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Classification

Assuming we have data with 2 or more groups, we may want to determine which group an observation belongs too

Thus the goal of classification analysis is to find a “good” rule to assign observations to an appropriate group…

But how do we do that?

Decision Theory

How does probability play a role in decision making?

Example: we want determine if a patient has lung cancer or not based on a CT-scan.

-inputs x: set of pixel intensities for our image

-outcome y: lung cancer (Y/N)

Consider probability of a subject’s class given their image using Bayes’ theorem

Intuitively we choose the class with the higher posterior probability

Definitions in Classification

We are looking for classification rules that minimize our total expected misclassification cost/loss

Minimizing Misclassification Rate

A simple cost function is to minimize the number of observations we misclassify

1. Develop a rule that assigns classes to each observation

2. Rule divides feature space X into regions Rk (decision

regions). Boundaries between these regions are called

decision boundaries.

3. Calculate the probability of a misclassification.

Consider a 2 class example, a mistake occurs when an input vector belonging to C1 is assigned to C2

The probability of a mistake is:

The minimum probability of making a mistake occurs if x assigned to class with the largest posterior probability

Generalized to k classes, it is often easier to maximize the probability of being correct

which is maximized when regions Rk chosen such that each x assigned to class with largest posterior probability

Minimizing Expected Loss

Often, minimizing misclassification is too simplistic

We may need to consider that some wrong decisions are worse than others

-lung cancer: incorrectly diagnosing a cancer patient as healthy can lead to premature death!

We can consider more formalized cost/loss functionswhich represent single measures of loss for each action

Goal then becomes to minimize the total cost/loss

In our cancer example consider a loss matrix:

The optimal solution minimizes the loss function

Problem is the loss function depends on true class, which is unknown (for new observations)

For a vector x, uncertainty on true class can be expressed as a joint probability

We then minimize the expected loss w.r.t. this distribution

Goal to choose regions Rj to minimize this expected loss- implies for each x assigned to class j should minimize

Reject Option

Classification errors can also result from:

-regions of input space where the largest posterior probabilities, , are small

-posterior probabilities for 2 or more classes are similar

It may be better to avoid making decisions for difficult cases in attempt to reduce error rate

Referred to as rejection option- set specific region where classifier does not make a decision

This does require there is some means (e.g. expert opinion) to make decision about those points within this region

Inference and Decision

Classification problem can be broken into two stages

-inference stage: use training data to learn a model for

-decision stage: use estimated posterior probability to make optimal class assignment

Alternatively we can solve both problems together an learn a function that maps inputs x directly into decisions

Such a function is called a discriminant function (this is what we will address initially)

Approaches

- Solve inference problem of finding posterior probabilities

for each class Ck individually

-use Bayes’ thm and find posterior class probabilities

-use posteriors to make optimal decision

-referred to as generative models

(2) Solve inference problem by determining

-use posteriors to make optimal decision

-referred to as discriminative models

(3) Find function f(x) (called a discriminant function) that maps each point x directly into class label

-NOTE: in this case, probabilities play no role

Approaches

There are merits and drawbacks to each of the approaches

First approach most computationally demanding especially if x is large… However, only approach that estimates prior class probabilitesp(Ck) and marginal probablities on data p(x)

If really only want to do classification, all this not necessary and only really need posterior probabilites provided in second approach

Third approach even more simple (most computationally easy) but we now do not have estimates of posterior probabilities on the classes which may be desirable

Approaches

We will initially focus on discriminant functions (approach 3 and on some level 2)

We still want to minimize total expected cost

In this case, our optimal classification rule classifies x0 to kth group if

smallest among the groups

Equivalently, for equal misclassification costs, classify x0 to k if

is the largest- from Bayes we obtain the posterior probabilities for the kth class

Other Considerations

So it seems like we have the information we need…

However, we don’t know what the fk(x)’s are

We must estimate these as well (and we will get to that)

We also want to evaluate the performance of any classifier

Apparent error rate: crosstabulate the actual class of observations in the training sample and the predicted classes for the same observations. Then count the number incorrectly classified.

Other Considerations

Apparent error rate not of greatest interest though

Much more interested in error on future observations since apparent error rate generally very optimistic due to overfitting

To avoid overfitting we can use some strategies discussed earlier

-Split data into test and training sets (in n large)

-K-fold cross-validation

-Generalized cross-validation

-Bootstrap re-sampling

To name a few…

Discriminant Functions

Takes input vector x and assigns it to one of K classes (CK)

-linear discriminantfunction means function is linear

Simplest linear discriminant function:

1st consider 2 class case:

-Input vector x assigned to class C1 if y(x)>0 and class C2 otherwise

-Decision boundary defined by y(x)>0

-b0 determines boundary location

Linear Discriminant Functions

In order to generalize this to K > 2 classes, we define one discriminant comprised of K linear functions:

Decision boundary between j and k is a (D-1)-dimensional hyperplane defined by:

Properties:

-Singly connected

-Convex

Least Squares Approach

Assuming data contains K classes, we can create a matrix of K vectors of binary indicators

Each class Ck has its own linear model

Vector of weights for the kthmodel determined using least squares solution for vector describing the kthclass

New input x assign to class for which is largest

Least Squares Approach

In this setting we view the regression model as an estimate of a conditional expectation

We can also think of this as an estimate of the posterior probability given our feature vector x

Question: how good is this approximation

-we can verify that

-but there are some issues…

Least Squares Approach

Positive aspects of this approach:

-Closed for solution for discriminant function parameters

-Familiarity in stats community

Some obvious problems with this approach:

-model output should represent probability but not constrained on (0,1) interval

-Poor separation of classes if a region assigned only a small posterior probability

-Lacks robustness to outliers (even worse for classification problems)

Small “posterior probability”…

So far we have made no assumptions about the distribution of the data

Decision theory suggests that we need to know the posterior class probability for optimal classification

This brings us back to Bayes…

We can see that having fk(x) can help us classify observations

Many techniques are based on models for the class densities

Linear (and quadratic) discriminant analysis assumes MVN

Linear Discriminant Analysis: 2 Class Case

First assume our features are MVN with equal covariances

Our fk(x)’s are:

In normal theory, we often resort to a likelihood ratio test… what if we consider something like that here?

The problem is we are interested in our posterior class probabilities rather than fk(x)…

2 Class Case

Assume covariances are equal

If the last quantity is <0, we classify an observation as belonging to class 1

2 Class Case

Instead of the ration of f1(x) to f2(x) , consider the ratio of the conditional probabilities

2 Class Case

We can think of this as the log odds of being in class 1 versus class 2

Therefore we make our decision about class assignment based on whether this ratio is larger or smaller than 0

Our decision boundary between the two classes occurs where

This boundary represents a hyperplane in p dimensions (based on the p features defined in x)

Linear Discriminant Analysis

We can generalize this to more than 2 classes

The more general form of the linear discriminant function for LDA is:

The decision boundary between classes is defined by multiple hyperplanes defined by each line where

LDA and Least Squares

In the 2 class case there is a correspondence between our least squares classifier and LDA

LDA classifies and observed x to class 2 if

It can be shown that the coefficients vector from LS is proportional to the LDA direction shown above

Decision rules only the same if N1 = N2

However for more than 2 classes, LDA and linear regression loose this correspondence

From a least squares standpoint, we can also think about classification via LDA as a dimensionality reduction problem

For 2 class case, we use p inputs (vector x) and project into one direction represented by:

Thresholding so that y > 0 classifies C1 same as least squares

Problem is loss of information… what may have been well separated in p-dimensional space isn’t in 1-d space

Instead adjust weights b to select a projection that maximizes separation

Quadratic Discriminant Analysis

One very strong assumption we’ve sort of glossed over in LDA is the assumption that all covariance matrices are equal

If this assumption does not hold, we can’t simplify our log ratio quite as easily

The first term can’t be dropped…

Quadratic Discriminant Analysis

We can simplify this a little bit…

Our final discriminant function in this case becomes

Quadratic Discriminant Analysis

Our discriminant function is no longer linear

Hence we refer to this as quadratic discriminant analysis

General notes

So how do we decide between linear and quadratic discriminant analysis?

We need to check if our covariances for our two (or more) populations are the same

-Bartlett’s test (as always, interpret with caution)

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