introduction to big data science

big data science

Albert Bifet (@abifet)Paris, 7 October 2015albert.bifet@telecom-paristech.fr

data science

Data Science is an interdisciplinary field focused onextracting knowledge or insights from large

volumes of data.

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data scientist

Figure 1: http://www.marketingdistillery.com/2014/11/29/is-data-science-a-buzzword-modern-data-scientist-defined/

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data science

Figure 2: Drew Convay’s Venn diagram

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classification

classification

DefinitionGiven nC different classes, a classifier algorithm builds amodel that predicts for every unlabelled instance I the class Cto which it belongs with accuracy.

ExampleA spam filter

ExampleTwitter Sentiment analysis: analyze tweets with positive ornegative feelings

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classification

Data set thatdescribes e-mailfeatures fordeciding if it isspam.

ExampleContains Domain Has Time“Money” type attach. received spam

yes com yes night yesyes edu no night yesno com yes night yesno edu no day nono com no day noyes cat no day yes

Assume we have to classify the following new instance:Contains Domain Has Time“Money” type attach. received spam

yes edu yes day ?6

k-nearest neighbours

k-NN Classifier

• Training: store all instances in memory• Prediction:

• Find the k nearest instances• Output majority class of these k instances

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bayes classifiers

Naïve Bayes

• Based on Bayes Theorem:

P(c|d) = P(c)P(d|c)P(d)

posterior=prior× likelikood

evidence• Estimates the probability of observing attribute a and theprior probability P(c)

• Probability of class c given an instance d:

P(c|d) = P(c)∏a∈dP(a|c)P(d)

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bayes classifiers

Multinomial Naïve Bayes

• Considers a document as a bag-of-words.• Estimates the probability of observing word w and the priorprobability P(c)

• Probability of class c given a test document d:

P(c|d) = P(c)∏w∈dP(w|c)nwdP(d)

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perceptron

Attribute 1

Attribute 2

Attribute 3

Attribute 4

Attribute 5

Output hw⃗(⃗xi)

w1

w2

w3

w4

w5

• Data stream: ⟨⃗xi,yi⟩• Classical perceptron: hw⃗(⃗xi) = sgn(⃗wT⃗xi),• Minimize Mean-square error: J(⃗w) = 1

2 ∑(yi−hw⃗(⃗xi))2

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perceptron

Attribute 1

Attribute 2

Attribute 3

Attribute 4

Attribute 5

Output hw⃗(⃗xi)

w1

w2

w3

w4

w5

• We use sigmoid function hw⃗ = σ (⃗wT⃗x) where

σ(x) = 1/(1+e−x)

σ ′(x) = σ(x)(1−σ(x))10

perceptron

• Minimize Mean-square error: J(⃗w) = 12 ∑(yi−hw⃗(⃗xi))2

• Stochastic Gradient Descent: w⃗= w⃗−η∇J⃗xi• Gradient of the error function:

∇J=−∑i(yi−hw⃗(⃗xi))∇hw⃗(⃗xi)

∇hw⃗(⃗xi) = hw⃗(⃗xi)(1−hw⃗(⃗xi))

• Weight update rule

w⃗= w⃗+η ∑i(yi−hw⃗(⃗xi))hw⃗(⃗xi)(1−hw⃗(⃗xi))⃗xi

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restricted boltzmann machines (rbms)

z4z3z2z1

x5x4x3x2x1

• Energy-based models, where

P(⃗x,⃗z) ∝ e−E(⃗x,⃗z).

• Manipulate a weight matrix W to find low-energy states andthus generate high probability P(⃗x,⃗z), where

E(⃗x,⃗z) =−W.

• RBMs can be stacked on top of each other to form so-calledDeep Belief Networks (DBNs)

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classification

Data set thatdescribes e-mailfeatures fordeciding if it isspam.

ExampleContains Domain Has Time“Money” type attach. received spam

yes com yes night yesyes edu no night yesno com yes night yesno edu no day nono com no day noyes cat no day yes

Assume we have to classify the following new instance:Contains Domain Has Time“Money” type attach. received spam

yes edu yes day ?12

classification

• Assume we have to classify the following new instance:Contains Domain Has Time“Money” type attach. received spam

yes edu yes day ?

Time

Contains “Money”

YES

Yes

NO

No

Day

YES

Night

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decision trees

Basic induction strategy:

• A← the “best” decision attribute for next node• Assign A as decision attribute for node• For each value of A, create new descendant of node• Sort training examples to leaf nodes• If training examples perfectly classified, Then STOP, Elseiterate over new leaf nodes

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bagging

ExampleDataset of 4 Instances : A, B, C, D

Classifier 1: B, A, C, BClassifier 2: D, B, A, DClassifier 3: B, A, C, BClassifier 4: B, C, B, BClassifier 5: D, C, A, C

Bagging builds a set of M base models, with a bootstrapsample created by drawing random samples withreplacement.

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random forests

• Bagging• Random Trees: trees that in each node only uses a randomsubset of the attributes

Random Forests is one of the most popular methods inmachine learning.

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boosting

The strength of Weak Learnability, Schapire 90

A boosting algorithm transforms a weak learnerinto a strong one

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boosting

A formal description of Boosting (Schapire)

• given a training set (x1,y1), . . . ,(xm,ym)• yi ∈ {−1,+1} correct label of instance xi ∈ X• for t= 1, . . . ,T

• construct distribution Dt• find weak classifier

ht : X→{−1,+1}

with small error εt = PrDt [ht(xi) ̸= yi] on Dt

• output final classifier

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boosting

AdaBoost1: Initialize D1(i) = 1/m for all i ∈ {1,2, ...,m}2: for t = 1,2,...T do3: CallWeakLearn, providing it with distribution Dt4: Get back hypothesis ht : X→ Y5: Calculate error of ht : εt = ∑i:ht(xi )̸=yi Dt(i)6: Update distribution

Dt : Dt+1(i) =Dt(i)Zt ×

{εt/(1− εt) if ht(xi) = yi1 otherwise

where Zt is a normalization constant (chosen so Dt+1 is aprobability distribution)

7: return hfin(x) = argmaxy∈Y ∑t:ht(x)=y− logεt/(1− εt)

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boosting

AdaBoost1: Initialize D1(i) = 1/m for all i ∈ {1,2, ...,m}2: for t = 1,2,...T do3: CallWeakLearn, providing it with distribution Dt4: Get back hypothesis ht : X→ Y5: Calculate error of ht : εt = ∑i:ht(xi )̸=yi Dt(i)6: Update distribution

Dt : Dt+1(i) =Dt(i)Zt ×

{εt if ht(xi) = yi1− εt otherwise

where Zt is a normalization constant (chosen so Dt+1 is aprobability distribution)

7: return hfin(x) = argmaxy∈Y ∑t:ht(x)=y− logεt/(1− εt)

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stacking

Use a classifier to combine predictions of base classifiers

Example

• Use a perceptron to do stacking• Use decision trees as base classifiers

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clustering

clustering

DefinitionClustering is the distribution of a set of instances of examplesinto non-known groups according to some common relationsor affinities.

ExampleMarket segmentation of customers

ExampleSocial network communities

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clustering

DefinitionGiven

• a set of instances I• a number of clusters K• an objective function cost(I)

a clustering algorithm computes an assignment of a clusterfor each instance

f : I→{1, . . . ,K}

that minimizes the objective function cost(I)

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clustering

DefinitionGiven

• a set of instances I• a number of clusters K• an objective function cost(C, I)

a clustering algorithm computes a set C of instances with|C|= K that minimizes the objective function

cost(C, I) = ∑x∈I

d2(x,C)

where

• d(x,c): distance function between x and c• d2(x,C) =minc∈Cd2(x,c): distance from x to the nearest pointin C

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k-means

• 1. Choose k initial centers C= {c1, . . . ,ck}• 2. while stopping criterion has not been met

• For i= 1, . . . ,N• find closest center ck ∈ C to each instance pi• assign instance pi to cluster Ck

• For k= 1, . . . ,K• set ck to be the center of mass of all points in Ci

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k-means++

• 1. Choose a initial center c1• For k= 2, . . . ,K

• select ck = p ∈ I with probability d2(p,C)/cost(C, I)• 2. while stopping criterion has not been met

• For i= 1, . . . ,N• find closest center ck ∈ C to each instance pi• assign instance pi to cluster Ck

• For k= 1, . . . ,K• set ck to be the center of mass of all points in Ci

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performance measures

Internal Measures

• Sum square distance• Dunn index D= dmin

dmax

• C-Index C= S−SminSmax−Smin

External Measures

• Rand Measure• F Measure• Jaccard• Purity

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density based methods

DBSCAN

• ε -neighborhood(p): set of points that are at a distance of pless or equal to ε

• Core object: object whose ε -neighborhood has an overallweight at least µ

• A point p is directly density-reachable from q if• p is in ε -neighborhood(q)• q is a core object

• A point p is density-reachable from q if• there is a chain of points p1, . . . ,pn such that pi+1 is directlydensity-reachable from pi

• A point p is density-connected from q if• there is point o such that p and q are density-reachable from o

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density based methods

DBSCAN

• A cluster C of points satisfies• if p ∈ C and q is density-reachable from p, then q ∈ C• all points p,q ∈ C are density-connected

• A cluster is uniquely determined by any of its core points• A cluster can be obtained

• choosing an arbitrary core point as a seed• retrieve all points that are density-reachable from the seed

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dbscan

Figure 3: DBSCAN Point Example with µ=3

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density based methods

DBSCAN

• select an arbitrary point p• retrieve all points density-reachable from p• if p is a core point, a cluster is formed• If p is a border point

• no points are density-reachable from p• DBSCAN visits the next point of the database

• Continue the process until all of the points have beenprocessed

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frequent pattern mining

frequent patterns

Suppose D is a dataset of patterns, t ∈D , and min_sup is aconstant.

DefinitionSupport (t): number ofpatterns in D that aresuperpatterns of t.

DefinitionPattern t is frequent ifSupport (t)≥ min_sup.

Frequent Subpattern ProblemGiven D and min_sup, find all frequent subpatterns of patternsin D .

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frequent patterns

Suppose D is a dataset of patterns, t ∈D , and min_sup is aconstant.

DefinitionSupport (t): number ofpatterns in D that aresuperpatterns of t.

DefinitionPattern t is frequent ifSupport (t)≥ min_sup.

Frequent Subpattern ProblemGiven D and min_sup, find all frequent subpatterns of patternsin D .

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frequent patterns

Suppose D is a dataset of patterns, t ∈D , and min_sup is aconstant.

DefinitionSupport (t): number ofpatterns in D that aresuperpatterns of t.

DefinitionPattern t is frequent ifSupport (t)≥ min_sup.

Frequent Subpattern ProblemGiven D and min_sup, find all frequent subpatterns of patternsin D .

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frequent patterns

Suppose D is a dataset of patterns, t ∈D , and min_sup is aconstant.

DefinitionSupport (t): number ofpatterns in D that aresuperpatterns of t.

DefinitionPattern t is frequent ifSupport (t)≥ min_sup.

Frequent Subpattern ProblemGiven D and min_sup, find all frequent subpatterns of patternsin D .

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pattern mining

Dataset ExampleDocument Patterns

d1 abced2 cded3 abced4 acded5 abcded6 bcd

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itemset mining

d1 abced2 cded3 abced4 acded5 abcded6 bcd

Support Frequentd1,d2,d3,d4,d5,d6 cd1,d2,d3,d4,d5 e,ced1,d3,d4,d5 a,ac,ae,aced1,d3,d5,d6 b,bcd2,d4,d5,d6 d,cdd1,d3,d5 ab,abc,abe

be,bce,abced2,d4,d5 de,cde

minimal support = 3

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itemset mining

d1 abced2 cded3 abced4 acded5 abcded6 bcd

Support Frequent6 c5 e,ce4 a,ac,ae,ace4 b,bc4 d,cd3 ab,abc,abe

be,bce,abce3 de,cde

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itemset mining

d1 abced2 cded3 abced4 acded5 abcded6 bcd

Support Frequent Gen Closed6 c c c5 e,ce e ce4 a,ac,ae,ace a ace4 b,bc b bc4 d,cd d cd3 ab,abc,abe ab

be,bce,abce be abce3 de,cde de cde

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itemset mining

d1 abced2 cded3 abced4 acded5 abcded6 bcd

Support Frequent Gen Closed Max6 c c c5 e,ce e ce4 a,ac,ae,ace a ace4 b,bc b bc4 d,cd d cd3 ab,abc,abe ab

be,bce,abce be abce abce3 de,cde de cde cde

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itemset mining

d1 abced2 cded3 abced4 acded5 abcded6 bcd

Support Frequent Gen Closed Max6 c c c5 e,ce e ce4 a,ac,ae,ace a ace4 b,bc b bc4 d,cd d cd3 ab,abc,abe ab

be,bce,abce be abce abce3 de,cde de cde cde

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itemset mining

d1 abced2 cded3 abced4 acded5 abcded6 bcd

e→ ce

Support Frequent Gen Closed Max6 c c c5 e,ce e ce4 a,ac,ae,ace a ace4 b,bc b bc4 d,cd d cd3 ab,abc,abe ab

be,bce,abce be abce abce3 de,cde de cde cde

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itemset mining

d1 abced2 cded3 abced4 acded5 abcded6 bcd

Support Frequent Gen Closed Max6 c c c5 e,ce e ce4 a,ac,ae,ace a ace4 b,bc b bc4 d,cd d cd3 ab,abc,abe ab

be,bce,abce be abce abce3 de,cde de cde cde

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itemset mining

d1 abced2 cded3 abced4 acded5 abcded6 bcd

Support Frequent Gen Closed Max6 c c c5 e,ce e ce4 a,ac,ae,ace a ace4 b,bc b bc4 d,cd d cd3 ab,abc,abe ab

be,bce,abce be abce abce3 de,cde de cde cde

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itemset mining

d1 abced2 cded3 abced4 acded5 abcded6 bcd

a→ ace

Support Frequent Gen Closed Max6 c c c5 e,ce e ce4 a,ac,ae,ace a ace4 b,bc b bc4 d,cd d cd3 ab,abc,abe ab

be,bce,abce be abce abce3 de,cde de cde cde

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itemset mining

d1 abced2 cded3 abced4 acded5 abcded6 bcd

Support Frequent Gen Closed Max6 c c c5 e,ce e ce4 a,ac,ae,ace a ace4 b,bc b bc4 d,cd d cd3 ab,abc,abe ab

be,bce,abce be abce abce3 de,cde de cde cde

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closed patterns

Usually, there are too many frequent patterns. We cancompute a smaller set, while keeping the same information.

ExampleA set of 1000 items, has 21000 ≈ 10301 subsets, that is morethan the number of atoms in the universe ≈ 1079

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closed patterns

A priori propertyIf t′ is a subpattern of t, then Support (t′)≥ Support (t).

DefinitionA frequent pattern t is closed if none of its propersuperpatterns has the same support as it has.

Frequent subpatterns and their supports can be generatedfrom closed patterns.

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maximal patterns

DefinitionA frequent pattern t is maximal if none of its propersuperpatterns is frequent.

Frequent subpatterns can be generated from maximalpatterns, but not with their support.

All maximal patterns are closed, but not all closed patterns aremaximal.

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non streaming frequent itemset miners

Representation:

• Horizontal layoutT1: a, b, cT2: b, c, eT3: b, d, e

• Vertical layouta: 1 0 0b: 1 1 1c: 1 1 0

Search:

• Breadth-first (levelwise): Apriori• Depth-first: Eclat, FP-Growth 41

the apriori algorithm

Apriori Algorithm1 Initialize the item set size k= 12 Start with single element sets3 Prune the non-frequent ones4 while there are frequent item sets5 do create candidates with one item more6 Prune the non-frequent ones7 Increment the item set size k= k+1

8 Output: the frequent item sets

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the eclat algorithm

Depth-First Search

• divide-and-conquer scheme : the problem is processed bysplitting it into smaller subproblems, which are thenprocessed recursively• conditional database for the prefix a• transactions that contain a

• conditional database for item sets without a• transactions that not contain a

• Vertical representation• Support counting is done by intersecting lists of transactionidentifiers

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the fp-growth algorithm

Depth-First Search

• divide-and-conquer scheme : the problem is processed bysplitting it into smaller subproblems, which are thenprocessed recursively• conditional database for the prefix a• transactions that contain a

• conditional database for item sets without a• transactions that not contain a

• Vertical and Horizontal representation : FP-Tree• prefix tree with links between nodes that correspond to thesame item

• Support counting is done using FP-Tree

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mining graph data

ProblemGiven a data set D of graphs, find frequent graphs.

Transaction Id Graph

1C C S N

O

O

2C C S N

O

C

3 C C S NN

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the gspan algorithm

gSpan(g,D,min_sup,S)Input: A graph g, a graph dataset D, min_sup.Output: The frequent graph set S.

1 if g ̸=min(g)2 then return S3 insert g into S4 update support counter structure5 C← /06 for each g′ that can be right-most

extended from g in one step7 do if support(g) ≥min_sup8 then insert g′ into C9 for each g′ in C

10 do S← gSpan(g′,D,min_sup,S)11 return S

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