Biometric Conference 2009, Taupo1 Maryann Pirie PhD candidate Department of Statistics and School of Environment University of Auckland

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


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


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


Method-issue

Assemble series into an array W is an array with elements wijt :

Problem: not all series are the same length


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,


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


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


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.


Simulated Results – Time series generated from normally distributed white noise


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


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




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



Normally distributed time series - Differences in level


Normally distributed time series - Difference in scale


Correlated time series - Differences in level


Other design issuesSensitivity to sample sizeRagged arrays

Adjust the overlap counts bxt, byt to proportions


A case study: Tree ring analysis using kauri from Northern New Zealand

Two subsets: small = 0-20cm from pith, large = 20-200cm from pith


Concordance indices


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


Thank youQuestions/comments


Documents

Biometric Conference 2009, Taupo1 Maryann Pirie PhD candidate Department of Statistics and School of Environment University of Auckland