Computational BioMedical Informatics
description
Transcript of Computational BioMedical Informatics
1
Computational BioMedical Informatics
SCE 5095: Special Topics Course
Instructor: Jinbo BiComputer Science and Engineering Dept.
2
Course Information
Instructor: Dr. Jinbo Bi – Office: ITEB 233– Phone: 860-486-1458– Email: [email protected]
– Web: http://www.engr.uconn.edu/~jinbo/– Time: Tue / Thur. 3:30pm – 4:45pm – Location: ITEB 127– Office hours: Tue/Thur. 4:45-5:30pm
HuskyCT– http://learn.uconn.edu– Login with your NetID and password– Illustration
3
Regression and classification
Both regression and classification problems are typically supervised learning problems
The main property of supervised learning– Training example contains the input variables
and the corresponding target label– The goal is to find a good mapping from the
input variables to the target variable
4
Classification: Definition
Given a collection of examples (training set )– Each example contains a set of variables
(features), and the target variable class. Find a model for class attribute as a function
of the values of other variables. Goal: previously unseen examples should be
assigned a class as accurately as possible.– A test set is used to determine the accuracy of the
model. Usually, the given data set is divided into training and test sets, with training set used to build the model and test set used to validate it.
5
Classification Application 1
Tid Refund MaritalStatus
TaxableIncome Cheat
1 Yes Single 125K No
2 No Married 100K No
3 No Single 70K No
4 Yes Married 120K No
5 No Divorced 95K Yes
6 No Married 60K No
7 Yes Divorced 220K No
8 No Single 85K Yes
9 No Married 75K No
10 No Single 90K Yes10
categorical
categorical
continuous
class
Refund MaritalStatus
TaxableIncome Cheat
No Single 75K ?
Yes Married 50K ?
No Married 150K ?
Yes Divorced 90K ?
No Single 40K ?
No Married 80K ?10
TestSet
Training Set Model
Learn Classifier
Past transaction records, label them
Current data, want to use the model to predict
Fraud detection – goals: Predict fraudulent cases in credit card transactions.
6
Classification: Application 2
Handwritten Digit Recognition Goal: Identify the digit of a handwritten number
– Approach:Align all images to derive the featuresModel the class (identity) based on these features
7
Illustrating Classification Task
Apply Model
Induction
Deduction
Learn Model
Model
Tid Attrib1 Attrib2 Attrib3 Class
1 Yes Large 125K No
2 No Medium 100K No
3 No Small 70K No
4 Yes Medium 120K No
5 No Large 95K Yes
6 No Medium 60K No
7 Yes Large 220K No
8 No Small 85K Yes
9 No Medium 75K No
10 No Small 90K Yes 10
Tid Attrib1 Attrib2 Attrib3 Class
11 No Small 55K ?
12 Yes Medium 80K ?
13 Yes Large 110K ?
14 No Small 95K ?
15 No Large 67K ? 10
Test Set
Learningalgorithm
Training Set
8
Classification algorithms
K-Nearest-Neighbor classifiers Naïve Bayes classifier Neural Networks Linear Discriminant Analysis (LDA) Support Vector Machines (SVM) Decision Tree Logistic Regression Graphical models
9
Regression: Definition
Goal: predict the value of one or more continuous target attributes give the values of the input attributes
Difference between classification and regression only lies in the target attribute– Classification: discrete or categorical target– Regression: continuous target
Greatly studied in statistics, neural network fields.
10
Regression application 1
categorical
categorical
continuous
Continuous ta
rget
Refund Marital Status
Taxable Income Loss
No Single 75K ?
Yes Married 50K ?
No Married 150K ?
Yes Divorced 90K ?
No Single 40K ?
No Married 80K ? 10
TestSet
Training Set Model
Learn Regressor
Past transaction records, label them
Current data, want to use the model to predict
goals: Predict the possible loss from a customer
Tid Refund MaritalStatus
TaxableIncome Loss
1 Yes Single 125K 100
2 No Married 100K 120
3 No Single 70K -200
4 Yes Married 120K -300
5 No Divorced 95K -400
6 No Married 60K -500
7 Yes Divorced 220K -190
8 No Single 85K 300
9 No Married 75K -240
10 No Single 90K 9010
11
Regression applications
Examples:– Predicting sales amounts of new product
based on advertising expenditure.– Predicting wind velocities as a function of
temperature, humidity, air pressure, etc.– Time series prediction of stock market indices.
12
Regression algorithms
Least squares methods Regularized linear regression (ridge regression) Neural networks Support vector machines (SVM) Bayesian linear regression
13
Practical issues in the training
Underfitting
Overfitting
Before introducing these important concept, let us study a simple regression algorithm – linear regression
14
Least squares
We wish to use some real-valued input variables x to predict the value of a target y
We collect training data of pairs (xi,yi), i=1,…N Suppose we have a model f that maps each x
example to a value of y’ Sum of squares function:
– Sum of the squares of the deviation between the observed target value y and the predicted value y’
N
iii
N
iii xfyyy
1
2
1
2 )('
15
Least squares
Find a function f such that the sum of squares is minimized
For example, your function is in the form of linear functions f (x) = wTx
Least squares with a linear function of parameters w is called “linear regression”
N
iii xfy
1
2)(min
N
ii
Ti xwyw
1
2min
16
Linear regression
Linear regression has a closed-form solution for w
The minimum is attained at the zero derivative
)(XwyXwy
min1
2
wE
wxyw
T
N
i
Tii
0)(2)(
XwyXw
wE T
yXXXw TT 1
17
x is evenly distributed from [0,1] y = f(x) + random error y = sin(2πx) + ε, ε ~ N(0,σ)
Polynomial Curve Fitting
18
Polynomial Curve Fitting
19
Sum-of-Squares Error Function
20
0th Order Polynomial
21
1st Order Polynomial
22
3rd Order Polynomial
23
9th Order Polynomial
24
Over-fitting
Root-Mean-Square (RMS) Error:
25
Polynomial Coefficients
26
Data Set Size:
9th Order Polynomial
27
Data Set Size:
9th Order Polynomial
28
Regularization
Penalize large coefficient values
Ridge regression
29
Regularization:
30
Regularization:
31
Regularization: vs.
32
Polynomial Coefficients
33
Ridge Regression
Derive the analytic solution to the optimization problem for ridge regression
22 |||||||| min wXwy
wwXwyXwy TT )()( min
wIXXwXwy TTT )(2- min Using KKT condition – first order derivative = 0
yXwIXX TT )(
wwXwXwXwy TTTT 2- min
yXIXXw TT 1)(
34
Neural networks
Introduction Different designs of NN Feed-forward Network (MLP) Network Training Error Back-propagation Regularization
35
Introduction
Neuroscience studies how networks of neurons produce intellectual behavior, cognition, emotion and physiological responses
Computer science studies how to simulate knowledge in cognitive science, including the way neurons process signals
Artificial neural networks simulate the connectivity in the neural system, the way it passes through signal, and mimic the massively parallel operations of the human brain
36
Common features
Dentrites
37
Different types of NN
Adaptive NN: have a set of adjustable parameters that can be tuned
Topological NN
Recurrent NN
38
Different types of NN
Feed-forward NN Multi-layer perceptron Linear perceptron
HiddenLayer
InputLayer Output
Layer
39
Different types of NN
Radial basis function NN (RBFN)
40
Multi-Layer Perceptron
Layered perceptron networks can realize any logical function, however there is no simple way to estimate the parameters/generalize the (single layer) Perceptron convergence procedure
Multi-layer perceptron (MLP) networks are a class of models that are formed from layered sigmoidal nodes, which can be used for regression or classification purposes.
They are commonly trained using gradient descent on a mean squared error performance function, using a technique known as error back propagation in order to calculate the gradients.
Widely applied to many prediction and classification problems over the past 15 years.
41
Linear perceptron
Input layer output layer
::
x1
wt
w2
w1
xt
x2 yΣ
y = w1*x1 + w2*x2 + … + wt*xt
Many functions can not be approximated using perceptron
42
Multi-Layer Perceptron
XOR (exclusive OR) problem
0+0=0 1+1=2=0 mod 2 1+0=1 0+1=1 Perceptron does not
work here!Single layer generates a linear decision boundary
43
Multi-Layer Perceptron
::
x1
wt1(1)
W21(1)
W11(1)
xt
x2 yf(Σ)
f(Σ)
f(Σ)
W11(2)
W22(1)
W21(2)
Each link is associated with a weight, and these weights are the tuning parameters to be learnedEach neuron except ones in the input layer receives inputs from the previous layer, and reports an output to next layer
Input layer Hidden layer output layer
44
Each neuron
wn
.
.
.
w2
S
w1
n
iiiout pwSUM
)exp(11
outj SUM
OUT
summation
f is Activation function
o The activation function f can beIdentity function f(x) = xSigmoid function Hyperbolic tangent
45
1st layer 2nd layer3rd layer
Universal Approximation: Three-layer network can in principleapproximate any function with any accuracy!
Universal Approximation of MLP
46
Feed-forward network function
The output from each hidden node
The final output
N
iiijj xwfo
1
)1( )1(
x1
xt
x2 y
N nodes M nodes
M
jjjkk owfy
1
)1()2(
Signal flows
47
Network Training
A supervised neural network is a function h(x;w) that maps from inputs x to target y
Usually training a NN does not involve the change of NN structures (such as how many hidden layers or how many hidden nodes)
Training NN refers to adjusting the values of connection weights so that h(x;w) adapts to the problem
Use sum of squares as the error metric
ijwwE
)(
L
iii wxhywE
1
2);()(
Use gradient descent
48
Gradient descent
Review of gradient descent Iterative algorithm containing many iterations Each iteration, the weights w receive a small
update
Terminate – until the network is stable (in other words, the
training error cannot be reduced further) E(wnew) < E(w) not hold– until the error on a validation set starts to
climb up (early stopping)
ij
ijnew
ij wEww
49
Error Back-propagation
The update of the weights goes backwards because we have to use the chain rule to evaluate the gradient of E(w)
Learning is backwards
x1
xt
x2 y=h(x;w)
M nodes N nodes
Signal flows forwards
W ij W jk
50
Error Back-propagation
Update the weights in the output layer first Propagate errors from the high layer to low layer Recall
Learning is backwardsx1
xt
x2 y=h(x;w)
M nodes N nodes
W ij W jk
N
iiijj xwfo
1
)1( )1(
M
jjj owfy
1
)1()2(
51
Evaluate gradient
First compute the partial derivatives for weights in the output layer
Second compute the partial derivatives for weights in the hidden layer
M
jjj owfy
1
)1()2(
N
iiijj xwfo
1
)1( )1(
L
pp
L
ppp EyywE
11
2 ~ˆ)(
)1(
1
)1(
)2()2(
)('ˆ2
ˆ2~
)2(
j
M
jjjii
j
iii
j
oowfyy
wyyy
wE
i
N
iiijj
M
jjjii
ij
jj
M
jjjii
ij
iii
ij
xxwfwowfyy
wo
wowfyy
wyyy
wE
)(')('ˆ2
)('ˆ2
ˆ2~
1
)2(
1
)1(
)1(
)1()2(
1
)1(
)1()1(
)1()2(
)2(
52
Back-propagation algorithm
Design the structure of NN Initialize all connection weights For t = 1, to T
– Present training examples, propagate forwards from input layer to output layer, compute y, and evaluate the errors
– Pass errors backwards through the network to recursively compute derivatives, and use them to update weights
– If termination rule is met, stop; or continue end
ij
tij
tij w
Eww 1
53
Notes on back-propagation
Note that these rules apply to different kinds of feed-forward networks. It is possible for connections to skip layers, or to have mixtures. However, errors always start at the highest layer and propagate backwards
54
Activation and Error back-propagation
55
Two schemes of training There are two schemes of updating weights
– Batch: Update weights after all examples have been presented (epoch).
– Online: Update weights after each example is presented.
Although the batch update scheme implements the true gradient descent, the second scheme is often preferred since – it requires less storage, – it has more noise, hence is less likely to get
stuck in a local minima (which is a problem with nonlinear activation functions). In the online update scheme, order of presentation matters!
56
Problems of back-propagation
It is extremely slow, if it does converge. It may get stuck in a local minima. It is sensitive to initial conditions. It may start oscillating.
57
Overfitting – number of hidden units
Over-fitting
Sinusoidal data set used in polynomial curve fitting example
58
Regularization (1)
How to adjust the number of hidden units to get the best performance while avoiding over-fitting
Add a penalty term to the error function
The simplest regularizer is the weight decay:
wwww T
2)()(~ EE
ij
ijnew
ij wEww )1(
59
Regularization (2)
A method to Early Stopping– obtain good generalization performance and– control the effective complexity of the network
Instead of iteratively reducing the error until a minimum error on the training data set has been reached
We have a validation set of data available Stop when the NN achieves the smallest error
w.r.t. the validation data set
60
Effect of early stopping
Validation Set
Training SetError vs. Number of iterations
A slight increase in the validation set error
61
Classification
Underfitting or Overfitting can also happen in classification approaches
We will illustrate these practical issues on classification problem
Before the illustration, we introduce a simple classification technique – K-nearest neighbor method
62
K-nearest neighbor (K-NN)
K-NN is one of the simplest machine learning algorithm
K-NN is a method for classifying test examples based on closest training examples in the feature space
An example is classified by a majority vote of its neighbors
k is a positive integer, typically small. If k = 1, then the example is simply assigned to the class of its nearest neighbor.
63
K-NN
K = 1K = 3
64
K-NN on real problem data
• Oil data set• K acts as a smoother, choosing K is model
selection• For , the error rate of the 1-nearest-neighbour
classifier is never more than twice the optimal error (obtained from the true conditional class distributions).
65
Limitation of K-NN
K-NN is a nonparametric model (no any particular function is fitted)
Nonparametric models requires storing and computing with the entire data set.
Parametric models, once fitted, are much more efficient in terms of storage and computation.
66
Probabilistic interpretation of K-NN
Given a data set with Nk data points from class Ck and , we have
and correspondingly
Since , Bayes’ theorem gives
67
Underfit and Overfit (Classification)
500 circular and 500 triangular data points.
Circular points:0.5 sqrt(x1
2+x22) 1
Triangular points:sqrt(x1
2+x22) > 1 or
sqrt(x12+x2
2) < 0.5
68
Underfit and Overfit (Classification)
500 circular and 500 triangular data points.
Circular points:0.5 sqrt(x1
2+x22) 1
Triangular points:sqrt(x1
2+x22) > 1 or
sqrt(x12+x2
2) < 0.5
69
Underfitting and Overfitting
Overfitting
Underfitting: when model is too simple, both training and test errors are large
Number of Iterations
70
Overfitting due to Noise
Decision boundary is distorted by noise point
71
Overfitting due to Insufficient Examples
Lack of data points in the lower half of the diagram makes it difficult to predict correctly the class labels of that region - Insufficient number of training records in the region causes the neural nets to predict the test examples using other training records that are irrelevant to the classification task
72
Notes on Overfitting
Overfitting results in classifiers (a neural net, or a support vector machine) that are more complex than necessary
Training error no longer provides a good estimate of how well the classifier will perform on previously unseen records
Need new ways for estimating errors
73
Occam’s Razor
Given two models of similar generalization errors, one should prefer the simpler model over the more complex model
For complex models, there is a greater chance that it was fitted accidentally by errors in data
Therefore, one should include model complexity when evaluating a model
74
How to Address Overfitting
Minimize training error no longer guarantees a good model (a classifier or a regressor)
Need better estimate of the error on the true population – generalization error Ppopulation( f(x) not equal to y )
In practice, design a procedure that gives better estimate of the error than training error
In theoretical analysis, find an analytical bound to bound the generalization error or use Bayesian formula
75
Model Evaluation (pp. 295—304 of data mining)
Metrics for Performance Evaluation– How to evaluate the performance of a model?
Methods for Performance Evaluation– How to obtain reliable estimates?
Methods for Model Comparison– How to compare the relative performance
among competing models?
76
Model Evaluation
Metrics for Performance Evaluation– How to evaluate the performance of a model?
Methods for Performance Evaluation– How to obtain reliable estimates?
Methods for Model Comparison– How to compare the relative performance
among competing models?
77
Metrics for Performance Evaluation
Regression– Sum of squares
– Sum of deviation
– Exponential function of the deviation
78
Metrics for Performance Evaluation
Focus on the predictive capability of a model– Rather than how fast it takes to classify or
build models, scalability, etc. Confusion Matrix:
PREDICTED CLASS
ACTUALCLASS
Class=Yes Class=No
Class=Yes a b
Class=No c d
a: TP (true positive)b: FN (false negative)c: FP (false positive)d: TN (true negative)
79
Metrics for Performance Evaluation…
Most widely-used metric:
PREDICTED CLASS
ACTUALCLASS
Class=Yes Class=No
Class=Yes a(TP)
b(FN)
Class=No c(FP)
d(TN)
FNFPTNTPTNTP
dcbada
Accuracy
80
Limitation of Accuracy
Consider a 2-class problem– Number of Class 0 examples = 9990– Number of Class 1 examples = 10
If model predicts everything to be class 0, accuracy is 9990/10000 = 99.9 %– Accuracy is misleading because model does
not detect any class 1 example
81
Cost Matrix
PREDICTED CLASS
ACTUALCLASS
C(i|j) Class=Yes Class=No
Class=Yes C(Yes|Yes) C(No|Yes)
Class=No C(Yes|No) C(No|No)
C(i|j): Cost of misclassifying class j example as class i
82
Computing Cost of Classification
Cost Matrix
PREDICTED CLASS
ACTUALCLASS
C(i|j) + -+ -1 100- 1 0
Model M1 PREDICTED CLASS
ACTUALCLASS
+ -+ 150 40- 60 250
Model M2 PREDICTED CLASS
ACTUALCLASS
+ -+ 250 45- 5 200
Accuracy = 80%Cost = 3910
Accuracy = 90%Cost = 4255
83
Cost vs Accuracy
Count PREDICTED CLASS
ACTUALCLASS
Class=Yes Class=No
Class=Yes a b
Class=No c d
Cost PREDICTED CLASS
ACTUALCLASS
Class=Yes Class=No
Class=Yes p q
Class=No q p
N = a + b + c + d
Accuracy = (a + d)/N
Cost = p (a + d) + q (b + c)
= p (a + d) + q (N – a – d)
= q N – (q – p)(a + d)
= N [q – (q-p) Accuracy]
Accuracy is proportional to cost if1. C(Yes|No)=C(No|Yes) = q 2. C(Yes|Yes)=C(No|No) = p
84
Cost-Sensitive Measures
baa
caa
(r) Recall
(p)Precision
Precision is biased towards C(Yes|Yes) & C(Yes|No) Recall is biased towards C(Yes|Yes) & C(No|Yes)
Count PREDICTED CLASS
ACTUALCLASS
Class=Yes
Class=No
Class=Yes
a b
Class=No
c d
A model that declares every record to be the positive class: b = d = 0
A model that assigns a positive class to the (sure) test record: c is small
Recall is high
Precision is high
85
Cost-Sensitive Measures (Cont’d)
cbaa
prrp
baa
caa
222(F) measure-F
(r) Recall
(p)Precision
F-measure is biased towards all except C(No|No)
dwcwbwawdwaw
4321
41Accuracy Weighted
Count PREDICTED CLASS
ACTUALCLASS
Class=Yes
Class=No
Class=Yes
a b
Class=No
c d
86
Model Evaluation
Metrics for Performance Evaluation– How to evaluate the performance of a model?
Methods for Performance Evaluation– How to obtain reliable estimates?
Methods for Model Comparison– How to compare the relative performance
among competing models?
87
Methods for Performance Evaluation
How to obtain a reliable estimate of performance?
Performance of a model may depend on other factors besides the learning algorithm:– Class distribution– Cost of misclassification– Size of training and test sets
88
Learning Curve
Learning curve shows how accuracy changes with varying sample size
Requires a sampling schedule for creating learning curve: Arithmetic sampling
(Langley, et al) Geometric sampling
(Provost et al)
Effect of small sample size:- Bias in the estimate- Variance of estimate
89
Methods of Estimation Holdout
– Reserve 2/3 for training and 1/3 for testing Random subsampling
– Repeated holdout Cross validation
– Partition data into k disjoint subsets– k-fold: train on k-1 partitions, test on the remaining one– Leave-one-out: k=n
Stratified sampling – oversampling vs undersampling
Bootstrap– Sampling with replacement
91
Methods of Estimation (Cont’d) Holdout method
– Given data is randomly partitioned into two independent sets Training set (e.g., 2/3) for model construction Test set (e.g., 1/3) for accuracy estimation
– Random sampling: a variation of holdout Repeat holdout k times, accuracy = avg. of the accuracies
obtained Cross-validation (k-fold, where k = 10 is most popular)
– Randomly partition the data into k mutually exclusive subsets, each approximately equal size
– At i-th iteration, use Di as test set and others as training set
– Leave-one-out: k folds where k = # of tuples, for small sized data– Stratified cross-validation: folds are stratified so that class dist. in
each fold is approx. the same as that in the initial data
92
Methods of Estimation (Cont’d) Bootstrap
– Works well with small data sets– Samples the given training tuples uniformly with replacement i.e., each time a tuple is selected, it is equally likely to be selected
again and re-added to the training set Several boostrap methods, and a common one is .632 boostrap
– Suppose we are given a data set of d examples. The data set is sampled d times, with replacement, resulting in a training set of d samples. The data points that did not make it into the training set end up forming the test set. About 63.2% of the original data will end up in the bootstrap, and the remaining 36.8% will form the test set (since (1 – 1/d)d ≈ e-1 = 0.368)
– Repeat the sampling procedure k times, overall accuracy of the model:
))(368.0)(632.0()( _1
_ settraini
k
isettesti MaccMaccMacc
93
Model Evaluation
Metrics for Performance Evaluation– How to evaluate the performance of a model?
Methods for Performance Evaluation– How to obtain reliable estimates?
Methods for Model Comparison– How to compare the relative performance
among competing models?
94
ROC (Receiver Operating Characteristic)
Developed in 1950s for signal detection theory to analyze noisy signals – Characterize the trade-off between positive
hits and false alarms ROC curve plots TPR (on the y-axis) against FPR
(on the x-axis) Performance of each classifier represented as a
point on the ROC curve If the classifier returns a real-valued prediction,
– changing the threshold of algorithm, sample distribution or cost matrix changes the location of the point
95
ROC Curve
At threshold t:TP=50, FN=50, FP=12, TN=88
PREDICTED CLASS
ACTUALCLASS
Class=Yes
Class=No
Class=Yes
a(TP)
b(FN)
Class=No
c(FP)
d(TN)
TPR = TP/(TP+FN)FPR = FP/(FP+TN)
96
ROC Curve
PREDICTED CLASS
ACTUALCLASS
Class=Yes
Class=No
Class=Yes
a(TP)
b(FN)
Class=No
c(FP)
d(TN)
TPR = TP/(TP+FN)FPR = FP/(FP+TN)
(TPR,FPR): (0,0): declare everything
to be negative class– TP=0, FP = 0
(1,1): declare everything to be positive class– FN = 0, TN = 0
(1,0): ideal– FN = 0, FP = 0
97
ROC Curve
(TPR,FPR): (0,0): declare everything
to be negative class (1,1): declare everything
to be positive class (1,0): ideal
Diagonal line:– Random guessing– Below diagonal line: prediction is opposite of the
true class
98
How to Construct an ROC curve
Instance P(+|A) True Class
1 0.95 +
2 0.93 +
3 0.87 -
4 0.85 -
5 0.85 -
6 0.85 +
7 0.76 -
8 0.53 +
9 0.43 -
10 0.25 +
• Use classifier that produces posterior probability for each test instance P(+|A)
• Sort the instances according to P(+|A) in decreasing order
• Apply threshold at each unique value of P(+|A)
• Count the number of TP, FP,
TN, FN at each threshold• TP rate, TPR = TP/(TP+FN)• FP rate, FPR = FP/(FP +
TN)
99
How to Construct an ROC curve
Instance P(+|A) True Class
1 0.95 +
2 0.93 +
3 0.87 -
4 0.85 -
5 0.85 -
6 0.85 +
7 0.76 -
8 0.53 +
9 0.43 -
10 0.25 +
• Use classifier that produces posterior probability for each test instance P(+|A)
• Sort the instances according to P(+|A) in decreasing order
• Pick a threshold 0.85• p>= 0.85, predicted to P• p< 0.85, predicted to N• TP = 3, FP=3, TN=2, FN=2• TP rate, TPR = 3/5=60%• FP rate, FPR = 3/5=60%
100
How to construct an ROC curveClass + - + - - - + - + +
P 0.25 0.43 0.53 0.76 0.85 0.85 0.85 0.87 0.93 0.95 1.00
TP 5 4 4 3 3 3 3 2 2 1 0
FP 5 5 4 4 3 2 1 1 0 0 0
TN 0 0 1 1 2 3 4 4 5 5 5
FN 0 1 1 2 2 2 2 3 3 4 5
TPR 1 0.8 0.8 0.6 0.6 0.6 0.6 0.4 0.4 0.2 0
FPR 1 1 0.8 0.8 0.6 0.4 0.2 0.2 0 0 0
Threshold >=
ROC Curve:
101
Using ROC for Model Comparison
No model consistently outperforms the other M1 is better for
small FPR M2 is better for
large FPR
Area Under the ROC curve (AUC) Ideal:
Area = 1 Random guess:
Area = 0.5
102
Data normalization
Example-wise normalization– Each example is normalized
and mapped to unit sphere Feature-wise normalization
– [0,1]-normalization: normalize each feature into a unit space
– Standard normalization: normalize each feature to have mean 0 and standard deviation 1
1
1
1
1
103
Training data is given. – Each object is associated with a class label Y {1, 2,
…, K} and a feature vector of d measurements: X = (X1, …, Xd).
Build a model from the training data.
Unseen objects are to be classified as belonging to one of a number of predefined classes {1, 2, …, K}.
Linear Discriminant Analysis / Fisher’s linear disciminant
Classification
104
Two classes
Variable 1
Varia
ble
2
Best a
xis
1u
2u
105
Three classes
106
Classifiers are built from a training set (TS) L = (X1, Y1), ..., (Xn,Yn)
Classifier C built from a learning set L:
C: X {1,2, ... ,K}
Bayes classifier base on conditional densities p(Ck | X), C(X) = arg maxk p(Ck | X) This is a maximum a posterior, and p(Ck | X) is a
posterior density
Classifiers
107
The Rules of Probability
Sum Rule
Product Rule
Bayes’ Rule
posterior likelihood × prior
= p(X|Y)p(Y)
is irrelevant to Y=C
)|( YXp)|( dataXCYp )(Yp
108
p(Ck | X) = p(X | Ck) p(Ck) /p(X)
Find a class label C(X) so that maxk p(Ck | X) = maxk p(X | Ck) p(Ck)
Naïve Bayes assumes independence among all features (last class)– p(X | Ck) = p(x1 | Ck) p(x2 | Ck) . . . p(xd | Ck)
Very strong assumption
Maximum a posterior
109
S
S
kkT
k
kdk XXCXp
1
21exp
)det()2(1)|(
Assume multivariate Gaussian (normal) class densities X|Y= k ~ N(k, Sk),
)(log)det()2(log
21)()|(log 1
kkd
kkT
kkk
Cp
XXCpCXp
S
S
C(X) = arg mink {(X - k)’ Sk-1
(X - k) + log| Sk | -2log(p(Ck))}
Multivariate normal dist for each class
Maximizing posterior is equivalent to maximizing p(X|CK)p(CK), and equivalent to maximizing the logorithm of p(X|CK)p(CK)
110
Two-class case
TXXXX TT SSSS 22
12211
111 loglog
1)()|()()|(
22
11 CpCXpCpCXp
)()|()()|( 2211 CpCXpCpCXp C(X) = C1If
otherwise C(X) = C2
))()(log(
)|()|(log
1
2
2
1
CpCp
CXpCXp
Equivalently, )()(
)|()|(
1
2
2
1
CpCp
CXpCXp
111
Guassian discriminant rule
For multivariate Gaussian (normal) class densities X|Y= k ~ N(k, Sk), the classification rule is
C(X) = arg mink {(X - k)’ Sk-1
(X - k) + log| Sk |}
In general, this is a quadratic rule (Quadratic discriminant analysis, or QDA)
In practice, population mean vectors k and covariance matrices Sk are estimated by corresponding sample quantities
TXXXX TT SSSS 22
12211
111 loglog
112
SiCx
Tii
ii xx
C))((1
iCxi
i xC ||1Class mean
Class covariance
Sample mean and variance
113
S
000021012
31
000011011
000000001
000010000
31
011011
001001
01001
0
31
))(())(())((31
111
31
.120
,112
,101
332211
321
321
TTT XXXXXX
XXX
XXX
Example
114
Two-class case
If the two classes have the same covariance matrix, Sk = S the discriminant rule is linear (Linear discriminant analysis, or LDA; FLDA for k = 2):
Quadratic rule
TXXXX TT SSSS 22
12211
111 loglog
cX T S )( 121
cwX T )( 121 S w
become
where
Usually, )(12211 SSS nn
n
115
Illustration
μ1 μ2
116
Two-class case
Maximize the signal-to-noise ratio
wwww
withinT
betweenT
w SSmax
Tbetween ))(( 1212 S
)(12211 SSS nn
nwithin
where
Solution is )( 121 S
withinw
Between-class separation
Within-class cohesion
117
Two-class case (illustration)
LDA gives the yellow direction
Two classes overlap
Two classes are separated
118
Two-class case (illustration)
1 2
2- 1
LDA axisBest Threshold
119
Multi-class case
Two approaches– Apply Fisher LDA to each “one-versus-rest”
class
120
Multi-class case
K
i Cx
Tiiw
K
k
Tkkkb
i
xxn
S
nn
S
1
1
))((1
))((1
Transformation matrix G that projects the data to be most separable is the matrix that maximizes
WSWWSW
wT
bT
maxW
Second approach:Similarly, find multiple directions that form a low dimensional space
Correct way to write it is
1
W))((tracemax WSWWSW w
Tb
T
Between-class matrix
Within-class matrix
121
The goal is to simultaneously maximize the between-class separation and minimize the within-class cohesion
The solution to is the generalized eigenvalue problem The generalized eigenvectors are eigenvectors by solving
Intuition
gSgS wb
bw SS 1
1
W))((tracemax WSWWSW w
Tb
T
122
Graphic view of the transformation (projection)
d
n
dnA Training data (matrix)
)1( knLA
n
K-1
Reduced training data
K-1
d
)1( kdWTransformation matrix
123
Graphical view of classificationd
n
dnA
n
K-1
)1( knLAK-1
d
)1( kdG
1K-1
)1(1 kLh
Find the nearest neighborOr nearest centroid
d1
dh 1
A test data point h
124
First applied by M. Barnard at the suggestion of R. A. Fisher (1936), Fisher linear discriminant analysis (FLDA):
Dimension reduction– Finds linear combinations of the features X=X1,...,Xd with
large ratios of between-groups to within-groups sums of squares - discriminant variables;
Classification– Predicts the class of an observation X by the class whose
mean vector is closest to X in terms of the discriminant variables
Summary