[SciPy-User] affine transformation - what's going on ?

[Warren Weckesser]

On Fri, Jun 3, 2011 at 8:20 AM, <josef.pktd at gmail.com> wrote:

> I'm puzzling for hours already what's going on, and I don't understand
> where my thinko or bug is.
>
> I *think* an affine transformation should return the same count in an
> inequality.
> x is (nobs, 3)
> a is (3)
> mu and A define an affine transformation
>
> Why do the following two not give the same result? The first is about
> 0.19, the second 0.169
>
> print (x<a).all(-1).mean()
>
> print (affine(x, mu, A) < affine(a, mu, A)).all(-1).mean()
>
>

An affine transformation will not necessarily preserve
the ordering of the components of two vectors.

Here's a counterexample:

In [92]: Q = array([[2, -1.5],[0,0.5]])

In [93]: Q
Out[93]:
array([[ 2. , -1.5],
       [ 0. ,  0.5]])

In [94]: x1 = array([1.0, 1.0])

In [95]: x2 = array([0.9, 0.1])

In [96]: x1 > x2
Out[96]: array([ True,  True], dtype=bool)

In [97]: dot(Q, x1)
Out[97]: array([ 0.5,  0.5])

In [98]: dot(Q, x2)
Out[98]: array([ 1.65,  0.05])

In [99]: dot(Q,x1) > dot(Q,x2)
Out[99]: array([False,  True], dtype=bool)



Warren



>
> full script below and in attachment
> -------------
> import numpy as np
>
> def affine(x, mu, A):
>    return np.dot(x-mu, A.T)
>
> cov3 = np.array([[ 1.  ,  0.5 ,  0.75],
>                   [ 0.5 ,  1.5 ,  0.6 ],
>                   [ 0.75,  0.6 ,  2.  ]])
>
> mu = np.array([-1, 0.0, 2.0])
>
> A = np.array([[ 1.22955725, -0.25615776, -0.38423664],
>               [-0.        ,  0.87038828, -0.26111648],
>               [-0.        , -0.        ,  0.70710678]])
>
> x = np.random.multivariate_normal(mu, cov3, size=1000000)
> print x.shape
>
> a = np.array([ 0. ,  0.5,  1. ])
>
> print (x<a).all(-1).mean()
> print (affine(x, mu, A) < affine(a, mu, A)).all(-1).mean()
>
> '''
> with 100000
> (100000, 3)
> 0.19185
> 0.16837
>
> with 1000000
>
> (1000000, 3)
> 0.191597
> 0.168814
> '''
> ------------------
>
> context: I'm transforming multivariate normal distributed random
> variables, and my cdf's don't match up.
>
> Can anyone help figuring out where my thinking or my calculations are
> wrong?
>
> Josef
>
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>
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