Create a dataset consisting of N = 100 two-dimensional samples

Question

How to create a data set consisting of N = 100 two-dimensional samples x = (x1,x2)T ∈ R2 , taken from a two-dimensional Gaussian distribution with an average value

 µ = (1,1)T 

and covariance matrix

 Σ = (0.3 0.2 0.2 0.2) 

I was told that you can use the Matlab randn function, but don’t know how to implement it in Python?

Answer 1

Just to answer @EamonNerbonne in more detail, answer as follows: Cholesky decomposition of the covariance matrix to generate correlated variables from uncorrelated normally distributed random variables.

 import numpy as np import matplotlib.pyplot as plt linalg = np.linalg N = 1000 mean = [1,1] cov = [[0.3, 0.2],[0.2, 0.2]] data = np.random.multivariate_normal(mean, cov, N) L = linalg.cholesky(cov) # print(L.shape) # (2, 2) uncorrelated = np.random.standard_normal((2,N)) data2 = np.dot(L,uncorrelated) + np.array(mean).reshape(2,1) # print(data2.shape) # (2, 1000) plt.scatter(data2[0,:], data2[1,:], c='green') plt.scatter(data[:,0], data[:,1], c='yellow') plt.show() 

Yellow dots were created by np.random.multivariate_normal . Green points were generated by multiplying the normally distributed points by the Cholesky factorization matrix L

Answer 2

You are looking for numpy.random.multivariate_normal

code

 >>> import numpy >>> print numpy.random.multivariate_normal([1,1], [[0.3, 0.2],[0.2, 0.2]], 100) [[ 0.02999043 0.09590078] [ 1.35743021 1.08199363] [ 1.15721179 0.87750625] [ 0.96879114 0.94503228] [ 1.23989167 1.13473083] [ 1.55917608 0.81530847] [ 0.89985651 0.7071519 ] [ 0.37494324 0.739433 ] [ 1.45121732 1.17168444] [ 0.69680785 1.2727178 ] [ 0.35600769 0.46569276] [ 2.14187488 1.8758589 ] [ 1.59276393 1.54971412] [ 1.71227009 1.63429704] [ 1.05013136 1.1669758 ] [ 1.34344004 1.37369725] [ 1.82975724 1.49866636] [ 0.80553877 1.26753018] [ 1.74331784 1.27211784] [ 1.23044292 1.18110192] [ 1.07675493 1.05940509] [ 0.15495771 0.64536509] [ 0.77409745 1.0174171 ] [ 1.20062726 1.3870498 ] [ 0.39619719 0.77919884] [ 0.87209168 1.00248145] [ 1.32273339 1.54428262] [ 2.11848535 1.44338789] [ 1.45226461 1.42061198] [ 0.33775737 0.24968543] [ 1.06982557 0.64674411] [ 0.92113229 1.0583153 ] [ 0.54987592 0.73198037] [ 1.06559727 0.77891362] [ 0.84371805 0.72957046] [ 1.83614557 1.40582746] [ 0.53146009 0.72294094] [ 0.98927818 0.73732053] [ 1.03984002 0.89426628] [ 0.38142362 0.32471126] [ 1.44464929 1.15407227] [-0.22601279 0.21045592] [-0.01995875 0.45051782] [ 0.58779449 0.44486237] [ 1.31335981 0.92875936] [ 0.42200098 0.6942829 ] [ 0.10714426 0.11083002] [ 1.44997839 1.19052704] [ 0.78630506 0.45877582] [ 1.63432202 1.95066539] [ 0.56680926 0.92203111] [ 0.08841491 0.62890576] [ 1.4703602 1.4924649 ] [ 1.01118864 1.44749407] [ 1.19936276 1.02534702] [ 0.67893239 0.8482461 ] [ 0.71537211 0.53279103] [ 1.08031573 1.00779064] [ 0.66412568 0.57121041] [ 0.96098528 0.72318386] [ 0.7690299 0.76058713] [ 0.77466896 0.77559282] [ 0.47906664 0.58602633] [ 0.52481326 0.78486453] [-0.40240438 0.17374116] [ 0.75730444 0.22365892] [ 0.67811008 1.17730408] [ 1.62245699 1.71775386] [ 1.12317847 1.04252136] [-0.06461117 0.23557416] [ 0.46299482 0.51585414] [ 0.88125676 1.23284201] [ 0.57920534 0.63765861] [ 0.88239858 1.32092112] [ 0.63500551 0.94788141] [ 1.76588148 1.63856465] [ 0.65026599 0.6899672 ] [ 0.06854287 0.29712499] [ 0.61575737 0.87526625] [ 0.30057552 0.54475194] [ 0.66578769 0.21034844] [ 0.94670438 0.7699764 ] [ 0.39870371 0.91681577] [ 1.37531351 1.62337899] [ 1.92350877 1.34382017] [ 0.56631877 0.77456137] [ 1.18702642 0.63700271] [ 0.74002244 1.04535471] [ 0.3272063 0.75097037] [ 1.57583435 1.55809705] [ 0.44325124 0.39620769] [ 0.59762516 0.58304621] [ 0.72253698 0.68302097] [ 0.93459597 1.01101948] [ 0.50139577 0.52500942] [ 0.84696441 0.68679341] [ 0.63483432 0.22205385] [ 1.43642478 1.34724612] [ 1.58663111 1.49941374] [ 0.73832806 0.95690866]] >>> 

Answer 3

Although numpy has convenient utility functions, you can always “rescale” several independent normally distributed variables according to your covariance matrix. Therefore, if you can generate a column vector x (or many vectors grouped into a matrix) in which each element is usually distributed and you scale with the matrix M , the result will have covariance MM^T And vice versa, if you analyze the covariance C in the form MM^T , then it is really simple to make such a distribution even without the utility function NumPy provides (only multiply your bunch of normally distributed vectors by M ).

This may not be the answer you are just looking for, but it is useful to remember, for example:

• If you ever find scaling of random output, you can combine scaling with initial covariance instead
• If you ever need to port code to libraries that do not directly support such a utility method, it is very easy to implement yourself.