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Lecture 3: Introduction to Classification

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Lecture 3: Introduction to Classification

CS 175, Fall 2007

Padhraic Smyth

Department of Computer Science

University of California, Irvine

- Overview of Classification:
- examples and applications of classification
- classification: mapping from features to a class label
- decision boundaries
- training and test data accuracy
- the nearest-neighbor classifier

- Assignments:
- Assignment 2 due Wednesday next week
- plotting classification data
- k-nearest-neighbor classifiers

- Assignment 2 due Wednesday next week

- Classification is an important component of intelligent systems
- We have a special discrete-valued variable called the class, C
- C takes values c, where c = 1, c = 2, …., c = m
- for now assume m=2, i.e., 2 classes: c= 1 or c= 2

- Problem is to decide what class an object is
- i.e., what value the class variable C is for a given object
- given measurements on the object, e.g., A, B, ….
- These measurements are called “features”
- we wish to learn a mapping from Features -> Class

- Notation:
- C is the class
- A, B, etc (the measurements) are called the “features” (sometimes also called “attributes” or “input variables”)

Feature Values (which

are known, measured)

Predicted Class Value

(true class is unknown

to the classifier)

a

b

c

Classifier

d

z

We want a mapping or function which takes any combination of

values x = (a, b, d, ..... z) and will produce a prediction c,

i.e., a function c = f(a, b, d, …. z) which produces a value c=1, c=2,…c=m

The problem is that we don’t know this mapping: we have to learn it from data!

- Medical Diagnosis
- classification of cancerous cells

- Credit card and Loan approval
- Most major banks

- Speech recognition
- IBM, Dragon Systems, AT&T, Microsoft, etc

- Optical Character/Handwriting Recognition
- Post Offices, Banks, Gateway, Motorola, Microsoft, Xerox, etc

- Email classification
- classify email as “junk” or “non-junk”

- Many other applications
- one of the most successful applications of AI technology

Class 2

Class 1

- Feature Vector:
- Say we have 2 features: we can think of the features as a 2-component vector
- i.e., a 2-dimensional vector, [a b]

- So the features correspond to a 2-dimensional space
- (clearly we can generalize to d-dimensional space)
- this is called the “feature space”

- Each feature vector represents the “coordinates” of a particular object in feature space
- If the feature-space is 2-dimensional (for example), and the features a and b are real-valued
- we can visually examine and plot the locations of the feature vectors

- Say we have 2 features: we can think of the features as a 2-component vector

- Now consider that we have data from m classes (e.g., m=2)
- We can imagine the data from each class being in a “cloud” in feature space
- data sets D1 and D2 (sets of points from classes 1 and 2)
- data are of dimension d (i.e., d-dimensional vectors)
- if d = 2 (2 features), we can plot the data
- we should see two “clouds” of data points, one cloud per class

Control Group

Anemia Group

- What is a Classifier?
- A classifier is a mapping from feature space (a d-dimensional vector) to the class labels {1, 2, … m}
- Thus, a classifier partitions the feature space into m decision regions
- The line or surface separating any 2 classes is the decision boundary

- Linear Classifiers
- a linear classifier is a mapping which partitions feature space using a linear function (a straight line, or a hyperplane)
- it is one of the simplest classifiers we can imagine
- in 2 dimensions the decision boundary is a straight line

- Consider two class case
- data from D1 and D2 may overlap
- features = {age, body temperature}, classes = {flu, not-flu}
- features = {income, savings}, classes = {good/bad risk}

- common in practice that the classes will naturally overlap
- this means that our features are usually not able to perfectly discriminate between the classes
- note: with more expensive/more detailed additional features (e.g., a specific test for the flu) we might be able to get perfect separation

- if there is overlap => classes are not linearly separable

- data from D1 and D2 may overlap

- Say we have N feature vectors
- Say we know the true class label for each feature vector
- We can measure how accurate a classifier is by how many feature vectors it classifies correctly
- Accuracy = percentage of feature vectors correctly classified
- training accuracy = accuracy on training data
- test accuracy = accuracy on new data not used in training

- Training data
- labeled data used to build a classifier

- Test data
- new data, not used in the training process, to evaluate how well a classifier does on new data

- Memorization versus Generalization
- better training_accuracy
- “memorizing” the training data:

- better test_accuracy
- “generalizing” to new data

- in general, we would like our classifier to perform well on new test data, not just on training data,
- i.e., we would like it to generalize well to new data
- Test accuracy is more important than training accuracy

- better training_accuracy

- Speech Recognition
- Training data
- words recorded and labeled in a laboratory

- Training data
- Test data
- words recorded from new speakers, new locations

- Training data
- zipcodes manually selected, scanned, labeled

- actual letters being scanned in a post office

- Training data
- historical database of loan applications with payment history or decision at that time

- you

- Training Data
- Dtrain = { [x(1), c(1)] , [x(2), c(2)] , …………[x(N), c(N)] }
- N pairs of feature vectors and class labels

- Feature Vectors and Class Labels:
- x(i) is the ith training data feature vector
- in MATLAB this could be the ith row of an N x d matrix
- c(i) is the class label of the ith feature vector
- in general, c(i) can take m different class values, e.g., c = 1, c = 2, ...
- Let y be a new feature vector whose class label we do not know, i.e., we wish to classify it.

- y is a new feature vector whose class label is unknown
- Search Dtrain for the closest feature vector to y
- let this “closest feature vector” be x(j)

- Classify y with the same label as x(j), i.e.
- y is assigned label c(j)

- How are “closest x” vectors determined?
- typically use minimum Euclidean distance
- dE(x, y) = sqrt(S (xi - yi)2)

- typically use minimum Euclidean distance
- Side note: this produces a “Voronoi tesselation” of the d-space
- each point “claims” a cell surrounding it
- cell boundaries are polygons

1

2

Feature 2

1

2

2

1

Feature 1

Each data point defines a “cell” of space that is closest to it. All points within that cell are assigned that class

1

2

Feature 2

1

2

2

1

Feature 1

Overall decision boundary = union

of cell boundaries where class

decision is different on each side

1

2

Feature 2

1

2

2

1

Feature 1

1

2

Feature 2

1

2

?

2

1

Feature 1

Boundary? Points that are equidistant

between points of class 1 and 2

Note: locally the boundary is

(1) linear (because of Euclidean distance)

(2) halfway between the 2 class points

(3) at right angles to connector

1

2

Feature 2

1

2

?

2

1

Feature 1

1

2

Feature 2

1

2

?

2

1

Feature 1

1

2

Feature 2

1

2

?

2

1

Feature 1

1

2

Feature 2

1

2

?

2

1

Feature 1

Decision Region

for Class 1

Decision Region

for Class 2

1

2

Feature 2

1

2

?

2

1

Feature 1

?

1

2

Feature 2

1

2

2

1

Feature 1

Feature 2

1

1

1

2

2

1

1

2

2

1

2

1

1

2

2

2

Feature 1

1

In general:

Nearest-neighbor classifier

produces piecewise linear

decision boundaries

1

1

Feature 2

2

2

1

1

2

2

1

2

1

1

2

2

2

Feature 1

- Find the k-nearest neighbors to y in Dtrain
- i.e., rank the feature vectors according to Euclidean distance
- select the k vectors which are have smallest distance to y

- Classification
- ranking yields k feature vectors and a set of k class labels
- pick the class label which is most common in this set (“vote”)
- classify y as belonging to this class

- Theoretical Considerations
- as k increases
- we are averaging over more neighbors
- the effective decision boundary is more “smooth”

- as N increases, the optimal k value tends to increase in proportion to log N

- as k increases

- Notes:
- In effect, the classifier uses the nearest k feature vectors from Dtrain to “vote” on the class label for y
- the single-nearest neighbor classifier is the special case of k=1
- for two-class problems, if we choose k to be odd (i.e., k=1, 3, 5,…) then there will never be any “ties”
- “training” is trivial for the kNN classifier, i.e., we just use Dtrain as a “lookup table” when we want to classify a new feature vector

- Extensions of the Nearest Neighbor classifier
- weighted distances
- e.g., if some of the features are more important
- e.g., if features are irrelevant

- fast search techniques (indexing) to find k-nearest neighbors in d-space

- weighted distances

Training Accuracy = 1/n SDtrain I( o(i), c(i) )

where I( o(i), c(i) ) = 1 if o(i) = c(i), and 0 otherwise

Where o(i) = the output of the classifier for training feature x(i)

c(i) is the true class for training data vector x(i)

Let Dtest be a set of new data, unseen in the training process: but

assume that Dtest is being generated by the same “mechanism” as generated Dtrain:

Test Accuracy = 1/ntestSDtest I( o(j), c(j) )

Test Accuracy is usually what we are really interested in: why?

Unfortunately test accuracy is often lower on average than train accuracy

Why is this so?

- Due Wednesday…..
- 4 parts
- Plot classification data in two-dimensions
- Implement a nearest-neighbor classifier
- Plot the errors of a k-nearest-neighbor classifier
- Test the effect of the value k on the accuracy of the classifier

simdata1 =

shortname: 'Simulated Data 1'

numfeatures: 2

classnames: [2x6 char]

numclasses: 2

description: [1x66 char]

features: [200x2 double]

classlabels: [200x1 double]

function classplot(data, x, y);

% function classplot(data, x, y);

%

% brief description of what the function does

% ......

% Your Name, CS 175, date

%

% Inputs

% data: (a structure with the same fields as described above:

% your comment header should describe the structure explicitly)

% Note that if you are only using certain fields in the structure

% in the function below, you need only define these fields in the input comments

-------- Your code goes here -------

function [class_predictions] = knn(traindata,trainlabels,k, testdata)

% function [class_predictions] = knn(traindata,trainlabels,k, testdata)

%

% a brief description of what the function does

% ......

% Your Name, CS 175, date

%

% Inputs

% traindata: a N1 x d vector of feature data (the "memory" for kNN)

% trainlabels: a N1 x 1 vector of classlabels for traindata

% k: an odd positive integer indicating the number of neighbors to use

% testdata: a N2 x d vector of feature data for testing the knn classifier

%

% Outputs

% class_predictions: N2 x 1 vector of predicted class values

-------- Your code goes here -------

function knn_plot(traindata,trainlabels,k,testdata,testlabels);

% function knn_plot(traindata,trainlabels,k,testdata,testlabels);

%

% Predicts class-labels for the data in testdata using the k nearest

% neighbors in traindata, and then plots the data (using the first

% 2 dimensions or first 2 features), displaying the data from each

% class in different colors, and overlaying circles on the points

% that were incorrectly classified.

%

% Inputs

% traindata: a N1 x d vector of feature data (the "memory" for kNN)

% trainlabels: a N1 x 1 vector of classlabels for traindata

% k: an odd positive integer indicating the number of neighbors to use

% testdata: a N2 x d vector of feature data for testing the knn classifier

% trainlabels: a N2 x 1 vector of classlabels for traindata

function [errors] = knn_error_rates(traindata,trainlabels, testdata, testlabels,kmax,plotflag)

% function [errors] = knn_error_rates(traindata,trainlabels, testdata, testlabels,kmax,plotflag)

%

% a brief description of what the function does

% ......

% Your Name, CS 175, date

%

% Inputs

% traindata: a N1 x d vector of feature data (the "memory" for kNN)

% trainlabels: a N1 x 1 vector of classlabels for traindata

% testdata: a N2 x d vector of feature data for testing the knn classifier

% testlabels: a N2 x 1 vector of classlabels for traindata

% kmax: an odd positive integer indicating the maximum number of neighbors

% plotflag: (optional argument) if 1, the error-rates versus k is plotted,

% otherwise no plot.

%

% Outputs

% errors: r x 1 vector of error-rates on testdata, where r is the

% number of values of k that are tested.

-------- Your code goes here -------

- The accuracy of our classifier on new unseen data is a fair/honest assessment of the performance of our classifier
- Why is training accuracy not good enough?
- Training accuracy is optimistic
- a classifier like nearest-neighbor can construct boundaries which always separate all training data points, but which do not separate new points
- e.g., what is the training accuracy of kNN, k = 1?

- A flexible classifier can “overfit” the training data
- in effect it just memorizes the training data, but does not learn the general relationship between x and C

- Generalization
- We are really interested in how our classifier generalizes to new data
- test data accuracy is a good estimate of generalization performance

6

5

4

3

2

1

0

-1

2

3

4

5

6

7

8

9

10

TWO-CLASS DATA IN A TWO-DIMENSIONAL FEATURE SPACE

Decision

Region 1

Decision

Region 2

Feature 2

Decision

Boundary

Feature 1

Y

X

Y = high-order polynomial in X

Y

X

Y = a X + b + noise

Y

X

Predictive

Error

Error on Training Data

Model Complexity

Predictive

Error

Error on Test Data

Error on Training Data

Model Complexity

Predictive

Error

Error on Test Data

Error on Training Data

Model Complexity

Ideal Range

for Model Complexity

Overfitting

Underfitting

- Important Concepts
- classification is an important component in intelligent systems
- a classifier = a mapping from feature space to a class label
- decision boundaries = boundaries between classes

- classification learning
- using training data to define a classifier

- the nearest-neighbor classifier
- training accuracy versus test accuracy