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-========= Assignment 3: Support Vector Machines ============
+====== Assignment 2 ======
-===== Part 1:  SVM with no bias term =====
+**Due:** 10/1 at 11:59pm.
-Formulate a soft-margin SVM without the bias term, i.e. $f(\mathbf{x}) = \mathbf{w}^{T} \mathbf{x}$.
+In this assignment you will explore ridge regression applied to the task of predicting wine quality.
-Derive the saddle point conditions, KKT conditions and the dual.
+You will use the [[http://archive.ics.uci.edu/ml/datasets/Wine+Quality | wine quality]] dataset from the UCI machine learning repository, and compare accuracy obtained using ridge regression to the results from a [[http://www.sciencedirect.com/science/article/pii/S0167923609001377# | recent publication]] (if you have trouble accessing that version of the paper, here's a link to a [[http://www3.dsi.uminho.pt/pcortez/wine5.pdf| preprint]].
-Compare it to the standard SVM formulation.
+The wine data is composed of two datasets - one for white wines, and one for reds.  Perform all your analyses on both.
-What is the implication of the difference on the design of SMO-like algorithms?
-Recall that SMO algorithms work by iteratively optimizing two variables at a time.
-Hint:  consider the difference in the constraints.
-===== Part 2:  Closest Centroid Algorithm =====
+=== Part 1 ===
+Implement ridge regression and functions for computing the following measures of error:
-Express the closest centroid algorithm in terms of kernels, i.e. determine how the coefficients $\alpha_i$ will be computed using a given labeled dataset.
+  * The Root Mean Square Error (RMSE).
+  * The Maximum Absolute Deviation (MAD).
-===== Part 3:  Soft-margin SVM for separable data =====
-Consider training a soft-margin SVM
+For a hypothesis $h$, they are defined as follows:
-with $C$ set to some positive constant. Suppose the training data is linearly separable.
-Since increasing the $\xi_i$ can only increase the objective of the primal problem (which
-we are trying to minimize), at the optimal solution to the primal problem, all the
-training examples will have $\xi_i$ equal
-to zero. True or false?  Explain!
-Given a linearly separable dataset, is it necessarily better to use a
-a hard margin SVM over a soft-margin SVM?
-===== Part 4:  Using SVMs =====
+$$RMSE(h) = \sqrt{\frac{1}{N}\sum_{i=1}^N (y_i - h(\mathbf{x}_i))^2}$$
-The data for this question comes from a database called SCOP (structural
+and
-classification of proteins), which classifies proteins into classes
-according to their structure (download it from {{assignments:scop_motif.data|here}}).
-The data is a two-class classification
-problem
-of distinguishing a particular class of proteins from a selection of
-examples sampled from the rest of the SCOP database.
-I chose to represent the proteins in
-terms of their motif composition.  A sequence motif is a
-pattern of nucleotides/amino acids that is conserved in evolution.
-Motifs are usually associated with regions of the protein that are
-important for its function, and are therefore useful in differentiating between classes of proteins.
-A given protein will typically contain only a handful of motifs, and
-so the data is very sparse.  It is also very high dimensional, since
-the number of conserved patterns in the space of all proteins is
-large.
-The data was constructed as part of the following analysis of detecting distant relationships between proteins:
-  * A. Ben-Hur and D. Brutlag. [[http://bioinformatics.oxfordjournals.org/content/19/suppl_1/i26.abstract | Remote homology detection: a motif based approach]]. In: Proceedings, eleventh international conference on intelligent systems for molecular biology. Bioinformatics 19(Suppl. 1): i26-i33, 2003.
+$$MAD(h) = \frac{1}{N}\sum_{i=1}^N |y_i - h(\mathbf{x}_i)|.$$
-In this part of the assignment we will explore the dependence of classifier accuracy on
+Compute these measures of error for ridge regression applied
-the kernel, kernel parameters, kernel normalization, and SVM parameter soft-margin parameter.
+to the wine dataset over a range of the regularization parameter, $\lambda$ (choose values on a logarithmic scale, e.g. 0.01, 0.1, 1, 10, 100, 1000) and plot the results (use a fixed test set for computing them!)
-The use of the SVM class is discussed in the PyML [[http://pyml.sourceforge.net/tutorial.html#svms|tutorial]], and by using help(SVM) in the python interpreter.
+The features for the wine dataset are not standardized, so make sure you do this, especially since we are going to consider the magnitude of the weight vector (recall that standardization entails subtracting the mean and then dividing by the standard deviation for each feature; you can use the [[http://docs.scipy.org/doc/numpy/reference/routines.statistics.html | Numpy statistics module]] to perform the required calculations).  What is the potential advantage of MAD over RMSE?
-By default, a dataset is instantiated with a linear kernel attached to it.
+In addition to RMSE and MAD, plot the Regression Error Characteristic (REC) curve of a representative classifier.
-To use a different kernel you need to attach a new kernel to the dataset:
+REC curves are described in the following [[http://machinelearning.wustl.edu/mlpapers/paper_files/icml2003_BiB03.pdf|paper]].
-<code python>
+What can you learn from this curve that you cannot learn from RMSE or MAD?
->>> from PyML import ker
->>> data.attachKernel(ker.Gaussian(gamma = 0.1))
-</code>
-or
-<code python>
->>> from PyML import her
->>> data.attachKernel(ker.Polynomial(degree = 3))
-</code>
-Alternatively, you can instantiate an SVM with a given kernel:
-<code python>
->>> classifier = SVM(ker.Gaussian(gamma = 0.1))
-</code>
-This will override the kernel the data is associated with.
-In this question we will consider both the Gaussian and polynomial kernels:
+Compare the results that you are getting with the published results in the paper.
-$$
-K_{gauss}(\mathbf{x}, \mathbf{x'}) = \exp(-\gamma || \mathbf{x} - \mathbf{x}' ||^2)
-$$
-$$
-K_{poly}(\mathbf{x}, \mathbf{x'}) = (1 + \mathbf{x}^T \mathbf{x}') ^{p}
-$$
-Plot the accuracy of the SVM, measured using the balnced success rate
-as a function of both the soft-margin parameter of the SVM, and the free parameter
-of the kernel function.
-Show a couple of representative cross sections of this plot for a given value
-of the ridge parameter, and for a given value of the kernel parameter.
-Comment on the results.  When exploring the values of a continuous
-classifier/kernel parameter it is
-useful to use values that are distributed on an exponential grid,
-i.e. something like 0.01, 0.1, 1, 10, 100 (note that the degree of the
-polynomial kernel is not such a parameter).
-For this type of sparse dataset it is useful to normalize the input features.
+=== Part 2 ===
-One way to do so is to divide each input example by its norm.  This is
-accomplished in PyML by:
-<code python>
->>> data.normalize()
-</code>
-Compare the results under this normalization with what you obtain
-without normalization.
-You can visualize the whole kernel matrix associated with the data using the following commands:
+As we discussed in class, the magnitude of the weight vector can be interpreted as a measure of feature importance.
-<code python>
+Train a ridge regression classifier on a subset of the dataset that you reserved for training.
->>> from PyML import ker
+We will explore the relationship between the magnitude of weight vector components and their relevance to the classification task in several ways.
->>> ker.showKernel(data)
+Each feature is associated with a component of the weight vector.  It can also be associated with the correlation of that feature with the vector of labels.
+Create a scatter plot of the weight vector component against the [[https://en.wikipedia.org/wiki/Pearson_product-moment_correlation_coefficient | Pearson correlation coefficient]] of a feature against the labels (again, you can use the [[http://docs.scipy.org/doc/numpy/reference/routines.statistics.html | Numpy statistics module]] to compute it).
+What can you conclude from this plot?
+The paper ranks features according to their importance using a different approach.  Compare your results with what they obtain.
+Next, perform the following experiment:
+Incrementally remove the feature with the lowest absolute value of the weight vector and retrain the ridge regression classifier.
+Plot RMSE and MAD as a function of the number of features that remain on the test set which you have set aside.
+===== Submission =====
+Submit your report via Canvas.  Python code can be displayed in your report if it is succinct (not more than a page or two at the most) or submitted separately.  The latex sample document shows how to display Python code in a latex document.  Code needs to be there so we can make sure that you implemented the algorithms and data analysis methodology correctly.  Canvas allows you to submit multiple files for an assignment, so DO NOT submit an archive file (tar, zip, etc).
+===== Grading =====
+Here is what the grade sheet will look like for this assignment.  A few general guidelines for this and future assignments in the course:
+  * Always provide a description of the method you used to produce a given result in sufficient detail such that the reader can reproduce your results on the basis of the description (UNLESS the method has been provided in class or is there in the book).  Your code needs to be provided in sufficient detail so we can make sure that your implementation is correct.  The saying that "the devil is in the details" holds true for machine learning, and is sometimes the makes the difference between correct and incorrect results.  If your code is more than a few lines, you can include it as an appendix to your report, or submit it as a separate file.  Make sure your code is readable!
+  * You can provide results in the form of tables, figures or text - whatever form is most appropriate for a given problem.
+  * In any machine learning paper there is a discussion of the results.  There is a similar expectation from your assignments that you reason about your results.  For example, for the learning curve problem, what can you say on the basis of the observed learning curve?
+  * Write succinct answers.  We will take off points for rambling answers that are not to the point, and and similarly, if we have to wade through a lot of data/results that are not to the point.
+<code>
+Grading sheet for assignment 2
+Part 1:  50 points.
+(20 points):  Plots of MAD and RMSE as a function of lambda are generated correctly.
+(20 points):  REC curves are generated correctly
+( 5 points):  discussion of REC curves
+( 5 points):  Discussion of the MAD and RMSE plots and comparison of results to the published ones.
+Part 2:  40 points.
+(30 points):  Weight vector analysis
+(10 points):  Comparison to the published results
+Report structure, grammar and spelling:  10 points
+(10 points):  Heading and subheading structure easy to follow and clearly divides report into logical sections.
+              Code, math, figure captions, and all other aspects of the report are well-written and formatted.
+              Grammar, spelling, and punctuation.
 </code>
-Explain the structure that you are seeing in the plot (it is more
-interesting when the data is normalized).

CS545 fall 2016

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