eigvec (loadings) [[-0.309 -0.839 -0.298 0.300] [-0.502 0.014 0.154 -0.615] [-0.470 -0.086 0.766 0.219] [-0.441 0.521 -0.236 0.615] [-0.487 0.128 -0.496 -0.326]] # R R> m2e <- empca(x=B2, w=B2wt, ncomp=4, seed=NULL) # # Un-sweep the eigenvalues to compare to python results R> round( sweep( m2e$scores, 2, m2e$eig, "*"), 3) PC1 PC2 PC3 PC4 G1 -2.791 0.216 -0.356 0.066 G2 -1.528 -0.942 0.187 -0.150 G3 -0.990 -0.620 0.101 0.207 G4 -0.159 1.472 -0.522 -0.032 G5 0.628 0.844 0.744 0.021 G6 1.738 0.351 0.161 0.050 G7 2.917 -0.808 -0.493 0.019 R> round( m2e$loadings, 3) PC1 PC2 PC3 PC4 E1 0.309 -0.839 0.298 -0.300 E2 0.502 0.014 -0.154 0.615 E3 0.470 -0.086 -0.766 -0.219 E4 0.441 0.521 0.236 -0.615 E5 0.487 0.128 0.496 0.326

Python code by @bailey2012principal, retrieved 1 Mar 2019 from https://github.com/sbailey/empca .

The Python code is difficult to read in places for a person [like me] not well-versed with Python. Three examples:

- It is not clear what values
`k`

takes in`for k in range(self.nvec)`

. - Gram-Schmidt orthogonalization is accomplished with a pair of nested
`for`

loops instead of a function. - The
`Model`

class structure makes it a bit tricky to figure out what objects have actually been modified inside a function.

The Python code iterates these two EM steps:

- Calculate the coefficient matrix C.
- Calculate ALL
`ncomp`

principal components P simultaneously (iterate each to convergence). Orthogonalize P.

For a complete-data problem, python and R give similar results. Note the `Coeff`

matrix in python does NOT have eigenvalues swept out of the columns.

For the missing-data problem, the python results are somewhat different from R.

Matlab code by Vicente Parot, retrieved 1 Mar 2019 from https://www.mathworks.com/matlabcentral/fileexchange/45353-empca.

The Matlab code feels similar to R.

The Matlab code calculates principal components sequentially, one at time. For each principal component, the algorithm iterates between these two steps:

- Calculate C[,h]
- Calculate P[,h]

While this is a type of EM algorithm, it is not the algorithm described by Bailey (2012) and is considered further.

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