oznur tastan 10601 machine learning recitation 3 sep 16 2009

Oznur Tastan

10601 Machine LearningRecitation 3Sep 16 2009

Outline

• A text classification example– Multinomial distribution– Drichlet distribution

• Model selection– Miro will be continuing in that topic

Text classification example

Text classification

• We are not into classification yet.• For the sake of example,

I’ll briefly go over what it is.

Classification Task:You have an input x, you classify which label it has y from some fixed set of labels y1,...,yk

Text classification spam filteringInput: document DOutput: the predicted class y from {y1,...,yk }

Spam filtering:

Classify email as ‘Spam’, ‘Other’.

P (Y=spam | X)

Text classificationInput: document DOutput: the predicted class y from {y1,...,yk }

Text classification examples:


What other text classification applications you can think of?

Text classificationInput: document xOutput: the predicted class y y is from {y1,...,yk }

Text classification examples:


Classify web pages as ‘Student’, ‘Faculty’, ‘Other’

Classify news stories into topics‘Sports’, ‘Politics’..

Classify business names by industry.

Classify movie reviews as ‘Favorable’, ‘Unfavorable’, ‘Neutral’ … and many more.

Text Classification: Examples Classify shipment articles into one 93 categories. An example category ‘wheat’

ARGENTINE 1986/87 GRAIN/OILSEED REGISTRATIONSBUENOS AIRES, Feb 26Argentine grain board figures show crop registrations of grains, oilseeds and their

products to February 11, in thousands of tonnes, showing those for future shipments month, 1986/87 total and 1985/86 total to February 12, 1986, in brackets:

Bread wheat prev 1,655.8, Feb 872.0, March 164.6, total 2,692.4 (4,161.0). Maize Mar 48.0, total 48.0 (nil). Sorghum nil (nil) Oilseed export registrations were: Sunflowerseed total 15.0 (7.9) Soybean May 20.0, total 20.0 (nil)The board also detailed export registrations for subproducts, as follows....

Representing text for classificationARGENTINE 1986/87 GRAIN/OILSEED REGISTRATIONSBUENOS AIRES, Feb 26Argentine grain board figures show crop registrations of grains, oilseeds and their


Bread wheat prev 1,655.8, Feb 872.0, March 164.6, total 2,692.4 (4,161.0). Maize Mar 48.0, total 48.0 (nil). Sorghum nil (nil) Oilseed export registrations were: Sunflowerseed total 15.0 (7.9) Soybean May 20.0, total 20.0 (nil)

The board also detailed export registrations for sub-products, as follows....

y

How would you represent the document?

Representing text: a list of words

argentine, 1986, 1987, grain, oilseed, registrations, buenos, aires, feb, 26, argentine, grain, board, figures, show, crop, registrations, of, grains, oilseeds, and, their, products, to, february, 11, in, …

Common refinements: remove stopwords, stemming, collapsing multiple occurrences of words into one….

y

Representing text for classificationARGENTINE 1986/87 GRAIN/OILSEED REGISTRATIONSBUENOS AIRES, Feb 26Argentine grain board figures show crop registrations of grains, oilseeds and their




y

How would you represent the document?

‘Bag of words’ representation of text





grain(s) 3

oilseed(s) 2

total 3

wheat 1

maize 1

soybean 1

tonnes 1

... ...

word frequency

Bag of word representation:

Represent text as a vector of word frequencies.

Bag of words representation

document i

Frequency (i,j) = j in document i

word j

A collection of documents

Bag of words

What simplifying assumption are we taking?

Bag of words

What simplifying assumption are we taking?

We assumed word order is not important.






grain(s) 3

oilseed(s) 2

total 3

wheat 1

maize 1

soybean 1

tonnes 1

... ...

word frequency

Pr( | )D Y y ?

1 1 1Pr( ,..., | )kW n W n Y y

Multinomial distribution• The multinomial distribution is a generalization of the binomial

distribution.

• The binomial distribution counts successes of an event (for example, heads in coin tosses).

• The parameters:– N (number of trials) – (the probability of success of the event)

• The multinomial counts the number of a set of events (for example, how many times each side of a die comes up in a set of rolls). – The parameters:– N (number of trials) – (the probability of success for each category)

1.. k

Multinomial DistributionFrom a box you pick k possible colored balls. You selected N balls randomly and put into your bag.

Let probability of picking a ball of color i is

For each color

Wi be the random variable denoting the number of balls selected in color i, can take values in {1…N}.

1,.., k i

Multinomial Distribution

W1,W2,..Wk are variables

1 2

11 1 1 1 21 2

!( ,..., | , ,.., ) ..

! !.. !knn n

k k kk

NP W n W n N

n n n

1

k

ii

n N

1

1k

ii

Number of possible orderings of N balls

order invariant selections

Note events are indepent

A binomial distribution is the multinomial distribution with k=2 and 1 2 2, 1






grain(s) 3

oilseed(s) 2

total 3

wheat 1

maize 1

soybean 1

tonnes 1

... ...

word frequency


grain(s) 3

oilseed(s) 2

total 3

wheat 1

maize 1

soybean 1

tonnes 1

... ...

word frequencyCan be represented as a multinomial distribution.

Words = colored balls, there are k possible typeof them

Document = contains N words, each wordoccurs ni times

The multinomial distribution of words is going to bedifferent for different document class.

In a document class of ‘wheat’, grain is more likely.where as in a hard drive shipment the parameter for ‘wheat’ is going to be smaller.

Multinomial distribution andbag of wordsRepresent document D as list of words w1,w2,..

For each category y, build a probabilistic model Pr(D|Y=y)

Pr(D={argentine,grain...}|Y=wheat) = ....Pr(D={stocks,rose,in,heavy,...}|Y=nonWheat) = ....

To classify, find the y which was most likely togenerate D

Conjugate distribution

• If the prior and the posterior are the same distribution, the prior is called a conjugate prior for the likelihood

• The Dirichlet distribution is the conjugate prior for the multinomial, just as beta is conjugate prior for the binomial.

Drichlet distribution

The Dirichlet parameter i can be thought of as a prior count of the ith class.

Dirichlet distribution generalizes the beta distributionjust like multinomial distribution generalizes the binomialdistribution

Gamma function

Dirichlet Distribution

Let’s say the prior for is

From observations we have the following counts The posterior distribution for given data

1( ,.., )kDir

1 1( ,.., )k kDir n n

1,.., k

1,.., kn n

1,.., k

So the prior works like a pseudo-counts.

Pseudo Count and prior• Let’s say you estimated the probabilities from a collection of

documents without using a prior.

• For all unobserved words in your document collection, you would assign zero probability to that word occurring in that document class. So whenever a document with that word comes in, the probability will be zero for that document being in that class. Which is probably wrong when you have only limited data.

• Using priors is a way of smoothing the probability distributions and leaving out some probability mass for the unobserved events in your data.

Generative model

C

w

ND

:

: Word: Document class generating the word: Number of word in the document: Collection of documentsmatrix of parameters for the multionomial specific to that document class

: Dirichlet( ) prior

W

C

N

D

for

Model Selection

Polynomial Curve FittingBlue: Observed data True: Green true distribution

Sum-of-Squares Error Function

0th Order PolynomialBlue: Observed data Red: Predicted curve True: Green true distribution

1st Order PolynomialBlue: Observed data Red: Predicted curve True: Green true distribution

3rd Order PolynomialBlue: Observed data Red: Predicted curve True: Green true distribution

9th Order PolynomialBlue: Observed data Red: Predicted curve True: Green true distribution

Which of the predicted curve is better?

Blue: Observed data Red: Predicted curve True: Green true distribution

What do we really want?

Why not choose the method with thebest fit to the data?



If we were to ask you the homework questions in the midterm, would we have a good estimate of how well you learned the concepts?



How well are you going to predictfuture data drawn from the samedistribution?

Example

General strategyYou try to simulate the real word scenario. Test data is your future data. Put it away as far as possible don’t look at it.

Validation set is like your test set. You use it to select your model.The whole aim is to estimate the models’ true error on the sample data you have.

!!! For the rest of the slides ..Assume we put the test data aldready away. Consider it as the validation data when it says test set.

Test set method• Randomly split some portion of your data Leave it aside as the test set• The remaining data is the training data

Test set method• Randomly split some portion of your data Leave it aside as the test set• The remaining data is the training data• Learn a model from the training set

This the model you learned.

How good is the prediction?• Randomly split some portion of your data Leave it aside as the test set• The remaining data is the training data• Learn a model from the training set• Estimate your future performance with the test data

Train test set split

It is simpleWhat is the down side ?

More data is betterWith more data you can learn better

Blue: Observed data Red: Predicted curve True: Green true distribution

Compare the predicted curves



1. You waste some portion of your data.



1. You waste some portion of your data.

What else?



1. You waste some portion of your data.2. You must be luck or unlucky with your test data



1. You waste some portion of your data.2. If you don’t have much data, you must be luck

or unlucky with your test data

How does it translate to statistics? Your estimator of performance has …?

Train/test set split


1. You waste some portion of your data.2. If you don’t have much data, you must be luck

or unlucky with your test data

How does it translate to statistics? Your estimator of performance has high variance

Cross Validation

Recycle the data!

LOOCV (Leave-one-out Cross Validation)

Your single test data

Let say we have N data pointsk be the index for data pointsk=1..N

Let (xk,yk) be the kth record

Temporarily remove (xk,yk) from the dataset

Train on the remaining N-1Datapoints

Test your error on (xk,yk)

Do this for each k=1..N and report the mean error.


There are N data points.. Do this N times. Notice thetest data is changing each time


There are N data points.. Do this N times. Notice thetest data is changing each time

MSE=3.33


Let say we have N data pointsk be the index for data pointsk=1..N

Let (xk,yk) be the kth record

Temporarily remove (xk,yk) from the dataset

Train on the remaining N-1datapoints

Test your error on (xk,yk)

Do this for each k=1..N and report the mean error.

K-fold cross validation

k-fold

traintest

Train on (k - 1) splitsTest

In 3 fold cross validation, there are 3 runs.In 5 fold cross validation, there are 5 runs.In 10 fold cross validation, there are 10 runs.

the error is averaged over all runs

Model SelectionIn-sample error estimates:

Akaike Information Criterion (AIC)Bayesian Information Criterion (BIC)Minimum Description Length Principle (MDL)Structural Risk Minimization (SRM)

Extra-sample error estimates: Cross-ValidationBootstrap

Most used method

References

• http://videolectures.net/mlas06_cohen_tc/• http://www.autonlab.org/tutorials/overfit.ht

ml

http://videolectures.net/mlas06_cohen_tc/

http://www.autonlab.org/tutorials/overfit.html

http://www.autonlab.org/tutorials/overfit.html

oznur tastan 10601 machine learning recitation 3 sep 16 2009

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