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Logistic Regression

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Logistic Regression

Ram Akella

Lecture 3

February 2, 2011

UC Berkeley

Silicon Valley Center/SC

- Motivating example
- Why not ordinary linear regression?
- The logistic formulation
- Probability of “success”
- Odds of “success”
- Logit of “success”

- The logistic regression model
- Running the model
- Interpreting the output
- Evaluating the goodness of fitting

0

- Similar to ordinary regression models, except the response, Y, is categorical.
- Y indicates the “group” membership of each observation (each category is a group). Y = C1, C2,…
- Predictors X1, X2,.., are continuous and/or categorical.
- Aims:
- Profiling (=Explanatory): What are the differences (in terms of X1, X2,…) between the various groups? (as indicated by Y)
- Classification (=Prediction): Predict Y (group membership) on the basis of X1, X2,…

0

- Financial analysts are interested in predicting the future solvency of firms. In order to predict whether a firm will go bankrupt in the near future, it is useful to look at different ratio measures of financial health such as :
- Cash_Debt:cash flow/total debt
- ROA:net income/total assets
- Current:current assets/current liabilities
- Assets_Sales:current assets/net sales
- Status: bankrupt / solvent

0

- A beer-maker would like to know the characteristics that distinguish customers who prefer light beer from those who prefer regular beer. The beer-maker would like to use this information to predict any customer’s beer preference based on:
- gender,
- marital status,
- income and age.

Consider the data on beer preferences (light or regular) of 100 customers along with their age, income, gender and marital status.

Suppose we code the response variable as

Now we fit the multiple regression model

Y = 0 + b1Gender + b2Married + b3Income + b4Age + e

- Model assumptions:
- Observations/residuals are independent
- Residuals are normally distributed
- Linear model is adequate
- Variance of residuals is constant

- Which assumptions are violated?
- What about predictions from this model?

- Let π = Prob(Y=1).
- In the beer example, π is the probability that a customer prefers _______ beer.
- It follows that Prob(Y=0) = _________ .
- In order to get rid of the 0/1 values, we can look at a function of π and treat it as the response model.

light

1- π

- Logistic regression learns the conditional distribution P(y | x)
- We will assume two classes y = 0 and y = 1
- The parametric form for P(y = 1 | x, w) and P(y=0|x,w) is:
were w is the parameter vector w=[w1, w2, …, wk]

- We can represent the probability ratio as a linear combination of features:
This is known as log odds

A linear function wx which ranges [-∞, ∞] can be transformed to a range [0,1] using the sigmoid function g(x,w)

Given P( y | x) we predict ŷ =1 if the expected loss function of predicting 0L(0,1) is greater than predicting 1L(1,0) (for now assume L(0,1) = L(1,0))

This assumed L(0,1)=L(1,0)

A similar derivation can be done for arbitrary L(0,1) and

L(1,0). (If we decide that one class is more important to be detected than the other)

- The likelihood function is the probability of the data (x,y)given the parameters w – p(x,y|w)
- It is a function of the parameters
- Maximum likelihood learning finds the parameters that maximize this likelihood function
- A common trick is to work with log-likelihood, i.e., take the logarithm of the likelihood function – log p(x,y|w)

- In our framework, we assume each training example (xi, yi) is drawn independently from the same (but unknown) distribution P( x ,y ) (the i. i. d assumption) hence, we can write:
- This is the function that we will maximize

- Further P(x|w)=P(x) because x because it does not depend on w, so:

- This can be written as:
- Then the objective learning function is:

- Choose class K to be the “reference class” and represent each of the other classes as a logistic function of the odds of k versus class K:

- Conditional probability for class k ≠ K can be computed as:
- For class K the conditional probability is:

A 1959 article presents data concerning the proportion of coal miners who exhibit the symptoms of severe pneumoconiosis and the number of years of exposure. y is the proportion of miners who have severe symptoms

- The fitted model is

- The covariance matrix is:

Logistic Regression Table

- Consider we have a single regressor xi
- If we increment the value of the regressor in one unit then:
- The difference between the two predicted values is:

- The odds ratio can be interpreted as the increase in probability of success associated with a one-unit change in the value of the predictor variable and it is defined as:

- Following the pneumoconiosis data we have the model equal to:
- The resulting odds ratio is
- This implies that every year of additional exposure increases the odds of contracting a severe case of pneumoconiosis by 10%

0

For maximum likelihood estimation, the fit of a model is measured by its deviance D (similar to sum-of-squared-errors in the case of least-squares estimation)

We compare the deviance of a model to the deviance of the naïve model (no explanatory variables: simply classify each observation as belonging to the majority class)

- If the ratio of
D/n-p,

where p is the number of predictors and n the number of samples

is much greater than unity, then the current model is not adequate

- Note:This test is similar in intent to the ____-test for overall usefulness of a linear regression model

F

0

- Each estimated coefficient, , has a standard error, sbj associated with it.
- To conduct the hypothesis test H0: ŵj = 0 vs. Ha: ŵj 0
- Use the test statistic / sbj , (called the Wald statistic)
- The associated p-value indicates the statistical significance of the predictor xi, or the significance of the contribution of this predictor beyond the other predictors.

0

- Evaluation of a classifier is necessary for two different purposes:
- To obtain the complete specification of a particular model i.e., to obtain numerical values of the parameters of a particular method.
- To compare two or more fully specified classifiers in order to select the “best” classifier.

- Useful criteria
- Reasonableness
- Accuracy
- Cost measures

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- As in regression, time series, and other models we expect the model to be reasonable:
- Based on the analyst’s domain knowledge, is there a reasonable basis for a causal relationship between the predictor variables and Y (group membership)?
- Are the predictor variables actually available for prediction in the future?
- If the classifier implies a certain order of importance among the predictor variables (indicated by p-values for specific predictors), is this order reasonable?

0

- The idea is to compare the predictions with the actual responses (like forecast errors in time series, or residuals in regression models).
- In regression/ time series etc. we displayed these as 3 columns (actual values, predicted/fitted values, errors) or plotted them on a graph.
- In classification the predictions and actual values are displayed in a compact format called a classification/confusion matrix.
- This can be done for the training and/or validation set.

Predicted

a

b

Actual

c

d

0

Example with two groups Y = C1 or C2

# of obs that were classified correctly as group C1

0

- The following classification matrix results from using a certain classifier on the data

0

- Based on the classification matrix there are 5 popular measures:
- The overall accuracy of a classifier is
- The overall error rate of a classifier is

0

- The base accuracy of a dataset is the accuracy of the naïve rule
- The base error rate is
- The lift of a classifier (aka its improvement) is

Proportion of majority class

1 – base accuracy

0

- Suppose the two groups are asymmetric in that it is more important to correctly predict membership in C1 than in C2. E.g., in a bankruptcy example, it may be more important to correctly predict a firm that is going bankrupt than to correctly predict a firm that is going to stay solvent. The classifier is essentially used as a system for detecting or signaling C1.
- In such a case, the overall accuracy is not a good measure for evaluating the classifier.

0

Predicted

- Sensitivity of a classifier = its ability to correctly detect the important group members
=% of C1 members correctly classified

- Specificity of a classifier = its ability to correctly rule out C2 members
=% of C2 members correctly classified

C1

C2

C1 = “important” group

C1

Actual

C2

0

- = false positive rate of classifier
- = false negative rate of classifier

Predicted

C1

C2

C1 = “important” group

C1

Actual

C2

- There are two types of errors
- Machine Learning methods usually minimize FP+FN
- Direct marketing maximizes TP

- In practice, true positive and false negative errors often incur different costs
- Examples:
- Medical diagnostic tests: does X have leukemia?
- Loan decisions: approve mortgage for X?
- Web mining: will X click on this link?
- Promotional mailing: will X buy the product?

- Most learning schemes do not perform cost-sensitive learning
- They generate the same classifier no matter what costs are assigned to the different classes
- Example: standard decision tree learner

- Simple methods for cost-sensitive learning:
- Re-sampling of instances according to costs
- Weighting of instances according to costs

- Some schemes are inherently cost-sensitive

- The lift charts help us to see what is the improvement of the classification vs. the random classification

The larger the area, the better the model is

Sort the samples by probability

Each point of the chart will consist in the cumulative sum of the actual class

For the random classification line, calculate the average of the classes (yes=1, no=0). Each point of the line is the multiplication of the average by the number of samples

- Stands for “receiver operating characteristic”
- Used in signal detection to show trade-off between hit rate and false alarm rate over noisy channel
- y axis shows percentage of true positives in sample
- x axis shows percentage of false positives in sample

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- Ideally if we have enough data, we’d set aside a validation set and use it to assess predictive performance of classifier.
- When data are scarce: K-fold validation
- Split data into K roughly equally-sized parts
- For each part fit a classifier to the other K-1 parts.
- Use the K-1 classifiers to classify the data in the left-out part.
- Combine the misclassification errors resulting from using the K-1 classifiers.
- Typical choice are K=5,10. K=N is known as “leave one out”.
- Smaller K will yield unbiased CV estimates, but with high variance. The opposite true for large K.

The case study involves the spring 1996 mailing campaign for an home equity loan (HEL) conducted by a major midwestern bank (MWB). We define as a responder (or a buyer) any customer that ended up taking an HEL and actually paying an interest on it. The continuous choice of concern is the YTD interest that the bank is expected to collect, which vary between customers based on the size and the terms of the HEL.

The models were evaluated and compared based on three criteria:

- Profitability
- Goodness of fit
- Prediction accuracy

- Profitability is measured in terms of the resulting total profit/return for the target mailing and audience and/or the average profit/return
- Goodness-of-fit exhibits how well the model is capable of discriminating between responders and non-responders and/or profitable and non-profitable people. In a binary model, it is measured by the actual response rate (the ratio of the number of buyers captured to the size of audience mailed), or the lift in the actual reponse rate attained by the model over a randomly selected mailing; in a continuous model by the average actual profit/return per customer.
- Prediction accuracy is measured by the difference in the predicted profit/response measures versus the actual results.

- The dataset of the study consist in the spring 1996 mailing audience
- The number of non-buyers is much greater than the number of buyers
- It was included only a sample of the non-buyers and all the buyers in the model training
- The log odds ratio should reflect the true proportion of the buyers and non-buyers

- A logistic regression model was fitted with different loss function for both classes.
- We change the loss function to detect correctly the buyers
- For each model fitted, the profit and the accuracy of the model was calculated

- The loss function will depend on:
- Which class is more important to detect (if a class is more important than the other the cost to no detect it correctly must be higher)
- The accuracy we want to achieve
- The profit we want to achieve (the maximum profit may not be achieved with the maximum accuracy)