Machine Learningin

Finance

Stefan Duprey / September 2013

Statistical learning scope

Data Mining

Exploration

UnivariatePie chart,

Histogram, etc…

MultivariateFeature

selection and transformation

Modelling

Clustering

Partitive

K-means

Gaussianmixture model

SOMHierarchical

Classification

Discriminant

Decision Tree

Neural Network

Support VectorMachine

Regression

Classifier for

Credit Scoring

Decision rule for Support Vector Machines

A quadratic optimization problem !

SVM non-linear case

SVM summary

avoid the plague of local minima the engineer’s expertise is in the appropriate

kernel (beware of overfitting, cross-validate and experiment your own kernels) only classify between 2 class (one vs all or one

vs one methodology) a reference in use cases in computer vision,

bio informatics

Neural Network : what are they ?

Neural Network summary

Gradient descent algorithm : stochastic, mini-batch, conjugate

plague of local minima : difficult to calibrate

the engineer’s expertise is in the appropriate architecture (beware of overfitting, cross-validate and experiment your own architecture ‘deeper learning’)

>> t = classregtree(X,Y);

>> Y_pred = t(X_new);

Regression Trees

Forests of Trees

predictors

updowndown

upupup

downup

downupup...

responseY

>> t = TreeBagger(nb_trees,X,Y);

>> [Y_pred,allpred] = predict(t,X_new);

Splitting criteria : information gain

Why a regression and what is a regression ?

A regression is a model to explain and predict a process :

supervised machine learning

Why regularizing ?• Terms are correlated

• The regression matrix becomes close to singular

• Badly conditioned matrix yield poor numerical results

• Bayesian interpretation

Likelihood

Regularisation term

Posterior

Prior

We rather minimize

Why Lasso and Elastic Net?• No method owns the truth

• Reduce the number of predictors in a regression model

• Identify important predictors

• Select among redundant predictors

• Produce shrinkage estimates with potentially lower predictive errors than ordinary least squares (cross validation)

Lasso :

Elastic Net :

Ensemble learning

Why ensemble learning ?

‘melding results from many weak learners into one high-quality ensemble predictor’

Main differences between Bagging and Boosting

BAGGING BOOSTING

Bagging is randomness Boosting is adaptative and deterministic

Bootstrapped sample Complete initial sample

Each model must perform well over the whole

sample

Each model has to perform better than the

previous one on outliers

Every model have the same weight Models are weighted according to their

performance

Defining features

Advantages and disadvantages

BAGGING BOOSTING

Reducing model variance Variance might rise

Not a simple model anymore Not a simple model anymore

Can be parallelized Can not be parallelized

Less noise over fitting : better than boosting

when noise

Models are weighted according to their

performance

Bagging is usually efficienter than boosting On specific cases, boosting might achieve a far

better accuracy

Big DataLearning

overDistributed Data

Distributed memory : MDCS & the MAP/REDUCE

paradigm

Big data & Machine learning

“It’s not who has the best algorithm that wins . It’s who

has the most data”

Quick overviewExploratory analysis

ClusteringClassification

Aims of this presentation

awareness of the range of methods for multivariate data

reasonable understanding of algorithms

Data Mining

• Exploratory Data Analysis

• Clustering

• Classification

• Regression

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• Categorical

• Ordinal

• Discontinuous

Exploratory Data Analysis

Why exploratory analysis ? Can be used to:o Graphical viewo “Pre filtering”: preliminary data trends and behaviour

• Means:• Multivariate Plots• Features transformation : principal component analysis, factor model• Features selection : stepwise optimization

Data Exploration: Getting an overview of individual variables

Basic Histogram>> hist(x(:,1))

Custom Number of Bins>> hist(x(:,1),50)

By Group>> hist(byGroup,20)

Gaussian fit>> histfit(x(:,2))

3D Histogram>> hist3(x(:,1:2))

Scatter Plot>>gscatter(x(:,1),x(:,2),groups)

Pie Chart>> pie3(proportions,groups)

>> X = [MPG,Acceleration,Displacement,Weight,Horsepower];

Box Plot>> boxplot(x(:,1),groups)

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Data Exploration: Getting an overview of multiple variables

Plot Matrix by Group

>> gplotmatrix(x,x,groups)

Parallel Coordinates Plot

>> parallelcoords(x,'Group',groups)

Andrews’ Plot

>> andrewsplot(x,'Group',groups)

Glyph Plot

>> glyphplot(x)Chernoff Faces

>> glyphplot(x,'Glyph','face')

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chevrolet chevelle malibubuick skylark 320 plymouth satellite

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Principal component analysis

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CommerzbankDeutscheBankInfineon

ThyssenKruppMANDaimlerHeidelbergerAllianzDeutscheBahnBMWSalzgitterSiemensDeutschePostLufthansaBASFAdidasMetroVWLindeEONMunichReBayerRWESAPMRKDeutscheTelekomBeiersdorf

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>>[pcs,scrs,variances]=princomp(stocks);

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Factor model Alternative to PCA to improve your components

>>[Lambda,Psi,T,stats,F]=factoran(stocks,3,'rotate','promax);

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SiemensDeutschePostCommerzbank

BASF

Adidas

LindeMunichReMetroHeidelberger

SAP

Bayer

Salzgitter

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EONRWE

VW

DeutscheTelekom

BeiersdorfMRKFresenius

Henkel

FreseniusMedical

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Paring predictors : stepwise optimization Some predictors might be correlated, other irrelevant

Requires Statistics Toolbox™>>[coeff,inOut]=stepwisefit(stocks, index);

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Cloud of randomly generated points• Each cluster center is randomly chosen inside specified bounds

• Each cluster contains the specified number of points per cluster

• Each cluster point is sampled from a gaussian distribution

• Multidimensionnal dataset

>>clusters = 8; % number of clusters.>>points = 30; % number of points in each cluster.>>std_dev = 0.05; % common cluster standard deviation>>bounds = [0 1]; % bounds for the cluster center

>>[x,vcentroid,proportions,groups] =cluster_generation(bounds,clusters,points,std_dev);

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Clustering Why clustering ?

o Segment populations into natural subgroupso Identify outlierso As a preprocessing method – build separate models on each

• Means• Hierarchical clustering• Clustering with neural network (self-organizer map, competitive layer)• Clustering with K-means nearest neighbours• Clustering with K-means fuzzy logic• Clustering using Gaussian mixture models

• Predictors: categorical, ordinal, discontinuous-0.1 0 0.1 0.2 0.3 0.4 0.5 0.60

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Hierarchical Cluster Analysis – what is it doing?

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Hierarchical Cluster Analysis – how do I do it ?

• Calculate pairwise distances between points

>> distances = pdist(x)

• Carry out hierarchical cluster analysis

>> tree = linkage(distances)

• Visualise as a dendrogram

>> dendrogram(tree)

• Assign points to clusters

>> assignments = cluster(tree,‘cutoff',0.1)

Assessing the quality of a hierarchical cluster analysis

• The cophenetic correlation coefficient measures how closely the length of the tree links match the original distances between points

• How ‘faithful’ the tree is to the original data

• 0 is poor, 1 is good

>> cophenet(tree,distances)

K-Means Cluster Analysis – what is it doing?

Randomly pick K cluster

centroids

Assign points to the

closest centroid

Recalculate positions of

cluster centroids

Reassign points to the

closest centroid

Recalculate positions of

cluster centroids

Repeat until centroid positions converge

………

K-Means Cluster Analysis – how do I do it ?

Running the K-mean algorithm for K fixed>> [memberships,centroids] = kmeans(x,K);

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Evaluating a K-Means analysis and choosing K

• Try a range of different K’s, and compare the point-centroid distances for each

>> for K=3:15

[clusters,centroids,distances] = kmeans(data,K);

totaldist(K-2)=sum(distances);

end

plot(3:15,totaldist);

• Create silhouette plots

>> silhouette(data,clusters)

Sidebar: Distance Metrics

• Measures of how similar datapoints are – different definitions make sense for different data

• Many built-in distance metrics, or define your own

>> doc pdist

>> distances = pdist(data,metric); %pdist = pairwise distances

>> squareform(distances)

>> kmeans(data,k,’distance’,’cityblock’) %not all metrics supported

Euclidean Distance

Default

Cityblock Distance

Useful for discrete variables

Cosine Distance

Useful for clustering variables

Fuzzy c-means Cluster Analysis – what is it doing?• Very similar to K-means

• Samples are not assigned definitively to a cluster, but have a ‘membership’ value relative to each cluster

Requires Fuzzy Logic Toolbox™

Running the fuzzy K-mean algorithm

for K fixed>> [centroids, memberships]=fcm(x,K);

Gaussian Mixture Models• Assume that data is drawn from a fixed number K of normal

distributions

• Fit these parameters using the EM algorithm

>> gmobj = gmdistribution.fit(x,8);

>> assignments = cluster(gmobj,x);

Plot the probability density>> ezsurf(@(x,y)pdf(gmobj,[x y]));

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Evaluating a Gaussian Mixture Model clustering

• Plot the probability density function of the model

>> ezsurf(@(x,y)pdf(gmobj,[x y]));

• Plot the posterior probabilities of observations

>> p = posterior(gmobj,data);

>> scatter(data(:,1),data(:,2),5,p(:,g)); % Do this for each group g

• Plot the Mahalanobis distances of observations to components

>> m = mahal(gmobj,data);

>> scatter(data(:,1),data(:,2),5,m(:,g)); % Do this for each group g

Choosing the right number of components in a Gaussian Mixture Model

• Evaluate for a range of K and plot AIC and/or BIC

• AIC (Akaike Information Criterion) and BIC (Bayesian Information Criterion) are measures of the quality of the model fit, with a penalty for higher K

>> for K=3:15

gmobj = gmdistribution.fit(data,K);

AIC(K-2) = gmobj.AIC;

end

plot(3:15,AIC);

Neural Networks – what are they?Input

variables

WeightsBias

Transfer

function

Output

variable

A two layer

feedforward

network

Build your

architecture

Self Organising Maps Neural Net – what are they?

• Start with a regular grid of ‘neurons’ laid over the dataset

• The size of the grid gives the number of clusters

• Neurons compete to recognise datapoints (by being close to them)

• Winning neurons are moved closer to the datapoints

• Repeat until convergence

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Summary: Cluster analysisNo method owns the truth

Use the diagnostic tools to assess your clusters

Beware of local minima : global optimization

Classification

Why classification ? Can be used to:

o Learning the way to classify from already classified observations

oClassify new observations

• Means:• Discriminant analysis classification

• Bootstrapped aggregated decision tree classifier

• Neural network classifier

• Support vector machine classifier

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Discriminant Analysis – how does it work?

• Fit a multivariate normal density to each class• linear — Fits a multivariate normal density to each group,

with a pooled estimate of covariance. This is the default.• diaglinear — Similar to linear, but with a diagonal

covariance matrix estimate (naive Bayes classifiers).• quadratic — Fits multivariate normal densities with

covariance estimates stratified by group.• diagquadratic — Similar to quadratic, but with a diagonal

covariance matrix estimate (naive Bayes classifiers).

• Classify a new point by evaluating its probability for each density function, and classifying to the highest probability

Discriminant Analysis – how do I do it?

• Linear Discriminant Analysis>> classes = classify(sample,training,group)

• Quadratic Discriminant Analysis>> classes = classify(x,x,y,’quadratic’)

• Naïve Bayes>> nbGau= NaiveBayes.fit(x, y);

>> y_pred= nbGau.predict(x);

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Interpreting Discriminant Analyses

• Visualise the posterior probability surfaces

>> [XI,YI] = meshgrid(linspace(4,8), linspace(2,4.5));

>> X = XI(:); Y = YI(:);

>> [class,err,P] = classify([X Y], meas(:,1:2), species,'quadratic');

>> for i=1:3

ZI = reshape(P(:,i),100,100);

surf(XI,YI,ZI,'EdgeColor','none');

hold on;

end

Interpreting Discriminant Analyses

• Visualise the probability density of sample observations

• An indicator of the region in which the model has support from training data

>> [XI,YI] = meshgrid(linspace(4,8), linspace(2,4.5));

>> X = XI(:); Y = YI(:);

>> [class,err,P,logp] = classify([X Y], meas(:,1:2), species, 'quadratic');

>> ZI = reshape(logp,100,100);

>> surf(XI,YI,ZI,'EdgeColor','none');

Classifying K-Nearest Neigbours – what does it do?

• One of the simplest classifiers – a sample is classified by taking the K nearest points from the training set, and choosing the majority class of those K points

• There is no real training phase – all the work is done during the application of the model

>> classes =

knnclassify(sample,training,group,K)

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Decision Trees – how do they work?• Threshold value for a variable

that partitions the dataset

• Threshold for all predictors

• Resulting model is a tree where each node is a logical test on a predictor (var1<thresh1, var2>thresh2)

Decision Trees – how do I build them ?

• Build tree model>> tree = classregtree(x,y);

>> view(tree)

• Evaluate the model on new data>> tree(x_new)

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Enhancing the model : bagged trees• Prune the decision tree>> [cost,secost,ntnodes,bestlevel] =test(t, 'test', x, y);

>> topt = prune(t, 'level', bestlevel);

• Bootstrapped aggregated trees forest>> [cost,secost,ntnodes,bestlevel] =test(t, 'test', x, y);

>> forest = TreeBagger(100, x, y);

>> y_pred = predict(forest,x);

• Visualise class boundaries as before

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Pattern Recognition Neural Network– what are they?

• Two-layer (i.e. one-hidden-layer) feed forward neural networks can learn any input-output relationship given enough neurons in the hidden layer.

• No restrictions on the predictors

Pattern Recognition Neural Network– how do I build them ?

• Build a neural network model>> net = patternnet(10);

• Train the net to classify observations

>> [net,tr] = train(net,x,y);

• Apply the model to new data>> y_pred = net(x);

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Support Vector Machines – what are they?

• The SVM algorithm finds a boundary between the classes that maximises the minimum distance of the boundary to any of the points

• No restrictions on the predictors

• 1 vs all to classify multiple classes

Support Vector Machines – how do I build them ?

• Build an SVM model>> svmmodel = svmtrain(x,y)

• Try different kernel functions>> svmmodel =

svmtrain(x,y,’kernel_function’,’rbf’)

• Apply the model to new data>> classes =

svmclassify(svmmodel,x_new);

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Evaluating a Classifying Model• Three main strategies

• Resubstitution – test the model on the same data that you trained it with

• Cross-Validation

• Holdout Test on a completely new dataset

• Use cross-validation to evaluate model parameters such as the number of leaf for a tree or the number of hidden neurons.

Apply cross validation to your classifying model>> cp = cvpartition(y,'k',10);

>> ldaFun= @(xtrain,ytrain,xtest)(classify(xtest,xtrain,ytrain));

>> ldaCVErr = crossval('mcr',x,y,'predfun',ldaFun,'partition',cp)

Summary: Classification algorithms

No absolute best methods

Simple does not mean inefficient

Decision trees produce models and neural network overfit the noise : use bootstrapping and cross-validation

Parallelize

RegressionWhy Regression ? Can be used to:

o Learn to model a continuous response from observationsoPredict the response for new observations

• Means:

• Linear regressions• Non-linear regressions• Bootstrapped regression tree• Neural network as a fitting tool

New data set with a continuous response from one predictor

• Non-linear function to fit

• A continuous response to fit from one continuous predictor

>>[x,t] = simplefit_dataset;

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Linear Regression – what is it?

• A collection of methods that find the best coefficients b such that y ≈ X*b

• Best b means minimising the least squares difference between the predicted and actual values of y

• “Linear” means linear in b –you can include extra variables to give a nonlinear relationship in X

Linear Regression – how do I do it ?

>> b = x\y

• Linear Regression>> b = regress(y, [ones(size(X,1),1) x])

>> stats = regstats(y, [ones(size(x,1),1) x])

• Robust Regression – better in the presence of outliers>> robust_b = robustfit(X,y) %NB (X,y) not (y,X)

• Ridge Regression – better if data is close to collinear>> ridge_b = ridge(y,X,k) %k is the ridge parameter

• Apply the model to new data>> y = newdata*b;

Interpreting a linear regression model• Examine coefficients to see

which predictors have a large effect on the response

>> [b,bint,r,rint,stats]=regress(y,X)

>> errorbar(1:size(b,1),b, b-bint(:,1),bint(:,2)-b)

• Examine residuals to check for possible outliers

>> rcoplot(r,rint)

• Examine R2 statistic and p-value to check overall model significance

>> stats(1)*100 %R2 as a percentage

>> stats(3) %p-value

• Additional diagnostics with regstats

Non linear curve fitting

Least square algorithm

>> model = @(b,x)(b(1)+b(2).*cos(b(3)*x+b(4))+b(5).*cos(b(6)*x+b(7))+b(8).*cos(b(9)*x+b(10)));

>> [ahat,r,J,cov,mse] = nlinfit(x,t,model,a0);

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Fit Neural Network– what are they?• Fitting networks are feedforward neural networks used to fit

an input-output relationship.

• This architecture can learn any input-output relationship given enough neurons.

• No restrictions on the predictors (categorical,ordinal,discontinuous)

Fit Neural Network– how do I build them ?

• Build a fit neural net model>> net = fitnet(10);

• Train the net to fit the target>> [net,tr] = train(net,x,t);

• Apply the model to new data>> y_pred = net(x);

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Regression trees– what are they?

• A decision tree with binary splits for regression. An object of class RegressionTree can predict responses for new data with the predict method.

• No restrictions on the predictors (categorical,ordinal,discontinuous)

Regression trees – how do I use them?

• Build a fit neural net model>> rtree = RegressionTree.fit(x,t);

• Train the net to fit the target>> y_tree = predict(rtree,x);

• Apply the model to new data>> y_pred = net(x);

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Summary

Data Mining

Exploration

UnivariatePie chart,

Histogram, etc…

MultivariateFeature

selection and transformation

Modelling

Clustering

Partitive

K-means

Gaussianmixture model

SOMHierarchical

Classification

Discriminant

Decision Tree

Neural Network

Support VectorMachine

Regression