As we saw in gs, to create a MultiNorm object, you pass a mean vector, a covariance matrix (both as Numpy arrays) and optionally a list of parameter names:
from multinorm import MultiNorm
mean = [10, 20, 30]
covariance = [[1, 0, 0], [0, 4, 0], [0, 0, 9]]
names = ["a", "b", "c"]
mn = MultiNorm(mean, covariance, names)But where do these things come from?
On this page, we look at the most common scenarios.
TODO: show example using scipy.optimise.curve_fit`
To use multinorm, we first need to fit some parameterised model to obtain a best-fit parameter vector and covariance matrix.
Let's use scipy.optimize_curve_fit to fit some data.
TODO: show example using iminuit
http://www.statsmodels.org/devel/examples/notebooks/generated/chi2_fitting.html https://github.com/cdeil/pyfit/blob/master/fitting_tutorial/src/tests/chi2_example.py https://lmfit.github.io https://iminuit.readthedocs.io https://sherpa.readthedocs.io
A common way to analyse likelihood or in Bayesian analyses the posterior probability distributions is to use MCMC methods that sample the distribution. E.g. emcee or pymc are Python packages that generate this kind of output.
Estimating the multivariate normal distribution from samples well can be difficult, there are many methods with different trade-offs. We recommend using a different package for this task, e.g. sklearn.covariance.
That said, there is a method MultiNorm.from_points that calls numpy.std to compute the mean vector, and numpy.cov to compute what's sometimes called the "empirical" multivariate normal estimate.
Points should always be given as 2-dimensional arrays with shape (n_dim, n_points).
>>> points = mn.sample(size=100, random_state=0)
>>> MultiNorm.from_points(points)
MultiNorm(n=3)
names: ['par_0', 'par_1', 'par_2']
mean: [ 9.87581591 20.21250462 30.30156153]
err: [0.98090098 1.97394775 3.09360932]
cov:
[[ 0.96216674 -0.04439635 0.33802118]
[-0.04439635 3.89646972 -0.45369176]
[ 0.33802118 -0.45369176 9.57041861]]TODO: show example how to take covar (or par errors) from a publication or blog post, i.e. as inputs.
TODO: document MultiNorm.from_product
TODO: document MultiNorm.make_example