Understanding Classification Techniques

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Discriminant Analysis

Suppose we have two or more different populations from which observations could come from. Discriminant analysis seeks to determine which of the possible population an observation comes from while making as few mistakes as possible

• This is an alternative to logistic approaches with the following advantages:

  • when there is clear separation between classes, the parameter estimates for the logic regression model can be surprisingly unstable, while discriminant approaches do not suffer

  • If X is normal in each of the classes and the sample size is small, then discriminant approaches can be more accurate

Notation

SImilar to MANOVA, let \(\mathbf{y}_{j1},\mathbf{y}_{j2},\dots, \mathbf{y}_{in_j} \sim iid f_j (\mathbf{y})\) for \(j = 1,\dots, h\)

Let \(f_j(\mathbf{y})\) be the density function for population j . Note that each vector \(\mathbf{y}\) contain measurements on all \(p\) traits

1. Assume that each observation is from one of \(h\) possible populations.
2. We want to form a discriminant rule that will allocate an observation \(\mathbf{y}\) to population j when \(\mathbf{y}\) is in fact from this population

Known Populations

The maximum likelihood discriminant rule for assigning an observation \(\mathbf{y}\) to one of the \(h\) populations allocates \(\mathbf{y}\) to the population that gives the largest likelihood to \(\mathbf{y}\)

Consider the likelihood for a single observation \(\mathbf{y}\), which has the form \(f_j (\mathbf{y})\) where j is the true population.

Since \(j\) is unknown, to make the likelihood as large as possible, we should choose the value j which causes \(f_j (\mathbf{y})\) to be as large as possible

Consider a simple univariate example. Suppose we have data from one of two binomial populations.

• The first population has \(n= 10\) trials with success probability \(p = .5\)

• The second population has \(n= 10\) trials with success probability \(p = .7\)

• to which population would we assign an observation of \(y = 7\)

• Note:

  • \(f(y = 7|n = 10, p = .5) = .117\)

  • \(f(y = 7|n = 10, p = .7) = .267\) where \(f(.)\) is the binomial likelihood.

  • Hence, we choose the second population

Another example

We have 2 populations, where

• First population: \(N(\mu_1, \sigma^2_1)\)

• Second population: \(N(\mu_2, \sigma^2_2)\)

The likelihood for a single observation is

\[ f_j (y) = (2\pi \sigma^2_j)^{-1/2} \exp\{ -\frac{1}{2}(\frac{y - \mu_j}{\sigma_j})^2\} \]

Consider a likelihood ratio rule

\[ \begin{aligned} \Lambda &= \frac{\text{likelihood of y from pop 1}}{\text{likelihood of y from pop 2}} \\ &= \frac{f_1(y)}{f_2(y)} \\ &= \frac{\sigma_2}{\sigma_1} \exp\{-\frac{1}{2}[(\frac{y - \mu_1}{\sigma_1})^2- (\frac{y - \mu_2}{\sigma_2})^2] \} \end{aligned} \]

Hence, we classify into

• pop 1 if \(\Lambda >1\)

• pop 2 if \(\Lambda <1\)

• for ties, flip a coin

Another way to think:

we classify into population 1 if the “standardized distance” of y from \(\mu_1\) is less than the “standardized distance” of y from \(\mu_2\) which is referred to as a quadratic discriminant rule.

(Significant simplification occurs in th special case where \(\sigma_1 = \sigma_2 = \sigma^2\))

Thus, we classify into population 1 if

\[ (y - \mu_2)^2 > (y - \mu_1)^2 \]

or

\[ |y- \mu_2| > |y - \mu_1| \]

and

\[ -2 \log (\Lambda) = -2y \frac{(\mu_1 - \mu_2)}{\sigma^2} + \frac{(\mu_1^2 - \mu_2^2)}{\sigma^2} = \beta y + \alpha \]

Thus, we classify into population 1 if this is less than 0.

Discriminant classification rule is linear in y in this case.

Multivariate Expansion

Suppose that there are 2 populations

• \(N_p(\mathbf{\mu}_1, \mathbf{\Sigma}_1)\)

• \(N_p(\mathbf{\mu}_2, \mathbf{\Sigma}_2)\)

\[ \begin{aligned} -2 \log(\frac{f_1 (\mathbf{x})}{f_2 (\mathbf{x})}) &= \log|\mathbf{\Sigma}_1| + (\mathbf{x} - \mathbf{\mu}_1)' \mathbf{\Sigma}^{-1}_1 (\mathbf{x} - \mathbf{\mu}_1) \\ &- [\log|\mathbf{\Sigma}_2|+ (\mathbf{x} - \mathbf{\mu}_2)' \mathbf{\Sigma}^{-1}_2 (\mathbf{x} - \mathbf{\mu}_2) ] \end{aligned} \]

Again, we classify into population 1 if this is less than 0, otherwise, population 2. And like the univariate case with non-equal variances, this is a quadratic discriminant rule.

And if the covariance matrices are equal: \(\mathbf{\Sigma}_1 = \mathbf{\Sigma}_2 = \mathbf{\Sigma}_1\) classify into population 1 if

\[ (\mathbf{\mu}_1 - \mathbf{\mu}_2)' \mathbf{\Sigma}^{-1}\mathbf{x} - \frac{1}{2} (\mathbf{\mu}_1 - \mathbf{\mu}_2)' \mathbf{\Sigma}^{-1} (\mathbf{\mu}_1 - \mathbf{\mu}_2) \ge 0 \]

This linear discriminant rule is also referred to as Fisher’s linear discriminant function

By assuming the covariance matrices are equal, we assume that the shape and orientation fo the two populations must be the same (which can be a strong restriction)

In other words, for each variable, it can have different mean but the same variance.

• Note: LDA Bayes decision boundary is linear. Hence, quadratic decision boundary might lead to better classification. Moreover, the assumption of same variance/covariance matrix across all classes for Gaussian densities imposes the linear rule, if we allow the predictors in each class to follow MVN distribution with class-specific mean vectors and variance/covariance matrices, then it is Quadratic Discriminant Analysis. But then, you will have more parameters to estimate (which gives more flexibility than LDA) at the cost of more variance (bias -variance tradeoff).

When \(\mathbf{\mu}_1, \mathbf{\mu}_2, \mathbf{\Sigma}\) are known, the probability of misclassification can be determined:

\[ \begin{aligned} P(2|1) &= P(\text{calssify into pop 2| x is from pop 1}) \\ &= P((\mathbf{\mu}_1 - \mathbf{\mu}_2)' \mathbf{\Sigma}^{-1} \mathbf{x} \le \frac{1}{2} (\mathbf{\mu}_1 - \mathbf{\mu}_2)' \mathbf{\Sigma}^{-1} (\mathbf{\mu}_1 - \mathbf{\mu}_2)|\mathbf{x} \sim N(\mu_1, \mathbf{\Sigma}) \\ &= \Phi(-\frac{1}{2} \delta) \end{aligned} \]

where

• \(\delta^2 = (\mathbf{\mu}_1 - \mathbf{\mu}_2)' \mathbf{\Sigma}^{-1} (\mathbf{\mu}_1 - \mathbf{\mu}_2)\)

• \(\Phi\) is the standard normal cdf

Suppose there are \(h\) possible populations, which are distributed as \(N_p (\mathbf{\mu}_p, \mathbf{\Sigma})\). Then, the maximum likelihood (linear) discriminant rule allocates \(\mathbf{y}\) to population j where j minimizes the squared Mahalanobis distance

\[ (\mathbf{y} - \mathbf{\mu}_j)' \mathbf{\Sigma}^{-1} (\mathbf{y} - \mathbf{\mu}_j) \]

Bayes Discriminant Rules

If we know that population j has prior probabilities \(\pi_j\) (assume \(\pi_j >0\)) we can form the Bayes discriminant rule.

This rule allocates an observation \(\mathbf{y}\) to the population for which \(\pi_j f_j (\mathbf{y})\) is maximized.

Note:

• Maximum likelihood discriminant rule is a special case of the Bayes discriminant rule, where it sets all the \(\pi_j = 1/h\)

Optimal Properties of Bayes Discriminant Rules

• let \(p_{ii}\) be the probability of correctly assigning an observation from population i

• then one rule (with probabilities \(p_{ii}\) ) is as good as another rule (with probabilities \(p_{ii}'\) ) if \(p_{ii} \ge p_{ii}'\) for all \(i = 1,\dots, h\)

• The first rule is better than the alternative if \(p_{ii} > p_{ii}'\) for at least one i.

• A rule for which there is no better alternative is called admissible

• Bayes Discriminant Rules are admissible

• If we utilized prior probabilities, then we can form the posterior probability of a correct allocation, \(\sum_{i=1}^h \pi_i p_{ii}\)

• Bayes Discriminant Rules have the largest possible posterior probability of correct allocation with respect to the prior

• These properties show that Bayes Discriminant rule is our best approach.

Unequal Cost

• We want to consider the cost misallocation Define \(c_{ij}\) to be the cost associated with allocation a member of population j to population i.

• Assume that

  • \(c_{ij} >0\) for all \(i \neq j\)

  • \(c_{ij} = 0\) if \(i = j\)

• We could determine the expected amount of loss for an observation allocated to population i as \(\sum_j c_{ij} p_{ij}\) where the \(p_{ij}s\) are the probabilities of allocating an observation from population j into population i

• We want to minimize the amount of loss expected for our rule. Using a Bayes Discrimination, allocate \(\mathbf{y}\) to the population j which minimizes \(\sum_{k \neq j} c_{ij} \pi_k f_k(\mathbf{y})\)

• We could assign equal probabilities to each group and get a maximum likelihood type rule. here, we would allocate \(\mathbf{y}\) to population j which minimizes \(\sum_{k \neq j}c_{jk} f_k(\mathbf{y})\)

Example:

Two binomial populations, each of size 10, with probabilities \(p_1 = .5\) and \(p_2 = .7\)

And the probability of being in the first population is .9

However, suppose the cost of inappropriately allocating into the first population is 1 and the cost of incorrectly allocating into the second population is 5.

In this case, we pick population 1 over population 2

In general, we consider two regions, \(R_1\) and \(R_2\) associated with population 1 and 2:

\[ R_1: \frac{f_1 (\mathbf{x})}{f_2 (\mathbf{x})} \ge \frac{c_{12} \pi_2}{c_{21} \pi_1} \]

\[ R_2: \frac{f_1 (\mathbf{x})}{f_2 (\mathbf{x})} < \frac{c_{12} \pi_2}{c_{21} \pi_1} \]

where \(c_{12}\) is the cost of assigning a member of population 2 to population 1.

Discrimination Under Estimation

Suppose we know the form of the distributions for populations of interests, but we still have to estimate the parameters.

Example:

we know the distributions are multivariate normal, but we have to estimate the means and variances

The maximum likelihood discriminant rule allocates an observation \(\mathbf{y}\) to population j when j maximizes the function

\[ f_j (\mathbf{y} |\hat{\theta}) \]

where \(\hat{\theta}\) are the maximum likelihood estimates of the unknown parameters

For instance, we have 2 multivariate normal populations with distinct means, but common variance covariance matrix

MLEs for \(\mathbf{\mu}_1\) and \(\mathbf{\mu}_2\) are \(\mathbf{\bar{y}}_1\) and \(\mathbf{\bar{y}}_2\)and common \(\mathbf{\Sigma}\) is \(\mathbf{S}\).

Thus, an estimated discriminant rule could be formed by substituting these sample values for the population values

Native Bayes

• The challenge with classification using Bayes’ is that we don’t know the (true) densities, \(f_k, k = 1, \dots, K\), while LDA and QDA make strong multivariate normality assumptions to deal with this.

• Naive Bayes makes only one assumption: within the kth class, the p predictors are independent (i.e,, for \(k = 1,\dots, K\)

\[ f_k(x) = f_{k1}(x_1) \times f_{k2}(x_2) \times \dots \times f_{kp}(x_p) \]

where \(f_{kj}\) is the density function of the j-th predictor among observation in the k-th class.

This assumption allows the use of joint distribution without the need to account for dependence between observations. However, this (native) assumption can be unrealistic, but still works well in cases where the number of sample (n) is not large relative to the number of features (p).

With this assumption, we have

\[ P(Y=k|X=x) = \frac{\pi_k \times f_{k1}(x_1) \times \dots \times f_{kp}(x_p)}{\sum_{l=1}^K \pi_l \times f_{l1}(x_1)\times \dots f_{lp}(x_p)} \]

we only need to estimate the one-dimensional density function \(f_{kj}\) with either of these approaches:

• When \(X_j\) is quantitative, assume it has a univariate normal distribution (with independence): \(X_j | Y = k \sim N(\mu_{jk}, \sigma^2_{jk})\) which is more restrictive than QDA because it assumes predictors are independent (e.g., a diagonal covariance matrix)

• When \(X_j\) is quantitative, use a kernel density estimator [Kernel Methods] ; which is a smoothed histogram

• When \(X_j\) is qualitative, we count the promotion of training observations for the j-th predictor corresponding to each class.

Comparison of Classification Methods

Assuming we have K classes and K is the baseline from (James , Witten, Hastie, and Tibshirani book)

Comparing the log odds relative to the K class

Logistic Regression

\[ \log(\frac{P(Y=k|X = x)}{P(Y = K| X = x)}) = \beta_{k0} + \sum_{j=1}^p \beta_{kj}x_j \]

LDA

\[ \log(\frac{P(Y = k | X = x)}{P(Y = K | X = x)} = a_k + \sum_{j=1}^p b_{kj} x_j \]

where \(a_k\) and \(b_{kj}\) are functions of \(\pi_k, \pi_K, \mu_k , \mu_K, \mathbf{\Sigma}\)

Similar to logistic regression, LDA assumes the log odds is linear in \(x\)

Even though they look like having the same form, the parameters in logistic regression are estimated by MLE, where as LDA linear parameters are specified by the prior and normal distributions

We expect LDA to outperform logistic regression when the normality assumption (approximately) holds, and logistic regression to perform better when it does not

QDA

\[ \log(\frac{P(Y=k|X=x}{P(Y=K | X = x}) = a_k + \sum_{j=1}^{p}b_{kj}x_{j} + \sum_{j=1}^p \sum_{l=1}^p c_{kjl}x_j x_l \]

where \(a_k, b_{kj}, c_{kjl}\) are functions \(\pi_k , \pi_K, \mu_k, \mu_K ,\mathbf{\Sigma}_k, \mathbf{\Sigma}_K\)

Naive Bayes

\[ \log (\frac{P(Y = k | X = x)}{P(Y = K | X = x}) = a_k + \sum_{j=1}^p g_{kj} (x_j) \]

where \(a_k = \log (\pi_k / \pi_K)\) and \(g_{kj}(x_j) = \log(\frac{f_{kj}(x_j)}{f_{Kj}(x_j)})\) which is the form of generalized additive model

Summary

• LDA is a special case of QDA

• LDA is robust when it comes to high dimensions

• Any classifier with a linear decision boundary is a special case of naive Bayes with \(g_{kj}(x_j) = b_{kj} x_j\), which means LDA is a special case of naive Bayes. LDA assumes that the features are normally distributed with a common within-class covariance matrix, and naive Bayes assumes independence of the features.

• Naive bayes is also a special case of LDA with \(\mathbf{\Sigma}\) restricted to a diagonal matrix with diagonals, \(\sigma^2\) (another notation \(diag (\mathbf{\Sigma})\) ) assuming \(f_{kj}(x_j) = N(\mu_{kj}, \sigma^2_j)\)

• QDA and naive Bayes are not special case of each other. In principal,e naive Bayes can produce a more flexible fit by the choice of \(g_{kj}(x_j)\) , but it’s restricted to only purely additive fit, but QDA includes multiplicative terms of the form \(c_{kjl}x_j x_l\)

• None of these methods uniformly dominates the others: the choice of method depends on the true distribution of the predictors in each of the K classes, n and p (i.e., related to the bias-variance tradeoff).

Compare to the non-parametric method (KNN)

• KNN would outperform both LDA and logistic regression when the decision boundary is highly nonlinear, but can’t say which predictors are most important, and requires many observations

• KNN is also limited in high-dimensions due to the curse of dimensionality

• Since QDA is a special type of nonlinear decision boundary (quadratic), it can be considered as a compromise between the linear methdos and KNN classification. QDA can have fewer training observations than KNN but not as flexible.

From simulation:

True decision boundary	Best performance
Linear	LDA + Logistic regression
Moderately nonlinear	QDA + Naive Bayes
Highly nonlinear (many training, p is not large)	KNN

• like linear regression, we can also introduce flexibility by including transformed features \(\sqrt{X}, X^2, X^3\)

Probabilities of Misclassification

When the distribution are exactly known, we can determine the misclassification probabilities exactly. however, when we need to estimate the population parameters, we have to estimate the probability of misclassification

• Naive method

  • Plugging the parameters estimates into the form for the misclassification probabilities results to derive at the estimates of the misclassification probability.

  • But this will tend to be optimistic when the number of samples in one or more populations is small.

• Resubstitution method

  • Use the proportion of the samples from population i that would be allocated to another population as an estimate of the misclassification probability

  • But also optimistic when the number of samples is small

• Jack-knife estimates:

  • The above two methods use observation to estimate both parameters and also misclassification probabilities based upon the discriminant rule

  • Alternatively, we determine the discriminant rule based upon all of the data except the k-th observation from the j-th population

  • then, determine if the k-th observation would be misclassified under this rule

  • perform this process for all \(n_j\) observation in population j . An estimate fo the misclassficaiton probability would be the fraction of \(n_j\) observations which were misclassified

  • repeat the process for other \(i \neq j\) populations

  • This method is more reliable than the others, but also computationally intensive

• Cross-Validation

Summary

Consider the group-specific densities \(f_j (\mathbf{x})\) for multivariate vector \(\mathbf{x}\).

Assume equal misclassification costs, the Bayes classification probability of \(\mathbf{x}\) belonging to the j-th population is

\[ p(j |\mathbf{x}) = \frac{\pi_j f_j (\mathbf{x})}{\sum_{k=1}^h \pi_k f_k (\mathbf{x})} \]

\(j = 1,\dots, h\)

where there are \(h\) possible groups.

We then classify into the group for which this probability of membership is largest

Alternatively, we can write this in terms of a generalized squared distance formation

\[ D_j^2 (\mathbf{x}) = d_j^2 (\mathbf{x})+ g_1(j) + g_2 (j) \]

where

• \(d_j^2(\mathbf{x}) = (\mathbf{x} - \mathbf{\mu}_j)' \mathbf{V}_j^{-1} (\mathbf{x} - \mathbf{\mu}_j)\) is the squared Mahalanobis distance from \(\mathbf{x}\) to the centroid of group j, and

  • \(\mathbf{V}_j = \mathbf{S}_j\) if the within group covariance matrices are not equal

  • \(\mathbf{V}_j = \mathbf{S}_p\) if a pooled covariance estimate is appropriate

and

\[ g_1(j) = \begin{cases} \ln |\mathbf{S}_j| & \text{within group covariances are not equal} \\ 0 & \text{pooled covariance} \end{cases} \]

\[ g_2(j) = \begin{cases} -2 \ln \pi_j & \text{prior probabilities are not equal} \\ 0 & \text{prior probabilities are equal} \end{cases} \]

then, the posterior probability of belonging to group j is

\[ p(j| \mathbf{x}) = \frac{\exp(-.5 D_j^2(\mathbf{x}))}{\sum_{k=1}^h \exp(-.5 D^2_k (\mathbf{x}))} \]

where \(j = 1,\dots , h\)

and \(\mathbf{x}\) is classified into group j if \(p(j | \mathbf{x})\) is largest for \(j = 1,\dots,h\) (or, \(D_j^2(\mathbf{x})\) is smallest).

Assessing Classification Performance

For binary classification, confusion matrix

	Predicted class
		- or Null	+ or Null	Total
True Class	- or Null	True Neg (TN)	False Pos (FP)	N
	+ or Null	False Neg (FN)	True Pos (TP)	P
	Total	N*	P*

and table 4.6 from [@james2013]

Name	Definition	Synonyms
False Pos rate	FP/N	Tyep I error, 1 0 SPecificity
True Pos. rate	TP/P	1 - Type II error, power, sensitivity, recall
Pos Pred. value	TP/P*	Precision, 1 - false discovery promotion
Neg. Pred. value	TN/N*

ROC curve (receiver Operating Characteristics) is a graphical comparison between sensitivity (true positive) and specificity ( = 1 - false positive)

y-axis = true positive rate

x-axis = false positive rate

as we change the threshold rate for classifying an observation as from 0 to 1

AUC (area under the ROC) ideally would equal to 1, a bad classifier would have AUC = 0.5 (pure chance)