Random Fields - A short introduction · Random Fields RANDOM FIELDS Measure-theoretic deﬁnition Deﬁnition Let (K,K,P) be a measure space.Let GN,d be the set of all Rd-valued functions

Random FieldsA short introduction

Abhishek Srivastav

Penn State Univ.(UP)

April 06, 2010

Srivastav (PSU) Random Fields April 06, 2010 1 / 23

Overview

OVERVIEW

• From random variables to random fields• Generalization of stochastic processes

• Random fields• Measure-theoretic and Kolmogorov’s definition

• Random fields on graphs• Markov and Gibbs random fields

• Equivalence Theorem

• Main questions and solutions• Problems of interest, sampling and approximate methods

• Approximate methods• Mean Field Theory

• Example application - sensor networks

• References


Random Fields

RANDOM FIELDSGeneralization of stochastic processes

Stochastic process

Let (Ω, Υ,P) be a probability space, then a stochastic process is z(t, ζ) is defined as a Υ −Rmeasurable map z : T × Ω → R; where the index set T ⊆ R serves as the time parameter

Definition (Random Field)

Let (K,K,P) be a complete probability space and T a topological space. Then a measurablemapping F : K → (RT)d is called a real-valued random field. Where, (RT)d is the space of allRd -valued functions on the topological space T

Examples:

• Crop yields and disease/infection over a spatially distributed region

• Geographical distribution of parameters such as temperature or rainfall

• Images

• Magnetic Resonance (MR) Images - Brain and other tissues

• Dynamics of complex networks


Random Fields

RANDOM FIELDSMeasure-theoretic definition

Definition

Let (K,K,P) be a measure space. Let GN,d be the set of all Rd -valued functions on RN ;N, d ∈ N, and GN,d be the corresponding σ-algebra. Then a measurable mapF : (K,K) → (GN,d ,GN,d ) is a called an N-dimensional random field.

• Random field F maps elements k of the sample space K to functions in GN,d . Equivalentlyit maps sets in K to sets in GN,d

• GN,d contains sets of the form {g ∈ GN,d : g(~ri ) ∈ Bi , i = 1, . . . , m}, where m is arbitrary,~ri ∈ R

N and Bi ∈ Bd

• Note: Sets of the form {g ∈ GN,d : g(~rα) ∈ Bα, α ∈ I}, where I is uncountable are notusually present in GN,d . Needs F to be separable.

• For a given k ∈ K, the corresponding function in GN,d is called a realization of the randomfield and is denoted as F(•, k). At a given point ~r ∈ RN the value of this function is writtenas F(~r , k)


Random Fields

RANDOM FIELDSKolmogorov’s definition

Definition

Let F be a family of random variables such that

F = {F(~ri , •) : (Ω, Υ) → (Rd , Bd ),~r ∈ RN} (1)

Then F is a random field if the distribution function P~r1,...,~rn(~x1, . . . , ~xn), ~x ∈ Rd satisfies the

following

1 Symmetry: P~r1,...,~rn (~x1, . . . , ~xn) is invariant under identical permutations of ~x and ~r .

2 Consistency: P~r1,...,~rn+m (B × Rmd ) = P~r1,...,~rn (B) for every n,m ≥ 1 and B ∈ B

nd

• Distribution function P~r1,...,~rn(~x1, . . . , ~xn) = P(. . . , (−∞, ~xi ], . . . ). Each semi-open interval(−∞, ~xi ] is d-dimensional.

• Distribution function P~r1,...,~rn(~x1, . . . , ~xn) is defined on Bnd


Random Fields

RANDOM FIELDS ON GRAPHS

• Let G , (S, E) be a graph; node set S = {s1, , s2, . . . , sN} and E is the edge set

• Let ∂ = {∂i}si∈S be the neighborhood system, where ∂i is the neighbor set for a node si .

• G = (S, E) and G = (S, ∂) are equivalent definitions. For a given edge set E ,∂i = {sk : (si , sk) ∈ E}

• Ω is the set of all states (or labels) that any node si can take

• K , ΩN is called the configuration space

Random field over graph

A random field over G = (S, E) is defined as

F = {Fi : Ω → Rd , i = 1, 2, . . . , N}

• k ∈ K is an ordered sequence (ω1, ω2, . . . , ωN) of N states


Random Fields

MARKOV RANDOM FIELDSRandom field on graphs

Definition

F is called a Markov Random Field (MRF), with respect to G(or equivalently ∂), if and only if itsatisfies the conditions:

1 Positivity: P(k) > 0,∀k ∈ K

2 Markov Property: P(ωi |kS\{i}) = P(ωi |∂i )

where kS\{i} is the configuration specified for the node set S \ {i} and P is the probabilitymeasure over the random field F .

• Markov random field evolves as a local process

• Allows to model contextual constraints or dependencies.


Random Fields

GIBBS RANDOM FIELDSRandom field on graphs

Definition

A random field F on a graph G is called a Gibbs random field(GRF) when the probabilitymeasure P on F follows the Gibbs distribution.

P(k) =1

Zexp(−βH(k)) (2)

where

• Z =∑

k∈K exp(−βH(k)) is the partition function and serves as a normalizing constant

• β is the inverse temperature

• H(k) is called the Hamiltonian and defines the energy of the configuration k ∈ K

• Gibbs Random Field evolves as a global process


Random Fields

MARKOV-GIBBS EQUIVALENCE

• The Hamiltonian or the energy function for k is defined as

H(k) =∑

A⊂S

VA(kA)

• VA are called a clique potentials if VA = 0 is A is not a clique

H(k) =∑

c∈C

Vc (kc )

• Corresponding Gibbs field it is called a neighbor Gibbs random field

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Theorem (MRF-GRF Equivalence)

Let the ∂ be the neighborhood system on a node-set S. Let F be a random field on S. Then Fis a Markov random field with respect to ∂ if and only if it is a neighbor Gibbs random field withrespect to ∂

Remarks

• Equivalence theorem essentially relates global and local behavior

• Provides a convenient way to model local Markov dependencies as clique potentials

• Joint density (global behavior) is given by the Gibbs distribution


Random Fields

MARKOV-GIBBS EQUIVALENCEProof

Proof that GRF =⇒ MRF

P(ωi |kS\{i}) =P(ωi , kS\{i})

P(kS\{i})=

P(k)∑

ω′i∈Ω P(k

′)(3)

P(ωi |kS\{i}) =exp (−βH(k))

∑

ω′i∈Ω exp (−βH(k

′))(4)

P(ωi |kS\{i}) =Λ(Ci , k)Λ(C̃i , k)

∑

ω′i∈Ω Λ(Ci , k

′)Λ(C̃i , k′)(5)

where

Λ(A, Q) = exp

−β∑

c∈A

Vc(Qc)

P(ωi |kS\{i}) =exp

(

−β∑

c∈CiVc (kc )

)

∑

ω′i∈Ω exp

(

−β∑

c∈CiVc(k′c )

) = P(ωi |k∂i ) (6)


Random Fields

MARKOV-GIBBS EQUIVALENCEProof (Contd . . . )

Proof that MRF =⇒ GRF [Besag]

P(k1)

P(k2)=

|S|∏

i=1

P(ω1i |ω11 , . . . , ω

1i−1, ω

2i+1, . . . , ω

2|S|

)

P(ω2i |ω11 , . . . , ω

1i−1, ω

2i+1, . . . , ω

2|S|

)(7)

requires the positivity condition P(k) > 0, ∀k ∈ KDefine

Q(k) ≡ ln

(

P(k)

P(0)

)

(8)

Theorem

For P satisfying the conditions of MRF there exists an expansion of Q(k) unique on K of thefollowing form:

Q(k) =∑

1≤i≤n

ωiGi (ωi ) +∑ ∑

1≤i

Random Fields

MAIN QUESTIONS & SOLUTIONSRandom Fields

Problems of interests

Problems of interest when dealing with a random field can be group as

• Sampling from the joint distribution P(k)Examples:

• Sensor networks• Multi-agent systems e.g. swarms

• Minimization of the Hamiltonian function H(~σ) over the configuration space KExamples:

• Optimization under local constraints• Image processing - MAP-MRF labeling for restoration of noisy images

• Expected values computation Examples:

• Physics - net magnetization• Social networks - opinion formation

Solution

• Exact solution of the joint density is usually difficult due to intractable Z

• Main solutions approaches are - Sampling methods and Variational approximation• Metropolis Algorithm• Simulated Annealing• Gibbs Sampling (Geman & Geman)• Variational approximations


Random Fields Approximate methods

APPROXIMATE METHODS

Definition (Kullback-Leibler (KL) divergence)

Let P and Q be two probability measures over K, then the relative entropy or theKullback-Leibler (KL) divergence is defined as:

DKL(Q||P) =∑

k

Q ln

(

Q

P

)

DKL(P||Q) ≥ 0; equality holds if and only if P and Q are identical

For a trial distribution Q and the Gibbs density P(k)

DKL(Q||P) = T lnZ + EQ[H] − TS(Q)

• EQ[H] is the variational energy• S(Q) is the entropy of Q• F ≡ −T lnZ is called the Helmholtz free energy• F (Q) = EQ[H] − TS(Q) is called the variational free energy

Optimization Problem

• Helmholtz free energy F ≤ F (Q), the variational free energy, since DKL(P||Q) ≥ 0

• True distribution can be recovered as P = arg minQ F (Q)


Random Fields Approximate methods

MEAN FIELD SOLUTIONSApproximate methods contd . . .

• For Q(k) =∏

i Qi (ωi ), approximation obtained is called the mean-field solution• All nodes in the system are assumed to be independent under mean-field Q• F (Q) can be optimized with respect to individual factors Qi to get mean-field equations

Qi (ωi ) =1

Ziexp

(

−βEQ[H(•|ωi )])

where Zi =∑

ωiexp(EQ[H(•|ωi )]) is the local partition function

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EQ [H(·|ω3)]

EQ [H(·|ω4)]

EQ [H(·|ω1)

EQ [H(·|ω2)

Remarks

• Mean-field solution gives the local update equations for a node si

• Local updates do not explicitly depend on the states of the rest of the network

• Local updates rely on the expectation (or mean-fields) EQ[H(•|ωi )] computed w.r.t. Q.

• Mean-field assumption transforms the underlying graph G to a new graph with no edges

• Iterative methods can be used to get fixed point or approximate solution Q∗

• Possibility of more than one fixed point due to non-linear local update equations


Random Fields Sensor networks

Collaboration in DSNsDeveloping the framework . . .

Objective

A framework for collaboration in DSNs based on statistical mechanics and random field approach, which is

• Robust

• Decentralized

• Local in Action, Global in Effect

• Scalable

• Resource-aware

• Dynamically adaptive and event-driven

• Real time

• Let P be a |Q|-simplex:

P =

p = (p1, . . . , p|Q|), pk ∈ R, pk ≥ 0,

|Q|∑

k=1

pk = 1

• Let the set of vertices V of the simplex be

V =

σ = (v1, . . . , v|Q|), vk ∈ {0, 1},

|Q|∑

k=1

vk = 1

• Let the sensor network be represented by a graph G

• Define a random field F = {Fi : Q → V, si ∈ S}

• E = {ǫ, e1, . . . , em−1} is the set of events

• ǫ = null event

• Let B = {b0, b1, . . . , bm−1} be the vertices of a |E|-simplex with b(ǫ) = b0




• Clique potentials are defined for inter-node and node-eventinteraction as follows:

H(~σ) = −∑

{i,j}∈C2

σT

i Jijσj −∑

{i}∈C1

σT

i Kibi

• Jij are |Q| × |Q| neighbor interaction matrices (NIM) forneighbor pairs {i , j} ∈ C2

• Ki are |Q| × |E| event response matrices (ERM) for nodes siσ

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89J

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Remarks

• Objective is to have local node dynamics that conform to the joint density

• Distributed sampling from the Gibbs distribution P is needed

• Methods such as Gibbs sampling require current states of neighbors

• Wireless communication problems prohibit such an approach

• A variational approximation is proposed for distributed sampling from P

• For independent updates, trial distribution Q =∏

i Qi (σi |bi ) is completely factorized

• Motivation is the graphical structure induced by the mean-field approximation




• Mean-field equations become

Qi (σi |bi ) =1

Ziexp

(

−βEQ[H(•|bi , σi )])

• Let Qi (σi |bi ) = pi be the state probability vector

• Since σi ∈ V , EQ[σi |bi ] = pi

• Mean-field equations are a map Ti (e, •) : P|∂i | → P

h(σi ; k) = −σT

i Kbi (τ) −∑

j∈∂i

σT

i Jijpj (k)

pi (σi ; k + 1) =exp(−βh(σi ; k))

∑

ℓ exp(−βh(σℓi ; k))

Remarks

• Node dynamics is assumed to be faster than the time-scale (τ) of events

• Ti (e, •) induce a continuous map TG(~e, •) : P|S| → P|S| on the entire sensor network

• Brouwer fixed point theorem states that TG(~e, •) has a fixed point TG(~e, ~p) = ~p

• Since the system is non-linear there maybe more than one fixed points

• For symmetric J, β is small enough ⇒ only fixed point attractors of TG(~e, •)

• Unique fixed point of TG(~e, •) for β < βc , a critical value



i -PFSAFramework for collaboration in DSNs

Definition (i-PFSA )

An interacting-Probabilistic Finite State Automata (i-PFSA ) is defined as the tupleAi = {Q, E, p, p

∗, Ti (e, {p}i )} where:

• Q is a finite set of states of the automata with Q = QC ∪ QNC ;

• E is a strictly partially ordered (using ≺) finite set of events;

• p ∈ P is defines the dynamics or the state visit probability vector;

• p∗ ∈ P is a vector of reference state visit probabilities; and

• Ti (e, {p}i )} define the local update dynamics

Remarks

• (A, G) is an i-PFSA network where A = {Ai , si ∈ S}

• i-PFSA runs at the top (application) layer of sensor nodes

• Probabilistic state transitions generate a state sequence

• Actions in each state may lead to the discovery of events

• Visiting set QC may give new dynamics of neighboring nodes

• New dynamics of i-PFSA are computed if required

q1

q2q3

Neighborhood interaction

State Sequence. . . q1q2 . . .

ActionsEvents


Adaptive Sensor Activity Scheduling Problem Description

Adaptive sensor activity schedulingProblem description

Problem description

• Sensor nodes are often operated on very low duty cycles

• Fixed duty-cycles good for data collection, not for detection of rare and random events

• Most power saving schemes available are geared towards communication

• Enabling detection of unpredictable events requires sensor nodes to be omni-active

• Energy costs for sensing are further aggravated if active sensors (e.g. radar) are in use

Methods in use

• Coordinated sleep/activity scheduling methods

• Un-coordinated or randomized scheduling

• A hierarchical approach of passive vigilance

• Switching between low resolution and a higher resolution sensing mode

Objective

Event driven adaptive scheduling of sensor activity to enable resource-aware detection andtracking using the i-PFSA framework


Adaptive Sensor Activity Scheduling Target Tracking

Adaptive Sensor Activity Scheduling (A-SAS)Target tracking

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Adaptive Sensor Activity Scheduling Target Tracking

DEMONSTRATIONAdaptive Sensor Activity Scheduling (A-SAS)

Click to start Click to start


SensorFieldMovie.aviMedia File (video/avi)

ContourMovie.aviMedia File (video/avi)

Adaptive Sensor Activity Scheduling Adaptive pattern tracking in multi-hop networks

Adaptive pattern tracking in multi-hop networks

• Q = {S + R, R, I}

• QC = {S + R, R}; QNC = {I}

• E = {ǫ,mEvent, tEvent}

• ǫ ≺ mEvent ≺ tEvent

• Dominance defined as pC ≥ pd

• p∗ = [0.3 0.1 0.6]T; p = [0.3 0.69 0.01]T; and

p = [0.99 0.005 0.005]T for mEvent and tEvent respectively

J =

w 0 00 w 00 0 w

• Local sinks determined via diffusion of sink-hop values

• mEvent is triggered at a node when it is the local sink

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REFERENCES

1 Markov Random Field Modeling in Computer Vision, Stan Z. Li, Springer-Verlag

2 Markov Random Fields and Their Applications, R. Kindermann and J.L. Snell,Contemporary Mathematics Series, American Mathematical Society (1980)

3 Spatial interaction and the statistical analysis of lattice systems, J. Besag, J. of the Royalsociety. Series B, Vol. 336, No. 2 (1974)

4 Stochastic relaxation, Gibbs distribution, and the Bayesian restoration of images, S.Geman and D. Geman, IEEE Trans. Pattern Analysis and Machine Intelligence, Vol. 6, No.6 (1984)

5 On the analysis of dirty pictures, J. Besag, J. of the Royal society. Series B, Vol. 48, No. 4(1986)


OverviewRandom FieldsApproximate methodsApplicationSensor networks

Adaptive Sensor Activity SchedulingProblem DescriptionTarget TrackingAdaptive pattern tracking in multi-hop networks

Documents

Random Fields - A short introduction · Random Fields RANDOM FIELDS Measure-theoretic deﬁnition Deﬁnition Let (K,K,P) be a measure space.Let GN,d be the set of all Rd-valued functions