MARKOV SWITCHING MODELS FOR HIGH-FREQUENCY TIME
SERIES FROM AUTOMATIC MONITORING OF ANIMALS
Luigi Spezia
Cecilia Pinto
Biomathematics & Statistics Scotland
School of Biological Sciences
Aberdeen, UK
University of Aberdeen
Aberdeen, UK
Spezia and Pinto –
Markov switching models for high-frequency time series
from automatic monitoring of animals
DYNSTOCH 2014 – University of Warwick, 10th-12th September,2014
Overview
• Flapper skate’s depth profile
• High-frequency time series
Long memory process
Non-linear time series
• Markov switching autoregressive models
• Results
• Current research
Spezia and Pinto –
Markov switching models for high-frequency time series
from automatic monitoring of animals
DYNSTOCH 2014 – University of Warwick, 10th-12th September,2014
Dipturus intermedia
Flapper skate’s depth profile
Cecilia Pinto’s PhD project:
Estimating the probability of
recolonization of endangered
marine species integrating
demographic and movement
parameters.
School of Biological Sciences,
University of Aberdeen
Spezia and Pinto –
Markov switching models for high-frequency time series
from automatic monitoring of animals
DYNSTOCH 2014 – University of Warwick, 10th-12th September,2014
Flapper skate’s depth profile
Spezia and Pinto –
Markov switching models for high-frequency time series
from automatic monitoring of animals
DYNSTOCH 2014 – University of Warwick, 10th-12th September,2014
Flapper skates’ depth profiles
Spezia and Pinto –
Markov switching models for high-frequency time series
from automatic monitoring of animals
DYNSTOCH 2014 – University of Warwick, 10th-12th September,2014
Flapper skates’ depth profiles
pressure levels every
two minutes → depth
(metres)
Spezia and Pinto –
Markov switching models for high-frequency time series
from automatic monitoring of animals
DYNSTOCH 2014 – University of Warwick, 10th-12th September,2014
Flapper skates’ depth profiles
Skate 7967
male
juvenile
12 months
T=254116
Skate 7972
male
adult
6 months
T=127360
Skate 7968
female adult
6 months
T=107891
Skate 8828
male
12 days
T=11404
Spezia and Pinto –
adult
Markov switching models for high-frequency time series
from automatic monitoring of animals
DYNSTOCH 2014 – University of Warwick, 10th-12th September,2014
Flapper skates’ depth profiles
Spezia and Pinto –
Markov switching models for high-frequency time series
from automatic monitoring of animals
DYNSTOCH 2014 – University of Warwick, 10th-12th September,2014
Flapper skates’ depth profiles
-150
-100
-50
0
-250
-200
-200
-150
-100
depth
Skate 7968
Skate 8828
-150
-100
-50
0
-50
0
-50
0
0.000 0.010 0.020 0.030
depth
probability
-250
Spezia and Pinto –
0.000 0.010 0.020 0.030
probability
-200
0.000 0.010 0.020 0.030
-250
probability
Skate 7972
0.000 0.010 0.020 0.030
probability
Skate 7967
-250
-200
-150
-100
Markov switching models for high-frequency time series
depth
from automatic monitoring of animals
DYNSTOCH 2014 – University of Warwick, 10th-12th September,2014
depth
Flapper skates’ depth profiles
30
40
0.8
50
0
10
20
30
time lag
Skate 7968
Skate 8828
50
40
50
0.4
0.8
time lag
40
0.0
0.4
0.0
0
10
Spezia and
Pinto
–
20
autocorrelation
10
0.8
0
autocorrelation
0.4
0.0
0.4
autocorrelation
0.8
Skate 7972
0.0
autocorrelation
Skate 7967
20
30
40
50
0
10
20
Markov
switching
models
for high-frequency
time series
timeautomatic
lag
from
monitoring of animals
DYNSTOCH 2014 – University of Warwick, 10th-12th September,2014
time lag
30
High-frequency time series
High-frequency time series exhibit sample ACF that persists for
a long time [i.e., long memory processes]
Modelling strategy 1: fractional differentiation (d)
et ∼ N(0; σe2)
(1-B)dyt = et
∞
(1-B)dyt
Γ(k-d)
= ∑
k=0
Γ(k+1) Γ(-d)
-0.5<d<0.5
Spezia and Pinto –
Markov switching models for high-frequency time series
from automatic monitoring of animals
DYNSTOCH 2014 – University of Warwick, 10th-12th September,2014
High-frequency time series
High-frequency time series exhibit sample ACF that persists for
a long time [i.e., long memory processes]
Modelling strategy 1: fractional differentiation
ARFIMA(p,d,q)
φ(B)(1-B)dyt = θ(B)et
-0.5<d<0.5
φ(B) = 1-φ1B-...-φpBp;
Bjyt=yt-j; j=1,..,p
θ(B) = 1+θ1B+...+θqBq
Spezia and Pinto –
Markov switching models for high-frequency time series
from automatic monitoring of animals
DYNSTOCH 2014 – University of Warwick, 10th-12th September,2014
High-frequency time series
Modelling strategy 2: non-linear time series with structural
breaks produce realizations that appear to
have long memory
“structural change” and “long memory” are
effectively different labels for the same
phenomenon
structural changes can be modelled as
stochastic regime switching
e.g., Markov switching autoregressive
models (non-Normal and non-linear
models)
Spezia and Pinto –
Markov switching models for high-frequency time series
from automatic monitoring of animals
DYNSTOCH 2014 – University of Warwick, 10th-12th September,2014
Markov switching autoregressive models
( {yt}; {xt} )
{xt}:
m-state hidden Markov chain
SX={1,2,…,m}
P(xt=i| xt-1=j) = γj,i
0< γj,i <1
∀ i,j ∈ SX
Γ = [γj,i ] (m×m)
{yt}:
conditional autoregressive process of order p
yt = µ(i) + φ1(i) yt-1 + φ2(i)yt-2 + … + φp(i)yt-p + et
Spezia and Pinto –
et ∼ N(0; λ(i)-1)
Markov switching models for high-frequency time series
from automatic monitoring of animals
DYNSTOCH 2014 – University of Warwick, 10th-12th September,2014
Non-homogeneous Markov switching autoregressive models
Γt = [γtj,i ](m×m)
γtj,i = P(xt=i| xt-1=j)
0< γj,i <1
∀ i,j ∈ SX; ∀ t=2,…,T
Z=(zp+1,…,zt,…, zT)’
zt=(zt,1,…, zt,n)’
α=[αj,i]
[αj,i]=(αj,i,0,αj,i,1,…,αj,i,n)
∀ t=p+1,…,T
logit( γtj,i ) = ln( γtj,i / γtj,j )
exp(ztαj,i)
γSpezia
j,i =
and+Pinto
– Markov
1
)
∑ exp(z
tαj,iswitching
t
i≠j
∀ i,j ∈ SX
∀ i,j ∈ SX
1
γ j,i =
models for high-frequency
timeαseries
1 + ∑ exp(z
t j,i)
t
from automatic monitoring of animals
i≠j
DYNSTOCH 2014 – University of Warwick, 10th-12th September,2014
Non-homogeneous Markov switching autoregressive models
γtj,i = P(xt=i| xt-1=j)
0< γj,i <1
Γt = [γtj,i ](m×m)
∀ i,j ∈ SX; ∀ t=2,…,T
At each time t:
Categorical covariate Dt defined on d categories
Gh = [ghj,i ](m×m)
d
Γt = ∑ Gh × I(Dt=h)
ghj,i = P(xt=i| xt-1=j, Dt=h)
h=1,…,d ∀ i,j ∈ SX; ∀ t=2,…,T
h=1
Spezia and Pinto –
Markov switching models for high-frequency time series
from automatic monitoring of animals
DYNSTOCH 2014 – University of Warwick, 10th-12th September,2014
Prior specification
m
p(Gh) = ∏ p(Ghj•)
Ghj•= (ghj,1,ghj,2,…,ghj,m)
j=1
Ghj• ~ D (•)
for all j=1,…,m and h=1,...,d
µ(i) ∼ N (•; •),
for all i=1,…,m
λ(i) ∼ G (•; •),
for all i=1,…,m
p
m
p(φ) = ∏ ∏ p(φj(i)),
j=1 i=1
φj(i) ∼ N (•; •),
Spezia and Pinto –
for all i=1,…,m and j=1,…,p
Markov switching models for high-frequency time series
from automatic monitoring of animals
DYNSTOCH 2014 – University of Warwick, 10th-12th September,2014
Gibbs sampling
• µ(i) ← Normal
for all i=1,…,m
• λ(i) ← Gamma
for all i=1,…,m
• φj(i) ← Normal
for all i=1,…,m and j=1,…,p
• Ghj• ← Dirichlet
for all j=1,…,m and h=1,...,d
• sample permutation
• (x1,...xT) ← forward filtering – backward sampling algorithm
Spezia and Pinto –
Markov switching models for high-frequency time series
from automatic monitoring of animals
DYNSTOCH 2014 – University of Warwick, 10th-12th September,2014
Label switching
Iteration k = 1,...,L: Burn-in
Iteration k = L+1,...,L+M: Posterior mode
{µ*, λ*, φ*, G*}=arg max p(µ(k), λ(k), φ(k), G(k))
Iteration k=L+M+1,…, L+M+N: Permutations
η*=arg min ||ηj(µ(k), λ(k), φ(k), G(k)) - (µ*, λ*, φ*, G*)||
ηj∈H
&
(µ(k), λ(k), φ(k), G(k)) = η*(µ(k), λ(k), φ(k), G(k))
Spezia and Pinto –
Markov switching models for high-frequency time series
from automatic monitoring of animals
DYNSTOCH 2014 – University of Warwick, 10th-12th September,2014
Bayesian inference
3 steps: model choice
variable selection
parameter estimation and hidden chain reconstruction
m = 1,…,4; p = 0,…,6; 3 covariates ⇒ 224 competing models
Spezia and Pinto –
Markov switching models for high-frequency time series
from automatic monitoring of animals
DYNSTOCH 2014 – University of Warwick, 10th-12th September,2014
Bayesian inference
3 steps: model choice
variable selection
parameter estimation and hidden chain reconstruction
m = 1,…,4; p = 0,…,6; 3 covariates ⇒ 224 competing models
Γt = [γtj,i ](m×m)
γtj,i = P(xt=i|1 xt-1=j)
3 covariates: Solar cycle S
Lunar cycle L 2
Lunar phase L4
s ∈ {0; 1}
l2 ∈ {0; 1}
l4 ∈ {1; 2; 3; 4}
Indicator Dt d ∈ {1,2,4,8,16}
Spezia and Pinto – Markov switching
models for high-frequency time series
t =
h × I(D =h)
ghj,i =monitoring
P(x
=i|
x
=j,
D
=h)
Γ
G
Gh = [ghj,i ](m×m)from automatic
t
t
of t-1
animals t
DYNSTOCH 2014 – University of Warwick, 10th-12th September,2014
Bayesian inference
3 steps: model choice
variable selection
parameter estimation and hidden chain reconstruction
m = 1,…,4; p = 0,…,6; 3 covariates ⇒ 224 competing models
Pragmatic alternative:
[1] best(m;p), with d=1
[2] best combination of covariates ⇒ 35 competing models
Spezia and Pinto –
Markov switching models for high-frequency time series
from automatic monitoring of animals
DYNSTOCH 2014 – University of Warwick, 10th-12th September,2014
Bayesian inference
Model choice and variable selection: Bayes factors
ln(marginal likelihood) computed by the method of Chib (2001)
Skate 7967 (ma; ju;12 months)
m=2;p=4; L4
Skate 7972 (ma; ad; 6 months)
m=2;p=3; L4
Skate 7968 (fe; ad; 6 months)
m=2;p=3; L4;L2
Skate 8828 (ma; ad; 12 days)
m=2;p=4; L4;L2;S
Spezia and Pinto –
Markov switching models for high-frequency time series
from automatic monitoring of animals
DYNSTOCH 2014 – University of Warwick, 10th-12th September,2014
Results
Spezia and Pinto –
(black
dots)forand
fitted (red time
line) series
Markovactual
switching
models
high-frequency
from automatic monitoring of animals
DYNSTOCH 2014 – University of Warwick, 10th-12th September,2014
Results
Spezia and Pinto –
of the
Markov switchingACF
models
for residuals
high-frequency time series
from automatic monitoring of animals
DYNSTOCH 2014 – University of Warwick, 10th-12th September,2014
Results
State 1 (68,337 visits, 54%) = low variability (resting, horizontal movement, slow ascending
and descending)
Spezia and Pinto –
Markov switching models for high-frequency time series
State 2 (59,019 visits, from
46%)automatic
= high variability
ascending and descending)
monitoring(fast
of animals
DYNSTOCH 2014 – University of Warwick, 10th-12th September,2014
Results
Spezia and Pinto –
Markov switching models for high-frequency time series
from automatic monitoring of animals
DYNSTOCH 2014 – University of Warwick, 10th-12th September,2014
Extending the model
( {yt}; {xt} )
State-dependent autoregressive orders:
p = (p1, p2,…, pm)
{xt}:
m-state hidden Markov chain
{yt}:
conditional autoregressive process of order px t
yt = µ(i) + φ1(i) yt-1 + φ2(i)yt-2 + … + φp (i)yt-p + et
i
Spezia and Pinto –
i
et ∼ N(0; λ(i)-1)
Markov switching models for high-frequency time series
from automatic monitoring of animals
DYNSTOCH 2014 – University of Warwick, 10th-12th September,2014
Extending the model
Markov switching ARCH noises:
ut ∼ N(0; 1)
et = ht ut
ht = η(i) +
2
α1(i)et-1
+
2
α2(i)et-2
+…+
2
αq(i)et-q
Local stationarity of each state-dependent ARCH process:
η(i) > 0; α1(i),…,αq(i)≥0;
q
∑ αj(i)≤1
j=1
Spezia and Pinto –
Markov switching models for high-frequency time series
from automatic monitoring of animals
DYNSTOCH 2014 – University of Warwick, 10th-12th September,2014
Extending the model
Markov switching ARCH noises:
ut ∼ N(0; 1)
et = ht ut
ht = η(i) +
2
α1(i)et-1
+
2
α2(i)et-2
+…+
2
αqi(i)et-qi
with state-dependent autoregressive orders:
q = (q1, q2,…, qm)
[auxiliary variable MCMC method]
Spezia and Pinto –
Markov switching models for high-frequency time series
from automatic monitoring of animals
DYNSTOCH 2014 – University of Warwick, 10th-12th September,2014
Extending the model
Model and variable selection:
Bayesian Information Criterion (BIC)
Deviance Information Criterion (DIC)
on a single subspace.
Spezia and Pinto –
Markov switching models for high-frequency time series
from automatic monitoring of animals
DYNSTOCH 2014 – University of Warwick, 10th-12th September,2014
Extending the model
Model and variable selection:
Bayesian Information Criterion (BIC)
Deviance Information Criterion (DIC)
on a single subspace.
Algorithm:
From Gibbs sampling to …
Spezia and Pinto –
Markov switching models for high-frequency time series
from automatic monitoring of animals
DYNSTOCH 2014 – University of Warwick, 10th-12th September,2014
Summary
Biologgers applied to animals produce long memory processes
due to a non-linear dynamics.
Flapper skate’s depth profile can be modelled effciently
by Markov switching autoregressive models with a
non-Homogeneous Markov chain, where the time-varying
transition probabilities depend on the dynamics of categorical
covariates.
Depth values can be classiefied into two regimes representing
two different classes of skates’ behaviours.
Spezia and Pinto –
Markov switching models for high-frequency time series
from automatic monitoring of animals
DYNSTOCH 2014 – University of Warwick, 10th-12th September,2014