cs401r_w2016:lab13
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| - | ====Objective:==== | ||
| - | To understand expectation maximization, and to explore how to learn the parameters of a Gaussian mixture model. | ||
| - | |||
| - | ---- | ||
| - | ====Deliverable:==== | ||
| - | |||
| - | For this lab, you will implement the Expectation Maximization algorithm on the Old Faithful dataset. This involves learning the parameters of a Gaussian mixture model. Your notebook should produce a visualization of the progress of the algorithm. The final figure could look something like this (they don't have to be arranged in subplots): | ||
| - | |||
| - | {{:cs401r_w2016:lab5_em.png?direct&600|}} | ||
| - | |||
| - | ---- | ||
| - | ====Grading:==== | ||
| - | Your notebook will be graded on the following elements: | ||
| - | |||
| - | * 10% Data is correctly mean-centered | ||
| - | * 20% Correctly updates responsibilities | ||
| - | * 20% Correctly updates means | ||
| - | * 20% Correctly updates covariances | ||
| - | * 20% Correctly updates mixing weights | ||
| - | * 10% Final plot(s) is tidy and legible | ||
| - | |||
| - | ---- | ||
| - | ====Description:==== | ||
| - | |||
| - | For this lab, we will be using the Expectation Maximization (EM) method to **learn** the parameters of a Gaussian mixture model. These parameters will reflect cluster structure in the data -- in other words, we will learn probabilistic descriptions of clusters in the data. | ||
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| - | For this lab, you will use the Old Faithful dataset, which you can download here: | ||
| - | |||
| - | [[http://hatch.cs.byu.edu/courses/stat_ml/old_faithful.mat|Old Faithful dataset]] | ||
| - | |||
| - | **The first thing you should is mean-center your data (ie, compute the mean of the data, then subtract that off from each datapoint).** (If you don't do this, you'll get zero probabilities for all of your responsibilities, given the initial conditions discussed below.) | ||
| - | |||
| - | The equations for implementing the EM algorithm are given in MLAPP 11.4.2.2 - 11.4.2.3. | ||
| - | |||
| - | The algorithm is: | ||
| - | |||
| - | - Compute the responsibilities $r_{ik}$ (Eq. 11.27) | ||
| - | - Update the mixing weights $\pi_k$ (Eq. 11.28) | ||
| - | - Update the covariances $\Sigma_k$ (Eq. 11.32) | ||
| - | - Update the means $\mu_k$ (Eq. 11.31) | ||
| - | |||
| - | Now, repeat until convergence. Note that if you change the order of operations, you may get slightly difference convergences than the reference image. | ||
| - | |||
| - | Since the EM algorithm is deterministic, and since precise initial conditions for your algorithm are given below, the progress of your algorithm should closely match the reference image shown above. | ||
| - | |||
| - | For your visualization, please print out at least nine plots. These should color each datapoint using $r_{ik}$, and they should plot the means and covariances of the Gaussians. See the hints section for how to plot an ellipse representing the 95% confidence interval of a Gaussian, given an arbitrary covariance matrix. | ||
| - | |||
| - | |||
| - | **Note: To help our TA better grade your notebook, you should use the following initial parameters:** | ||
| - | |||
| - | <code python> | ||
| - | |||
| - | # the Gaussian means (as column vectors -- ie, the mean for Gaussian 0 is mus[:,0] | ||
| - | mus = np.asarray( [[-1.17288986, -0.11642103], | ||
| - | [-0.16526981, 0.70142713]]) | ||
| - | |||
| - | # the Gaussian covariance matrices | ||
| - | covs = list() | ||
| - | covs.append( | ||
| - | np.asarray([[ 0.74072815, 0.09252716], | ||
| - | [ 0.09252716, 0.5966275 ]]) ) | ||
| - | covs.append( | ||
| - | np.asarray([[ 0.39312776, -0.46488887], | ||
| - | [-0.46488887, 1.64990767]]) ) | ||
| - | |||
| - | # The Gaussian mixing weights | ||
| - | mws = [ 0.68618439, 0.31381561 ] # called alpha in the slides | ||
| - | |||
| - | </code> | ||
| - | |||
| - | ---- | ||
| - | ====Hints:==== | ||
| - | |||
| - | In order to visualize a covariance matrix, you should plot an ellipse representing the 95% confidence bounds of the corresponding Gaussian. Here is some code that accepts as input a covariance matrix, and returns a set of points that define the correct ellipse; these points can be passed directly to the ''plt.plot()'' command as the ''x'' and ''y'' parameters. | ||
| - | |||
| - | <code python> | ||
| - | import numpy as np | ||
| - | |||
| - | def cov_to_pts( cov ): | ||
| - | circ = np.linspace( 0, 2*np.pi, 100 ) | ||
| - | sf = np.asarray( [ np.cos( circ ), np.sin( circ ) ] ) | ||
| - | [u,s,v] = np.linalg.svd( cov ) | ||
| - | pmat = u*2.447*np.sqrt(s) # 95% confidence | ||
| - | return np.dot( pmat, sf ) | ||
| - | |||
| - | </code> | ||
| - | |||
| - | Here are some additional python functions that may be helpful to you: | ||
| - | |||
| - | <code python> | ||
| - | |||
| - | # compute the likelihood of a multivariate Gaussian | ||
| - | scipy.stats.multivariate_normal.pdf | ||
| - | |||
| - | # scatters a set of points; check out the "c" keyword argument to change color, and the "s" arg to change the size | ||
| - | plt.scatter | ||
| - | plt.xlim # sets the range of values for the x axis | ||
| - | plt.ylim # sets the range of values for the y axis | ||
| - | |||
| - | # to check the shape of a vector, use the .shape member | ||
| - | foo = np.random.randn( 100, 200 ) | ||
| - | foo.shape # an array with values (100,200) | ||
| - | |||
| - | # to transpose a vector, you can use the .T operator | ||
| - | foo = np.atleast_2d( [42, 43] ) # this is a row vector | ||
| - | foo.T # this is a column vector | ||
| - | |||
| - | import numpy as np | ||
| - | np.atleast_2d | ||
| - | np.sum | ||
| - | |||
| - | </code> | ||
cs401r_w2016/lab13.txt ยท Last modified: 2021/06/30 23:42 (external edit)
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