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Estimation of Spatially Correlated Random Fields
in Heterogeneous Wireless Sensor Networks
Ido Nevat
Sense & Sense-Abilities (S&S)
I2R
A*STAR
Joint work with Gareth Peters (UCL), Francois Spetier (Telecom1 Lille) and
Tomoko Matsui (ISM)
July 29, 2014
Ido Nevat
Random Field Reconstruction in WSN
Outline
1
Introduction to random processes
2
Wireless sensor network system model
3
Estimation goals and criteria
4
Algorithms development
5
Simulations
6
Conclusions
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Stochastic Processes and Random Fields
Definition (Stochastic process)
Given a parameter space X , a stochastic process f over X is a
collection of random variables
{f (x) : x ∈ X } .
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{f (x) : x ∈ X } .
Definition (Gaussian Random Field)
A random field f on a parameter set X for which the (finite
dimensional) distributions of (f (x1 ) , · · · , f (xk )) are multivarite
Gaussian for each 1 ≤ k ≤ ∞ and each (x1 , . . . , xk ) ∈ X k .
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{f (x) : x ∈ X } .
Definition (Gaussian Random Field)
A random field f on a parameter set X for which the (finite
dimensional) distributions of (f (x1 ) , · · · , f (xk )) are multivarite
Gaussian for each 1 ≤ k ≤ ∞ and each (x1 , . . . , xk ) ∈ X k .
Gaussian random fields are determined by their mean and
covariance functions:
µ (·; θ) , E [f (·)] : Rn 7→ R
C (·, ·; Ψ) , E [(f (·) − µ (·; θ)) (f (·) − µ (·; θ))] : Rn × Rn 7→ R
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Covariance functions
The covariance function is a measure of similarity and smoothness
of the random field.
Some common covariance functions:
1
2
3
C (x1 , x2 ; Ψ) = xT
1 x2
θ |x2 −x1 | 2
Exponential: C (x1 , x2 ; Ψ) = exp −
θ1
Linear:
Matérn:
C (x1 , x2 ; Ψ) =
21−ν
Γ(ν)
√
2ν|x2 −x1 |
l
ν
Kν
√
2ν|x2 −x1 |
l
James Mercer
Bertil Matérn
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Example: Gaussian Processes with exponential kernel
θ |x2 −x1 | 2
C (x1 , x2 ; Ψ) = exp −
θ1
1
Correlation
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Example: 2-D Gaussian Processes
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Why Gaussian ?
A few good reasons for using Gaussian Random Fields:
Good approximation for many physical phenomena found in
nature (ecology, geology, epidemiology, geography, image
analysis, meteorology, forestry, geosciences....)
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Why Gaussian ?
Fully characterized with two moments
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Why Gaussian ?
Likelihood accessible (conjugate model)
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Why Gaussian ?
Conditional expectation is linear
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Why Gaussian ?
Conditional expectation is linear
Stability under linear combinations, marginalization and
conditioning
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The problem
The Random Field Reconstruction problem:
Given observations from sensors which are deployed in the
field, to perform estimation regarding some attributes of the
field at un-monitored locations.
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The problem
If the observations are “Analog” (linear transformation of the
intensity of the field + additive Gaussian noise), inference via
Gaussian Process regression is trivial to perform.
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The problem
In many cases, it’s impossible to place “Analog” sensors in
locations of interest, due to transmission power constraint etc.
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The problem
In many cases, it’s impossible to place “Analog” sensors in
locations of interest, due to transmission power constraint etc.
Instead, it is possible to place “Digital” sensors in problematic
locations.
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Heterogeneous sensor network deployment
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Heterogeneous sensor network deployment
Our goal is to develop a new approach to fuse mixed
analog/digital observations in order to perform spatial field
reconstruction.
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System Model
A1 A random spatial phenomenon defined over a 2-dimensional space
X ∈ R2 . The mean of the physical process is modelled by a smooth
continuous spatial function f(·) : X 7→ R, modelled a-priori as a
Gaussian Process:
F := f (·) : R2 7→ R s.t. f (·) ∼ GP (µ (·; θ) , C (·, ·; Ψ)) ,
with µ (·; θ) : R2 7→ R, and C (·, ·; Ψ) : R2 × R2 7→ R .
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System Model
Gaussian Process:
F := f (·) : R2 7→ R s.t. f (·) ∼ GP (µ (·; θ) , C (·, ·; Ψ)) ,
A2 Let N be the number of sensors that are deployed over a 2-D region
X ⊆ R2 , with xn ∈ X , n = {1, · · · , N}, the physical location of the
n-th sensor, assumed known by the FC. The number of analog and
digital sensors is NA and ND , respectively, so that N = NA + ND .
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System Model
Gaussian Process:
F := f (·) : R2 7→ R s.t. f (·) ∼ GP (µ (·; θ) , C (·, ·; Ψ)) ,
A2 Let N be the number of sensors that are deployed over a 2-D region
X ⊆ R2 , with xn ∈ X , n = {1, · · · , N}, the physical location of the
n-th sensor, assumed known by the FC. The number of analog and
digital sensors is NA and ND , respectively, so that N = NA + ND .
A3 Sensors measurement model: each sensor collects a noisy
observation of the spatial phenomenon f (·). At the n-th sensor, the
observation is expressed as:
Z (xn ) = f (xn ) + Wn , n = {1, · · · , N}
2
.
where Wn is i.i.d Gaussian noise Wn ∼ N 0, σW
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System Model
A4 Analog sensors processing: each of the NA analog sensors
transmits its noisy observation to the FC over AWGN channels:
YnA = Z (xn ) + Vn , n = {1, . . . , NA } ,
where Vn is i.i.d Gaussian noise Vn ∼ N 0, σV2 .
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System Model
A4 Analog sensors processing: each of the NA analog sensors
transmits its noisy observation to the FC over AWGN channels:
YnA = Z (xn ) + Vn , n = {1, . . . , NA } ,
where Vn is i.i.d Gaussian noise Vn ∼ N 0, σV2 .
A5 Digital Sensors processing:
1 Thresholding: at the n-th digital sensor, n = {1, . . . , ND }, a
thresholding process is given as follows:
B(xn )=1
Z (xn )
≥
<
λ,
B (xn ) = −1
2
where λ is a pre-defined threshold.
Wireless Communications to Fusion Centre Model: the
decision B (xn ) is transmitted to the FC over imperfect binary
wireless channels, with transition probabilities
p0,0 , p0,1 , p1,0 , p1,1 .
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Example
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Example
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Realisation from a 1-D Gaussian Process
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Example
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Example
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Example
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Field reconstruction
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Estimation Objectives
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1 Objective I: MMSE spatial random field reconstructionAccurately reconstruct (i.e. estimate) the spatial random field at
un-monitored locations, x∗ ∈ Ω, from samples collected by the N
sensors . The Minimum Mean Squared Error (MMSE) utilises the
following distortion metric:
2 b
b
D f∗ , f∗ := E f∗ − f∗
.
The optimal solution in the sense of minimising this distortion
metric is the posterior predictive mean, given by the solution to the
following integral:
Z
b
f∗ = E [f∗ |xN , x∗ , YN ] = f∗ p (f∗ |x∗ , xN , YN ) df∗ .
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2 Objective II: spatial exeedance mapConstruct a spatial exceedance map estimation is quantified by the
following metric: find a region De ⊂ Ω such that, with a certain
given probability, f (x) ≥ T for all x ∈ De for a given level T :
De = {x : P (f∗ ≥ T |xN , x∗ , YN ) ≥ 1 − α}
Z ∞
p (f∗ |xN , x∗ , YN ) df∗ ≥ 1 − α ,
= x:
T
where T is a pre-defined threshold and α is the confidence level and
De is the domain or set of x values satisfying the exceedance of the
spatial field.
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3 Objective III: Spatial ClassificationPredict the confidence for each class at un-monitored
locations, x∗ ∈ Ω. That means that we find the classifier
b∗ : Ω ↔ {0, 1} that minimizes the error probability
B
b∗ , at an arbitrary location x∗ ∈ X .
P B∗ 6= B
This requires the calculation of the binary conditional
predictive distribution in closed form, given by:
P (B∗ = 0|x∗ , xN , YN , λ) =
P (B∗ = 1|x∗ , xN , YN , λ) =
Z
Z
P (B∗ = 0|f∗ , x∗ , xN , YN , λ) p (f∗ |x∗ , xN , YN ) df∗ ,
P (B∗ = 1|f∗ , x∗ , xN , YN , λ) p (f∗ |x∗ , xN , YN ) df∗ .
and the classifier
b∗ =
B
(
1
0
, P (B∗ |x∗ , xN , YN ) ≥ λ
, P (B∗ |x∗ , xN , YN ) < λ.
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The common feature of Objectives 1 − 3 is the posterior
predictive distribution p (f∗ |x∗ , xN , YN ).
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The common feature of Objectives 1 − 3 is the posterior
predictive distribution p (f∗ |x∗ , xN , YN ).
Z Z
p (f∗ |x∗ , xN , YN ) = . . .
p (f∗ |fN , x∗ , xN ) p (fN |xN , YN ) dfN
N
Z ZR
= ...
p (f∗ |fN , x∗ , xN ) p (fA |fD , xN , YN ) p (fD |xN , YN ) dfN
RN
1
p (f∗ |fN , x∗ , xN ): conditional predictive prior distribution.
2
p (fA |fD , xN , YN ): posterior distribution for the spatial phenomenon at
the analog sensor locations given observations.
3
p (fD |xN , YN ): posterior distribution for the spatial phenomenon at the
digital sensor locations given observations.
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Z Z
p (f∗ |x∗ , xN , YN ) = . . .
p (f∗ |fN , x∗ , xN )p (fA |fD , xN , YN ) p (fD |xN , YN ) dfN
RN
Lemma
The conditional predictive prior distribution, p (f∗ |fN , x∗ , xN ), is
given by:
p (f∗ |fN , x∗ , xN ) = N f∗ ; µf∗ |fN , σf2∗ |fN
µf∗ |fN : = µ (x∗ ) + k (x∗ , xN ) K−1 (xN , xN ) (fN − µ (xN ))
σf2∗ |fN : = k (x∗ , x∗ ) − k (x∗ , xN ) K−1 (xN , xN ) k (xN , x∗ )
Ido Nevat
Z Z
p (f∗ |x∗ , xN , YN ) = . . .
p (f∗ |fN , x∗ , xN ) p (fA |fD , xN , YN )p (fD |xN , YN ) dfN
RN
Lemma
The conditional distribution, p (fA |fD , xN , YN ), is given by:
p (fA |fD , xN , YN ) = N fA ; µfA |fD ,YN , ΣfA |fD ,YN
−1 −2
−1
−2
µfA |fD ,YN := Σ−1
+
σ
µ
I
Σ
+
σ
Y
A
f
|f
W
W
fA |fD
fA |fD A D
−1
−2
.
ΣfA |fD ,YN := Σ−1
fA |fD + σW I
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The posterior distribution
Z Z
p (f∗ |x∗ , xN , YN ) = . . .
p (f∗ |fN , x∗ , xN ) p (fA |fD , xN , YN ) p (fD |xN , YN )dfN
RN
Using Bayes’ law, the posterior distribution for the spatial
phenomenon at the digital sensor locations is given by
p (fD |xN , YN ) =
Ido Nevat
Z Z
p (f∗ |x∗ , xN , YN ) = . . .
RN
p (fD |xN , YN ) =
P (YN |xN , fD ) p (fD |xN )
P (YN |xN )
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Z Z
p (f∗ |x∗ , xN , YN ) = . . .
RN
P (YN |xN )
=R R
. . . RN P (YN |xN , fD ) p (fD |xN ) dfD
p (fD |xN , YN ) =
The numerator can be easily evaluated.
However, the denominator cannot be evaluated pointwise.
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Saddle Point Approximations for the predictive distribution
We approximate p (fD |xN , YN ) using using a series expansion of
the Saddle-point (Laplace) type via a Gaussian basis.
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We approximate p (fD |xN , YN ) using using a series expansion of
the Saddle-point (Laplace) type via a Gaussian basis.
This transforms the intractable multiple integrals to produce
simple closed form expressions. Based on these expressions we
derive new algorithms and provide closed form solutions.
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The series expansion becomes:
p (fD |xN , YN )
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p (fD |xN , YN )
= exp
log (p(fD |xN ,YN ))
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p (fD |xN , YN )
= exp
= exp
T
MAP + 1 f −b
fD
g (b
) 2 ( D fDMAP ) ∇2 g |bf MAP (fD −bfDMAP )
D
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e
expR3 (fD )
p (fD |xN , YN )
= exp
= exp
=
T
MAP + 1 f −b
fD
g (b
D
1
N
1/2
1
bMAP
exp− 2 (fD −fD )
T
e
expR3 (fD )
MAP
fD
H −1 (fD −b
)
(2π) |H|
N
1/2
e
bMAP
× exp(g (fD )+R3 (fD )+log((2π) |H| ))
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p (fD |xN , YN )
= exp
= exp
=
T
MAP + 1 f −b
fD
g (b
D
1
N
1/2
1
bMAP
T
e
expR3 (fD )
MAP
fD
H −1 (fD −b
)
(2π) |H|
N
1/2
e
bMAP
× exp(g (fD )+R3 (fD )+log((2π) |H| ))
where H −1 := −∇2 g |bf MAP .
D
Ido Nevat
p (fD |xN , YN )
= exp
= exp
=
T
MAP + 1 f −b
fD
g (b
D
1
N
1/2
1
bMAP
T
e
expR3 (fD )
MAP
fD
H −1 (fD −b
)
(2π) |H|
N
1/2
e
bMAP
× exp(g (fD )+R3 (fD )+log((2π) |H| ))
where H −1 := −∇2 g |bf MAP .
D
We obtain that the posterior distribution can be expressed as
N
1/2
bMAP
e
MAP
p (fD |xN , YN ) = N fD ; bfD
, H exp(g (fD )+R3 (fD )+log((2π) |H| ))
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Theorem
The posterior distribution at the digital sensors, p (fD |xN , YN ):
MAP
log p (fD |xN , YN ) = log q fD ; bfD
, H + R3 (fD ) .
where
MAP
MAP
q fD ; bfD
, H = N fD ; bfD
, H −1 ,
bf MAP = arg max p (fD |xN , YN ) ,
D
fD
∂2
p (fD |xN , YN ) |bf MAP ,
[H]i,j = −
D
∂fi ∂fj
MAP
e3 (fD ) + log (2π)n |H|1/2
+R
R3 (fD ) = g bfD
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Obtaining the MAP estimate
The MAP estimate is given by:
bf MAP = arg max p (fD |YN , xN )
D
fD
= arg max P (YN |fD , xN ) p (fD )
fD
= arg max P (YD |fD , YA , xN ) p (YA |fD , xN ) p (fD )
fD
= arg max
fD
X
ND
n=1
1
X
P YnD |Bn = l P (Bn = l|fn )
log
l=0
!
2
INA + ΣfA |fD
+ log N YA ; µfA |fD , σV2 + σW
+ log N (fD ; µ (xD ) , K (xD , xD )) .
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To solve this N-dimensional optimisation problem, we show that
the objective function is quasi-convex and can therefore be solved
exactly using any gradient based approach.
We utilse the Iterated Conditional on the Modes (ICM) of Besag
to solve this problem.
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Using ICM algorithm, the
MAP estimate of the n-th component of
MAP
fD , bfn = arg maxfn p fn |xN , bf1:ND \n , YN , can be evaluated by
solving the following one-dimensional non-linear equation:
2 (P (Y |B = 0) − P (Y |B = 1))
φ λ, f (xn ) , σW
n n
n n
2
P (Yn |Bn = 1) + Φ λ, f (xn ) , σW (P (Yn |Bn = 0) − P (Yn |Bn = 1))
−1
T
2
K (xA , xD ) K −1 (xD , xD )
σV2 + σW
INA + ΣfA |fD
= µfA |fD − YA
f (xn ) − µxn|fD \n
+
σx2n|f \n
D
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The posterior predictive distribution
Putting it all together.
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The posterior predictive distribution is approximated by
Z Z
p (f∗ |x∗ , xN , YN ) = . . .
RN
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Z Z
p (f∗ |x∗ , xN , YN ) = . . .
RN
Z Z
= ...
RN
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Z Z
p (f∗ |x∗ , xN , YN ) = . . .
RN
Z Z
= ...
N
Z ZR
MAP
, H −1 dfN
≈ ...
N f∗ ; µf∗ |fN , σf2∗ |fN N fA ; µfA |fD ,YN , ΣfA |fD ,YN N fD ; bfD
RN
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Z Z
p (f∗ |x∗ , xN , YN ) = . . .
RN
Z Z
= ...
N
Z ZR
MAP
, H −1 dfN
≈ ...
N f∗ ; µf∗ |fN , σf2∗ |fN N fA ; µfA |fD ,YN , ΣfA |fD ,YN N fD ; bfD
N
R
= N f∗ ; µf∗ |YN , σf2∗ |YN
where
µf∗ |YN = µ (x∗ ) + k (x∗ , xN ) K −1 (xN , xN ) µfN |YN − µ (xN ) ,
σf2∗ |YN = Σ22
f∗ ,fN |YN .
Ido Nevat
Spatial field reconstruction, exceedance level estimation
and spatial classification
Objective I: spatial MMSE random field reconstructionb
f∗ = E [f∗ |xN , x∗ , YN ]
Z
p (f∗ |x∗ , xN , YN ) df∗
≃ f∗ b
= µ (x∗ ) + k (x∗ , xN ) K −1 (xN , xN ) µfN |YN − µ (xN ) .
Objective II: spatial exeedence map:
b
f∗ = P (f∗ ≥ λ|xN , x∗ , YN ) ≃ 1 − Φ λ, µf∗ |YN , σf2∗ |YN .
Spatial Classification:
2
+ σf2∗ |YN ,
P (B∗ = 0|x∗ , xN , YN , λ) = Φ λ, µf∗ |YN , σW
2
+ σf2∗ |YN .
P (B∗ = 1|x∗ , xN , YN , λ) = 1 − Φ λ, µf∗ |YN , σW
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Simulations
Simulations
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Simulations
Spatial field reconstruction
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Spatial Field Reconstruction
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50
x
in WSN
0
100
10
90
20
80
30
70
40
60
50
50
y
y
60
40
70
30
80
20
90
100
10
0
10
20
30
40
50
60
70
80
90
100
0
0
10
20
30
40
0
10
10
20
20
30
30
40
40
50
50
60
60
70
70
80
80
90
100
50
60
70
80
90
100
60
70
80
90
100
x
0
y
y
x
90
0
10
20
30
40
50
x
60
70
80
90
Ido Nevat
100
100
0
10
20
30
40
50
x
in WSN
Simulations
Spatial Classification
Ido Nevat
0
10
20
30
y
40
50
60
70
80
90
100
0
10
20
30
40
50
60
70
80
90
100
x
Ido Nevat
0
0
10
10
20
20
30
30
40
40
50
50
y
y
60
60
70
70
80
80
90
100
90
0
10
20
30
40
50
60
x
Ido Nevat
70
100
80 0
90 10 10020
30
40
50
60
70
80
x
90
100
0
0
10
10
20
20
20
30
30
30
40
40
40
50
50
50
y
0
10
y
y
60
60
60
70
70
70
80
80
80
90
90
100
0
10
20
30
40
50
60
70
100
80 0
90
90 10 10020
x
30
40
50
60
70
100
80 0
90 10 10020
30
40
x
Ido Nevat
50
x
60
70
80
90
100
0
0
10
10
20
20
20
30
30
30
40
40
40
50
50
50
y
0
10
y
y
60
60
60
70
70
70
80
80
80
90
90
100
0
10
20
30
40
50
60
70
100
80 0
90
90 10 10020
x
30
40
50
60
70
100
80 0
90 10 10020
30
40
x
Ido Nevat
50
x
60
70
80
90
100
0
0
10
10
20
20
20
30
30
30
40
40
40
50
50
50
y
0
10
y
y
60
60
60
70
70
70
80
80
80
90
90
100
0
10
20
30
40
50
60
70
100
80 0
90
90 10 10020
x
30
40
50
60
70
100
80 0
90 10 10020
30
40
x
Ido Nevat
50
x
60
70
80
90
100
0
0
10
10
20
20
20
30
30
30
40
40
40
50
50
50
y
0
10
y
y
60
60
60
70
70
70
80
80
80
90
90
100
0
10
20
30
40
50
60
70
100
80 0
90
90 10 10020
x
30
40
50
60
70
100
80 0
90 10 10020
30
40
x
Ido Nevat
50
x
60
70
80
90
100
Simulations
Real deployment in Singapore
Ido Nevat
Field Reconstruction
Wireless sensor network deployed in Clementi to monitor acoustic
intensity (”noise”)
Ido Nevat
Sensors deployment
Ido Nevat
Sensors deployment
68
53
52
63
46
52
67
44
52
44
41
68
37
57
55
50
58
49
54
72
52
59
46
58
55
54
51
52
68
55
63
46
47
47
Ido Nevat
65
61
55
62
55
46
Random field reconstruction
35 analog sensors
Ido Nevat
35 analog + 5 digital sensors
Conclusions
1
Developed a new model for sensors networks with mixed
analog and digital (binary) sensors.
2
Derived the Laplace approximation to obtain the predictive
posterior density.
3
Developed the spatial field reconstruction, spatial
classification and spatial exceedance algorithms.
4
Simulations show the benefits of using digital sensors.
Ido Nevat
Questions?
Thanks very much!
Questions?
Ido Nevat

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