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Return Interval Distribution of Extreme Events in Long

memory Time Series With Two Scaling Exponents

Smrati Kumar Katiyar

Department of PhysicsIISER, Pune

May 3, 2011

Katiyar S K (IISER Pune) Thesis Presentation May 3, 2011 1 / 27

1 Explanation for the title

2 Statistical test for long memory

3 Foundation stone for our work

4 our workAnalytical approachNumerical approach to the problemComparison of analytical and numerical results

5 Long memory probability process with two scaling exponents

6 conclusion

7 future direction

Katiyar S K (IISER Pune) Thesis Presentation May 3, 2011 2 / 27

What are the key terms?

Return interval distribution of extreme events in long memory time serieswith two scaling exponents.

Katiyar S K (IISER Pune) Thesis Presentation May 3, 2011 3 / 27

What are the key terms?

Return interval distribution of extreme events in long memory time serieswith two scaling exponents.

1 Return interval and extreme events

Katiyar S K (IISER Pune) Thesis Presentation May 3, 2011 3 / 27

What are the key terms?

Return interval distribution of extreme events in long memory time serieswith two scaling exponents.

1 Return interval and extreme events

2 Long memory time series

Katiyar S K (IISER Pune) Thesis Presentation May 3, 2011 3 / 27

What are the key terms?

Return interval distribution of extreme events in long memory time serieswith two scaling exponents.

1 Return interval and extreme events

2 Long memory time series

3 Scaling exponents

Katiyar S K (IISER Pune) Thesis Presentation May 3, 2011 3 / 27

Return interval and extreme events

Given a time series X (t)

Katiyar S K (IISER Pune) Thesis Presentation May 3, 2011 4 / 27

Return interval and extreme events

Given a time series X (t)

0 20 40 60 80 100t

-4

-3

-2

-1

0

1

2

3x(

t)

threshold

r1 r2 r3

Figure: Return intervals and extreme events

Katiyar S K (IISER Pune) Thesis Presentation May 3, 2011 4 / 27

Aim of our project work

Example : let say we are given a time series X(t) and there are total 11time instants at which the value of X is more than the threshold(q).Those time instants are,t = 0, 1, 3, 5, 6, 7, 10, 11, 12, 14, 16

Katiyar S K (IISER Pune) Thesis Presentation May 3, 2011 5 / 27

Aim of our project work

Example : let say we are given a time series X(t) and there are total 11time instants at which the value of X is more than the threshold(q).Those time instants are,t = 0, 1, 3, 5, 6, 7, 10, 11, 12, 14, 16So the return intervals will be :return intervals = 1, 2, 2, 1, 1, 3, 1, 1, 2, 2out of these 10 return intervals we have5 return intervals of length 14 return intervals of length 2and 1 return interval of length 3

Katiyar S K (IISER Pune) Thesis Presentation May 3, 2011 5 / 27

Aim of our project work

Example : let say we are given a time series X(t) and there are total 11time instants at which the value of X is more than the threshold(q).Those time instants are,t = 0, 1, 3, 5, 6, 7, 10, 11, 12, 14, 16So the return intervals will be :return intervals = 1, 2, 2, 1, 1, 3, 1, 1, 2, 2out of these 10 return intervals we have5 return intervals of length 14 return intervals of length 2and 1 return interval of length 3so the probability of occurance of return interval of length 1 will beP(1) = 510 ,similarly P(2) = 410 and P(3) =

110

Katiyar S K (IISER Pune) Thesis Presentation May 3, 2011 5 / 27

Long memory time series

Katiyar S K (IISER Pune) Thesis Presentation May 3, 2011 6 / 27

Long memory time series

Plot of sample autocorrelation function (ACF) k against lag k is one ofthe most useful tool to analyse a given time series.

Katiyar S K (IISER Pune) Thesis Presentation May 3, 2011 6 / 27

Long memory time series

Plot of sample autocorrelation function (ACF) k against lag k is one ofthe most useful tool to analyse a given time series.

k =n

t=k+1(xtx)(xtkx)nt=1(xtx)

2

Katiyar S K (IISER Pune) Thesis Presentation May 3, 2011 6 / 27

Long memory time series

Plot of sample autocorrelation function (ACF) k against lag k is one ofthe most useful tool to analyse a given time series.

k =n

t=k+1(xtx)(xtkx)nt=1(xtx)

2

For long memory processes

k Ck as k

where C > 0 and (0, 1)

Katiyar S K (IISER Pune) Thesis Presentation May 3, 2011 6 / 27

Long memory time series

k =n

t=k+1(xtx)(xtkx)nt=1(xtx)

2

For long memory processes

k Ck as k

where C > 0 and (0, 1)

0 10 20 30 40 50 60

0.0

0.2

0.4

0.6

0.8

1.0

Lag

AC

F

Series a

Katiyar S K (IISER Pune) Thesis Presentation May 3, 2011 6 / 27

Long memory time series

k =n

t=k+1(xtx)(xtkx)nt=1(xtx)

2

For long memory processes

k Ck as k

where C > 0 and (0, 1)

0 10 20 30 40 50 60

0.0

0.2

0.4

0.6

0.8

1.0

Lag

AC

F

Series a

A Long memory process is trend reinforcing, which means the direction(up or down compared to the last value) of the next value is more likelythe same as current value.

Katiyar S K (IISER Pune) Thesis Presentation May 3, 2011 6 / 27

Scaling exponents

Katiyar S K (IISER Pune) Thesis Presentation May 3, 2011 7 / 27

Scaling exponents

The most common power laws relate two variables and have the form

f (x) x

Katiyar S K (IISER Pune) Thesis Presentation May 3, 2011 7 / 27

Scaling exponents

The most common power laws relate two variables and have the form

f (x) x

Here is called the scaling exponent. where the word scaling denotesthe fact that a power-law function satisfies

f (cx) = cf (x) f (x)

Here c is a constant.

Katiyar S K (IISER Pune) Thesis Presentation May 3, 2011 7 / 27

Statistical test for long memory

How to find whether a given time series x(t) has long memory or not?

Katiyar S K (IISER Pune) Thesis Presentation May 3, 2011 8 / 27

Statistical test for long memory

How to find whether a given time series x(t) has long memory or not?Detrended fluctuation analysis

Katiyar S K (IISER Pune) Thesis Presentation May 3, 2011 8 / 27

Statistical test for long memory

How to find whether a given time series x(t) has long memory or not?Detrended fluctuation analysis

x(t) is the time series. (t = 1, 2, 3, .......Nmax )

Katiyar S K (IISER Pune) Thesis Presentation May 3, 2011 8 / 27

Statistical test for long memory

How to find whether a given time series x(t) has long memory or not?Detrended fluctuation analysis

x(t) is the time series. (t = 1, 2, 3, .......Nmax )y(k) =

ki=1(xi x) cumulative sum or profile

Katiyar S K (IISER Pune) Thesis Presentation May 3, 2011 8 / 27

Statistical test for long memory

How to find whether a given time series x(t) has long memory or not?Detrended fluctuation analysis

x(t) is the time series. (t = 1, 2, 3, .......Nmax )y(k) =

ki=1(xi x) cumulative sum or profile

Divide y(k) into time window of length n samples. In each box of lengthn, we fit y(k), using a polynomial function of order l , which represents thetrend in that box. The y coordinate of the fit line in each box is denotedby yn(k). Since we use a polynomial fit of order l , we denote the algorithmas DFA-l .

0 100 200 300 400 500 600 700 800 900 10000

50

100

150

200

250

300

350

K

Yk

Katiyar S K (IISER Pune) Thesis Presentation May 3, 2011 8 / 27

The integrated signal y(k) is detrended by subtracting the local trendyn(k) in each box of length n.

For a given box size n, the root-mean-square (rms) fluctuation for thisintegrated and detrended signal is calculated:

F (n) =

1Nmax

Nmaxk=1 [y(k) yn(k)]

2

Katiyar S K (IISER Pune) Thesis Presentation May 3, 2011 9 / 27

The integrated signal y(k) is detrended by subtracting the local trendyn(k) in each box of length n.

For a given box size n, the root-mean-square (rms) fluctuation for thisintegrated and detrended signal is calculated:

F (n) =

1Nmax

Nmaxk=1 [y(k) yn(k)]

2

The above computation is repeated for a broad range of scales (box sizen) to provide a relationship between F (n) and the box size n.

Katiyar S K (IISER Pune) Thesis Presentation May 3, 2011 9 / 27

The integrated signal y(k) is detrended by subtracting the local trendyn(k) in each box of length n.

For a given box size n, the root-mean-square (rms) fluctuation for thisintegrated and detrended signal is calculated:

F (n) =

1Nmax

Nmaxk=1 [y(k) yn(k)]

2

The above computation is repeated for a broad range of scales (box sizen) to provide a relationship between F (n) and the box size n.

A power-law relation between the average root-meansquare fluctuationfunction F (n) and the box size n indicates the presence of scaling

F (n) n

Katiyar S K (IISER Pune) Thesis Presentation May 3, 2011 9 / 27

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