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Claire Miller1, Adrian Bowman1, Marian Scott1 and Laurence Carvalho2
1Statistics, University of Glasgow2Centre for Ecology & Hydrology, Edinburgh
A Case Study of Loch Leven
http://www.tulbol.demon.co.uk/art/banatvewe.jpg
Regression Case StudyAdditive and Varying-Coefficient Models
Dataset:
1968 - 2002
150 variables measured,
covering…
• Physics
• Lake chemistry
• Lake biology
• Weather
Loch Leven (LL)
Log SRP
Years
Lo
g(S
RP
, m
ug
/l)
1970 1980 1990 2000
-10
12
34
Log TP
Years
Lo
g(T
P,
mu
g/l)
1970 1980 1990 20003
.54
.04
.55
.05
.5
Log Chlorophyll
Years
Lo
g(C
hlo
rop
hyl
l, m
ug
/l)
1970 1980 1990 2000
12
34
5
Water Temperature
Years
Wa
ter
Tem
pe
ratu
re,
oC
1970 1980 1990 2000
05
10
15
20
Log Daphnia
Years
Lo
g(D
ap
hn
ia,
ind
/l)
1970 1980 1990 2000
-4-2
02
4
Log NO3N
Years
Lo
g(N
O3
N,
mg
/l)
1970 1980 1990 2000
-4-2
0
Loch Leven – The Data
The EC Water Framework Directive (WFD) 2000 states that there should be ‘good status’ in all shallow waters by 2016.
Research:
Investigating statistical techniques to explore the combined impacts of climate and nutrients on water quality.
Motivation
tt3t
ttt
ta
)N)-(NO)temp(water
)(SRP)()(month)(year
lchlorophyl
(log))(log(
log
)log(
654
321
mmDaphniam
mmm
t
chlorophylla
Response
year, month, SRP, Daphnia, water temp, NO3-N
Covariates
Additive Model 1988 - 2002
a) estimate of m1(year)
year
m1
(ye
ar)
1990 1994 1998 2002
-10
12
b) estimate of m2(month)
month
m2
(mo
nth
)
2 4 6 8 10 12-1
01
2
c) estimate of m3(log SRP)
log(SRP,mug/l)
m3
(lo
g(S
RP
))
1 2 3 4
-10
12
d) estimate of m4(log Daphnia)
log(Daphnia, individuals/l)
m4
(lo
g(D
ap
hn
ia))
-4 -2 0 2 4
-10
12
e) estimate of m5(water temp)
water temp,oC
m5
(wa
ter
tem
p)
5 10 15 20
-10
12
f) estimate of m6(log NO3-N)
log(NO3-N,mg/l)
m6
(lo
g(N
O3
-N))
-4 -3 -2 -1 0 1
-10
12
Chlorophyll
Mainly linear relationships between variables.
Little evidence of trend and seasonality.
Important ecological relationships identified.
The nature of trends and relationships can be explored simultaneously.
For chlorophylla (in this time period)-
Summary of Additive Models
Linear Regression Model
p
jijiji xY
1
Varying-coefficient Model (VCM)
p
jijiji xmmY
1)()(
Varying-coefficient Model
m = month
Do relationships change throughout the year?
VCM LL Case Study 1988-2007
ttt mm )SRPlog()()()chlalog(
0 1 2 3 4
2.0
2.5
3.0
3.5
4.0
4.5
5.0
log(SRP, g l-1
)
log
(ch
loro
ph
yll,
g l-1
)