Date post: | 29-Mar-2015 |
Category: |
Documents |
Upload: | tessa-chester |
View: | 218 times |
Download: | 1 times |
Biometric Conference 2009, Taupo 1
Maryann PiriePhD candidate
Department of Statistics and School of EnvironmentUniversity of Auckland
OverviewKey questionData setsDeveloping the methodsConclusions from simulationsApplication to tree-ring
dataset
Biometric Conference 2009, Taupo 2
To investigate a possible failure of the uniformitarianism principle in the use of kauri
ring-widths to investigate past climates
Contains rings from the inner of the core, formed when tree was smaller
Contains rings from the outer of cores, formed when tree was larger
3Biometric Conference 2009, Taupo
The key question:
Data
Biometric Conference 2009, Taupo 4
MethodsFor a given core we have a series of ring
widths, wijt t = 1, … , T
We may have several cores from the same tree, j = 1, … , Ci – typically Ci = 2
We have many trees, i = 1, … , L
Biometric Conference 2009, Taupo 5
Method-issue
Assemble series into an array W is an array with elements wijt :
Problem: not all series are the same length
Biometric Conference 2009, Taupo 6
Method-issue
Tree 1 Tree i
7
wijt = width tree, core, index
Biometric Conference 2009, Taupo
Method-issueAssume times Tij are all equal, We have two matrices of time series, X,YWhere X,
Biometric Conference 2009, Taupo 8
For each time we average
To give:
And, for a similar matrix of time series for Y
Statistical QuestionHow do we formalise
the difference between the two series;
and ? This will be termed the
concordance
These are not stationary series
We do not want to use correlation coefficients
Biometric Conference 2009, Taupo 9
Method-ideafor the common period m=nProduce bootstrapped
replicates of:
For each time, t sort the averaged bootstrapped time series,
Count the number of bootstrap replicates that overlap at each time, t
Biometric Conference 2009, Taupo 10
0.0 0.5 1.0 1.5 2.0
0.0
0.2
0.4
0.6
0.8
1.0
Picture illustrating bxt, byt and R
Index
valu
e
*
*
*
*
*
*
######
*
*
*
*
*
*
#
#
#
#
#
#
Bxt = 3R = 6
Byt = 6R = 6
Bxt = 5R = 6
Byt = 5R = 6
Concordance, PThe concordance at time, t, can be defined
as:
t lies between 0 and 1The overall concordance of how similar the
two time seriesCombines concordances for all (common)
times.
Biometric Conference 2009, Taupo 11
Simulated Results – Time series generated from normally distributed white noise
Biometric Conference 2009, Taupo 12
Average time series for matrix X (black) and Y (red), ~N(0,1)
Time
x.m
aste
r
1860 1880 1900 1920 1940
-0.2
-0.1
0.0
0.1
0.2
Biometric Conference 2009, Taupo 13
5 10 15 20
-0
.4-0
.20
.00
.20
.4
sorted bootstrapped replicated for X (black), and Y (red), for the first 20 series
series index
va
lue
Simulated Results – Time series generated from normally distributed white noise
Simulated Results – Time series generated from normally distributed white noise
Biometric Conference 2009, Taupo 14
Histogram of prec$x.over.y
prec$x.over.y
Fre
qu
en
cy
0 20 40 60 80 100
02
04
0
Histogram of prec$y.over.x
prec$y.over.x
Fre
qu
en
cy
0 20 40 60 80 100
01
03
0
Simulated Results – Time series generated from normally distributed white noise
Biometric Conference 2009, Taupo 15
Normally distributed time series - Differences in level
Biometric Conference 2009, Taupo 16
Normally distributed time series - Difference in scale
Biometric Conference 2009, Taupo 17
Correlated time series - Differences in level
Biometric Conference 2009, Taupo 18
Other design issuesSensitivity to sample sizeRagged arrays
Adjust the overlap counts bxt, byt to proportions
Biometric Conference 2009, Taupo 19
A case study: Tree ring analysis using kauri from Northern New Zealand
Two subsets: small = 0-20cm from pith, large = 20-200cm from pith
22Biometric Conference 2009, Taupo
Concordance indices
23Biometric Conference 2009, Taupo
ConclusionConcordance indices are able to identify
periods of similarity/dissimilarity between two matrices of time series
The Concordance tends to zero when there is little or no overlap between matrices of time series
There was a difference detected between the subsets ‘small’ and ‘large’ for Huapai
Suggesting failure of uniformitarianism principle
Biometric Conference 2009, Taupo 25
Thank youQuestions/comments
26Biometric Conference 2009, Taupo