Can I fix the mean of one component of a Gaussian mixture model in python before fitting?

Benjamin Doughty

I am interested in fitting a 2-component Gaussian mixture model to the data shown below. However, since I'm plotting log-transformed counts here, normalized to be between 0-1, the maximum value my data will take is 0. When I try to do a naive fit using sklearn.mixture.GaussianMixture (code below), I get results that fit, which is obviously not what I want.Log-transformed count rate data, cannot exceed 0

from sklearn.mixture import GaussianMixture
import numpy as np

# start with some count data in (0,1]
logged_counts = np.log(counts)
model = GaussianMixture(2).fit(logged_counts.reshape(-1,1))

# plot resulting fit
x_range = np.linspace(np.min(logged_counts), 0, 1000)
pdf = np.exp(model.score_samples(x_range.reshape(-1, 1)))
responsibilities = model.predict_proba(x_range.reshape(-1, 1))
pdf_individual = responsibilities * pdf[:, np.newaxis]

plt.hist(logged_counts, bins='auto', density=True, histtype='stepfilled', alpha=0.5)
plt.plot(x_range, pdf, '-k', label='Mixture')
plt.plot(x_range, pdf_individual, '--k', label='Components')
plt.legend()
plt.show()

Fitting using sklearn's two-component GMMI would love it if I could fix the mean of the top component at 0 and only optimize the other mean, the two variances, and the mixing fractions. (Additionally I would love to be able to use a half-normal for the component on the right.) Is there a simple way to do this with built-in functions in python/sklearn, or will I have to build that model myself using some probabilistic programming language?

Drey

Afaik, you cannot do exactly what you want in sklearn.

Imho, basically there are multiple strategies: (i) implement GMM yourself, (ii) switch to another language/framework, (iii) adapt GMM code, or (iv) adapt.

(i) You probably do not want to do this unless you want to learn for yourself.

(ii) You could use stan and adapt the code in the last paragraph to have a fixed component of your choice (distribution type and parameters)

(iii) You could do (i) but slightly adapt the sklearn code or simply use the methods for estimation but with you own slight modifications.

(iv)

  • Gaussian Mixture model will not work here (as you mentioned) because you require a truncated Normal distribution for the "first" (fixed) component.
  • If you would not require to fit for the variance of the fixed component then you can always just substract your fixed component from the data. (i.e. for each point subtract the point's quantile-value from the point value)
  • If you don't mind the accuracy of the estimate, you can do this by two things: First identify the two components using the GMM. Then, only look at the data in the component you want to fix. Fit a truncated Gaussian model (using .fit(data)). The resulting parameter (as in option 2) is then subtracted from the original data. Then install a GMM. Find the next component.

Hope this helps :-)

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