Can I create a multivariate_normal matrix using dask?

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Somewhat related to this post, I am trying to replicate multivariate_normal in dask: Using numpy I can create a multivariate normal matrix with a specified covariance using:

import numpy as np
n_dim = 5
size = 300
A = np.random.randn(n_dim, n_dim) # a matrix
covm = A.dot(A.T) # A*A^T is positive semi-definite, as a covariance matrix
x = np.random.multivariate_normal(size=300, mean=np.zeros(len(covm)),cov=covm) # generate data

I however need a significantly large matrix with n_dim = 4_500_000 and size = 100000. This will be expensive to compute both with respective to CPU and memory. Fortunately, I have access to a Cloudera DataScience Workbench Cluster and was trying to solve this using dask:

import dask.array as da
n_dim = 4_500_000
size = 100000
A = da.random.standard_normal((n_dim, n_dim))  
covm = A.dot(A.T)
#x = da.random.multivariate_normal(size=300, mean=np.zeros(len(covm)),cov=covm) # generate data

In the documentation, I cannot find any function that seem to do what I need it to. Does anyone know a solution / workarround, possibly using xarray or any other module that runs on clusters?

dask-distributed python-xarray dask pyspark python
1 Answers

An work arround for now, is to use a cholesky decomposition. Note that any covariance matrix C can be expressed as C=G*G'. It then follows that x = G'*y is correlated as specified in C if y is standard normal (see this excellent post on StackExchange Mathematic). In code:

Numpy

n_dim =4
size = 100000
A = np.random.randn(n_dim, n_dim)
covm = A.dot(A.T)

x=  np.random.multivariate_normal(size=size, mean=np.zeros(len(covm)),cov=covm)
## verify numpys covariance is correct
np.cov(x, rowvar=False)
covm

Dask

## create covariance matrix
A = da.random.standard_normal(size=(n_dim, n_dim),chunks=(2,2))
covm = A.dot(A.T)

## get cholesky decomp
L = da.linalg.cholesky(covm, lower=True)

## drawn standard normal 
sn= da.random.standard_normal(size=(size, n_dim),chunks=(100,100))

## correct for correlation
x =L.dot(sn.T)
x.shape

## verify
covm.compute()
da.cov(x, rowvar=True).compute()
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