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# mblondel/lda_gibbs.py

Last active October 24, 2022 15:48
Latent Dirichlet Allocation with Gibbs sampler
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 """ (C) Mathieu Blondel - 2010 License: BSD 3 clause Implementation of the collapsed Gibbs sampler for Latent Dirichlet Allocation, as described in Finding scientifc topics (Griffiths and Steyvers) """ import numpy as np import scipy as sp from scipy.special import gammaln def sample_index(p): """ Sample from the Multinomial distribution and return the sample index. """ return np.random.multinomial(1,p).argmax() def word_indices(vec): """ Turn a document vector of size vocab_size to a sequence of word indices. The word indices are between 0 and vocab_size-1. The sequence length is equal to the document length. """ for idx in vec.nonzero(): for i in xrange(int(vec[idx])): yield idx def log_multi_beta(alpha, K=None): """ Logarithm of the multinomial beta function. """ if K is None: # alpha is assumed to be a vector return np.sum(gammaln(alpha)) - gammaln(np.sum(alpha)) else: # alpha is assumed to be a scalar return K * gammaln(alpha) - gammaln(K*alpha) class LdaSampler(object): def __init__(self, n_topics, alpha=0.1, beta=0.1): """ n_topics: desired number of topics alpha: a scalar (FIXME: accept vector of size n_topics) beta: a scalar (FIME: accept vector of size vocab_size) """ self.n_topics = n_topics self.alpha = alpha self.beta = beta def _initialize(self, matrix): n_docs, vocab_size = matrix.shape # number of times document m and topic z co-occur self.nmz = np.zeros((n_docs, self.n_topics)) # number of times topic z and word w co-occur self.nzw = np.zeros((self.n_topics, vocab_size)) self.nm = np.zeros(n_docs) self.nz = np.zeros(self.n_topics) self.topics = {} for m in xrange(n_docs): # i is a number between 0 and doc_length-1 # w is a number between 0 and vocab_size-1 for i, w in enumerate(word_indices(matrix[m, :])): # choose an arbitrary topic as first topic for word i z = np.random.randint(self.n_topics) self.nmz[m,z] += 1 self.nm[m] += 1 self.nzw[z,w] += 1 self.nz[z] += 1 self.topics[(m,i)] = z def _conditional_distribution(self, m, w): """ Conditional distribution (vector of size n_topics). """ vocab_size = self.nzw.shape left = (self.nzw[:,w] + self.beta) / \ (self.nz + self.beta * vocab_size) right = (self.nmz[m,:] + self.alpha) / \ (self.nm[m] + self.alpha * self.n_topics) p_z = left * right # normalize to obtain probabilities p_z /= np.sum(p_z) return p_z def loglikelihood(self): """ Compute the likelihood that the model generated the data. """ vocab_size = self.nzw.shape n_docs = self.nmz.shape lik = 0 for z in xrange(self.n_topics): lik += log_multi_beta(self.nzw[z,:]+self.beta) lik -= log_multi_beta(self.beta, vocab_size) for m in xrange(n_docs): lik += log_multi_beta(self.nmz[m,:]+self.alpha) lik -= log_multi_beta(self.alpha, self.n_topics) return lik def phi(self): """ Compute phi = p(w|z). """ V = self.nzw.shape num = self.nzw + self.beta num /= np.sum(num, axis=1)[:, np.newaxis] return num def run(self, matrix, maxiter=30): """ Run the Gibbs sampler. """ n_docs, vocab_size = matrix.shape self._initialize(matrix) for it in xrange(maxiter): for m in xrange(n_docs): for i, w in enumerate(word_indices(matrix[m, :])): z = self.topics[(m,i)] self.nmz[m,z] -= 1 self.nm[m] -= 1 self.nzw[z,w] -= 1 self.nz[z] -= 1 p_z = self._conditional_distribution(m, w) z = sample_index(p_z) self.nmz[m,z] += 1 self.nm[m] += 1 self.nzw[z,w] += 1 self.nz[z] += 1 self.topics[(m,i)] = z # FIXME: burn-in and lag! yield self.phi() if __name__ == "__main__": import os import shutil N_TOPICS = 10 DOCUMENT_LENGTH = 100 FOLDER = "topicimg" def vertical_topic(width, topic_index, document_length): """ Generate a topic whose words form a vertical bar. """ m = np.zeros((width, width)) m[:, topic_index] = int(document_length / width) return m.flatten() def horizontal_topic(width, topic_index, document_length): """ Generate a topic whose words form a horizontal bar. """ m = np.zeros((width, width)) m[topic_index, :] = int(document_length / width) return m.flatten() def save_document_image(filename, doc, zoom=2): """ Save document as an image. doc must be a square matrix """ height, width = doc.shape zoom = np.ones((width*zoom, width*zoom)) # imsave scales pixels between 0 and 255 automatically sp.misc.imsave(filename, np.kron(doc, zoom)) def gen_word_distribution(n_topics, document_length): """ Generate a word distribution for each of the n_topics. """ width = n_topics / 2 vocab_size = width ** 2 m = np.zeros((n_topics, vocab_size)) for k in range(width): m[k,:] = vertical_topic(width, k, document_length) for k in range(width): m[k+width,:] = horizontal_topic(width, k, document_length) m /= m.sum(axis=1)[:, np.newaxis] # turn counts into probabilities return m def gen_document(word_dist, n_topics, vocab_size, length=DOCUMENT_LENGTH, alpha=0.1): """ Generate a document: 1) Sample topic proportions from the Dirichlet distribution. 2) Sample a topic index from the Multinomial with the topic proportions from 1). 3) Sample a word from the Multinomial corresponding to the topic index from 2). 4) Go to 2) if need another word. """ theta = np.random.mtrand.dirichlet([alpha] * n_topics) v = np.zeros(vocab_size) for n in range(length): z = sample_index(theta) w = sample_index(word_dist[z,:]) v[w] += 1 return v def gen_documents(word_dist, n_topics, vocab_size, n=500): """ Generate a document-term matrix. """ m = np.zeros((n, vocab_size)) for i in xrange(n): m[i, :] = gen_document(word_dist, n_topics, vocab_size) return m if os.path.exists(FOLDER): shutil.rmtree(FOLDER) os.mkdir(FOLDER) width = N_TOPICS / 2 vocab_size = width ** 2 word_dist = gen_word_distribution(N_TOPICS, DOCUMENT_LENGTH) matrix = gen_documents(word_dist, N_TOPICS, vocab_size) sampler = LdaSampler(N_TOPICS) for it, phi in enumerate(sampler.run(matrix)): print "Iteration", it print "Likelihood", sampler.loglikelihood() if it % 5 == 0: for z in range(N_TOPICS): save_document_image("topicimg/topic%d-%d.png" % (it,z), phi[z,:].reshape(width,-1))

### cdfox commented May 28, 2012

I don't think numpy/scipy are making this code slower rather faster. I did a quick test and found that a pure python implementation of sampling from a multinomial distribution with 1 trial (i.e. a discrete distribution)

```import random
def draw(p):
r = random.random()
for i in range(len(p)):
r = r - p[i]
if r < 0:
return i
return len(p) - 1```

is much faster than using numpy.random's multinomial function

```def draw(p):
return np.random.multinomial(1,p).argmax()```

### flaviotruzzi commented Dec 12, 2012

I'll try to make a cython version when I have time.

### marcmaxson commented Apr 3, 2013

Not sure why the import failed to include the misc subfolder, but I was able to get the demonstration running with this tweak to save_document_image():

``````def save_document_image(filename, doc, zoom=2):
"""
Save document as an image.

doc must be a square matrix
"""
height, width = doc.shape
zoom = np.ones((width*zoom, width*zoom))
# imsave scales pixels between 0 and 255 automatically
sp.misc.imsave(filename, np.kron(doc, zoom))
try:
sp.misc.imsave(filename, np.kron(doc, zoom))
except:
import scipy.misc
scipy.misc.imsave(filename, np.kron(doc, zoom))
``````

### svanschalkwyk commented Apr 18, 2013

@marc - still not able to find misc even though it is imported as part of sp (have verified). Is sp overwritten somewhere? Wil llook when I have time.
@Mathiue - thanks for the code!

### whille commented May 15, 2013

hi @Mathiue, here const doc_length is used for training samples. According to Gregor's Parameter estimation for text analysis, Fig.7 defination, Nm(doc length) should follow Poisson distribution, E=λ.
so in line 210, const number should better modified to

``````for n in np.random.poisson(length, DOC_NUM):
``````

### elplatt commented Nov 24, 2013

Thanks for posting this. I'd like to use this code for a project. If that's ok, can you add a license?

### corydolphin commented Dec 12, 2013

Quick note to anyone struggling with the scipy.misc.imsave import, you need to have PIL installed for this import to work. Python dependency management is crazy!

### jnothman commented May 10, 2014

Re @cdfox's comment, you're much better off doing a `searchsorted(cumsum(p), rand())` than `np.random.multinomial(1,p).argmax()` which is efficient only when you're taking a large sample.

### ChangUk commented Jun 9, 2014

I implemented Gibbs sampler for standard LDA inference. My program updates alpha(vector) and beta(scalar) during the iterative sampling process by using Minka's fixed-point iteration.
Visit here: https://gist.github.com/ChangUk/a741e0ccf5737954956e