Supplemental Web Material: Hierarchical
Bayesian Spatio-Temporal Conway-Maxwell
Poisson Models with Dynamic Dispersion
Guohui Wu1 , Scott H. Holan2 , and Christopher K. Wikle2
Appendix A: Markov chain Monte Carlo Algorithm
We describe the full conditionals and the Markov chain Monte Carlo (MCMC) algorithm for
the hierarchical Bayesian spatio-temporal CMP model with dynamic dispersion. Note that
migration to other models is analogous and, therefore, not presented here. Recall that the
model can be summarized using the following formulation:
• Data model
Y t |λt , wt ∼ CMP(Kt λt , exp(wt )).
• Process model
log(λt ) = µ + Ψαt + t ,
αt




H1 αt−1 + ηt



=
H2 αt−1 + ηt





H3 αt−1 + ηt
if ct < dL
if dL ≤ ct ≤ dU .
(A.1)
if ct > dU
• Parameter model
µ|β, σµ2 ∼ Gau(Xβ, σµ2 In ),
wt = φ0 + φ1 wt−1 + ξt .
1
(to whom correspondence should be addressed) Department of Statistics, University of Missouri, 146
Middlebush Hall, Columbia, MO 65211-6100, [email protected]
2
Department of Statistics, University of Missouri-Columbia,146 Middlebush Hall, Columbia, MO 652116100
Using Baye’s theorem and assuming conditional independence of the log intensity function
and the parameters, the joint posterior distribution of the processes and parameters given
the observed data can be expressed as
[λ1 , . . . , λT , H1 , H2 , H3 , α0 , . . . , αT , Ση , µ, β, w1 , . . . , wT , φ0 , φ1 , σ2 , σµ2 , σξ2 ]
∝
T
Y
[Y t |λt , wt ][λt |µ, αt , σ2 ][αt |αt−1 , H1 , H2 , H3 , Ση ][wt |wt−1 , φ0 , φ1 , σξ2 ]
t=1
× [µ|β, σµ2 ][H1 ][H2 ][H3 ][Ση ][α0 ][β][φ0 ][φ1 ][σ2 ][σµ2 ][σξ2 ].
Let Lt = log(λt ), then the full-conditional distributions and sampling algorithms are as
follows:
• [Lt |·], t = 1, . . . , T .
Note that we have missing data at some locations at each time t. Therefore, sampling
of Lt requires two steps.
– Sample LSt t |·, t = 1, . . . , T , where St is the set of indices for locations with observed data at time t. Assuming si ∈ St , the MH algorithm reads:
1. Generate a candidate L(si ; t) ∼ Gau(L(si ; t)(k−1) , θL2 ) at the k-th MCMC iteration (θL2 is a tuning parameter chosen such that the acceptance rate for the MH
algorithm is between 20% and 40%), and compute the ratio
R=
[Y (si ; t)|exp(L(si ; t)), exp(wt )][L(si ; t)|µ(si ), αt , σ2 ]
.
[Y (si ; t)|exp(L(si ; t)(k−1) ), exp(wt )][L(si ; t)(k−1) |µ(si ), αt , σ2 ]
2. Accept L(si ; t)(k) = L(si ; t) with probability min(1, R); otherwise set L(si ; t)(k) =
L(si ; t)(k−1) .
†
– Sample Lt t |·, where †t refers to the set of indices for locations with missing data
at time t. For this part, we update using the predictive distribution, i.e.,
†
Lt t |· ∼ Gau(µ†t + Ψα†t t , σ2 In†t ),
where n†t = |†t | is the cardinality of the set †t .
2
• [µ|·] ∝
o
2
]
[µ|β, σµ2 ].
[L
|µ,
α
,
σ
t
t
t=1
nQ
T
It then follows that µ|· ∼ Gau(µµ|· , Σµ|· ), where
Σµ|· =
−1
σ2 σµ2
=
In ,
T σµ2 + σ2
!
(L
−
Ψα
)
Xβ
t
t
t=1
+ 2 .
σ2
σµ
T In
In
+ 2
2
σ
σµ
PT
µµ|· = Σµ|·
• [β|·] ∝ [µ|β, σµ2 ][β|β 0 , Σβ ].
Therefore, β|· ∼ Gau(µβ |· , Σβ |· ), where
−1
X0 X
−1
+ Σβ
,
Σβ|· =
σµ2
0
Xµ
−1
µβ |· = Σβ |·
+ Σ β0 .
β
σµ2
• [σ2 |·] ∝
Q
T
2
t=1 [Lt |µ, αt , σ ]
[σ2 ].
It follows that σ2 |· ∼ IG(Aσ2 , Bσ2 ), where
nT
Aσ2 = q +
,
2
PT
(Lt − µ − Ψαt )0 (Lt − µ − Ψαt )
+ r .
Bσ2 = t=1
2
• [σµ2 |·] ∝ [µ|β, σµ2 ][σµ2 ].
Thus, σµ2 |· ∼ IG(Aσµ2 , Bσµ2 ), where
n
,
2
(µ − Xβ)0 (µ − Xβ)
=
+ rµ .
2
Aσµ2 = qµ +
Bσµ2
• [αt |·], t = 0, 1, . . . , T .
3
For simplicity of notation, denote
Gt,∗




H1



= H2





H 3
if ct < dL
if dL ≤ ct ≤ dU ,
if ct > dU
the sampling algorithm can then be expressed as
– [α0 |·] ∝ [α1 |α0 ][α0 ].
This implies that α0 |· ∼ Gau(µα0 |· , Σα0 |· ), where
−1
−1
e
Σα0 |· = G01,∗ Σ−1
G
+
Σ
,
1,∗
η
α0
−1
e
µα0 |· = Σα0 |· G01,∗ Σ−1
α
+
Σ
u
1
η
α0 α0 .
– [αt |·] ∝ [αt |αt−1 ] [αt+1 |αt ] [Lt |αt , σ2 ] , t = 1, 2, . . . , T − 1.
It follows that αt |· ∼ Gau(µαt |· , Σαt |· ) with
−1
Ψ0 Ψ
−1
0
−1
Σαt |· = Ση + Gt+1,∗ Ση Gt+1,∗ + 2
,
σ
Ψ0 (Lt − µ)
−1
0
−1
µαt |· = Σαt |· Ση Gt,∗ αt−1 + Gt+1,∗ Ση αt+1 +
.
σ2
– [αT |·] ∝ [αT |αT −1 ] [LT |αT , σ2 ].
It then follows αT |· ∼ Gau(µαT |· , ΣαT |· ) with
−1
Ψ0 Ψ
,
=
+ 2
σ
Ψ0 (LT − µ)
−1
= ΣαT |· Ση GT,∗ αT −1 +
.
σ2
ΣαT |·
µαT |·
Σ−1
η
• [vec(Hi )|·], i = 1, 2, 3.
4
Let
χH1 = {t : ∀t such that ct < dL } ,
χH2 = {t : ∀t such that dL ≤ ct ≤ dU } ,
χH3 = {t : ∀t such that dU < ct } ,
Ai1 = {αt : t ∈ χHi } ,
Ai0 = {αt−1 : t ∈ χHi } ,
with ni = |Ai1 | denoting the cardinality of Ai1 . Then, (A.1) can be rewritten as
vec(Ai1 ) = (Ai0 )0 ⊗ Ip vec(Hi ) + vec(η t1 , . . . , η tni ),
where Cov(vec(η t1 , . . . , η tni )) = Ini ⊗Ση . Therefore, [vec(Hi )|·] ∝ [Ai1 |Hi , Ση ][vec(Hi )].
It then follows that vec(Hi )|· ∼ Gau(µvec(Hi )|· , Σvec(Hi )|· ), where
o−1
0
e −1
(Ai0 )0 ⊗ Ip [Ini ⊗ Ση ]−1 (Ai0 )0 ⊗ Ip + Σ
,
hi
n
o0
−1 i 0
i 0
0 e −1
e
= Σvec(Hi )|· vec(A1 ) [Ini ⊗ Ση ]
(A0 ) ⊗ Ip + (hi ) Σhi .
Σvec(Hi )|· =
µvec(Hi )|·
•
[Σ−1
η |·]
∝
Q
T
t=1
n
[αt |αt−1 , Ση ] [Ση ].
∗ ∗
Thus, Σ−1
η |· ∼ Wishart(S , vη ), where
S∗ =
3
X
!−1
(Ai1 − Hi Ai0 )(Ai1 − Hi Ai0 )0 + vη Sη
i=1
vη∗ = T + vη .
• [wt |·], t = 1, . . . , T .
1) [w1 |·] ∝ [Y 1 |λ1 , exp(w1 )][w2 |w1 ][w1 ].
2) [wt |·] ∝ [Y t |λt , exp(wt )][wt |wt−1 ][wt+1 |wt ], t = 2, 3, . . . , T − 1.
3) [wT |·] ∝ [Y T |λT , exp(wT )][wT |wT −1 ].
5
,
Therefore updating wt , t = 1, . . . , T requires a MH algorithm and can be performed
similar to LSt t |·.
• [φ0 |·] ∝
QT
2
t=2 [wt |φ0 , φ1 , wt−1 , σξ ][φ0 ].
Therefore φ0 |· ∼ N (µφ0 |· , σφ2 0 |· ), where
σφ2 0 |·
µφ0 |·
• [φ1 |·] ∝
!−1
T −1
1
,
=
+ 2
σξ2
σφ0
!
PT
(w
−
φ
w
)
µ
t
1
t−1
φ
0
2
t=2
= σφ0 |·
+ 2 .
σξ2
σ φ0
QT
2
t=2 [wt |φ0 , φ1 , wt−1 , σξ ][φ1 ].
Therefore φ1 |· ∼ N (µφ1 |· , σφ2 1 |· )I(−1,1) (φ1 ), where
PT
µφ1 |·
σφ2 1 |·
• [σξ2 |·] ∝
Q
t=2 (wt − φ0 )wt−1
= σφ2 1 |·
σξ2
!−1
PT
2
w
t=2 t−1
.
=
σξ2
T
2
t=2 [wt |φ0 , φ1 , wt−1 , σξ ]
!
,
[σξ2 ].
Therefore σξ2 |· ∼ IG(Aξ , Bξ ), where
T −1
Aξ = qξ +
,
2
PT
(wt − φ0 − φ1 wt )2
Bξ = t=2
+ rξ .
2
Appendix B: Additional Simulation Results
Section 4 of the manuscript presents simulation results for the hierarchical Bayesian spatiotemporal CMP model with dynamic dispersion. The additional results (figures) provided here
correspond to the simulation studies presented in the manuscript. Recall, simulation studies
6
were presented to assess the performance of our proposed model in terms of prediction and
inference, relative to the number of time replicates available. We considered two simulated
examples (for T = 56 and T = 240) with model choices similar to what we expect from
the analysis in the mallard duck data application. (See Section 5 of the main manuscript.)
For both simulations we constructed the basis function expansion matrix Ψ for p = 8 and
n = 2, 171 using kernel PCA.
As described in the manuscript, redistribution matrices Hi (i = 1, 2, 3) were chosen
such that the {αt } process is not explosive (i.e., the maximum eigenvalues for each Hi
iid
(i = 1, 2, 3) are less than one in modulus). That is, we set eigen(H1 ) ∼ Unif[−0.2, 0.3],
iid
iid
eigen(H2 ) ∼ Unif[0.2, 0.6] and eigen(H3 ) ∼ Unif[0.1, 0.4], where eigen(Hi ) denotes the
eigenvalues of the matrix Hi .
Again, the dispersion parameters wt = log(νt ) were simulated according to wt =
φ0 + φ1 wt−1 + ξt , t = 2, . . . , T with φ0 = 0.01, φ1 = 1.0 and w1 = −0.5.
The co-
iid
variance matrix Ση was simulated such that eigen(Ση ) ∼ Unif(0.1, 0.3). Finally, we set
β = (0.78, −0.03, −0.08, −0.007, −0.002, −0.012)0 , σ2 = 0.02, σµ2 = 0.04, and σξ2 = 0.02.
For the simulation with T = 56, Figure 1 presents the posterior summary statistics for the
dispersion parameters νt along with the corresponding true values. The posterior pointwise
95% CIs and true values for 50 randomly selected elements in Lt = log(λt ), t = 12, 27, 55 are
presented in Figure 2. The pointwise posterior mean and true values for α series are plotted
in Figure 3. In addition, plots for posterior summary statistics and true values of precision
matrix Σ−1
η are presented in Figure 4.
For the simulation with T = 240, Figure 5 displays the posterior mean and standard
deviation, 95% CIs, as well as the corresponding true values for Hi (i = 1, 2, 3). In addition, Figure 6 presents the posterior summary statistics for dispersion parameters νt with
the corresponding true values. The posterior pointwise 95% CIs and true values for 50 randomly selected elements in Lt = log(λt ), t = 12, 27, 239, are presented in Figure 7. Finally,
7
2.5
pointwise posterior mean and true values for α series are plotted in Figure 8.
1.5
1.0
νt
2.0
q.975
post. mean
true
q.025
0
10
20
30
40
50
Time t
Figure 1: Plot of the posterior mean and pointwise 95% credible interval for νt , t = 1, . . . , T -1
in the simulation study with T = 56.
8
−2
−1
0
1
2
q.975
true
q.025
0
10
20
30
40
50
(a) Plot for elements in L12
−2
−1
0
1
2
3
q.975
true
q.025
0
10
20
30
40
50
(b) Plot for elements in L27
−4
−2
0
2
4
q.975
true
q.025
0
10
20
30
40
50
(c) Plot for elements in L55
Figure 2: Plot of the pointwise 95% credible intervals and true values for 50 randomly chosen
elements in Lt , t = 12, 27, 55 for the simulation study T = 56.
9
4
20
−6
−2 0
α2
2
10
0
−20
α1
0
10
20
30
40
50
0
10
20
40
50
40
50
40
50
40
50
Time t
−10
−2
−5
0
α4
0
−1
α3
1
5
2
Time t
30
0
10
20
30
40
50
0
10
20
30
Time t
1
0
−2
−3
−1
α6
−1 0
α5
1
2
2
Time t
0
10
20
30
40
50
0
10
20
Time t
−4
−5
0
α8
0
−2
α7
2
5
Time t
30
0
10
20
30
40
50
0
Time t
10
20
30
Time t
Figure 3: Plot of the posterior mean and true values for the α series in the simulation study
with T = 56 (purple dashed line: posterior mean; red dotted line: truth).
10
1
1
8
8
2
2
3
3
6
4
6
4
4
4
5
5
2
6
2
6
7
7
0
0
8
8
1
2
3
4
5
6
7
8
1
2
3
(a) posterior mean
q.975
true
4
5
6
7
8
(b) true
q.025
15
1
3.0
2
10
2.8
3
2.6
5
4
2.4
5
2.2
0
6
2.0
−5
7
1.8
8
0
10
20
30
40
50
1
60
(c) 95% CIs
2
3
4
5
6
7
8
(d) SD
Figure 4: Plot of the posterior statistics and true values for the precision matrix Σ−1
η in the
simulation study with T = 56. This figure contains the posterior mean (a), true values (b),
95% pointwise credible intervals (c), and posterior standard deviation (d).
11
3
q.975
true
post. mean
q.025
1
1
1.5
1.5
2
2
2
3
3
1.0
1
4
1.0
4
0.5
0.5
5
0
5
6
0.0
−1
7
6
0.0
7
−0.5
8
0
10
20
30
40
50
1
60
(a) H1 : 95% C.I.
2
3
4
5
6
7
8
1
2
(b) H1 : Posterior Mean
q.975
true
post. mean
q.025
4
−0.5
8
4
5
6
7
8
(c) H1 : True
1
1
1.5
2
3
3
3
3
1.0
2
1.0
1.5
2
4
4
1
0.5
5
0.5
5
0
0.0
−1
6
−0.5
7
−2
8
0
10
20
30
40
50
(d) H2 : 95% C.I.
2
3
4
5
6
7
8
1
2
(e) H2 : Posterior Mean
q.975
true
post. mean
q.025
4
−0.5
7
8
1
60
0.0
6
3
4
5
6
7
8
(f) H2 : True
1
1
1.5
0
2
2
1.5
2
3
1.0
3
1.0
4
0.5
4
0.5
5
5
0.0
6
0.0
6
−0.5
−2
−0.5
7
7
−1.0
8
0
10
20
30
40
50
(g) H3 : 95% C.I.
60
−1.0
8
1
2
3
4
5
6
7
8
(h) H3 : Posterior Mean
1
2
3
4
5
6
7
8
(i) H3 : True
Figure 5: Plot of the posterior summary statistics for redistribution matrix Hi , (i = 1, 2, 3)
for simulation with T = 240.
12
1
2
3
νt
4
5
6
q.975
post. mean
true
q.025
0
50
100
150
200
Time t
Figure 6: Plot of the posterior mean and pointwise 95% credible interval for νt , t = 1, . . . , T -1
in the simulation study with T = 240.
13
−2
−1
0
1
2
q.975
true
q.025
0
10
20
30
40
50
(a) Plot for elements in L12
−2
−1
0
1
2
q.975
true
q.025
0
10
20
30
40
50
(b) Plot for elements in L27
−2
−1
0
1
2
q.975
true
q.025
0
10
20
30
40
50
(c) Plot for elements in L239
Figure 7: Plot of the pointwise 95% credible intervals and true values of 50 randomly chosen
elements in Lt , t = 12, 27, 239 for the simulation study T = 240.
14
6
4
−4
−20
−10
0
α2
2
0 5
α1
0
50
100
150
200
250
0
50
100
250
150
200
250
150
200
250
150
200
250
−2
2 4 6 8
α4
1
0
−2
−6
−1
α3
200
Time t
2
Time t
150
0
50
100
150
200
250
0
50
100
Time t
α6
−2
−2
−1
−1
0
α5
0
1
1
2
Time t
0
50
100
150
200
250
0
50
100
2
α8
1
−2
−4
0
α7
0 2 4 6 8
Time t
3
Time t
0
50
100
150
200
250
0
Time t
50
100
Time t
Figure 8: Plot of the posterior mean and true values of the α series in the simulation study
with T = 240 (purple dashed line: posterior mean; red dotted line: truth).
15
1
1
12
12
2
2
10
3
10
3
8
8
4
6
4
6
5
4
5
4
6
2
6
2
7
0
7
0
−2
8
1
2
3
4
5
6
7
−2
8
8
1
2
3
20
(a) posterior mean
q.975
true
4
5
6
7
8
(b) true
q.025
15
1
2
3.0
10
3
4
5
2.5
5
0
6
2.0
−5
7
8
0
10
20
30
40
50
1
60
(c) 95% CIs
2
3
4
5
6
7
8
(d) SD
Figure 9: Plot of the pointwise posterior statistics and true values for the precision matrix
Σ−1
η in the simulation study with T = 240.
16