To understand Gaussian process regression, and to be able to generate nonparametric regressions with confidence intervals. Also to understand the interplay between a kernel (or covariance) function, and the resulting confidence intervals of the regression.
You will turn in an iPython notebook that performs Gaussian process regression on a simple dataset. You will explore multiple kernels and vary their parameter settings.
When you are done, you should produce visualizations like the following (for noiseless observations):
and like this (for noisy observations):
Your notebook will be graded on the following:
The data that we will use for this lab is simple:
data_xvals = numpy.atleast_2d( [ 1.0, 3.0, 5.0, 6.0, 7.0, 8.0 ] ) data_yvals = numpy.sin( data_xvals )
You must perform Gaussian process regression on this dataset, and produce visualizations for both noiseless and noisy observations. Your notebook should produce one visualization for each of the following kernel types:
Therefore, your notebook should produce six different visualizations: two for each kernel type.
For the noisy observation case, use \sigma_n^2=0.1.
For the polynomial kernel use a degree of 3.
For the Gaussian / RBF kernel, set all parameters to 1.0
You should also answer the following questions:
Your visualizations should be done on the range
[-2 10] of the x-axis.
For the errorbars, you can just plot the mean +/- the variance. This isn't really a statistically meaningful quantity, but it makes the plots look nice. :)
Hint: a Gaussian process only allows you to make a prediction for a single query point. So how do you generate the smoothly varying lines in the example images?
The following functions may be useful to you:
plt.gca().fill_between plt.scatter numpy.linalg.inv numpy.eye