0
10
20
30
40
50
cul$semana[cul$Id == trapID_2[i]]
44
20
26
30
40
50
cul$semana[cul$Id == trapID_2[i]]
31
0
10
20
30
40
50
cul$semana[cul$Id == trapID_2[i]]
47
10
20
30
40
50
cul$semana[cul$Id == trapID_2[i]]
32
0
10
20
30
40
50
cul$semana[cul$Id == trapID_2[i]]
49
30
20
10
0
0.8
0.4
m
0
0
10
20
33
0
10
20
51
40
50
cul$semana[cul$Id == trapID_2[i]]
53
10
20
30
40
50
cul$semana[cul$Id == trapID_2[i]]
54
10
20
30
40
50
cul$semana[cul$Id == trapID_2[i]]
10
20
30
40
50
cul$semana[cul$Id == trapID_2[i]]
50
0
0
0
0
120
80
40
0
10
20
30
40
50
cul$semana[cul$Id == trapID_2[i]]
56
Hilborn & Ludwig. 1993. The limits of applied ecological
research. Ecol. Appl. 3:550-552.
Kate Searle
(and many other stressed out ecological modellers)
0
0
15
10
5
0
0
10
20
30
40
50
cul$semana[cul$Id == trapID_2[i]]
8 10
30
0
60
40
20
0
20
50
6
200 400 600
52
10
40
4
cul$semana[cul$Id == trapID_2[i]]
0
30
cul$semana[cul$Id == trapID_2[i]]
2
50
50
0
40
150
30
100
20
2000 4000 6000
0.0 0.5 1.0
-1.0
10
40
cul$semana[cul$Id == trapID_2[i]]
Analysis of ecological data:
"ecology isn't rocket science, it's
harder”
0
30
300
20
0.0
15
10
10
0 100
30
cul$NUM_CULICOIDES[cul$Id == trapID_2[i]] cul$NUM_CULICOIDES[cul$Id == trapID_2[i]] cul$NUM_CULICOIDES[cul$Id == trapID_2[i]] cul$NUM_CULICOIDES[cul$Id == trapID_
body
cul$semana[cul$Id == trapID_2[i]]
cond
0
20
50
15
40
10
30e
5
20
0
Y
F
Mule
0deer10
cul$NUM_CULICOIDES[cul$Id == trapID_2[i]] cul$NUM_CULICOIDES[cul$Id == trapID_2[i]] cul$NUM_CULICOIDES[cul$Id == trapID_2[i]] cul$NUM_CULICOIDES[cul$Id == trapID_
W
T
5
P
0
N
0.8
N
W
T
25
 2n  k
 2n 
log(  jk )  b0k   ank sin 
t jk   bn cos 
t jk 
 52

 52

n
k
k
  cm M m (t jk )   jk   jk
0.4
N
N
22
0.0
20 40 60 80
A
cul$NUM_CULICOIDES[cul$Id == trapID_2[i]] cul$NUM_CULICOIDES[cul$Id == trapID_2[i]] cul$NUM_CULICOIDES[cul$Id == trapID_2[i]] cul$NUM_CULICOIDES[cul$Id == trapID_
A
g
σ
P
E
0
σN
P
15
S
G
P
10
P
W
A
5
E
0
cul$NUM_CULICOIDES[cul$Id == trapID_2[i]] cul$NUM_CULICOIDES[cul$Id == trapID_2[i]] cul$NUM_CULICOIDES[cul$Id == trapID_2[i]] cul$NUM_CULICOIDES[cul$Id == trapID_
Gr
16
0
10
20
30
40
50
cul$semana[cul$Id == trapID_2[i]]
Ecological processes and systems are multi-faceted and multi-scaled, such that an
understanding of any individual part of the system requires recognition of drivers and
constraints resulting from many interconnected processes
Behaviour
Populations
Spatial and temporal
heterogeneity
Communities and
Ecosystems
Moreover, states and variables within ecological systems are often not able to be measured
directly, but must be inferred from surrogate observations.
How do we observe the animal
we are interested in?
How do we measure habitat
quality for the animal?
• It is often difficult to design experiments to adhere to standard statistical assumptions
• This means that ecological data typically confound simple statistical approaches due to
factors such as:
• detectability
• sampling or measurement error
• unequal and irregular sampling effort over space and time
Most common issues encountered:
• Detectability – structural zeros, design error, observer error, animal error, naughty noughts
(sampled outside habitat range)
• Zero-inflated models, hurdle models
• multi-state mark-recapture models
• Hierarchical models – states and processes are measured at multiple scales
Yi,t
What we measured
The “true” process
o
Ei,t
 p , p
P( E , p ,  p ,  o | Y ) 
[Y | E ,  o ] 
[E |  p , p ]
[ p ,  p ,  o ]
• Spatial and temporal autocorrelation, or both
1. Hierarchical path analysis of the effect of habitat phenology on
deer body condition
2. Seasonal abundance models for Culicoides insects
1. Hierarchical path analysis of the effect of habitat phenology on deer
body condition
Asynchrony in vegetation phenology
• herbivores are able to prolong the period during which
they have access to forage of peak nutritional value
p*
Digestible Energy
Intake
• spatially and temporally asynchronous pulses of plant
growth
a
de*
Plant Biomass
b
Plant Biomass
b
c
d
p*
c
Digestible Energy
Intake
direct link to consumer fitness
a
t*
de*
de*
a
b
c
Time
d
PREDICTIONS:
More asynchronous phenology = longer ‘green-up’ periods prolonged access = better winter body
condition
Shorter ‘green-up’ periods = compression in the time period over = poorer body condition
Vegetation metrics:
Integrative NDVI (INDVI): productivity and biomass – correlates well with ANPP. Higher INDVI = higher
body condition.
Maximal or mean slope of NDVI during green-up: fastness of greening up in the Spring – e.g., how
elongated or compressed is the phenological development of plants in each individual’s home range.
Elongated green-up – higher body condition.
Onset of vegetation emergence: earlier vegetation onset = higher body condition.
DATA:
• GPS location data, home ranges
• Data model for Mule deer body
condition (% fat)
• NDVI
• Climate
Path analysis diagram for how performance (percent body fat) of mule deer is affected directly and indirectly
by climate and plant phenology in western Colorado. All lines in diagram represent a specific linear model.
Green-up
precipitation
Elevation
Winter
precipitation
Aspect
PWTN
PSN
σobs1
PEN
PWPN
PAN
PWPF
NDVI
indices
σ proc1
Green-up
temperature
PWTF
PNF
Path coefficients for
effect of e.g., N
(NDVI) on F (%FAT)
PNF
Year
Age
PAF
Capture
month
PCF
Mule deer body
condition
(percent fat)
σ obs2
PYF
Body fat
measurement
regression
equation
PRF
σ proc2
Range
σ
Error (exogenous independent
variables) reflecting error in
measurement or process variance
Data model:
Mean slope during vegetation green-up:
Green-up
precipitation
Elevation
0.22
(0.13,0.30)
Aspect
-0.072
(-0.15,0.0037)
Winter
precipitation
0.21
(0.14,0.30)
Green-up
temperature
-0.26
(-0.36,-0.16)
0.12
(0.013,0.22)
Mean slope
0.049
(-0.041,0.14)
-0.10
(-0.31,0.093)
Age
Mule deer body
condition
-0.036
(-0.086,0.013)
Mean slope adjusted R2: 0.28
BODY CONDITION adjusted R2: 0.62
(percent fat)
Path analysis diagram for how performance (percent fat) of adult, female mule deer is affected directly and indirectly by climate in
western Colorado in 2008,2009 and 2010. Indirect linkages are manifested through a measure of the speed of vegetation green-up in the
spring derived from NDVI measurements (‘mean slope’). All lines in the diagram represent a specific linear model. Thick solid lines
represent strong evidence for an effect (95% credible interval does not overlap zero). Dotted lines represent no clear effect. Regression
coefficient estimates are given with 95% credible intervals. ‘+’ predicted positive relationship, ‘-‘ predicted negative relationship.
2. Seasonal abundance models for Culicoides insects
We know that the European
distribution of Culicoides disease
vectors is driven by climatic, host and
land cover variation – how can we use
phenology to better understand
disease risk?
Need to understand the spatial and
temporal patterns of abundance
C. obsoletus
complex
C. pulicaris
complex
C. dewulfi
200 400 600
-1.0
0.0 0.5 1.0
0
5
10
15
0
0
0
0
10
Orders of magnitude
variation in abundance
10
10
10
20
20
20
20
30
30
30
30
40
40
40
40
50
cul$semana[cul$Id == trapID_2[i]]
30
50
cul$semana[cul$Id == trapID_2[i]]
44
cul$semana[cul$Id == trapID_2[i]]
50
52
50
cul$semana[cul$Id == trapID_2[i]]
0
20 40 60 80
2000 4000 6000
0
20
40
60
0.0
0.4
0.8
0
5
10
15
20
22
0
0
0
0
10
10
10
10
20
20
20
20
30
30
30
30
40
40
40
40
50
cul$semana[cul$Id == trapID_2[i]]
31
50
cul$semana[cul$Id == trapID_2[i]]
47
cul$semana[cul$Id == trapID_2[i]]
50
53
50
cul$semana[cul$Id == trapID_2[i]]
50
100
150
0
5
10
15
0
5
10
15
20
0.0
0.4
0.8
25
0
0
0
0
10
10
10
10
20
cul$semana[cul$Id == trapID_2[i]]
20
20
20
Lots of zeros
30
30
30
30
40
40
40
40
50
32
50
cul$semana[cul$Id == trapID_2[i]]
49
cul$semana[cul$Id == trapID_2[i]]
50
54
50
cul$semana[cul$Id == trapID_2[i]]
0
2
4
6
8 10
0
40
80
120
0 100
300
0
10
20
30
cul$NUM_CULICOIDES[cul$Id == trapID_2[i]] cul$NUM_CULICOIDES[cul$Id == trapID_2[i]] cul$NUM_CULICOIDES[cul$Id == trapID_2[i]] cul$NUM_CULICOIDES[cul$Id == trapID_2[i]]
0
cul$NUM_CULICOIDES[cul$Id == trapID_2[i]] cul$NUM_CULICOIDES[cul$Id == trapID_2[i]] cul$NUM_CULICOIDES[cul$Id == trapID_2[i]] cul$NUM_CULICOIDES[cul$Id == trapID_2[i]]
0
cul$NUM_CULICOIDES[cul$Id == trapID_2[i]] cul$NUM_CULICOIDES[cul$Id == trapID_2[i]] cul$NUM_CULICOIDES[cul$Id == trapID_2[i]] cul$NUM_CULICOIDES[cul$Id == trapID_2[i]]
0
cul$NUM_CULICOIDES[cul$Id == trapID_2[i]] cul$NUM_CULICOIDES[cul$Id == trapID_2[i]] cul$NUM_CULICOIDES[cul$Id == trapID_2[i]] cul$NUM_CULICOIDES[cul$Id == trapID_2[i]]
16
26
0
0
0
0
10
10
10
10
20
20
20
20
30
30
30
30
40
40
40
40
50
cul$semana[cul$Id == trapID_2[i]]
33
50
cul$semana[cul$Id == trapID_2[i]]
51
Messy
cul$semana[cul$Id == trapID_2[i]]
50
56
50
cul$semana[cul$Id == trapID_2[i]]
•
Modelling seasonal dynamics of Culiciodes spp. to generate vector abundance
predictions for use in a BTV-1 spread model for the 2007 outbreak
• 6 years of weekly trapping data from the whole of Spain
•
GLMM (Poisson –log link) with overdispersion, temporal autocorrelation (AR-1) and
hierarchical structure for between site differences
jth trap catch for site k (yjk) collected in week tjk:
Seasonality in population with
site-specific parameters
 2n  k
 2n 
log(  jk )  b   a sin 
t jk   bn cos 
t jk  
52
52




n
100 150
50
Y[158:208, 5]
20
30
40
Index
10
20
30
Index
40
80
30
40
50
Index
0
40
Y[53:104, 5]
20
10
0
20
10
30
20 50
30
40
Index
40
2
Index
-2
50
50
120
-2
-6
2
-2
-6
0
0
100 150
10
50
-6
120
80
40
0
0
5000
40
10
50
50
30
Index
10
20
30
40
50
Index
0
Index
40
5000 0
30
20
0
Y[158:208, 5]
M (t jk )
20
Y[105:157, 5]
k
m
10
10
0 2000
Y[53:104, 5]
5 10
0
2000
Corresponding meteorological variables:
0
overdispersion
Y[105:157, 5]
Influence of meteorological
parameters with site specific
parameters
Y[1:52, 5]
20
0
2
m
-2
5 10
M mk (t jk )   jk   jk
0
k
m
Y[1:52, 5]
c
Temporal
autocorrelation
2
b0k a nk bnk
k
n
20
k
0
-6
Poisson(  jk )
b.bugs$mean$seas[105:157]
b.bugs$mean$seas[105:157]b.bugs$mean$seas[1:52]
b.bugs$mean$seas[1:52]
y jk
0
10
20
30 40 50
0
10
20 30
Index
Index
40
0
10
20
30
40
50
cul$semana[cul$Id == trapID_2[i]]
0 100
300
33
0
10
20
30
40
50
cul$semana[cul$Id == trapID_2[i]]
51
80
120
and g is a fixed function; there appear to be two natural choices for g:
• the triangular function g(w) = max(0, |1 − w|); or
• the density function φ(w) of a standard normal distribution.
0
40
λ : background midge abundance when not in a peak
sk : width of peak (assumed to be the same for all sites, so s1 represents the
longest peak
and
S20K the30shortest
peak at each site)
0
10
40
50
mik : magnitude of the k-th longest peak at site i
cul$semana[cul$Id == trapID_2[i]]
pik : timing of the k-th longest peak at site i
φij : impact of time-varying covariates in modifying magnitude of the peak
56
8 10
0
0
6
0
ES[cul$Id == trapID_2[i]] cul$NUM_CULICOIDES[cul$Id == trapID_2[i]] cul$NUM_CULICOIDES[cul$Id == trapID_2[i]] cul$NUM_CU
0
Conclusions
• Multi-site spatio-temporal models
• extreme events – droughts and flood
• detection of long-term trends in multifaceted variable times-series (sampling methods)
Thank you
Adam Butler (BioSS)
Beth Purse (CEH)
Mindy Rice (Colorado Division of Wildlife)
Tom Hobbs (Colorado State University)
Simon Carpenter (Institute of Animal Health)