[Numpy-discussion] Sampling from the multivariate normal

[Robert Kern]

On Wed, Nov 9, 2011 at 16:40, Joshua Anthony Reyes
<joshua.reyes at gmail.com> wrote:
> Hi List,
>
> I'm new to Numpy and I'm a little confused about the behavior of
> numpy.random.multivariate_normal(). I'm not sure I'm passing the
> variances correctly. My goal is to sample from a bivariate normal, but
> the kooky behavior shows up when I sample from a univariate distribution.
>
> In short, the multivariate normal function doesn't seem to give me
> values in the appropriate ranges. Here's an example of what I mean.
>
> In[1]: from numpy.random import normal, multivariate_normal
>
> In [2]: normal(100, 100, 10)
> Out[2]:
> array([ 258.62344586,   70.16378815,   49.46826997,   49.58567724,
>         182.68652256,  226.67292034,   92.03801549,   18.2686146 ,
>          94.09104313,   80.35697507])

Here, you are asking for a Gaussian distribution with a mean of 100
and a stdev of 100 (or equivalently, a variance of 10000).

> The samples look about right to me. But then when I try to do the same
> using the multivariate_normal, the values it draws look too close to the
> mean.
>
> In [3]: multivariate_normal([100], [[100]], 10)

Here you are asking for a Gaussian with a mean of 100 and a variance
of 100 (or equivalently, a stdev of 10). Note the differences between
the two signatures:

  np.random.normal(mean, stdev)
  np.random.multivariate_normal(mean, covariance)

> Out[3]:
> array([[ 109.10083984],
>        [  97.43526716],
>        [ 108.43923772],
>        [ 97.87345947],
>        [ 103.405562  ],
>        [ 110.2963736 ],
>        [ 103.96445585],
>        [ 90.58168544],
>        [  91.20549222],
>        [ 104.4051761 ]])
>
> These values all fall within 10 units of the mean.
>
> In [4]: multivariate_normal([100], [[1000]], 10)
> Out[4]:
> array([[  62.04304611],
>        [ 123.29364557],
>        [ 83.53840083],
>        [ 64.67679778],
>        [ 127.82433157],
>        [  11.3305656 ],
>        [  95.4101701 ],
>        [ 126.53213908],
>        [ 104.68868736],
>        [  32.45886112]])
>
> In [5]: normal(100, 1000, 10)
> Out[5]:
> array([ 1052.93998938, -1254.12576419,   258.75390045,   688.32715327,
>           -2.36543806, -1570.54842269,   472.90045029,   394.62908014,
>          137.10320437,  1741.85017871])
>
> And just to exaggerate things a little more:
>
> In [6]: multivariate_normal([100], [[10000]], 10).T][0]
> Out[6]:
> array([ 274.45446694,   85.79359682,  245.03248721,  120.10912405,
>         -34.76526896,  134.93446664,   47.6768889 ,   93.34140984,
>          80.27244669,  229.64700591])
>
> Whereas
> In [7]: normal(100, 10000, 10)
> Out[7]:
> array([  -554.68666687,   3724.59638363, -14873.55303901,  -3111.22162495,
>        -10813.66412901,   4688.81310356,  -9510.92470735, -12689.02667559,
>        -10379.01381925,  -4534.60480031])
>
> I'm shocked that I don't get some negative values in Out[4]. And Out[6]
> really ought to have some numbers in the thousands.
>
> I'd totally be willing to believe that I don't understand the
> multivariate normal and/or variance. Can someone tell me whether these
> numbers look sane?
>
> For the bivariate case I do something like this:
>
> means = [100, 100]
> variances = [100, 1000]
> Sx, Sy = variances
> sx, sy = map(sqrt, variances)
> cor = 0.7
> cov = [[Sx, cor*sx*sy], [cor*sy*sx, Sy]]
> draws = 10
> samples = multivariate_normal(means, cov, draws)
>
> As mentioned before, the samples are shockingly close to their means.

They look right to me.

[~]
|19> means = [100, 100]

[~]
|20> variances = [100, 1000]

[~]
|21> Sx, Sy = variances

[~]
|22> sx, sy = map(sqrt, variances)

[~]
|23> cor = 0.7

[~]
|24> cov = np.array([[Sx, cor*sx*sy], [cor*sy*sx, Sy]])

[~]
|26> samples = np.random.multivariate_normal(means, cov, 10000)

[~]
|27> cov
array([[  100.        ,   221.35943621],
       [  221.35943621,  1000.        ]])

[~]
|28> np.cov(samples.T)
array([[  101.16844481,   222.00301056],
       [  222.00301056,  1001.58403922]])

-- 
Robert Kern

"I have come to believe that the whole world is an enigma, a harmless
enigma that is made terrible by our own mad attempt to interpret it as
though it had an underlying truth."
  -- Umberto Eco