Description
Introduction
In this project, you will design three classifiers: a naive Bayes classifier and
a perceptron classifier. You will test your classifiers on two image data sets:
a set of scanned handwritten digit images and a set of face images in which
edges have already been detected. Even with simple features, your
classifiers will be able to do quite well on these tasks when given enough
training data.
Optical character recognition (OCR) is the task of extracting text from image
sources. The first data set on which you will run your classifiers is a
collection of handwritten numerical digits (0-9). This is a very commercially
useful technology, similar to the technique used by the US post office to
route mail by zip codes.
There are systems that can perform with over 99%
classification accuracy (see LeNet-5 for an example system in action).
Face detection is the task of localizing faces within video or still images. The
faces can be at any location and vary in size.
There are many applications
for face detection, including human computer interaction and surveillance.
You will attempt a simplified face detection task in which your system is
presented with an image that has been pre-processed by an edge detection
algorithm. The task is to determine whether the edge image is a face or not.
There are several systems in use that perform quite well at the face
detection task. One good system is the Face Detector by Schneiderman and
Kanade. You can even try it out on your own photos in this demo.
The code for this project includes the following files and data, available as
a zip archive (classification.zip) on Canvas.
Data file
classification.zip Data file, including the digit and face data.
Files you will edit
NaiveBayes.py The location where you will write your naive
Bayes classifier.
perceptron.py The location where you will write your
perceptron classifier.
answers.py Answers to Question 2 and Question 4 go here.
Files you should read but NOT edit
samples.py
I/O code to read in the classification data. (You
will see how raw pixels are represented for the
digit and the face pictures.)
dataClassifier.py
The wrapper code that will call your classifiers.
You will also use this code to analyze the
behavior of your classifier. (See the
basicFeatureExtractorFace method for
turning raw data to data used for training.)
classificationMethod.py
Abstract super class for the classifiers you will
write.
(You should read this file carefully to see how
the infrastructure is set up.)
util.py
Code defining some useful tools. You may be
familiar with some of these by now, and they
will save you a lot of time. (Understand the
data structure Counter.)
mostFrequent.py A simple baseline classifier that just labels
every instance as the most frequent class.
What to submit: You will fill in portions
of answers.py, naiveBayes.py, and perceptron.py (only) during the
assignment, and submit them.
Evaluation: Your code will be autograded for technical correctness.
Please do not change the names of any provided functions or classes within
the code, or you will wreak havoc on the autograder.
Academic Dishonesty: We will be checking your code against other
submissions in the class for logical redundancy. If you copy someone else’s
code and submit it with minor changes, we will know. These cheat detectors
are quite hard to fool, so please don’t try. We trust you all to submit your
own work only; please don’t let us down. Instead, contact the course staff if
you are having trouble.
Getting Started
To try out the classification pipeline, run dataClassifier.py from the
command line. This will classify the digit data using the default classifier
(mostFrequent) which blindly classifies every example with the most frequent
label.
python dataClassifier.py
As usual, you can learn more about the possible command line options by
running:
python dataClassifier.py -h
We have defined some simple features for you. Later you will design some
better features. Our simple feature set includes one feature for each pixel
location, which can take values 0 or 1 (off or on). The features are encoded
as a Counter where keys are feature locations (represented as (column,row))
and values are 0 or 1. (See util.py for details. Actually, I highly
recommend you study this data structure and its provided methods before
going any further.)
The face recognition data set has value 1 only for those
pixels identified by a Canny edge detector.
Implementation Note: You’ll find it easiest to hard-code the binary feature
assumption. If you do, make sure you don’t include any non-binary features.
Or, you can write your code more generally, to handle arbitrary feature
values, though this will probably involve a preliminary pass through the
training set to find all possible feature values (and you’ll need an “unknown”
option in case you encounter a value in the test data you never saw during
training).
Naive Bayes
A skeleton implementation of a naive Bayes classifier is provided for you
in naiveBayes.py. You will fill in the trainAndTune function,
the calculateLogJointProbabilities function and the
findHighOddsFeatures function.
Theory
A naive Bayes classifier models a joint distribution over a label and a set
of observed random variables, or features, , using the
assumption that the full joint distribution can be factored as follows (features
are conditionally independent given the label):
To classify a datum, we can find the most probable label given the feature
values for each pixel, using Bayes theorem:
Because multiplying many probabilities together often results in underflow,
we will instead compute log probabilities which have the same argmax:
To compute logarithms, use math.log(), a built-in Python function.
Parameter Estimation
Our naive Bayes model has several parameters to estimate. One parameter
is the prior distributionover labels (digits, or face/not-face), .
We can estimate directly from the training data:
where is the number of training instances with label y and n is the
total number of training instances.
The other parameters to estimate are the conditional probabilities of our
features given each label y: . We do this for each possible
feature value ( ).
where is the number of times pixel took value in the
training examples of label y.
Smoothing
Your current parameter estimates are unsmoothed, that is, you are using
the empirical estimates for the parameters . These estimates are
rarely adequate in real systems. Minimally, we need to make sure that no
parameter ever receives an estimate of zero, but good smoothing can boost
accuracy quite a bit by reducing overfitting.
In this project, we use Laplace smoothing, which adds k counts to every
possible observation value:
If k=0, the probabilities are unsmoothed. As k grows larger, the probabilities
are smoothed more and more. You can use your validation set to determine
a good value for k. Note: don’t smooth P(Y).
Question 1 (60
points) Implement trainAndTune and calculateLogJointProbabilities in naiveB
ayes.py. In trainAndTune, estimate conditional probabilities from the training
data for each possible value of k given in the list kgrid. Evaluate accuracy on
the held-out validation set for each k and choose the value with the highest
validation accuracy. In case of ties, prefer the lowest value of k. Test your
classifier with:
python dataClassifier.py -c naiveBayes –autotune
Hints and observations:
• The method calculateLogJointProbabilities uses the conditional probability
tables constructed bytrainAndTune to compute the log posterior probability
for each label y given a feature vector. The comments of the method
describe the data structures of the input and output.
• You can add code to the analysis method in dataClassifier.py to
explore the mistakes that your classifier is making. This is optional.
• When trying different values of the smoothing parameter k, think about
the number of times you scan the training data. Your code should save
computation by avoiding redundant reading.
• To run your classifier with only one particular value of k, remove the —
autotune option. This will ensure that kgrid has only one value, which you
can change with -k.
• Using a fixed value of k=2 and 100 training examples, you should get a
validation accuracy of about 69% and a test accuracy of 55%.
• Using –autotune, which tries different values of k, you should get a
validation accuracy of about 74% and a test accuracy of 65%.
• Accuracies may vary slightly because of implementation details. For
instance, ties are not deterministically broken in
the Counter.argMax() method.
• To run on the face recognition dataset, use -d faces (optional).
Odds Ratios
One important tool in using classifiers in real domains is being able to
inspect what they have learned. One way to inspect a naive Bayes model is
to look at the most likely features for a given label.
Another, better, tool for understanding the parameters is to look at odds
ratios. For each pixel feature and classes , consider the odds
ratio:
This ratio will be greater than one for features which cause belief in to
increase relative to .
The features that have the greatest impact at classification time are those
with both a high probability (because they appear often in the data) and a
high odds ratio (because they strongly bias one label versus another).
To run the autograder for this question:
python autograder.py -q q1
Question 2 (5 points) Fill in the function findHighOddsFeatures(self, label1,
label2). It should return a list of the 100 features with highest odds ratios
for label1 over label2. The option -o activates an odds ratio analysis. Use the
options -1 label1 -2 label2 to specify which labels to compare. Running the
following command will show you the 100 pixels that best distinguish
between a 3 and a 6.
python dataClassifier.py -a -d digits -c naiveBayes -o -1 3 -2
6
Use what you learn from running this command to answer the following
question. Which of the following images best shows those pixels which have
a high odds ratio with respect to 3 over 6? (That is, which of these is most
like the output from the command you just ran?)
(a) (b) (c) (d) (e)
To answer: please return ‘a’, ‘b’, ‘c’, ‘d’, or ‘e’ from the
function q2 in answers.py.
Hints:
• To avoid divide-zero errors you may want to use smoothing when
computing the odds ratio.
Perceptron
A skeleton implementation of a perceptron classifier is provided for you
in perceptron.py. You will fill in the train function, and
the findHighWeightFeatures function.
Unlike the naive Bayes classifier, a perceptron does not use probabilities to
make its decisions. Instead, it keeps a weight vector of each
class ( is an identifier, not an exponent). Given a feature list , the
perceptron compute the class whose weight vector is most similar to the
input vector .
Formally, given a feature vector (in our case, a map from
pixel locations to indicators of whether they are on), we score each class
with:
Then we choose the class with highest score as the predicted label for that
data instance. In the code, we will represent as a Counter.
Learning weights
In the basic multi-class perceptron, we scan over the data, one instance at a
time. When we come to an instance , we find the label with highest
score:
We compare to the true label . If , we’ve gotten the instance
correct, and we do nothing. Otherwise, we guessed but we should have
guessed . That means that should have scored higher,
and should have scored lower, in order to prevent this error in the
future.
We update these two weight vectors accordingly:
Using the addition, subtraction, and multiplication functionality of
the Counter class in util.py, the perceptron updates should be relatively
easy to code.
Certain implementation issues have been taken care of for you
in perceptron.py, such as handling iterations over the training data and
ordering the update trials. Furthermore, the code sets up the weights data
structure for you. Each legal label needs its own Counter full of weights.
Question 3 (30 points) Fill in the train method in perceptron.py. Run
your code with:
python dataClassifier.py -c perceptron
Hints and observations:
• The command above should yield validation accuracies in the range
between 40% to 70% and test accuracy between 40% and 70% (with the
default 3 iterations). These ranges are wide because the perceptron is a
lot more sensitive to the specific choice of tie-breaking than naive Bayes.
• One of the problems with the perceptron is that its performance is
sensitive to several practical details, such as how many iterations you
train it for, and the order you use for the training examples (in practice,
using a randomized order works better than a fixed order). The current
code uses a default value of 3 training iterations.
You can change the
number of iterations for the perceptron with the -i iterations option. Try
different numbers of iterations and see how it influences the
performance. In practice, you would use the performance on the
validation set to figure out when to stop training, but you don’t need to
implement this stopping criterion for this assignment.
To run the autograder for this question and visualize the output:
python autograder.py -q q3
Visualizing weights
Perceptron classifiers, and other discriminative methods, are often criticized
because the parameters they learn are hard to interpret. To see a
demonstration of this issue, we can write a function to find features that are
characteristic of one class. (Note that, because of the way perceptrons are
trained, it is not as crucial to find odds ratios.)
Question 4 (5 point) Fill in findHighWeightFeatures(self,
label) in perceptron.py. It should return a list of the 100 features with
highest weight for that label.
You can display the 100 pixels with the largest
weights using the command:
python dataClassifier.py -c perceptron -w
Use this command to look at the weights, and answer the following
true/false question. Which of the following sequence of weights is most
representative of the perceptron?
Answer the question answers.py in the method q4, returning either ‘a’ or ‘b’.
