CiteSeerX — Expected Value

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Information Fusion and Integration

Masatoshi Ishikawa Carson Reynolds

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Last Week

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What is the largest prime factor of the next integer

after the largest known factorial prime?

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Weekly Puzzle

Answer via twitter with a direct message to

@CarsonReynolds

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Which of these shapes is

unlike the others?

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⁴

A casino has a game which

gives players 100 yen for each dot on a rolled die. They

charge 550 yen per roll. On average, how much does the

casino make per game?

Image Credit: Nanami Kamimura

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Expected Value of a Function

5 Figure Credit: Wikipedia Expected Value

N → ∞

Bishop (2006) PR&ML Chapter 1

E(x) =

� N i=1

P (x _i )x _i

E(x) � 1 N

� N i=1

x _i

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Expected Value of a Function

5 Figure Credit: Wikipedia Expected Value

N → ∞

Bishop (2006) PR&ML Chapter 1

Average

Sample Mean

First Statistical Moment Unbiased Estimator of Mean

Expectation E(x) =

� N i=1

P (x _i )x _i

E(x) � 1 N

� N i=1

x _i

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Variance

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Press et al. (1992) NR in C

The variance is a measure of the spread of a random variable. Particularly how far it varies from the

expected or mean value.

Figure Credit:

IPCC Third Assessment Report Client Change 2001

var(x) = 1

N − 1

� N i=1

(x _i − E(x)) ² σ(x) = �

var(x)

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Skew(x) = 1 N

� N i=1

( x _i − E(x) σ ) ³

Kurt(x) = ( 1 N

� N i=1

( x _i − E(x)

σ ) ⁴ ) − 3

Press et al. (1992) Numerical Recipes in C

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⁸

Suppose I have two discrete random variables each with their own mean and variance.

Figure Credit:

Maxfield and Lyon (1983)

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Demo:

Multivariate Gaussian

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Covariance

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We can represent

covariance using the marginal variances of the variables and a

rotation parameter (the correlation).

If x and y are

independent their covariance is 0.

Σ(x, y) = E[(x − E[x])(y − E[y])]

Σ = E �

(X − E[X]) (X − E[X]) ^� � Σ =

� σ _x ² ρσ _x σ _y ρσ _x σ _y σ _y ²

�

ρ = E[(x − E[x])(y − E[y])]

σ _x σ _y

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Figure Credit: DHS Figure A.4

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More about Covariance

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• The diagonal of a

covariance matrix is the variance.

• If all the rows of the input matrix are

identical, then the

covariance matrix is

zero.

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¹³

Equal Distance Contours

K. Van Laerhoven (2004)

�p − q� ₁

Euclidean

Manhattan Chebyshev Mahalanobis

�p − q� ₂ �p − q� _∞ ?

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Mahalanobis Distance

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∆ ² (x, µ) =

�

(x − µ) ^� Σ ⁻¹ (x − µ)

• Mahalanobis Distance is defined for an observation vector x, a mean vector μ, and a covariance matrix associated with the mean.

• When the covariance is the identity matrix, equivalent to Euclidean distance.

• It provides a measure of similarity between data

points in a multidimensional Gaussian space.

• Root of sum of square

differences divided out by the covariance.

Bradski and Kaehler (2008)

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The slope of a linear regression line is:

cov(x,y)/var(x)

Where would the regression line appear in the each of the above examples?

Barton et al. (2007) Evolution

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Using Mahalanobis Distance

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Suppose you want to train a system to

recognized eye blinks using just a few examples measuring distance between the components, and width and height of each component.

Grauman et al. (2001)

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Blink Tracker

Blink Tracker with Chye Connsynn

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Bhattacharyya distance

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D _B (p, q) = − ln( �

x ∈X

� p(x)q(x))

Figure Credit:

B. Mak and E. Bernard (1996)

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What is information and how do we fuse it?

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Transmission of Information

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R.V.L. Hartley (1928) Transmission of Information

Researchers at Bell Laboratories grew

interested in how signals such as morse code could be interpreted in the

presence of noise.

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²¹

Figure Credit: A.B.C. Telegraphic Code (1901)

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Measure of Information

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R.V.L. Hartley (1928) Transmission of Information

Hartley sought to make a quantitative measure of information. His model analyzed messages of length n composed of symbols that could take on s different values.

However each symbol value was equally likely.

H = n log s H = log s ⁿ

2 3 4 5 n

10 20 30

log s ⁿ

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Messages, Signals & Noise

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C. E. Shannon (1948)

A Mathematical Theory of Communication

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Shannon’s Entropy

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H = −

� m i

p _i log p _i Shannon instead

considered the list of symbols as different

observations of a discrete random variable. When all the probabilities are equal it is exactly Hartley’s H.

C. E. Shannon (1948)

� m i=1

p _i = 1

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Units of

Measure for Entropy

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Base Units Conversion

2 bits

e nats 1 bit = ln 2

10 bans 1 bit = log

10

2 “measures how much one random variables tells us about another”

P. E. Latham and Y. Roudi (2009) Scholarpedia

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Probability-Weighted Information Gain

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The “entropy term is the average amount of

information to be gained from a certain set of

events.”

Information gain is

inversely related to the probability of an event.

Pluim et al. (2003)

H =

� m i

p _i log 1

p _i

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Computing Entropy

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If we have a list n of

discrete probabilities, we can compute n log n for each, and then sum each of the entries

Pluim et al. (2003)

� f (n)

“...a 1-yr old child uses the words ‘mummy,’

‘daddy,’ ‘cat,’ and ‘uh-oh.’”

Word Probability

mummy 0.35

daddy 0.2

cat 0.2

uh-oh 0.25

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What is the Entropy?

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n = (0.35, 0.2, 0.2, 0.25)

Word Probability

mummy 0.35

daddy 0.2

cat 0.2

uh-oh 0.25

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What is the Entropy?

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n = (0.35, 0.2, 0.2, 0.25)

f (n) = ( −0.5301, −0.4644, −0.4644, −0.5)

Word Probability

mummy 0.35

daddy 0.2

cat 0.2

uh-oh 0.25

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What is the Entropy?

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n = (0.35, 0.2, 0.2, 0.25)

f (n) = ( −0.5301, −0.4644, −0.4644, −0.5)

� f (n) = 1.9589

Word Probability

mummy 0.35

daddy 0.2

cat 0.2

uh-oh 0.25

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What is the Entropy of a 2 year old?

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Word Probability

mummy 0.05

daddy 0.05

cat 0.02

train 0.02

car 0.02

cookie 0.02

telly 0.02

no 0.80

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What is the Entropy of a 2 year old?

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Word Probability

mummy 0.05

daddy 0.05

cat 0.02

train 0.02

car 0.02

cookie 0.02

telly 0.02

no 0.80

� f (n) = 1.25412

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Entropy Minimized

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�1.0 �0.5 0.5 1.0

5 10 15

“A distribution with a

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single sharp peak

corresponds to a low entropy value...”

Entropy is minimized by the Dirac Delta function.

Pluim et al. (2003)

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Entropy Maximized

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1.5 2.0 2.5 3.0

0.1 0.2 0.3 0.4 0.5

“When all messages are

0.6

equally likely to occur, the entropy is maximal, because you are completely

uncertain which message you will receive.”

Entropy is maximized by the Uniform Distribution.

Pluim et al. (2003)

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Demo: Minimizing and Maximizing Distributions

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�1.0 �0.5 0.5 1.0

5 10 15 20

1.5 2.0 2.5 3.0

0.1 0.2 0.3 0.4 0.5 0.6

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Comparing Entropy

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Kullback-Leibler Divergence

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D _KL (p(x), q(x)) = �

x

p(x) log p(x) q(x)

DHS Appendix A7.2

If two distributions cover the same discrete range, then we can compare

how far one diverges from the other. KL

divergence expresses

how many additional bits are needed to encode p using the distribution q.

Suppose we wanted to

compare a random variable with itself? What would the KL divergence be in this

case?

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Kullback-Leibler Divergence

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D _KL (p(x), q(x)) = �

x

p(x) log p(x) q(x)

DHS Appendix A7.2

If two distributions cover the same discrete range, then we can compare

how far one diverges from the other. KL

divergence expresses

how many additional bits are needed to encode p using the distribution q.

Suppose we wanted to

compare a random variable with itself? What would the KL divergence be in this

case?

KL divergence

is not a a distance metric

because it does not satisfy the triangle inequality:

D KL (p,q) != D KL (q,p)

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Mutual Information

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I(A, B) = H(B) − H(B|A) I(A, B) = �

a,b

p(a, b) log p(a, b) p(a)p(b)

Pluim et al. (2003)

Conditionally

As a distance

metric

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Kullback-Leibler and Mutual Information

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D _KL (p(x), q(x)) = �

x

p(x) log p(x) q(x) Mutual Information is the

Kullback-Leibler

divergence between the joint distribution and the product of the

independent ones. ^{I(A, B) =}

�

a,b

p(a, b) log p(a, b) p(a)p(b)

P. E. Latham and Y. Roudi (2009) Scholarpedia

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I(A, B) ≤ I(A, A)

Properties of

Mutual Information

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I(A, B) = I(B, A) I(A, A) = H(A)

Pluim et al. (2003)

Symmetry

Relationship to Entropy

Mutual Information is Less than Self Information

I(A, B) = 0 ⇐⇒ A & B are

Independent

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Image Registration

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A. Goshtasby & M. Satter (1999)

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³⁹

N-bin

Histogram of Image

N-bin

Histogram of Image

Joint Distribution Independent

Distributions

Kullback Liebler Divergence

Photo Credit: M. Farmer (2003)

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CT & MR

Image Registration

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Pluim et al. (2003)

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⁴¹

Figure Credit: M. Farmer (2003)

Image Registration

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A. Dame and E. Marchand (2010)

Map/Aerial View Matching

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Applications:

Augmented Reality

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A. Dame and E. Marchand (2010)

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Case Study:

Sensor-Generated Social Networks

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Figure 2: The hardware implementation of a motion detector node. Left: early MITes-based version, Right: An ultra-low-power, standards-based prototype.

we ask that you cite this report[23] in all publications that derive from your work on this data set.

4. DATA DESCRIPTION

The dataset is comprised of several segments:

• raw motion data

• calibration data

• calendar data

• solar and weather data

• intermediate track analytics

The section contains detailed descriptions of each of these subsets.

4.1 Raw Motion Data

The primary data stream is the output of the motion detector network. The sensors are ceiling mounted at approximately two meter intervals along hallways and in grids covering public spaces such as lobbies and meeting rooms. There are no sensors in individual oﬃces. The sensors are installed with the intention of covering the floor area completely with little or no overlap between sensor fields of view. The ceiling height varies, but is approximately three meters in most areas. Figure 1 shows the tiled arrangement of the sensors and a snapshot of motion activity.

The sensors use the MITes platform[15] for processing and communication coupled to a modified KC7783R sensor board. Two versions of the hardware are shown in Figure 2.

The MITes-based node used to collect the data in this re- lease is pictured on the left. The re-engineered node on the right is used in new installations and it features an 802.15.4 radio, a modular design, and ultra-low power consumption for drastically extended battery life.

The MITes employ an unreliable network protocol that does not contain checksum information. As a result packets are sometimes lost, duplicated, or garbled. An attempt has been made to filter out the duplicated and garbled packets.

However it is certain that there are packets missing from in the data and any analysis should take that into account. The

loss rate is low, but is diﬃcult to measure since it depends on the network load.

The sensor boards are the type commonly found in motion activated lighting fixtures and security sensors. They are passive infrared motion detectors. They work by sensing light emitted in the far-infrared by warm objects and signal on high-frequency changes in the scene at those frequencies.

They were modified to reduce their adaption rate from min- utes to seconds. Since the timing of the detections depends on the analog characteristics of the circuit, the minimum inter-detection time varies, but is observed to typically be around 1.5 seconds.

The raw data is available as compressed ASCII text files in five parts:

0114.txt.gz Mar 21 23:00:25 2006 – Jun 11 00:26:40 2006 0115.txt.gz Jun 11 00:26:40 2006 – Oct 4 18:13:20 2006 0116.txt.gz Oct 4 18:13:20 2006 – Jan 28 11:00:00 2007 0117.txt.gz Jan 28 11:00:00 2007 – May 24 05:46:40 2007 0118.txt.gz May 24 05:46:40 2007 – Jul 2 15:41:50 2007

The filename refers to the high-order bits of the timestamps on the data contained in each file.

The files contain data like this:

470 01179980510828 01179980511853 1.0 469 01179980512169 01179980513193 1.0 467 01179980513580 01179980514609 1.0 468 01179980514573 01179980515598 1.0

The first element is the sensor identification number. The second and third numbers are the timestamps of the begin- ning of the event. The fourth number is a meaningless place holder value.

The map in Figure 3 depicts the test area. Executives and administrators occupy the wing on the right right side of the eighth floor map. Researchers occupy the bottom and left wings, and most of the 7th floor. The central core of the building contains restrooms, lobbies, elevators, and on the eighth floor, the mail room and the kitchen. There are several stairwells that connect the floors.

We have been collecting data at this facility since October of 2005. Data from the entire area depicted on that map has been continuously recorded since March 2006. The system generates approximately two million motion detections per month.

4.1.1 Time

The data is timestamped with the number of milliseconds since the epoch: January 1, 1970 UTC. Like the windows system clock, this number becomes larger than 2³² after 50 days, on February 19, 1970, so you must take care to use 64- bit integer representations when manipulating timestamps:

• __int64 or long long in C

• use bignum in PERL

• java.math.BigInteger in Java

11

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⁴⁵

Kautz and Selman

(1998) developed

ReferralWeb to mine bibliographical

references and

automatically model the implied social network

Social Networks from

Data Mining

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⁴⁶

Choudhury and

Pentland (2002) used wearable sensors to assess social network and interpersonal

interaction.

Social Networks from Wearable Hardware

A similar analysis of the larger dataset shows clustering of groups and a reduction of interaction with increasing physical separation. Subject IDs 2-9 belong to group 1, IDs 10,12-15 belong to group 2, IDs 16-19 to group 3 and 21- 24 to group 4, IDs 20 and 25 were physically co-located with groups 1&2 (no one was assigned ID# 1 or 11). Note, that there are few individuals that have broad connections across groups (ID 3, 8, and 13) - this type of individuals usually have an important effect on the information flow within the community.

Figure 9 The connectivity matrix of interaction duration. Each row is a different individual and each column depicts the fraction of his/her interaction with others. Image(i,j) depicts person i’s interaction with person j. Dark region signify absence of interaction.

These connectivity graph or network graph can then be used to estimate centrality measures as traditionally done in social network analysis. Centrality measures seek to quantify an individual’s prominence within a network by summarizing the relationships among the different individuals in the network. There are different measures of centrality e.g. degree, betweenness, eigenvector etc.[15, 16]. Here we use the eigenvector centrality where the status of a person is recursively related to the statuses of the people he/she is connected to. If an individual is chosen by a popular person it should add to the person’s popularity. If A is the adjacency matrix where a_ij means that I contributes to the status of j and x is the vector of centrality score – the most general form of eigenvector centrality is:

1 1 2 2 ...

i i i ni n

x =a x +a x + +a x (1) In matrix representation:

A xt = (2) x The eigenvector centrality is the eigenvector of the adjacency matrix corresponding to an eigenvalue of 1.

Normalizing the rows of A to sum to 1 ensures that

equation 2 is solvable. The eigenvector centrality measure for the larger group based on the adjacency matrix is shown below (Figure 10), ID 3 and 8 with highest centrality scores also are individuals who had most connection across groups.

Figure 10 Eigenvector centrality measures of the 23 individual participating in the larger study calculated from proximity data

4. Conclusion and Future Work

In this paper, we present a method for analyzing the connectivity of interacting groups using data gathered from wearable sensors. We have presented initial results from our efforts in sensor-based modeling of human communication networks. We show that we can automatically and reliably estimate when people are in close proximity and when they are talking. We demonstrate the advantage of continuous sensing of interactions that allows us to measure the structure of communication networks along various dimensions – duration, frequency, ratio of interaction etc. We also present centrality scores for each individuals computed automatically from raw sensor data. Centrality measures are often used in social network analysis as a measure of influence and embeddedness of a person in his/her community. In many studies it has been shown that topology of people’s connectivity is the most important feature and the actual interaction content is not as crucial in understanding a person’s role within the community[1, 17- 19]. We are currently obtaining quantitative results for our algorithms by comparing the accuracy of our techniques to hand-labeled ground truth data of the interactions. We are also incorporating our work on modeling the dynamics of the network as a whole that will in the future allow us to quantitatively measure influences people have on each other [20].

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Photo: C. Wren

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Photo: C. Wren

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Photo: C. Wren

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Photo:

K. Ryall

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⁵¹

Sensor Locations

(% ##% $ " # ' +$%!"' % & % !#&' , #" ' #('& # ' ( "

%& #&%) , &"&#%& & % *,& #& " !'" %##!&

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?84I8 5HG G;8E8 <F AB J4L GB 7<FG<A:H<F; BA8 9EB@ G;8 BG;8E

!AFG847 B9 E8CE8F8AG<A: 846; CBFF<5?8 ;LCBG;8G<64? ?458?<A:

4F 4 7<FG<A6G 8?8@8AG <A BHE 74G454F8 J8 <AFG847 E86BE7 G;8 GE4AF<G<BAF G;4G J8 4E8 FHE8 B9 4A7 ?<A> G;8@ GB:8G;8E 4G CB<AGF B9 4@5<:H<G<8F GB 9BE@ 4 7<E86G87 :E4C; ,;<F :E4C;

J<?? 6BAG4<A G;8 GEH8 GE46> 4F 4 FH5:E4C; 5HG 4?FB 6BAG4<AF 4?? 4?G8EA4G<I8 6BAF<FG8AG ;LCBG;8G<64? GE46>F 4F J8??

,;8 GE46>?8GF 6BH?7 58 E8:8A8E4G87 9EB@ G;8 E4J @BG<BA 74G4 5HG J8 <A6?H78 BHE <AG8ECE8G4G<BA B9 G;8 GE46>?8G :E4C;

9BE 6BAI8A<8A68 ,;8 E8CE8F8AG4G<BA B9 GE46>?8G :E4C; <F 8K C?4<A87 <A 78G4<? <A G;8 ?8 151

5. DATA MANAGEMENT

/8 FGBE8 BHE 74G4 <A 4 74G454F8 GB 946<?<G4G8 86<8AG 4668FF 4A7 F84E6; %BFG B9 G;8 74G4 ?8F 4E8 9BE@4GG87 GB 58 E847 7<E86G?L <AGB 4 74G454F8 FC86<64??L %L+)$ I8EF<BA ,;8

0.) ?8F 4E8 8K4@C?8F B9 6E84G<A: G45?8F ?B47<A: 74G4 4A7 C8E9BE@<A: 8K4@C?8 DH8E<8F !G <F ABG A868FF4EL GB HF8 4 74G454F8 @BFG B9 G;8 ?8F J<?? 4?FB ?B47 <AGB %4G?45 BE 64A 58 E847 84F<?L <AGB 6HFGB@ FB9GJ4E8 8J4E8 ;BJ8I8E G;4G G;8 74G4F8G <F F<M45?8

6. BIBLIOGRAPHY

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A8GJBE>F 4A7 G;8 C8E68CG<BA B9 46G<I<GL

7. ACKNOWLEDGMENTS

,;8 4HG;BEF JBH?7 ?<>8 GB 8KCE8FF G;8<E :E4G<GH78 9BE G;8 HA7L<A: FHCCBEG 4A7 HAE8?8AG<A: 8A6BHE4:8@8AG B9 E "BF8C;

%4E>F <E86GBE B9 G;8 *8F84E6; $45BE4GBEL B9 G;8 %<G FH5<F;< ?86GE<6 *8F84E6; $45BE4GBE<8F J;8A G;<F F8AFBE A8G JBE> J4F 6BA68<I87 78F<:A87 5H<?G 4A7 <AFG4??87 /8 4?FB J<F; GB G;4A> G;8 6HEE8AG @4A4:8@8AG B9 %*$ 9BE G;8<E 6BAG<AH87 FHCCBEG B9 G;<F CEB=86G A7 A4??L J8 @HFG G;4A>

G;8 8@C?BL88F <AG8EAF 78?<I8EL C8BC?8 6?84A<A: 6E8J 4A7 I<F<GBEF J;B 6BAGE<5HG87 74G4 GB G;<F F8G 5L @BI<A: 4EBHA7

EB47J4L 58GJ88A %4E6; 4A7 %4E6;

8. REFERENCES

1 3 5BJ7 B5<6> ! FF4 %LA4GG 4A7 / *B:8EF ,;8 4J4E8 ;B@8 8I8?BC<A: G86;AB?B:<8F 9BE FH668FF9H? 4:<A: !A 41%''&+0)5 1(

"14-5*12 10 761/#6+10 #5 # #4' +8'4 13 * <CC8EFC46; B;8A 4A7 " 4AAL %B78?<A:

;H@4A 58;4I<BE 9EB@ F<@C?8 F8AFBEF <A G;8 ;B@8 !A

41%''&+0)5 ( *' 10('4'0%' 0 '48#5+8'

1/276+0)

13 4H@58E: 4A7 B:: A 86<8AG @8G;B7 9BE 6BAGBHE GE46><A: HF<A: 46G<I8 F;4C8 @B78?F !A

41%''&+0) 1( 6*' "14-5*12 10 16+10 1( 104+)+&

#0& 46+%7.#6'& $,'%65 ! B@CHG8E +B6<8GL

13 B5<6> %BI8@8AG 46G<I<GL 4A7 46G<BA G;8 EB?8 B9 >ABJ?87:8 <A G;8 C8E68CG<BA B9 @BG<BA

*+.1512*+%#. 4#05#%6+105 +1.1)+%#. %+'0%'5

N

13 , ;BH7;HEL 4A7 (8AG?4A7 ;4E46G8E<M<A: FB6<4?

A8GJBE>F HF<A: G;8 FB6<B@8G8E !A 41%''&+0)5 1( 6*'

146* /'4+%#0 551%+#6+10 1( 1/276#6+10#. 1%+#.

#0& 4)#0+;#6+10#. %+'0%'

13

(57)

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Tracklets

52

(58)

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⁵³

Binary Adjacency Matrices



 

P

_1,1

· · · P

_1,30

... . .. ...

P

_1,30

· · · P

^30,30



= 0.

• x = 1101011111

• y = 0001110001

• δ[x − y] = 1100101110

• hd(x, y) = 6

• nhd(x, y) =

₁₀⁶

= 0.6

• ! 0 1 1 0

" 

 1 1 1 1 1 1 1 1 1







 



0 1 1 0 1 0 1 0 1 1 0 1 0 0 1 0



 

= 0.

• x = 1101011111

• y = 0001110001

• δ[x − y] = 1100101110

• hd(x, y) = 6

• nhd(x, y) =

₁₀⁶

= 0.6

• ! 0 1 1 0

" 

 1 1 1 1 1 1 1 1 1







 



0 1 1 0 1 0 1 0 1 1 0 1 0 0 1 0



 

= 0.

• x = 1101011111

• y = 0001110001

• δ[x − y] = 1100101110

• hd(x, y) = 6

• nhd(x, y) =

₁₀⁶

= 0.6

• ! 0 1 1 0

" 

 1 1 1 1 1 1 1 1 1







 



0 1 1 0 1 0 1 0 1 1 0 1 0 0 1 0



 



1

1 2

1

2 3

1

2 3

4

(60)

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⁵⁵

Hamming Distance

8 PRIVACY AND SENSOR-GENERATED SOCIAL NETWORKS

The second row indicates that the 1-year sensor data is in fact rather similar to the survey graph. However, third second indicates that the email graph is not very similar to the survey graph. We can illustrate this visually with the following scaled Venn Diagrams as in Figure 2.

In short, the sensors provide a more complete picture of the social network connections than the email alone. Moreover, there are an order of magnitude fewer false-positive network links in the sensor graph data (0.6%) than in the email graph data (2%).

E.2 Hamming Distance

The next graph comparison metric which we applied was Hamming distance. A Ham- ming distance hd(x, y) can be defined as an operation on two vectors x and y of the same length n. The Hamming distance is the sum of the number of elements which are equal at the index location in x

_i

and y

_i

. First suppose we have a delta function which is similar to Kronecker’s delta, but which emits a 0 when the elements are equal:

δ[n] =

 



0, if n = 0

1, if n �= 0 (2)

By making use of this delta function as defined in Equation (2) the Hamming distance hd(x, y) may be computed as follows:

hd(x, y) =

�

n i=1

δ[n] =

 



0, if n = 0

1, if n �= 0 (2)

By making use of this delta function as defined in Equation (2) the Hamming distance hd(x, y) may be computed as follows:

hd(x, y) =

�

n i=1

δ[x

_i

− y

ⁱ

]. (3)

That is the sum of the delta of all elements from 1 to n in the vectors x and y. Since the maximal Hamming distance varies with diﬀerent vector lengths n it is convenient to have a normalized Hamming distance nhd(x, y):

REYNOLDS, WREN AND IVANOV: PRIVACY AND SENSOR-GENERATED SOCIAL NETWORKS 9

nhd(x, y) =

�

_n

i=0

δ[x

_i

− y

ⁱ

]

n (4)

When this normalized Hamming distance is zero, the graphs are identical and no edits are required. As the number increases, a greater proportion of the edges must be edited in order to make the two graphs identical. When the number is one, every edge in the graph has to be altered to make the graphs equal. In Table I normalized Hamming distances are listed for our data sets.

Ordering by normalized Hamming distance edits, we see that the network constructed by fusing the email and sensor data is most similar to the survey network. These results also suggest the sensors provide a better approximation of the social network indicated by the survey data. On this metric, the email network is somewhat dissimilar to the survey network.

TABLE I

Similarity Metrics

Social Network Jaccard Edge Coeﬃcient Normalized Hamming Distance

Fused 0.883 0.117

Sensors 0.863 0.137

Email 0.161 0.838

E.3 Degree Distribution

= 0.

• x = 1101011111

• y = 0001110001

• δ[x − y] = 1100101110

• hd(x, y) = 6

• nhd(x, y) =

₁₀⁶

= 0.6

1 • If person x emailed y then P

^x,y

= 1 otherwise P

_x,y

= 0.

• If person x visited y then P

^x,y

= 1 otherwise P

_x,y

= 0.

• If person x said they knew y then P

^x,y

= 1 otherwise P

_x,y

= 0.

• x = 1101011111

• y = 0001110001

• δ[x − y] = 1100101110

• hd(x, y) = 6

• nhd(x, y) =

₁₀⁶

= 0.6

1 • If person x emailed y then P

^x,y

= 1 otherwise P

_x,y

= 0.

• If person x visited y then P

^x,y

= 1 otherwise P

_x,y

= 0.

• If person x said they knew y then P

^x,y

= 1 otherwise P

_x,y

= 0.

• x = 1101011111

• y = 0001110001

• δ[x − y] = 1100101110

• hd(x, y) = 6

• nhd(x, y) =

₁₀⁶

= 0.6

1 • If person x emailed y then P

^x,y

= 1 otherwise P

_x,y

= 0.

• If person x visited y then P

^x,y

= 1 otherwise P

_x,y

= 0.

• If person x said they knew y then P

^x,y

= 1 otherwise P

_x,y

= 0.

• x = 1101011111

• y = 0001110001

• δ[x − y] = 1100101110

• hd(x, y) = 6

• nhd(x, y) =

₁₀⁶

= 0.6

1

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⁵⁶

Jaccard Edges

REYNOLDS, WREN AND IVANOV: PRIVACY AND SENSOR-GENERATED SOCIAL NETWORKS 7

As is typical, we define a graph G to be an ordered pair of sets denoting verticies and edges G = (V, E). For convenience we use the notion V (G) to indicate a graph’s verticies and E(G) to denote its edges.

E.1 Jaccard Similarity Coeﬃcient

The Jaccard similarity coeﬃcient (a measure of similarity) is defined as the cardinality (or number of elements) of the intersection of sets A and B the divided by the cardinality of the union.

Fig. 2. A visual comparison of the overlap in Jaccard edges using Venn diagrams scaled to network size. On the left the survey data set (blue) and the sensor data set (orange) are compared. On the right the survey data set (blue again) and email data set (green) are compared. Note that the sensor data set covers nearly all of the survey data.

As we were more interested in the relationships, we used the edges of the social graph which denote some sort of communicative relationship. Thus we used an extended version of the Jaccard similarity coeﬃcient which instead looked the cardinality of edges in the graphs.

For convenience we refer to the result of this function application as Jaccard edges. Equation (1) defines Jaccard edges J _E (A, B) by considering the intersection over the union of the edges of graphs A and B.

J _E (A, B) = |E(A) �

E(B) |

|E(A) �

E(B) | (1)

In Table I Jaccard edge similarity coeﬃcients are listed. Firstly we see that the fused

network drawing from both email and sensor data was most similar to the survey graph.

(62)

http://www.k2.t.u-tokyo.ac.jp Ishikawa Oku Laboratory We hypothesized that:

1. social network structure (as reported through an interface) can be estimated by sensor network

II. a fused network will be more similar to the survey data than either email or sensor data alone

57 REYNOLDS, WREN AND IVANOV: PRIVACY AND SENSOR-GENERATED SOCIAL NETWORKS 9

nhd(x, y) =

�

n

i=0

δ[x

_i

− y

ⁱ

]

n (4)

When this normalized Hamming distance is zero, the graphs are identical and no edits are required. As the number increases, a greater proportion of the edges must be edited in order to make the two graphs identical. When the number is one, every edge in the graph has to be altered to make the graphs equal. In Table I normalized Hamming distances are listed for our data sets.

Ordering by normalized Hamming distance edits, we see that the network constructed by fusing the email and sensor data is most similar to the survey network. These results also suggest the sensors provide a better approximation of the social network indicated by the survey data. On this metric, the email network is somewhat dissimilar to the survey network.

TABLE I

Similarity Metrics

Social Network Jaccard Edge Coeﬃcient Normalized Hamming Distance

Fused 0.883 0.117

Sensors 0.863 0.137

Email 0.161 0.838

E.3 Degree Distribution

Another line of analysis to judge the similarities of potential graphs is to compare their

degree distribution. This approach is used by Newman et al. in examining empirical social

network data sets [9]. For any vertex v in a graph, its degree k indicates how many adjacent

vertices exist. For an undirected graph A, the average degree ¯ k can be defined in terms of

the cardinality of graph edges |E(A)| and the cardinality of graph verticies |V (A)|.

(63)

CiteSeerX — Expected Value

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Information Fusion and Integration

Masatoshi Ishikawa Carson Reynolds

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Last Week

What is the largest prime factor of the next integer

after the largest known factorial prime?

103007

http://bit.ly/kmdcw7

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Weekly Puzzle

Answer via twitter with a direct message to

@CarsonReynolds

Which of these shapes is

unlike the others?

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A casino has a game which

gives players 100 yen for each dot on a rolled die. They

charge 550 yen per roll. On average, how much does the

casino make per game?

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Expected Value of a Function

N → ∞

Bishop (2006) PR&ML Chapter 1

E(x) =

� N i=1

P (x i )x i

E(x) � 1 N

� N i=1

x i

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Expected Value of a Function

N → ∞

Bishop (2006) PR&ML Chapter 1

Average

Sample Mean

First Statistical Moment Unbiased Estimator of Mean

Expectation E(x) =

� N i=1

P (x i )x i

E(x) � 1 N

� N i=1

x i

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Variance

Press et al. (1992) NR in C

The variance is a measure of the spread of a random variable. Particularly how far it varies from the

expected or mean value.

var(x) = 1

N − 1

� N i=1

(x i − E(x)) 2 σ(x) = �

var(x)

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Skew(x) = 1 N

� N i=1

( x i − E(x) σ ) 3

Kurt(x) = ( 1 N

� N i=1

( x i − E(x)

σ ) 4 ) − 3

Press et al. (1992) Numerical Recipes in C

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Suppose I have two discrete random variables each with their own mean and variance.

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Demo:

Multivariate Gaussian

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Covariance

We can represent

covariance using the marginal variances of the variables and a

rotation parameter (the correlation).

If x and y are

independent their covariance is 0.

Σ(x, y) = E[(x − E[x])(y − E[y])]

Σ = E �

(X − E[X]) (X − E[X]) � � Σ =

� σ x 2 ρσ x σ y ρσ x σ y σ y 2

�

P (x _i )x _i

x _i

P (x _i )x _i

x _i

(x _i − E(x)) ² σ(x) = �

( x _i − E(x) σ ) ³

( x _i − E(x)

σ ) ⁴ ) − 3

(X − E[X]) (X − E[X]) ^� � Σ =

� σ _x ² ρσ _x σ _y ρσ _x σ _y σ _y ²

σ _x σ _y

�p − q� ₁

�p − q� ₂ �p − q� _∞ ?

∆ ² (x, µ) =

(x − µ) ^� Σ ⁻¹ (x − µ)

D _B (p, q) = − ln( �

H = n log s H = log s ⁿ

log s ⁿ