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Bayesian Transformed GaussianRandom Field: A Review
Benjamin Kedem
Department of Mathematics & ISR
University of Maryland
College Park, MD
(Victor De Oliveira, David Bindel, Boris and
Sandra Kozintsev)
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Berger, De Oliveira, Sanso (2001). Objective
Bayesian Analysis of Spatially Correlated Data.
JASA, 96, 1361-1374.
(•) Bindel, De Oliveira, Kedem (1997). An
Implementation of the Bayesian Transformed
Gaussian Spatial Prediction Model.
http://www.math.umd.edu/ bnk/btg page.html.
Box, Cox (1964). An Analysis of Transforma-
tions (with discussion). JRSS, B 26, 211-252.
De Oliveira (2000). Bayesian Prediction of
Clipped Gaussian Random Fields. Comp. Data
Analysis, 34, 299-314.
(•) De Oliveira, Kedem, Short (1997). Bayesian
Prediction of Transformed Gaussian Random
Fields. JASA, 92, 1422-1433.
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Handcock, Stein (1993). A Bayesian Analysis
of Kriging. Technometrics, 35, 403-410.
(•) De Oliveira, Ecker (2002). Bayesian Hot
Spot Detection in the Presence of a Spatial
Trend: Application to Total Nitrogen Con-
centration in the Chesapeake Bay. Environ-
metrics, 13, 85-101.
(•) Kedem, Fokianos (2002). Regression Mod-
els for Time Series Analysis, New York: Wiley.
Kozintsev, Kedem (2000). Generation of ”Sim-
ilar” Images From a Given Discrete Image. J.
Comp. Graphical Stat., 2000, Vol. 9, No. 2,
286-302.
Kozintseva (1999). Comparison of Three Meth-
ods of Spatial Prediction. M.A. Thesis, De-
partment of Mathematics, University of Mary-
land, College Park.
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Spatial/temporal geostatistical data often dis-
play:
• Non-Gaussian skewed sampling distributions.
• Positive continuous data.
• Heavy right tails.
• Bounded support.
• Small data sets observed irregularly (gaps).
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Possible remedies:
• Bayesian Transformed Gaussian (BTG): A
Bayesian approach combined with paramet-
ric families of nonlinear transformations to
Gaussian data.
• BTG provides a unified framework for in-
ference and prediction/interpolation in a
wide variety of models, Gaussian and non-
Gaussian.
• Will describe BTG and illustrate it using
spatial and temporal data.
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Stationary isotropic Gaussian random field.
Let Z(s), s ∈ D ⊂ Rd, be a spatial process or
a random field.
A random field Z(s) is Gaussian if for all
s1, ..., sn ∈ D, the vector (Z(s1), ..., Z(sn)) has
a multivariate normal distribution.
Z(s) is (second order) stationary when for
s, s + h ∈ D we have
(•) E(Z(s)) = µ,
(•) Cov(Z(s + h), Z(s)) ≡ C(h).
The function C(·) is called the covariogram or
covariance function.
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We shall assume that C(h) depends only on
the distance ‖h‖ between the locations s + h
and s but not on the direction of h.
In this case the covariance function as well as
the process are called isotropic.
The corresponding isotropic correlation func-
tion is given by K(l) = C(l)/C(0), where l is
the distance between points.
Useful special case: Matern correlation
Kθ(l) =
12θ2−1Γ(θ2)
(
lθ1
)θ2κθ2
(
lθ1
)
if l 6= 0
1 if l = 0
where θ1 > 0, θ2 > 0, and κθ2 is a modified
Bessel function of the third kind of order θ2.
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Kθ(l) =
12θ2−1Γ(θ2)
(
lθ1
)θ2κθ2
(
lθ1
)
if l 6= 0
1 if l = 0
Matern (θ1 = 8, θ2 = 3).
2040
6080
100120
X20
40
60
80
100
120
Y
-4-3
-2-1
01
23
Z
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Spherical correlation:
Kθ(l) =
1 − 32
(
lθ
)
+ 12
(
lθ
)3if l ≤ θ
0 if l > θ
where θ > 0 controls the correlation range.
Spherical (θ = 120).
2040
6080
100120
X20
40
60
80
100
120
Y
-2-1
01
2Z
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Exponential correlation:
Kθ(l) = exp(lθ2 log θ1)
θ1 ∈ (0,1), θ2 ∈ (0,2]
Exponential (θ1 = 0.5, θ2 = 1).
2040
6080
100120
X20
40
60
80
100
120
Y
-4-2
02
46
Z
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Rational quadratic:
Kθ(l) =
(
1 + l2
θ21
)−θ2
θ1 > 0, θ2 > 0.
Rational quadratic (θ1 = 12, θ2 = 8).
2040
6080
100120
X20
40
60
80
100
120
Y
-4-2
02
4Z
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Clipped, at 3 levels, realizations from Gaussian
random fields. Top left: Matern (8,3). Top
right: spherical (120). Bottom left: exponen-
tial (0.5,1). Bottom right: rational quadratic
(12,8).
www.math.umd.edu/~bnk/bak/ generate.cgi?4
Kozintsev(1999), Kozintsev and Kedem(2000).
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Ordinary Kriging.
Given the data
Z ≡ (Z(s1), . . . , Z(sn))′
observed at locations s1, . . . , sn in D, the prob-
lem is to predict (or estimate) Z(s0) at loca-
tion s0 using the best linear unbiased predictor
(BLUP) obtained by minimizing
E(Z(s0) −n∑
i=1
λiZ(si))2 subject to
n∑
i=1
λi = 1
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Define
1 = (1,1, . . . ,1)′, 1 × n vector
c = (C(s0 − s1), . . . , C(s0 − sn))′
C = (C(si − sj)), i, j = 1, ..., n
λ = (λ1, λ2, . . . , λn)′
Then
λ = C−1
(
c +1 − 1′C−1c
1′C−111
)
.
The ordinary kriging predictor is then
Z(s0) = λ′Z.
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Define,
m =1 − 1′C−1c
1′C−11
and the kriging variance
σ2k(s0) = E(Z(s0) − Z(s0))
2 = C(0) − λ′c + m.
Under the Gaussian assumption,
Z(s0) ± 1.96σk(s0)
is a 95% prediction interval for Z(s0). For non-
Gaussian fields this may not hold.
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Bayesian Spatial Prediction: The BTG Model.
RF Z(s), s ∈ D observed at
s1, . . . , sn ∈ D
Parametric family of monotone transformations
G = gλ(.) : λ ∈ Λ.
⋆ Assumption: Z(.) can be transformed into
a Gaussian random field by a member of G.
A useful parametric family of transformations
often used in applications to ‘normalize’ posi-
tive data is the Box-Cox (1964) family of power
transformations,
gλ(x) =
xλ−1λ if λ 6= 0
log(x) if λ = 0.
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For some unknown ‘transformation parameter’
λ ∈ Λ, gλ(Z(s)), s ∈ D is a Gaussian random
field with
Egλ(Z(s)) =p∑
j=1
βjfj(s),
covgλ(Z(s)), gλ(Z(u)) = τ−1Kθ(s,u),
Regression parameters: β = (β1, . . . , βp)′
Covariates: f(s) = (f1(s), . . . , fp(s))
Variance: τ−1 = vargλ(Z(s))
Simplifying assumption: Isotropy,
Kθ(s,u) = Kθ(||s− u||)
θ = (θ1, . . . , θq) ∈ Θ ⊂ Rq.
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Data: Zobs = (Z1,obs, . . . , Zn,obs)
gλ(Zi,obs) = gλ(Z(si)) + ǫi ; i = 1, . . . , n,
ǫ1, . . . , ǫn are i.i.d. N(0, ξτ ).
Parameters: η = (β, τ, ξ, θ, λ).
Prediction problem:
Predict Z0 = (Z(s01), . . . , Z(s0k)) from the pre-
dictive density function, defined by
p(zo|zobs) =
∫
Ωp(zo, η|zobs)dη
=
∫
Ωp(zo|η, zobs)p(η|zobs)dη,
where Ω = Rp × (0,∞)2 × Θ × Λ.
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Notation: For a = (a1, . . . , an), we write
gλ(a) ≡ (gλ(a1), . . . , gλ(an)).
The Likelihood:
L(η; zobs) =
(
τ
2π
)n2∣
∣
∣Ψξ,θ
∣
∣
∣
−12 exp
−τ
2Q
Jλ,
Q =(
gλ(zobs) − Xβ)′
Ψ−1ξ,θ
(
gλ(zobs) − Xβ)
.
X n × p design matrix, Xij = fj(si).
Ψξ,θ = Σθ + ξI, n × n matrix.
Σθ;ij = Kθ(si, sj).
I identity matrix.
Jλ =∏n
i=1 |g′(zi,obs)|, the Jacobian.
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The Prior.
Insightful arguments in Box and Cox(1964),
De Oliveira, Kedem, Short (1997), as well as
practical experience lead us to the prior
p(η) ∝p(ξ)p(θ)p(λ)
τJpnλ
,
where p(ξ), p(θ) and p(λ) are the prior marginals
of ξ, θ and λ, respectively, which are assumed
to be proper.
Unusual prior: it depends on the data through
the Jacobian.
For more on prior selection see Berger, De
Oliveira, Sanso (2001).
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Simplifying assumption: No measurement noise
(ξ = 0).
gλ(Zi,obs) = gλ(Z(si)), i = 1, . . . , n,
p(β, τ, θ, λ) ∝p(θ)p(λ)
τJp/nλ
(1)
η = (β, τ, θ, λ)′.
Also, write
z = zobs
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The Posterior.
p(η|z) = p(β, τ, θ, λ|z) = p(β, τ |θ, λ, z)p(θ, λ|z).
To get the first factor:
(β|τ, θ, λ, z) ∼ Np(βθ,λ,1
τ(X′
Σ−1θ
X)−1)
(τ |θ, λ, z) ∼ Ga(n − p
2,
2
qθ,λ
)
where
βθ,λ = (X′Σ
−1θ
X)−1X
′Σ
−1θ
gλ(z)
qθ,λ = (gλ(z) − Xβθ,λ)′Σ
−1θ
(gλ(z) − Xβθ,λ).
and
p(β, τ |θ, λ, z) = p(β|τ, θ, λ, z)p(τ |θ, λ, z)
is Normal-Gamma.
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To get the second factor:
p(θ, λ|z) ∝
|Σθ|−1/2|X′
Σ−1θ
X|−1/2q−n−p
2
θ,λJ1−p
nλ p(θ)p(λ)
In addition to the joint posterior distribution
p(η|z) derived above, the predictive density p(zo|z)
also requires p(zo|η, z). We have
p(zo|η, z) = (τ
2π)k/2|Dθ|
−1/2k∏
j=1
|g′λ(zoj)|
× exp
−τ
2(gλ(zo) − Mβ,θ,λ)
′D
−1θ
(gλ(z) − Mβ,θ,λ)
where Mβ,θ,λ,Dθ are known.
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We now have the integrand p(zo|η, z)p(η|z) needed
for p(zo|z). By integrating out β and τ we ob-
tain the simplified form of the predictive den-
sity:
p(zo|z) =
∫
Λ
∫
Θp(zo|θ, λ, z)p(θ, λ|z)dθdλ
=
∫
Λ
∫
Θ p(zo|θ, λ, z)p(z|θ, λ)p(θ)p(λ)dθdλ∫
Λ
∫
Θ p(z|θ, λ)p(θ)p(λ)dθdλ
where
p(zo|θ, λ, z) =Γ(n−p+k
2 )∏k
j=1 |g′λ(zoj)|
Γ(n−p2 )πk/2|qθ,λCθ|
1/2
×[1 + (gλ(zo) − mθ,λ)′(qθ,λCθ)−1
×(gλ(zo) − mθ,λ)]−n−p+k
2
and from Bayes theorem,
p(z|θ, λ) ∝ |Σθ|−1/2|X′
Σ−1θ
X|−1/2q−n−p
2
θ,λJ1−p
nλ
where mθ,λ,Cθ are known.
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BTG Algorithm:
Predictive Density Approximation
1. Let S = z(j)o : j = 1, . . . , r be the set
of values obtained by discretizing the effective
range of Z0.
2. Generate independently θ1, . . . , θm i.i.d. ∼
p(θ) and λ1, . . . , λm i.i.d. ∼ p(λ).
3. For zo ∈ S, the approximation to p(zo|z) is
given by
pm(zo|z) =
∑mi=1 p(zo|θi, λi, z)p(z|θi, λi)
∑mi=1 p(z|θi, λi)
p(zo|θ, λ, z) and p(z|θ, λ) given above.
(⋆) Z0 = Median of (Z0|Z)
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Software: tkbtg application. Hybrid of C++,
Tcl/Tk, and FORTRAN 77 (Bindel et al (1997)).
http://www.math.umd.edu/~bnk/btg_page.html
The tkbtg Interface Layout
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Example: Spatial Rainfall Prediction
Rain gauge positions and weekly rainfall to-
tals in mm, Darwin, Australia, 1991.
x
y
0 2 4 6 8 10 12
02
46
810
12
29.55 41.85
26.7633.98
31.27
54.35 30.57
22.8266.76 33.68 19.31
23.6963.1 46.06
24.47
28.9868.2526.83 18.24
31.14
21.717.72
29.92 23.6
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1. Use the Box-Cox transformation family.
2. λ ∼ Unif(−3,3).
3. m = 500.
4. Correlation: Matern and exponential.
5. No covariate information. Assume constant
regression: Egλ(Z(s)) = β1.
6. Data apparently not normal.
10 20 30 40 50 60 70
0.0
0.01
0.02
0.03
0.04
Total Rainfall in mm
Mean=33.94
Median=29.73
SD=15.06
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No. z z 95% PI
1 29.55 33.74 (10.70, 56.78)2 41.85 34.32 (15.29, 53.35)3 26.76 36.71 (20.93, 52.49)4 33.98 33.39 (18.49, 48.29)5 31.27 32.99 (9.53, 56.45)6 54.35 39.13 (16.46, 61.79)7 30.57 24.45 (15.40, 33.59)8 22.82 23.09 (11.52, 34.66)9 66.76 64.12 (28.25, 100)
10 33.68 35.16 (18.01, 52.30)11 19.31 24.51 (15.62, 33.40)12 23.69 26.45 (15.35, 37.54)13 63.10 72.07 (44.14, 100)14 46.06 40.60 (18.58, 62.63)15 24.47 22.32 (14.00, 30.63)16 28.98 21.62 (13.43, 29.81)17 68.25 46.54 (19.25, 73.84)18 26.83 29.52 (16.57, 42.46)19 18.24 19.00 (10.79, 27.21)20 31.14 37.36 (20.33, 54.39)21 21.70 22.97 (14.71, 31.22)22 17.72 22.69 (11.64, 33.74)23 29.92 26.56 (12.21, 40.91)24 23.60 21.83 (10.85, 32.81)
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Spatial prediction and contour maps from theDarwin data using Matern correlation.
0
2
4
6
8
10
X
0
2
4
6
8
10
Y
1020
3040
5060
7080
Z
x
y
0 2 4 6 8 10 12
02
46
810
12
20
25
30
30
30
30
30
35 35
40
40
45
50
55
60
65
70
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Spatial prediction and contour maps from theDarwin data using exponential correlation.
0
2
4
6
8
10
X
0
2
4
6
8
10
Y
1020
3040
5060
7080
Z
x
y
0 2 4 6 8 10 12
02
46
810
12
20
25
25
25
30
30
35 3540
40
40
45
45
50
50
55
55
6060
65
70
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Predictive densities, 95% prediction intervals,
and cross-validation: Predicting a true value
from the remaining 23 observations using Matern
correlation. The vertical line marks the loca-
tion of the true value.
x
p(x)
0 20 40 60 80 100
0.0
0.04
0.08
0.12
True: 41.85Median: 34.31(15.60, 53.02)
x
p(x)
0 20 40 60 80 100
0.0
0.04
0.08
0.12
True: 63.10Median: 70.76(41.52, 100)
x
p(x)
0 20 40 60 80 100
0.0
0.04
0.08
0.12
True: 28.98Median: 22.1(13.26, 30.94)
x
p(x)
0 20 40 60 80 100
0.0
0.04
0.08
0.12
True: 21.70Median: 23.05(14.84, 31.26)
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Comparison of BTG With Kriging and Trans-
Gaussian kriging (Kozintseva (1999)).
Cross validation results using artificial data on
50 × 50 grid.
Data obtained by transforming a Gaussian (0,1)
RF using inverse Box-Cox transformation.
In Kriging and TG kriging λ, θ, were known.
Not in BTG (!)
λ = 0: Log-Normal.
λ = 1: Normal.
λ = 0.5: Between Normal and Log-Normal.
In most cases BTG has more reliable but larger
prediction intervals.
BTG predicts at the original scale. TG kriging
does not.
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Matern(1,10)
λ 0 0.5 1
KRG MSE 68397.48 7.15 0.58TGK MSE 55260.90 7.08 0.58BTG MSE 64134.30 7.31 0.56
KRG AvePI 2.42 2.51 2.42TGK AvePI 291.80 8.21 2.42BTG AvePI 330.68 10.23 2.87
KRG % out 100% 48% 6%TGK % out 18% 8% 6%BTG % out 12% 6% 6%
Exponential(e−0.03,1)λ 0 0.5 1
KRG MSE 12212.32 1.83 0.13TGK MSE 11974.73 1.84 0.13BTG MSE 12520.70 1.89 0.14
KRG AvePI 1.45 1.43 1.45TGK AvePI 267.92 5.24 1.45BTG AvePI 466.69 6.10 1.63
KRG % out 98% 64% 2%TGK % out 20% 4% 2%BTG % out 6% 2% 2%
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Application of BTG to Time Series Prediction.
Short time series observed irregularly.
Set: s = (x, y) = (t,0).
Can predict/interpolate as in state space pre-
diction: k–step prediction forward, backward,
and “in the middle”.
Example 1: Monthly data of unemployed women
20 years of age and older, 1997–2000. Data
source: Bureau of Labor Statistics. N = 48.
Example 2: Monthly airline passenger data,
1949–1960. Data source: Box-Jenkins (1976).
Use only N = 36 out of 144 observations,
t = 51, ...,86.
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Example: Prediction of Monthly number of un-
employed women (Age ≥ 20), 1997–2000. Data
in hundreds of thousands.
Cross validation and 95% prediction intervals.
Observations at times t = 12,13,36 are out-
side their 95% PI’s. N = 48 − 1 = 47.
t
No.
Une
mpl
oyed
Wom
en
0 10 20 30 40
1015
2025
3035
TruePredictedLowerUpper
Unemployed Women 1997--2000
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Forward and backward one and two step pre-
diction in the unemployed women series.
In 2–step higher dispersion.
x
p(x)
10 20 30 40 50
0.0
0.05
0.10
0.15
0.20
True: 18.34Median: 21.78(17.24, 26.33)
One step forward
x
p(x)
10 20 30 40 50
0.0
0.05
0.10
0.15
0.20
True: 18.34Median: 22.20(16.36, 28.05)
Two step forward
x
p(x)
10 20 30 40 50
0.0
0.05
0.10
0.15
0.20
True: 28.98Median: 26.17(21.94, 30.40)
One step backward
x
p(x)
10 20 30 40 50
0.0
0.05
0.10
0.15
0.20
True: 28.98Median: 24.69(19.77, 29.61)
Two step backward
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Example: Time series of monthly international
airline passengers in thousands, January 1949-
December 1960. N=144. Seasonal time se-
ries.
t
No.
Pas
seng
ers
0 20 40 60 80 100 120 140
100
200
300
400
500
600
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BTG cross validation and prediction intervals
for the monthly airline passengers series, t =
51, ...,86, using Matern correlation. Observa-
tions at t = 62,63 are outside the PI. N =
36 − 1 = 35.
t
No.
Pas
seng
ers
50 60 70 80
150
200
250
300
350
400
TruePredictedLowerUpper
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Application to Rainfall: Heuristic Argument.
Let Xn represent the area average rain rate
over a region such that
Xn = mXn−1 + λ + ǫn, n = 1,2,3, · · · ,
where the noise ǫn is a martingale difference.
It can be argued that necessary conditions for
Xn → LogNormal, n → ∞,
are that m → 1− and λ → 0+.
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The monotone increase in m (Curve a) and the
monotone decrease in λ (Curve b) as a func-
tion of the square root of the area. Source:
Kedem and Chiu(1987).
This suggests that the lognormal distribution
as a model for averages or rainfall amounts
over large areas or long periods.
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It is interesting to obtain the posterior p(λ | z)
of λ, the transformation parameter in the Box-
Cox family, given the data, where the data are
weekly rainfall totals from Darwin, Australia.
With a uniform prior for λ, the medians of
p(λ | z) in 5 different weeks are:
Week 1 median = − 0.45 (to the left of 0).
Week 2 median = 0.45 (to the right of 0).
Week 3 median = 0.15 (Not far from 0).
Week 4 median = 0.20 (Not far from 0).
Week 5 median = 0.95 (Close to 1).
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Weekly posterior p(λ | z) of λ given rainfall
totals from Darwin, Australia, for 5 different
weeks.
(a)
lambda
dens
ity
-3 -2 -1 0 1 2 3
0.0
0.5
1.0
1.5
2.0
MAE = -0.45
(b)
lambda
dens
ity
-3 -2 -1 0 1 2 3
0.0
0.5
1.0
1.5
2.0
MAE = 0.45
(c)
lambda
dens
ity
-3 -2 -1 0 1 2 3
0.0
0.5
1.0
1.5
2.0
MAE = 0.15
(d)
lambda
dens
ity
-3 -2 -1 0 1 2 3
0.0
0.5
1.0
1.5
2.0
MAE = 0.2
(e)
lambda
dens
ity
-3 -2 -1 0 1 2 3
0.0
0.5
1.0
1.5
2.0
MAE = 0.95
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