Random Fields
A short introduction
Abhishek Srivastav
Penn State Univ.(UP)
April 06, 2010
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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
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RANDOM FIELDS
Generalization 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 − R
measurable 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 measurable
mapping F : K → (RT )d is called a real-valued random field. Where, (RT )d is the space of all
Rd -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
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RANDOM FIELDS
Measure-theoretic definition
Definition
Let (K, K, P) be a measure space. Let G N,d be the set of all Rd -valued functions on RN ;
N, d ∈ N, and G N,d be the corresponding σ-algebra. Then a measurable map
F : (K, K) → (G N,d , G N,d ) is a called an N-dimensional random field.
• Random field F maps elements k of the sample space K to functions in G N,d . Equivalently
it maps sets in K to sets in G N,d
• G N,d contains sets of the form {g ∈ G N,d : g (~ri ) ∈ Bi , i = 1, . . . , m}, where m is arbitrary,
~ri ∈ RN and Bi ∈ B d
• Note: Sets of the form {g ∈ G N,d : g (~rα ) ∈ Bα , α ∈ I }, where I is uncountable are not
usually present in G N,d . Needs F to be separable.
• For a given k ∈ K, the corresponding function in G N,d is called a realization of the random
field and is denoted as F (•, k). At a given point ~r ∈ RN the value of this function is written
as F (~r , k)
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Random Fields
RANDOM FIELDS
Kolmogorov’s definition
Definition
Let F be a family of random variables such that
F = {F (~ri , •) : (Ω, Υ) → (Rd , B d ),~r ∈ RN }
Then F is a random field if the distribution function P~r1 ,...,~rn (~x1 , . . . , ~xn ), ~x ∈
following
(1)
Rd
satisfies the
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 B nd
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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
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MARKOV RANDOM FIELDS
Random field on graphs
Definition
F is called a Markov Random Field (MRF), with respect to G(or equivalently ∂), if and only if it
satisfies 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 probability
measure over the random field F .
• Markov random field evolves as a local process
• Allows to model contextual constraints or dependencies.
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GIBBS RANDOM FIELDS
Random field on graphs
Definition
A random field F on a graph G is called a Gibbs random field(GRF) when the probability
measure P on F follows the Gibbs distribution.
P(k) =
1
exp(−βH(k))
Z
(2)
where
• Z =
P
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
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MARKOV-GIBBS EQUIVALENCE
• The Hamiltonian or the energy function for k is defined as
H(k) =
X
VA (kA )
A⊂S
• VA are called a clique potentials if VA = 0 is A is not a clique
H(k) =
X
Vc (kc )
S1
S3
S1
S3
S1
S3
S2
S4
S2
S4
S2
S4
S1
S1
S3
S2
S4
S4
S3 S1
S2
S1
S2
S4
S3
S4 S2
S3
S1
S3
S2
S4
c∈C
• Corresponding Gibbs field it is called a neighbor Gibbs random field
Theorem (MRF-GRF Equivalence)
Let the ∂ be the neighborhood system on a node-set S. Let F be a random field on S. Then F
is a Markov random field with respect to ∂ if and only if it is a neighbor Gibbs random field with
respect 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
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Random Fields
MARKOV-GIBBS EQUIVALENCE
Proof
Proof that GRF =⇒ MRF
P(ωi |kS\{i } ) =
P(ωi , kS\{i } )
P(kS\{i } )
P(ωi |kS\{i } ) = P
P(ωi |kS\{i } ) = P
where
= P
P(k)
P(k′ )
(3)
ωi′ ∈Ω
exp (−βH(k))
exp (−βH(k′ ))
(4)
Λ(Ci , k)Λ(C˜i , k)
Λ(Ci , k′ )Λ(C˜i , k′ )
(5)
ωi′ ∈Ω
ωi′ ∈Ω

Λ(A, Q) = exp  −β
X
c∈A

Vc (Qc )
P
exp −β c∈Ci Vc (kc )
= P(ωi |k∂i )
P(ωi |kS\{i } ) = P
P
′
ω ′ ∈Ω exp −β
c∈Ci Vc (kc )
(6)
i
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MARKOV-GIBBS EQUIVALENCE
Proof (Contd . . . )
Proof that MRF =⇒ GRF [Besag]
|S|
2 )
P(ωi1 |ω11 , . . . , ωi1−1 , ωi2+1 , . . . , ω|S|
Y
P(k1 )
=
2 )
P(k2 )
P(ωi2 |ω11 , . . . , ωi1−1 , ωi2+1 , . . . , ω|S|
i =1
(7)
requires the positivity condition P(k) > 0, ∀k ∈ K
Define
P(k)
Q(k) ≡ ln
P(0)
(8)
Theorem
For P satisfying the conditions of MRF there exists an expansion of Q(k) unique on K of the
following form:
X
XX
Q(k) =
ωi Gi (ωi ) +
ωi ωj Gi ,j (ωi , ωj ) + · · ·
(9)
1≤i ≤n
1≤i <j≤n
the function Gi ,j,... may be arbitrary except that it maybe non-null if and only if the nodes i , j, . . .
form a clique.
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Random Fields
MAIN QUESTIONS & SOLUTIONS
Random 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 K
Examples:
• 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
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Approximate methods
APPROXIMATE METHODS
Definition (Kullback-Leibler (KL) divergence)
Let P and Q be two probability measures over K, then the relative entropy or the
Kullback-Leibler (KL) divergence is defined as:
X
Q
DKL (Q||P) =
Q ln
P
k
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 ln Z + EQ [H] − TS(Q)
•
•
•
•
EQ [H] is the variational energy
S(Q) is the entropy of Q
F ≡ −T ln Z 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)
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Approximate methods
MEAN FIELD SOLUTIONS
Approximate methods contd . . .
Q
• 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 ) =
where Zi =
P
ωi
1
exp −βEQ [H(•|ωi )]
Zi
exp(EQ [H(•|ωi )]) is the local partition function
S1
S3
S2
S4
EQ [H(·|ω1)
EQ [H(·|ω2)
S1
S3
S2
S4
EQ [H(·|ω3)]
EQ [H(·|ω4)]
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
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Random Fields
Sensor networks
Collaboration in DSNs
Developing the framework . . .
Objective
A framework for collaboration in DSNs based on statistical mechanics and random field approach, which is
• Robust
• Resource-aware
• Decentralized
• Dynamically adaptive and event-driven
• Local in Action, Global in Effect
• Real time
• Scalable
• Let P be a |Q|-simplex:


|Q|


X
P = p = (p1 , . . . , p|Q| ), pk ∈ R, pk ≥ 0,
pk = 1


k=1
• Let the set of vertices V of the simplex be


|Q|


X
V = σ = (v1 , . . . , v|Q| ), vk ∈ {0, 1},
vk = 1


k=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
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Sensor networks
Collaboration in DSNs
Developing the framework . . .
• Clique potentials are defined for inter-node and node-event
b4
b3
interaction as follows:
K
H(~
σ) = −
X
{i ,j}∈C2
σiT Jij σj
−
X
σiT Ki bi
K2
neighbor pairs {i , j} ∈ C2
• Ki are |Q| × |E| event response matrices (ERM) for nodes si
J12
σ4
34
J23
J25
K12
K9
σ6
J16
J47
J45
σ5
J58
J56
J69
σ10
σ7
J
σ1
b12
9
6
K6
σ2
K11
K8
b
b
σ3
• Jij are |Q| × |Q| neighbor interaction matrices (NIM) for
K10
b11
b8
K5
b1
b10
K7
b5
K1
{i }∈C1
b7
K4
3
b2
J78
σ8
σ9
J89
J811
J710
σ11
σ12
J1011
J1112
J912
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 =
Q
i
Qi (σi |bi ) is completely factorized
• Motivation is the graphical structure induced by the mean-field approximation
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Random Fields
Sensor networks
Collaboration in DSNs
Developing the framework . . .
• Mean-field equations become
Qi (σi |bi ) =
1
exp −βEQ [H(•|bi , σi )]
Zi
• 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)
=
−σiT Kbi (τ ) −
X
σiT Jij pj (k)
j∈∂i
pi (σi ; k + 1)
=
exp(−βh(σi ; k))
P
ℓ
ℓ exp(−βh(σi ; 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
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Random Fields
Sensor networks
i-PFSA
Framework for collaboration in DSNs
Definition (i-PFSA )
An interacting-Probabilistic Finite State Automata (i-PFSA ) is defined as the tuple
Ai = {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
Neighborhood interaction
• (A, G) is an i-PFSA network where A = {Ai , si ∈ S}
q1
• i-PFSA runs at the top (application) layer of sensor nodes
• Probabilistic state transitions generate a state sequence
q2
q3
State Sequence
. . . q1 q2 . . .
• Actions in each state may lead to the discovery of events
• Visiting set QC may give new dynamics of neighboring nodes
Events
Actions
• New dynamics of i-PFSA are computed if required
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Adaptive Sensor Activity Scheduling
Problem Description
Adaptive sensor activity scheduling
Problem 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 and
tracking using the i-PFSA framework
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Adaptive Sensor Activity Scheduling
Target Tracking
Adaptive Sensor Activity Scheduling (A-SAS)
Target tracking
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Adaptive Sensor Activity Scheduling
Target Tracking
DEMONSTRATION
Adaptive Sensor Activity Scheduling (A-SAS)
Click to start
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Adaptive Sensor Activity Scheduling
Adaptive pattern tracking in multi-hop networks
Adaptive pattern tracking in multi-hop networks
• 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


w
0
0

0
w
0 
J=
0
0
w
• Q = {S + R, R, I }
• QC = {S + R, R}; QNC = {I }
• E = {ǫ, mEvent, tEvent}
• ǫ ≺ mEvent ≺ tEvent
• Dominance defined as p C ≥ pd
• 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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Adaptive Sensor Activity Scheduling
Adaptive pattern tracking in multi-hop networks
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 Royal
society. 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)
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