Master's Thesis defence presentation

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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

    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)

    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

    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)

    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