internet of things data science

Albert Bifet@abifetParis, 7 October 2015albert.bifet@telecom-paristech.fr

internet of things data science architecture

real time analytics

introduction: data streams

Data Streams

• Sequence is potentially infinite• High amount of data: sublinear space• High speed of arrival: sublinear time per example• Once an element from a data stream has been processed itis discarded or archived

ExamplePuzzle: Finding Missing Numbers• Let π be a permutation of {1, . . . ,n}.• Let π−1 be π with one element missing.• π−1[i] arrives in increasing orderTask: Determine the missing number

Data Streams

Use a n-bitvector tomemorize all thenumbers (O(n)space)

Data Streams

Data Streams:O(log(n)) space.

Data Streams

Data Streams:O(log(n)) space.Store

n(n+1)2

−∑j≤i

π−1[j].

data streams

Data Streams

Tools:

• approximation• randomization, sampling• sketching

data streams

Data Streams

Approximation algorithms

• Small error rate with high probability• An algorithm (ε,δ )−approximates F if it outputs F̃ for whichPr[|F̃−F|> εF]< δ .

data streams approximation algorithms

1011000111 1010101

Sliding WindowWe can maintain simple statistics over sliding windows, usingO(1ε log

2N) space, where

• N is the length of the sliding window• ε is the accuracy parameter

M. Datar, A. Gionis, P. Indyk, and R. Motwani.Maintaining stream statistics over sliding windows. 2002

10110001111 0101011

2N) space, where

101100011110 1010111

2N) space, where

1011000111101 0101110

2N) space, where

10110001111010 1011101

2N) space, where

101100011110101 0111010

2N) space, where

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

data stream classification cycle

1 Process an example at a time,and inspect it only once (atmost)

2 Use a limited amount of memory3 Work in a limited amount of time4 Be ready to predict at any point

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 ?

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)

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)

perceptron

Attribute 1

Attribute 2

Attribute 3

Attribute 4

Attribute 5

Output hw⃗(⃗xi)

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

2 ∑(yi−hw⃗(⃗xi))2

perceptron

Attribute 1

Attribute 2

Attribute 3

Attribute 4

Attribute 5

Output hw⃗(⃗xi)

• We use sigmoid function hw⃗ = σ (⃗wT⃗x) whereσ(x) = 1/(1+e−x)

σ ′(x) = σ(x)(1−σ(x))13

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

perceptron

Perceptron Learning(Stream,η)

1 for each class2 do Perceptron Learning(Stream,class,η)

Perceptron Learning(Stream,class,η)

1 � Let w0 and w⃗ be randomly initialized2 for each example (⃗x,y) in Stream3 do if class= y4 then δ = (1−hw⃗(⃗x)) ·hw⃗(⃗x) · (1−hw⃗(⃗x))5 else δ = (0−hw⃗(⃗x)) ·hw⃗(⃗x) · (1−hw⃗(⃗x))6 w⃗= w⃗+η ·δ ·⃗x

Perceptron Prediction(⃗x)1 return argmaxclasshw⃗class

(⃗x)14

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 ?

classification

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

yes edu yes day ?

Contains “Money”

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

hoeffding trees

Hoeffding Tree : VFDT

Pedro Domingos and Geoff Hulten.Mining high-speed data streams. 2000

• With high probability, constructs an identical model that atraditional (greedy) method would learn

• With theoretical guarantees on the error rate

YESYes

YESNight

hoeffding bound inequality

Probability of deviation of its expected value.

hoeffding bound inequality

Let X= ∑iXi where X1, . . . ,Xn are independent and indenticallydistributed in [0,1]. Then

1 Chernoff For each ε < 1

Pr[X> (1+ ε)E[X]]≤ exp(−ε2

)2 Hoeffding For each t> 0

Pr[X> E[X]+ t]≤ exp(−2t2/n

)3 Bernstein Let σ2 = ∑i σ2

i the variance of X. If Xi−E[Xi]≤ b foreach i ∈ [n] then for each t> 0

Pr[X> E[X]+ t]≤ exp

(− t2

2σ2+ 23bt

hoeffding tree or vfdt

HT(Stream,δ )1 � Let HT be a tree with a single leaf(root)2 � Init counts nijk at root3 for each example (x,y) in Stream4 do HTGrow((x,y),HT,δ )

HTGrow((x,y),HT,δ )1 � Sort (x,y) to leaf l using HT2 � Update counts nijk at leaf l3 if examples seen so far at l are not all of the same class4 then� Compute G for each attribute

5 if G(Best Attr.)−G(2nd best) >√

R2 ln1/δ2n

6 then� Split leaf on best attribute7 for each branch8 do� Start new leaf and initiliatize counts

hoeffding tree or vfdt

HT(Stream,δ )1 � Let HT be a tree with a single leaf(root)2 � Init counts nijk at root3 for each example (x,y) in Stream4 do HTGrow((x,y),HT,δ )

HTGrow((x,y),HT,δ )1 � Sort (x,y) to leaf l using HT2 � Update counts nijk at leaf l3 if examples seen so far at l are not all of the same class4 then� Compute G for each attribute

5 if G(Best Attr.)−G(2nd best) >√

R2 ln1/δ2n

6 then� Split leaf on best attribute7 for each branch8 do� Start new leaf and initiliatize counts

hoeffding trees

HT features

• With high probability, constructs an identical model that atraditional (greedy) method would learn

• Ties: when two attributes have similar G, split if

G(Best Attr.)−G(2nd best)<√

R2 ln1/δ2n

• Compute G every nmin instances• Memory: deactivate least promising nodes with lower pl×el

• pl is the probability to reach leaf l• el is the error in the node

hoeffding naive bayes tree

Hoeffding TreeMajority Class learner at leaves

Hoeffding Naive Bayes Tree

G. Holmes, R. Kirkby, and B. Pfahringer.Stress-testing Hoeffding trees, 2005.

• monitors accuracy of a Majority Class learner• monitors accuracy of a Naive Bayes learner• predicts using the most accurate method

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.

bagging

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

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

bagging

Classifier 1: A, B, B, C: A(1) B(2) C(1) D(0)Classifier 2: A, B, D, D: A(1) B(1) C(0) D(2)Classifier 3: A, B, B, C: A(1) B(2) C(1) D(0)Classifier 4: B, B, B, C: A(0) B(3) C(1) D(0)Classifier 5: A, C, C, D: A(1) B(0) C(2) D(1)

Each base model’s training set contains each of the originaltraining example K times where P(K= k) follows a binomialdistribution.

bagging

Figure 1: Poisson(1) Distribution.

Each base model’s training set contains each of the originaltraining example K times where P(K= k) follows a binomialdistribution.

oza and russell’s online bagging for m models

1: Initialize base models hm for all m ∈ {1,2, ...,M}2: for all training examples do3: for m= 1,2, ...,M do4: Set w= Poisson(1)5: Update hm with the current example with weight w

6: anytime output:7: return hypothesis: hfin(x) = argmaxy∈Y ∑T

t=1 I(ht(x) = y)

Evolving Stream Classification

data mining algorithms with concept drift

No Concept Drift

-input output

DM Algorithm

Counter1

Counter2

Counter3

Counter4

Counter5

Concept Drift

-input output

DM Algorithm

Static Model

Change Detect.-

data mining algorithms with concept drift

No Concept Drift

-input output

DM Algorithm

Counter1

Counter2

Counter3

Counter4

Counter5

Concept Drift

-input output

DM Algorithm

Estimator1

Estimator2

Estimator3

Estimator4

Estimator5

optimal change detector and predictor

• High accuracy• Low false positives and false negatives ratios• Theoretical guarantees

• Fast detection of change• Low computational cost: minimum space and time needed

• No parameters needed

algorithm adaptive sliding window

ExampleW= 101010110111111

ADWIN: Adaptive Windowing Algorithm1 Initialize Window W2 for each t> 03 do W←W∪{xt} (i.e., add xt to the head of W)4 repeat Drop elements from the tail of W5 until |µ̂W0− µ̂W1 | ≥ εc holds6 for every split of W into W=W0 ·W17 Output µ̂W

ExampleW= 101010110111111

W0= 1 W1 = 01010110111111

ExampleW= 101010110111111

W0= 10 W1 = 1010110111111

ExampleW= 101010110111111

W0= 101 W1 = 010110111111

ExampleW= 101010110111111

W0= 1010 W1 = 10110111111

ExampleW= 101010110111111

W0= 10101 W1 = 0110111111

ExampleW= 101010110111111

W0= 101010 W1 = 110111111

ExampleW= 101010110111111

W0= 1010101 W1 = 10111111

ExampleW= 101010110111111

W0= 10101011 W1 = 0111111

ExampleW= 101010110111111 |µ̂W0− µ̂W1 | ≥ εc : CHANGE DET.!

W0= 101010110 W1 = 111111

ExampleW= 101010110111111 Drop elements from the tail of W

W0= 101010110 W1 = 111111

ExampleW= 01010110111111 Drop elements from the tail of W

W0= 101010110 W1 = 111111

TheoremAt every time step we have:

1 (False positive rate bound). If µt remains constant within W,the probability that ADWIN shrinks the window at this step is atmost δ .

2 (False negative rate bound). Suppose that for some partitionof W in two parts W0W1 (where W1 contains the most recentitems) we have |µW0−µW1 |> 2εc. Then with probability 1−δADWIN shrinks W to W1, or shorter.

ADWIN tunes itself to the data stream at hand, with no need forthe user to hardwire or precompute parameters.

ADWIN using a Data Stream Sliding Window Model,

• can provide the exact counts of 1’s in O(1) time per point.• tries O(logW) cutpoints• uses O(1ε logW)memory words• the processing time per example is O(logW) (amortized andworst-case).

Sliding Window Model

1010101 101 11 1 1

Content: 4 2 2 1 1

Capacity: 7 3 2 1 1

vfdt / cvfdt

Concept-adapting Very Fast Decision Trees: CVFDT

G. Hulten, L. Spencer, and P. Domingos.Mining time-changing data streams. 2001

• It keeps its model consistent with a sliding window ofexamples

• Construct “alternative branches” as preparation for changes• If the alternative branch becomes more accurate, switch oftree branches occurs

YESYes

YESNight

decision trees: cvfdt

YESYes

YESNight

No theoretical guarantees on the error rate of CVFDT

CVFDT parameters :

1 W: is the example window size.2 T0: number of examples used to check at each node if thesplitting attribute is still the best.

3 T1: number of examples used to build the alternate tree.4 T2: number of examples used to test the accuracy of thealternate tree.

decision trees: hoeffding adaptive tree

Hoeffding Adaptive Tree:

• replace frequency statistics counters by estimators• don’t need a window to store examples, due to the fact that wemaintain the statistics data needed with estimators

• change the way of checking the substitution of alternatesubtrees, using a change detector with theoreticalguarantees

Advantages over CVFDT:

1 Theoretical guarantees2 No Parameters

adwin bagging (kdd’09)

ADWINAn adaptive sliding window whose size is recomputed onlineaccording to the rate of change observed.

ADWIN has rigorous guarantees (theorems)

• On ratio of false positives and negatives• On the relation of the size of the current window and changerates

ADWIN BaggingWhen a change is detected, the worst classifier is removedand a new classifier is added.

Randomization as a powerful tool to increase accuracy anddiversity

There are three ways of using randomization:

• Manipulating the input data• Manipulating the classifier algorithms• Manipulating the output targets

leveraging bagging for evolving data streams

Leveraging Bagging

• Using Poisson(λ )

Leveraging Bagging MC

• Using Poisson(λ ) and Random Output Codes

Fast Leveraging Bagging ME

• if an instance is misclassified: weight = 1• if not: weight = eT/(1−eT),

empirical evaluation

Accuracy RAM-HoursHoeffding Tree 74.03 0.01Online Bagging 77.15 2.98ADWIN Bagging 79.24 1.48Leveraging Bagging 85.54 20.17Leveraging Bagging MC 85.37 22.04Leveraging Bagging ME 80.77 0.87

Leveraging Bagging

• Leveraging Bagging• Using Poisson(λ )

• Leveraging Bagging MC• Using Poisson(λ ) and Random Output Codes

• Leveraging Bagging ME• Using weight 1 if misclassified, otherwise eT/(1−eT)

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

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)

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)

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

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

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

performance measures

Internal Measures

• Sum square distance• Dunn index D= dmin

• C-Index C= S−SminSmax−Smin

External Measures

• Rand Measure• F Measure• Jaccard• Purity

Balanced Iterative Reducing and Clustering usingHierarchies

• Clustering Features CF= (N,LS,SS)• N: number of data points• LS: linear sum of the N data points• SS: square sum of the N data points• Properties:• Additivity: CF1+CF2 = (N1+N2,LS1+LS2,SS1+SS2)• Easy to compute: average inter-cluster distanceand average intra-cluster distance

• Uses CF tree• Height-balanced tree with two parameters• B: branching factor• T: radius leaf threshold

Balanced Iterative Reducing and Clustering usingHierarchies

Phase 1: Scan all data and build an initial in-memory CFtree

Phase 2: Condense into desirable range by building asmaller CF tree (optional)

Phase 3: Global clusteringPhase 4: Cluster refining (optional and off line, as requires

more passes)

clu-stream

Clu-Stream

• Uses micro-clusters to store statistics on-line• Clustering Features CF= (N,LS,SS,LT,ST)• N: numer of data points• LS: linear sum of the N data points• SS: square sum of the N data points• LT: linear sum of the time stamps• ST: square sum of the time stamps

• Uses pyramidal time frame

clu-stream

On-line Phase

• For each new point that arrives• the point is absorbed by a micro-cluster• the point starts a new micro-cluster of its own• delete oldest micro-cluster• merge two of the oldest micro-cluster

Off-line Phase

• Apply k-means using microclusters as points

streamkm++: coresets

Coreset of a set P with respect to some problemSmall subset that approximates the original set P.

• Solving the problem for the coreset provides an approximatesolution for the problem on P.

(k,ε)-coresetA (k,ε)-coreset S of P is a subset of P that for each C of size k

(1− ε)cost(P,C)≤ costw(S,C)≤ (1+ ε)cost(P,C)

streamkm++: coresets

Coreset Tree

• Choose a leaf l node at random• Choose a new sample point denoted by qt+1 from Placcording to d2

• Based on ql and qt+1, split Pl into two subclusters and createtwo child nodes

StreamKM++

• Maintain L= ⌈log2( nm)+2⌉ buckets B0,B1, . . . ,BL−1

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 .

frequent patterns

pattern mining

Dataset ExampleDocument Patterns

d1 abced2 cded3 abced4 acded5 abcded6 bcd

itemset mining

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

itemset mining

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

be,bce,abce3 de,cde

itemset mining

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

itemset mining

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

itemset mining

e→ ce

itemset mining

a→ ace

itemset mining

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

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.

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.

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

mining patterns over data streams

Requirements: fast, use small amount of memory and adaptive

• Type:• Exact• Approximate

• Per batch, per transaction• Incremental, Sliding Window, Adaptive• Frequent, Closed, Maximal patterns

moment

• Computes closed frequents itemsets in a sliding window• Uses Closed Enumeration Tree• Uses 4 type of Nodes:

• Closed Nodes• Intermediate Nodes• Unpromising Gateway Nodes• Infrequent Gateway Nodes

• Adding transactions: closed items remains closed• Removing transactions: infrequent items remains infrequent

fp-stream

• Mining Frequent Itemsets at Multiple Time Granularities• Based in FP-Growth• Maintains

• pattern tree• tilted-time window

• Allows to answer time-sensitive queries• Places greater information to recent data• Drawback: time and memory complexity

tree and graph mining: dealing with time changes

• Keep a window on recent stream elements• Actually, just its lattice of closed sets!

• Keep track of number of closed patterns in lattice, N• Use some change detector on N• When change is detected:

• Drop stale part of the window• Update lattice to reflect this deletion, using deletion rule

Alternatively, sliding window of some fixed size

Summary

overview of big data science

Short Course Summary

1 Introduction to Big Data2 Big Data Science3 Real Time Big Data Management4 Internet of Things Data Science

Open Source Software

1 MOA: http://moa.cms.waikato.ac.nz/2 SAMOA: http://samoa-project.net/

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