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# GaelVaroquaux/mutual_info.py

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Estimating entropy and mutual information with scikit-learn: visit https://github.com/mutualinfo/mutual_info
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 ''' Non-parametric computation of entropy and mutual-information Adapted by G Varoquaux for code created by R Brette, itself from several papers (see in the code). This code is maintained at https://github.com/mutualinfo/mutual_info Please download the latest code there, to have improvements and bug fixes. These computations rely on nearest-neighbor statistics ''' import numpy as np from scipy.special import gamma,psi from scipy import ndimage from scipy.linalg import det from numpy import pi from sklearn.neighbors import NearestNeighbors __all__=['entropy', 'mutual_information', 'entropy_gaussian'] EPS = np.finfo(float).eps def nearest_distances(X, k=1): ''' X = array(N,M) N = number of points M = number of dimensions returns the distance to the kth nearest neighbor for every point in X ''' knn = NearestNeighbors(n_neighbors=k + 1) knn.fit(X) d, _ = knn.kneighbors(X) # the first nearest neighbor is itself return d[:, -1] # returns the distance to the kth nearest neighbor def entropy_gaussian(C): ''' Entropy of a gaussian variable with covariance matrix C ''' if np.isscalar(C): # C is the variance return .5*(1 + np.log(2*pi)) + .5*np.log(C) else: n = C.shape[0] # dimension return .5*n*(1 + np.log(2*pi)) + .5*np.log(abs(det(C))) def entropy(X, k=1): ''' Returns the entropy of the X. Parameters =========== X : array-like, shape (n_samples, n_features) The data the entropy of which is computed k : int, optional number of nearest neighbors for density estimation Notes ====== Kozachenko, L. F. & Leonenko, N. N. 1987 Sample estimate of entropy of a random vector. Probl. Inf. Transm. 23, 95-101. See also: Evans, D. 2008 A computationally efficient estimator for mutual information, Proc. R. Soc. A 464 (2093), 1203-1215. and: Kraskov A, Stogbauer H, Grassberger P. (2004). Estimating mutual information. Phys Rev E 69(6 Pt 2):066138. ''' # Distance to kth nearest neighbor r = nearest_distances(X, k) # squared distances n, d = X.shape volume_unit_ball = (pi**(.5*d)) / gamma(.5*d + 1) ''' F. Perez-Cruz, (2008). Estimation of Information Theoretic Measures for Continuous Random Variables. Advances in Neural Information Processing Systems 21 (NIPS). Vancouver (Canada), December. return d*mean(log(r))+log(volume_unit_ball)+log(n-1)-log(k) ''' return (d*np.mean(np.log(r + np.finfo(X.dtype).eps)) + np.log(volume_unit_ball) + psi(n) - psi(k)) def mutual_information(variables, k=1): ''' Returns the mutual information between any number of variables. Each variable is a matrix X = array(n_samples, n_features) where n = number of samples dx,dy = number of dimensions Optionally, the following keyword argument can be specified: k = number of nearest neighbors for density estimation Example: mutual_information((X, Y)), mutual_information((X, Y, Z), k=5) ''' if len(variables) < 2: raise AttributeError( "Mutual information must involve at least 2 variables") all_vars = np.hstack(variables) return (sum([entropy(X, k=k) for X in variables]) - entropy(all_vars, k=k)) def mutual_information_2d(x, y, sigma=1, normalized=False): """ Computes (normalized) mutual information between two 1D variate from a joint histogram. Parameters ---------- x : 1D array first variable y : 1D array second variable sigma: float sigma for Gaussian smoothing of the joint histogram Returns ------- nmi: float the computed similariy measure """ bins = (256, 256) jh = np.histogram2d(x, y, bins=bins)[0] # smooth the jh with a gaussian filter of given sigma ndimage.gaussian_filter(jh, sigma=sigma, mode='constant', output=jh) # compute marginal histograms jh = jh + EPS sh = np.sum(jh) jh = jh / sh s1 = np.sum(jh, axis=0).reshape((-1, jh.shape[0])) s2 = np.sum(jh, axis=1).reshape((jh.shape[1], -1)) # Normalised Mutual Information of: # Studholme, jhill & jhawkes (1998). # "A normalized entropy measure of 3-D medical image alignment". # in Proc. Medical Imaging 1998, vol. 3338, San Diego, CA, pp. 132-143. if normalized: mi = ((np.sum(s1 * np.log(s1)) + np.sum(s2 * np.log(s2))) / np.sum(jh * np.log(jh))) - 1 else: mi = ( np.sum(jh * np.log(jh)) - np.sum(s1 * np.log(s1)) - np.sum(s2 * np.log(s2))) return mi ############################################################################### # Tests def test_entropy(): # Testing against correlated Gaussian variables # (analytical results are known) # Entropy of a 3-dimensional gaussian variable rng = np.random.RandomState(0) n = 50000 d = 3 P = np.array([[1, 0, 0], [0, 1, .5], [0, 0, 1]]) C = np.dot(P, P.T) Y = rng.randn(d, n) X = np.dot(P, Y) H_th = entropy_gaussian(C) H_est = entropy(X.T, k=5) # Our estimated entropy should always be less that the actual one # (entropy estimation undershoots) but not too much np.testing.assert_array_less(H_est, H_th) np.testing.assert_array_less(.9*H_th, H_est) def test_mutual_information(): # Mutual information between two correlated gaussian variables # Entropy of a 2-dimensional gaussian variable n = 50000 rng = np.random.RandomState(0) #P = np.random.randn(2, 2) P = np.array([[1, 0], [0.5, 1]]) C = np.dot(P, P.T) U = rng.randn(2, n) Z = np.dot(P, U).T X = Z[:, 0] X = X.reshape(len(X), 1) Y = Z[:, 1] Y = Y.reshape(len(Y), 1) # in bits MI_est = mutual_information((X, Y), k=5) MI_th = (entropy_gaussian(C[0, 0]) + entropy_gaussian(C[1, 1]) - entropy_gaussian(C) ) # Our estimator should undershoot once again: it will undershoot more # for the 2D estimation that for the 1D estimation print((MI_est, MI_th)) np.testing.assert_array_less(MI_est, MI_th) np.testing.assert_array_less(MI_th, MI_est + .3) def test_degenerate(): # Test that our estimators are well-behaved with regards to # degenerate solutions rng = np.random.RandomState(0) x = rng.randn(50000) X = np.c_[x, x] assert np.isfinite(entropy(X)) assert np.isfinite(mutual_information((x[:, np.newaxis], x[:, np.newaxis]))) assert 2.9 < mutual_information_2d(x, x) < 3.1 def test_mutual_information_2d(): # Mutual information between two correlated gaussian variables # Entropy of a 2-dimensional gaussian variable n = 50000 rng = np.random.RandomState(0) #P = np.random.randn(2, 2) P = np.array([[1, 0], [.9, .1]]) C = np.dot(P, P.T) U = rng.randn(2, n) Z = np.dot(P, U).T X = Z[:, 0] X = X.reshape(len(X), 1) Y = Z[:, 1] Y = Y.reshape(len(Y), 1) # in bits MI_est = mutual_information_2d(X.ravel(), Y.ravel()) MI_th = (entropy_gaussian(C[0, 0]) + entropy_gaussian(C[1, 1]) - entropy_gaussian(C) ) print((MI_est, MI_th)) # Our estimator should undershoot once again: it will undershoot more # for the 2D estimation that for the 1D estimation np.testing.assert_array_less(MI_est, MI_th) np.testing.assert_array_less(MI_th, MI_est + .2) if __name__ == '__main__': # Run our tests test_entropy() test_mutual_information() test_degenerate() test_mutual_information_2d()

### Striderl commented Jul 26, 2019

Thank you for your work. Just one quick comment about the nearest_distances function.
Shouldn't line 31 be "knn = NearestNeighbors(n_neighbors=k+1)" ?

--g

I agree. according to the original paper, http://citeseerx.ist.psu.edu/viewdoc/download?doi=10.1.1.422.5607&rep=rep1&type=pdf, this should calculate the Euclidean distances to the k-nn of xi in X \xi.

### GaelVaroquaux commented Jul 30, 2019

What if I want the MI of two sets that don't have the same amount of samples? like X.shape = (30, 10) and Y.shape = (1,10). My impression is that the np.hstack in line 103 is not enough.

I don't really see how it would be possible to compute non-parametric mutual information between two sets where there is no correspondence between the samples. By construction to estimate MI one needs this correspondence.

### GaelVaroquaux commented Jul 30, 2019

Shouldn't line 31 be "knn = NearestNeighbors(n_neighbors=k+1)" ?

I agree that using n_neighbors=k gives the k-1 nearest neighbor, as the first is the point itself. The code was wrong. Maybe that error was introduced in an evolution of scikit-learn. I've fixed it.

### Sandy4321 commented Nov 4, 2019

thanks
PS
it would be very kind of you to share some links to understand
why
https://scikit-learn.org/stable/modules/generated/sklearn.feature_selection.mutual_info_classif.html#sklearn.feature_selection.mutual_info_classif
Kozachenko, N. N. Leonenko,
is good for feature selection
why it impossible just calculate similarity/mutual information for each feature and target?
Then if similarity/mutual information for given feature and target is high then this feature is good to use??

I've noticed that scaling samples changes the output. I suspect this is a small bug in the scaling somewhere. Will post back if I trace it down.

Compare the following:

In [126]: np.random.seed(12)
...: for i in range(20):
...:     print(mi.mutual_information([np.random.randn(100, 1), 1 * np.random.randn(100, 1)], k=10))
...:
-0.020132950207217615
0.0007764531823655219
-0.10097362008313171
0.008234538277922088
-0.019989602665604345
-0.07712501107587677
0.02317718497197019
-0.01624389693603323
-0.037344436692854366
0.04950941718866808
-0.05049871422680052
-0.028624985257037494
-0.004130417096971595
-0.09652205195754915
0.07427403221566875
-0.0040393263534936885
-0.058616537991666995
0.09200872016468775
0.04512428420750236
-0.07361872725115903

In [127]: np.random.seed(12)
...: for i in range(20):
...:     print(mi.mutual_information([np.random.randn(100, 1), 100 * np.random.randn(100, 1)], k=10))
...:
-2.0021553153700378
-1.7289150378782132
-1.938608386083576
-1.999900260449417
-1.7849647677622498
-1.8527331847233786
-2.0258220171874264
-2.00130782835749
-2.017433687721022
-1.8716648526808015
-2.159052646410464
-2.090519859781173
-2.1565500325401965
-1.9789886099442322
-1.9400487388101713
-2.004367515829771
-1.9796955922060349
-2.016857734129518
-2.1187142194709843
-1.9679714350429087


UPDATE: I think this is just the result of using a differential entropy throughout. Probably (not sure yet) but using some copula style normalization (see pd.rank for example) is a decent way of getting something invariant to some transformations. Might destroy some structure depending on the use case though.

Also, there are few forks kicking around now. I think I've got one that fixed some small issue with the case of mutual information with itself.

https://gist.github.com/cottrell/770b811c03c846386b19e9c4bd18772f/revisions

I'm wondering if we should try to coral everyone together and make a repo. Viewing diffs is a little less convenient from the gist. Though I like the single discussion thread.

### AbdolvakilFazli commented Apr 17, 2020

Thank you for the work,

Assume that I have a node with some features(f1,f2,f3). And another nodes from features (f1',f2',f3'). how can I find the mutual information of the two?

### Sandy4321 commented Apr 19, 2020

Good question
Really, how it will be?

If I use k = [1,2,3,4,5,6] and then I test mutual_info_2d(k,k), I get as a result: 1.7456376329342476.
Shouldn't it be 1 when I test the mutual information between two identical vectors?

I would assume the natural result would be -6*(1/6)*log(1/6)=log(6) = 1.79175946923 .
This is assuming each value has the same probability of 1/6 .
Looks good given the that it is an estimation.

### thismartian commented Feb 2, 2021

I noted two errors in this code (based on the reference paper from Krasov), and this also fixed the negative MI values I was getting.

1. volume_unit_ball is missing a small expression. It should be : (pi**(d/2)/gamma(d/2 +1)/2**d)
2. volume_unit_ball values for every variable need to be multiplied to each other before calculating the log. So line 75, volume_unit_ball = (pi**(.5d)) / gamma(.5d + 1) should actually be replaced by something like this:

volume_unit_ball = 1
for dim in range(1, d+1):
volume_unit_ball = volume_unit_ball * (pi**(dim/2)/gamma(dim/2 +1)/2**dim)

### cottrell commented Feb 5, 2021

@thismartion I haven't yet found it in another library where it should probably sit. Have created this and added you and @GaelVaroquaux . You should have admin privileges if I did it right.

### GaelVaroquaux commented Feb 5, 2021

Creating a library for collective maintenance was really the right move. I'll update the gist to point there.

### cottrell commented Feb 5, 2021

volume_unit_ball = 1
for dim in range(1, d+1):
volume_unit_ball = volume_unit_ball * (pi**(dim/2)/gamma(dim/2 +1)/2**dim)

I've copy pasta'd your change into this branch. https://github.com/mutualinfo/mutual_info/tree/thismartian

There is some test failure but I haven't looked at it yet. I suspect if you still have it in your head it is an easy test change. I'm not ever sure what is correct since it's been a few months since I've used this stuff.

### thismartian commented Feb 6, 2021

Hi, cottrell. Thanks a lot for this. I figured out what the problem is. The distance used to calculate the entropy should be 2x the distance to the nearest neighbor. Not sure I'm doing it right but I don't seem to have the permission to make changes to the file, perhaps you could try this: in the entropy function: return d * np.mean(np.log(2*r + np.finfo(X.dtype).eps)) + np.log(volume_unit_ball) + psi(n) - psi(k)

### cottrell commented Feb 6, 2021

Hi, cottrell. Thanks a lot for this. I figured out what the problem is. The distance used to calculate the entropy should be 2x the distance to the nearest neighbor. Not sure I'm doing it right but I don't seem to have the permission to make changes to the file, perhaps you could try this: in the entropy function: return d * np.mean(np.log(2*r + np.finfo(X.dtype).eps)) + np.log(volume_unit_ball) + psi(n) - psi(k)

Hmmm, I can past it in and try but let's try to fix the permissions. Having a look now ...

### cottrell commented Feb 6, 2021

Ok, I think I've now added @thismartian and @GaelVaroquaux as admins on the org and the repo now. For some reason it only sets you up as members initially when I punched them in. Let me know if you can't access it.

### cottrell commented Feb 6, 2021

@thismartian The tests now pass after I pasted the change in. It's still just on that branch if you want to review it. If there is an obvious test that would have failed before your change and passed after, maybe we can add that?

### thismartian commented Feb 9, 2021

@cottrell the changes look fine. I think the entropy test was failing (as expected). Is that what you mean? Sorry it's not clear to me...

### cottrell commented Feb 10, 2021

@cottrell the changes look fine. I think the entropy test was failing (as expected). Is that what you mean? Sorry it's not clear to me...

No I mean before the change, no tests were failing but you noticed something was wrong. What is a missing test that would have shown something was wrong?

Ah, for me the main indication was just that for some random combinations the mutual info was negative (as you had pointed out earlier too). I think in the test itself, if you change the covariance matrix to unit variance, it shows that the test failed. Honestly, I am not sure I understand how the covariance matrix is generated in the tests (I need to read about it). What I would do is: define a correlation coefficient (r), standard deviation (s), and based on these two it's possible to have a unit variance, zero mean covariance matrix:



s = 0.6
r = 0.9

C= np.array([[0.5*s**2, r*0.5*s**2], [r*0.5*s**2, 0.5*s**2]])

mean = [0,0]

x,y = np.random.multivariate_normal(mean, C, size = n).T

X = x.reshape(len(x), 1)

Y = y.reshape(len(y), 1)


### thismartian commented Feb 11, 2021

I was probably not clear, let me try to rephrase:

• In the mutual info test written by @GaelVaroquaux, the covariance matrix does not have a unit variance. If you change it to a unit variance matrix, the test fails. The gaussian reference used in the paper is based on a zero mean, unit variance covariance matrix.
• I don't have an expertise in statistics so I don't really understand the approach used in the code to generate the correlated XY pair. I have instead generated the XY pair based on a numpy function (previous comment), which I understand better.

### cottrell commented Feb 12, 2021

Thanks @thismartian I will take a look a try to jam a test in shortly.

### cottrell commented Mar 3, 2021

@thismartian Yeah I can't remember where I left it. I think I put some new tests in and have a branch there. I think I was looking at differential entropy in general and how it sort of has a dimensionality issue. The branch should run and we could just merge. Likely I'll dig into it more next time I am looking for entropy of continuous random variables.

friends

### thismartian commented Mar 15, 2021

Yes, I let cottrell manage the merge after reviewing thismartian branch.

### cottrell commented Mar 16, 2021

friends

I can merge it in. There isn't a ton of testing or anything because it's just the gist copied over so we can break and fix as we go along. I should probably set up my notifications for that org ... am not noticing these often enough.

### cottrell commented Mar 16, 2021

friends

I can merge it in. There isn't a ton of testing or anything because it's just the gist copied over so we can break and fix as we go along. I should probably set up my notifications for that org ... am not noticing these often enough.

I think my reply got dropped. It is all merged in now. I had some minor refactors and added some tests. I can't remember the details but was trying to make in shift and scale invariant.

### EtienneCmb commented Apr 2, 2021

Hi everyone, thanks for the shared code, very interesting.

I've a question regarding the estimation of the entropy for gaussian variables. What is the reason for taking the absolute value of the determinant? Is it for numerical reasons?

### cottrell commented Apr 16, 2021

Hi everyone, thanks for the shared code, very interesting.

I've a question regarding the estimation of the entropy for gaussian variables. What is the reason for taking the absolute value of the determinant? Is it for numerical reasons?