Exercises in factor analysis   Gottfried Helms D-Kassel 2010- intro princcomp minres

Principal components by rotation

Definitions

| x1    y1    z1 |
L =     | x2    y2    z2 |
| x3    y3    z3 |

Here the factors are represented by the columns and the variables by the rows.

Actually this notation is only a mnemonic for the illustration of the general method of computation; the dimension for the factormatrix-rotation is meant as arbitrary.

Method:

iteratively rotate planes (all pairs of columns/factors in one iteration) until convergence (this is the jacobi-method for principal components)

Criterion for one plane-rotation:

maximize sum-of-squares of loadings ("ssl") in the left column of the pair and minimize it in the right column of the pair.

Example for rotation of pair of factors X,Y

squares of loadings in factors X and Y: x1²  x2²,  x3²    and     y1²  ,  y2² ,   y3²

ssl(X) =  x1² + x2² +  x3²
ssl(Y) =   y1² + y2 ² + y3²

Notation for rotated X and Y:

c = cos(φ), s = sin(φ)
X° = Xc – Ys        Y° = Yc + Xs

Criterion for each rotation:

 ssl(X°)=

is maximum and

 ssl(Y°)=

is minimum.

Derivation of the formula for determination of value of required φ

We can smooth the above formulae if we look for the maximum of the difference ssl(X°) – ssl(Y°). We have then

is maximum

For the inner term Ti for some index i we get with some rewriting:

Writing c2 for c²–s² = cos(φ)²–sin(φ)² = cos(2φ) and s2 for 2cs = 2cos(φ)sin(φ) = sin(2φ) we have

Reintroducing the sum over all indexes i and separating c2 and s2 we can now write:

For convenience we reduce notation further:

and arrive at a short notation for the function of the optimization-criterion

The first derivative:

and the second derivative

A maximum occurs where

f '(φ ) = 0                            f "(φ ) < 0

The formula for determination of the required rotation-angle φ :

The derivative                 f '(φ ) = 0

separated for the trigonometrics of the arc φ reads

or, where the expressions critx and crity are made explicite:

The arc φ (and then the required values for cos(φ) and sin(φ)) can be found by

φ = arctan( s2/c2) /2 = arctan( –crity/critx) /2
c = cos(φ)           s = sin(φ)

But we need a bit more of discussion here because of the ambiguity of signs for s2 and c2 : -s2/-c2 lead to the same t2 as +s2/+c2 but drawn in the euclidean plane (x,y) the half angle is different for both cases. So I give a more useful derivation.

From f '(φ ) = 0 follows, that with a constant m = sqrt(critx²+crity²)  we'll have with two choices for the signing of c2 and s2:

1.1)       c2 = + critx/m                    s2 = – crity/m
1.2)       c2 = – critx/m                    s2 = + crity/m

from which the formula for the second derivative can be rewritten

2.1)       f "(φ ) = –4(+ critx² + crity²)/m = –4 m
2.2)       f "(φ ) = –4(– critx² – crity²)/m = + 4 m

Since m is positive and independent of φ the choice 2.1) is required and thus from 1.1) we have now unconditionally:

c2 =    critx/m
s2 = – crity/m

If we locate a vector v2 in the euclidean plane (x,y) from the origin pointing to the coordinates (c2,s2) and halve the angle 2 φ  of v2 with the x-axis to get the vector v1 pointing to (c,s) we have s always positive and c negative if s2 was negative.

If we actually compute the final values cos(φ) and sin(φ) it happens, that we have a rotation φ a little less than π which reflects the signs of the x-coordinates. This is unnecessary – the sign of a complete factor can be chosen arbitrarily. If such a case occurs, we can simply invert the sign of the rotation-parameters cos(φ) and sin(φ) (or add π to the rotation angle).

Thus we have (in Pari/GP-notation)

{princcomp_rotplane(MAT,x,y) = local(crit,phi,c2,s2,c1,s1,m,NEWMAT);
crit = princcomp_crit(MAT,x,y); \\ this gives the vector [critx,crity]
m = sqrt( crit[1]^2 + crit[2]^2);
c2 = crit[1]/m;  s2 = - crit[2]/m ;
phi  = arg2(c2 + I*s2)/2;   \\ arg2 gives range 0..2*pi for phi
if(phi>Pi/2, phi+=Pi);   \\ prevent flipping signs in X for small effective rotation-angles
c1 = cos(phi);  s1 = sin(phi);
NEWMAT = RotatePlane(MAT,x,y,c1,s1); \\ this rotates two columns in a matrix given cos/sin of an angle
return(NEWMAT); }

The computation of the principal-components-criterion is

{princcomp_crit(MAT,x,y) = local(rs=rows(M),X,Y,crit_x=0,crit_y=0);
X = MAT[,x]; Y = MAT[,y];
for(i=1,rs,
crit_x += X[i]^2 – Y[i]^2;
crit_y += 2*X[i]*Y[i];
);
return([crit_x,crit_y]) }

The computation of the rotation of a plane in a matrix M given the cos and sin for the rotation is

{ RotatePlane(M, x, y, rotcos, rotsin) = local(X, Y);
X = M[,x]; Y=M[,y];
M[,x] = X*rotcos - Y*rotsin;
M[,y] = Y*rotcos + X*rotsin;
return(M); }

Gottfried Helms