cs401r_w2016:lab13
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cs401r_w2016:lab13 [2016/01/13 17:26] admin |
cs401r_w2016:lab13 [2021/06/30 23:42] |
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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 should look something like this: | ||
| - | |||
| - | {{: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 is tidy and legible | ||
| - | |||
| - | ---- | ||
| - | ====Description:==== | ||
| - | |||
| - | 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 ] | ||
| - | |||
| - | </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 | ||
| - | |||
| - | # 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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