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Given the following text description, write Python code to implement the functionality described below step by step
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Description:
Step1:  Multi-threaded Parallelism
author
Step2:  1. Naive Bayes
Let's start off with a simple model like naive Bayes.
Step3:  It looks like scikit-learn starts off as faster on a single thread, but that adding multiple threads will improve pomegranate's performance, typically past that of scikit-learn on a single thread. The reason this is the case is likely because pomegranate has to create one object per feature per class, whereas scikit-learn will just define the means and variances in a matrix. It's efficient when you know what distribution you're implementing in the naive Bayes, but not as modular.
2. Bayes' Classifier
The next complex model is the Bayes' classifier. This model is similar to the naive Bayes' model, but does not require that the features are independent from each other.
Step4:  We see a nice spead improvement here as well. There is no model to compare against in scikit-learn because their naive Bayes' model assumes that all features are independent.
3. General Mixture Models
Next, lets take a look at Gaussian mixture models.
Step5:  It looks like pomegranate can be faster than sklearn at performing 15 iterations of EM on this data set, and that parallelization can speed things up even further. We can ignore the convergence warnings. Here we're just timing a fixed amount of computational work. Different models might converge after a different number of iterations, and we don't want the pseedup to come just from that.
Lets now take a look at the time it takes to make predictions using GMMs. Lets fit the model to a small amount of data, and then predict a larger amount of data drawn from the same underlying distributions.
Step6:  It looks like pomegranate can be slightly slower than sklearn when using a single processor, but that it can be parallelized to get faster performance.
To ensure that we're getting the exact same results, only faster, lets subtract the predictions from each other and make sure that the sum is equal to 0.
Step7:  Great, no difference between the two.
Lets now make sure that pomegranate and sklearn are learning basically the same thing. Lets fit both models to some 2 dimensional 2 component data and make sure that they both extract the underlying clusters by plotting them.
Step8:  It looks like we're getting the same basic results for the two. The two algorithms are initialized a bit differently, and so it can be difficult to directly compare the results between them, but it looks like they're getting roughly the same results.
3. Multivariate Gaussian HMM
Now let's move on to training a hidden Markov model with multivariate Gaussian emissions with a diagonal covariance matrix. We'll randomly generate some Gaussian distributed numbers and use pomegranate with either one or four threads to fit our model to the data.
Step9:  All we had to do was pass in the n_jobs parameter to the fit function in order to get a speed improvement. It looks like we're getting a really good speed improvement, as well! This is mostly because the HMM algorithms perform a lot more operations than the other models, and so spend the vast majority of time with the GIL released. You may not notice as strong speedups when using a MultivariateGaussianDistribution because BLAS uses multithreaded operations already internally, even when only one job is specified.
Now lets look at the prediction function to make sure the we're getting speedups there as well. You'll have to use a wrapper function to parallelize the predictions for a HMM because it returns an annotated sequence rather than a single value like a classic machine learning model might.
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Python Code:
%matplotlib inline
import numpy
import matplotlib.pyplot as plt
import seaborn; seaborn.set_style('whitegrid')
from pomegranate import *
numpy.random.seed(0)
numpy.set_printoptions(suppress=True)
%load_ext watermark
%watermark -m -n -p numpy,scipy,pomegranate
def create_dataset(n_samples, n_dim, n_classes, alpha=1):
    Create a random dataset with n_samples in each class.
    
    X = numpy.concatenate([numpy.random.normal(i*alpha, 1, size=(n_samples, n_dim)) for i in range(n_classes)])
    y = numpy.concatenate([numpy.zeros(n_samples) + i for i in range(n_classes)])
    idx = numpy.arange(X.shape[0])
    numpy.random.shuffle(idx)
    return X[idx], y[idx]
Explanation: Multi-threaded Parallelism
author: Jacob Schreiber <br>
contact: jmschreiber91@gmail.com
pomegranate supports parallelization through a set of built in functions based off of joblib. All computationally intensive functions in pomegranate are implemented in cython with the global interpreter lock (GIL) released, allowing for multithreading to be used for efficient parallel processing. 
These functions can all be simply parallelized by passing in n_jobs=X to the method calls. This tutorial will demonstrate how to use those calls. First we'll look at a simple multivariate Gaussian mixture model, and compare its performance to sklearn. Then we'll look at a hidden Markov model with Gaussian emissions, and lastly we'll look at a mixture of Gaussian HMMs. These can all utilize the build-in parallelization that pomegranate has.
Let's dive right in!
End of explanation
from sklearn.naive_bayes import GaussianNB
n, d, k = 50000, 1000, 2
X, y = create_dataset(n, d, k)
print("sklearn GNB")
%timeit GaussianNB().fit(X, y)
print() 
print("pomegranate GNB")
%timeit NaiveBayes.from_samples(NormalDistribution, X, y)
print()
print("pomegranate GNB (2 jobs)")
%timeit NaiveBayes.from_samples(NormalDistribution, X, y, n_jobs=2)
print()
print("pomegranate GNB (4 jobs)")
%timeit NaiveBayes.from_samples(NormalDistribution, X, y, n_jobs=4)
Explanation: 1. Naive Bayes
Let's start off with a simple model like naive Bayes.
End of explanation
print("pomegranate GBC")
%timeit BayesClassifier.from_samples(MultivariateGaussianDistribution, X, y)
print()
print("pomegranate GBC (2 jobs)")
%timeit BayesClassifier.from_samples(MultivariateGaussianDistribution, X, y, n_jobs=2)
print()
print("pomegranate GBC (4 jobs)")
%timeit BayesClassifier.from_samples(MultivariateGaussianDistribution, X, y, n_jobs=4)
Explanation: It looks like scikit-learn starts off as faster on a single thread, but that adding multiple threads will improve pomegranate's performance, typically past that of scikit-learn on a single thread. The reason this is the case is likely because pomegranate has to create one object per feature per class, whereas scikit-learn will just define the means and variances in a matrix. It's efficient when you know what distribution you're implementing in the naive Bayes, but not as modular.
2. Bayes' Classifier
The next complex model is the Bayes' classifier. This model is similar to the naive Bayes' model, but does not require that the features are independent from each other.
End of explanation
from sklearn.mixture import GaussianMixture
n, d, k = 1000000, 5, 3
X, y = create_dataset(n, d, k)
print("sklearn GMM")
%timeit GaussianMixture(n_components=k, covariance_type='full', max_iter=15, tol=1e-10).fit(X)
print()
print("pomegranate GMM")
%timeit GeneralMixtureModel.from_samples(MultivariateGaussianDistribution, k, X, max_iterations=15, stop_threshold=1e-10)
print()
print("pomegranate GMM (2 jobs)")
%timeit GeneralMixtureModel.from_samples(MultivariateGaussianDistribution, k, X, n_jobs=2, max_iterations=15, stop_threshold=1e-10)
print
print("pomegranate GMM (4 jobs)")
%timeit GeneralMixtureModel.from_samples(MultivariateGaussianDistribution, k, X, n_jobs=4, max_iterations=15, stop_threshold=1e-10)
Explanation: We see a nice spead improvement here as well. There is no model to compare against in scikit-learn because their naive Bayes' model assumes that all features are independent.
3. General Mixture Models
Next, lets take a look at Gaussian mixture models.
End of explanation
X, _ = create_dataset(1000, 25, 2)
a = GaussianMixture(k, n_init=1, max_iter=25).fit(X)
b = GeneralMixtureModel.from_samples(MultivariateGaussianDistribution, k, X, max_iterations=25)
X, _ = create_dataset(1000000, 25, 2)
print("sklearn GMM")
%timeit -n 1 a.predict_proba(X)
print()
print("pomegranate GMM")
%timeit -n 1 b.predict_proba(X)
print()
print("pomegranate GMM (4 jobs)")
%timeit -n 1 b.predict_proba(X, n_jobs=4)
Explanation: It looks like pomegranate can be faster than sklearn at performing 15 iterations of EM on this data set, and that parallelization can speed things up even further. We can ignore the convergence warnings. Here we're just timing a fixed amount of computational work. Different models might converge after a different number of iterations, and we don't want the pseedup to come just from that.
Lets now take a look at the time it takes to make predictions using GMMs. Lets fit the model to a small amount of data, and then predict a larger amount of data drawn from the same underlying distributions.
End of explanation
print((b.predict_proba(X) - b.predict_proba(X, n_jobs=4)).sum())
Explanation: It looks like pomegranate can be slightly slower than sklearn when using a single processor, but that it can be parallelized to get faster performance.
To ensure that we're getting the exact same results, only faster, lets subtract the predictions from each other and make sure that the sum is equal to 0.
End of explanation
X, y = create_dataset(1000, 2, 2, alpha=2)
a = GaussianMixture(2, n_init=1, max_iter=25).fit(X)
b = GeneralMixtureModel.from_samples(MultivariateGaussianDistribution, 2, X, max_iterations=25)
y1, y2 = a.predict(X), b.predict(X)
plt.figure(figsize=(16,6))
plt.subplot(121)
plt.title("sklearn clusters", fontsize=14)
plt.scatter(X[y1==0, 0], X[y1==0, 1], color='m', edgecolor='m')
plt.scatter(X[y1==1, 0], X[y1==1, 1], color='c', edgecolor='c')
plt.subplot(122)
plt.title("pomegranate clusters", fontsize=14)
plt.scatter(X[y2==0, 0], X[y2==0, 1], color='m', edgecolor='m')
plt.scatter(X[y2==1, 0], X[y2==1, 1], color='c', edgecolor='c')
plt.show()
Explanation: Great, no difference between the two.
Lets now make sure that pomegranate and sklearn are learning basically the same thing. Lets fit both models to some 2 dimensional 2 component data and make sure that they both extract the underlying clusters by plotting them.
End of explanation
X = numpy.random.randn(1000, 500, 50)
print("pomegranate Gaussian HMM (1 job)")
%timeit -n 1 -r 1 HiddenMarkovModel.from_samples(NormalDistribution, 5, X, max_iterations=5)
print()
print("pomegranate Gaussian HMM (2 jobs)")
%timeit -n 1 -r 1 HiddenMarkovModel.from_samples(NormalDistribution, 5, X, max_iterations=5, n_jobs=2)
Explanation: It looks like we're getting the same basic results for the two. The two algorithms are initialized a bit differently, and so it can be difficult to directly compare the results between them, but it looks like they're getting roughly the same results.
3. Multivariate Gaussian HMM
Now let's move on to training a hidden Markov model with multivariate Gaussian emissions with a diagonal covariance matrix. We'll randomly generate some Gaussian distributed numbers and use pomegranate with either one or four threads to fit our model to the data.
End of explanation
model = HiddenMarkovModel.from_samples(NormalDistribution, 5, X, max_iterations=2, verbose=False)
print("pomegranate Gaussian HMM (1 job)")
%timeit predict_proba(model, X)
print()
print("pomegranate Gaussian HMM (2 jobs)")
%timeit predict_proba(model, X, n_jobs=2)
Explanation: All we had to do was pass in the n_jobs parameter to the fit function in order to get a speed improvement. It looks like we're getting a really good speed improvement, as well! This is mostly because the HMM algorithms perform a lot more operations than the other models, and so spend the vast majority of time with the GIL released. You may not notice as strong speedups when using a MultivariateGaussianDistribution because BLAS uses multithreaded operations already internally, even when only one job is specified.
Now lets look at the prediction function to make sure the we're getting speedups there as well. You'll have to use a wrapper function to parallelize the predictions for a HMM because it returns an annotated sequence rather than a single value like a classic machine learning model might.
End of explanation
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