110

CHAPTER-5

ENTROPY APPROACH TO DATA ANALYSIS

111

5.1 Introduction

Nonlinear dynamical analysis is a powerful approach to understanding

financial systems. The calculations, however usually require very long data

sets that can be difficult or impossible to obtain. Pincus devised the theory and

method for a measure of regularity closely related to the kolmogorov entropy,

the rate of generation of new information that can be applied to the typically

short and noisy time series of clinical data. This family of statistics, named

approximate entropy (ApEn) [4] is rooted in the work of Grassberger and

Procaccia and Eckmann and Ruelle and has been widely applied in clinical

cardiovascular studies.

The method examines time series for similar epochs: more frequent and

more similar epochs lead to lower values of ApEn . Informally, given N points,

the family of statistics ApEn (m , r , N) is approximately equal to negative

average natural logarithm of the conditional probability that the two sequences

that are similar for m points remains similar , that is within a tolerance r [3] ,

the next point. Thus a low value of ApEn reflects a high degree of regularty.

Importantly, the ApEn algorithm counts each sequence as matching itself, a

practice carried over from the work of Eckmann and Ruelle [5] to avoid the

occurrence of ln(0) in the calculations. This step has led to discussion of the

bias of ApEn. In practice, it is found that this bias causes ApEn to lack two

important expected properties. First, ApEn is heavily dependent on the record

length and is uniformly lower than expected for short records. Second, it lacks

relative consistency. That is, if ApEn of one data set is higher than that of

another, it should, but does not, remain higher for all conditions tested. This

shortcoming is particularly important, because ApEn has been repeatedly

recommended as a relative measure for comparing data sets [6].

112

The terminology and notation of Grassberger and Procaccia, Eckmann

and Ruelle, and Pincus can be employed in

describing techniques for

estimating the Kolmogorov entropy [21, 22] of a process represented by a time

series and the related statistics ApEn and SampEn. The parameters N, m, and r

must be fixed for each calculation. N is the length of the time series, m is the

length of sequences to be compared, and r is the tolerance for accepting

matches. It is convenient to set the tolerance as r × SD, the standard deviation

of the data set, allowing measurements on data sets with different amplitudes

to be compared. Throughout this work, all time series have been normalized to

have SD = 1.

Sample Entropy is a statistical measure proposed by Richman and

Moorman (2000) [18] which quantifies the variability of time-series by

comparing sequences of consecutive data points [17, 19, 20]. It provides a

measure of the regularity or predictability of a time-series (high sample

entropy is related to low predictability / high complexity)

Sample entropy is derived from the conditional probability that a

sequence of data points is within a certain tolerance range r for m steps. This

tolerance r is usually measured in units of the standard deviation (SD) of the

series. Hence sample entropy depends on the length of the data series N, the

length m of sequences to be compared and the tolerance range r to be

specified. However SampEn can‟t be used to distinguish between signals of

similar form but different frequency. A signal contains noise and has a certain

period is no more complex than the same quantity of data but with a different

periodicity. SampEn calculation was applied to methane (CH4) for a model

run of the chemistry-climate model (CCM).

SampEn statistics is developed to be free of the bias caused by self-

matching. The name refers to the applicability to time series data sampled from

113

a continuous process. In addition, the algorithm suggests ways to employ

sample statistics to evaluate the results, as explained

below.

There are two major differences between SampEn and ApEn statistics.

First, SampEn does not count self-matches. Discounting self-matches are

justified on the grounds that entropy is conceived as a measure of the rate of

information production, and in this context comparing data with themselves is

meaningless. Furthermore, self-matches are explicitly dismissed in the later

work of Grassberger and co-workers. Second,

SampEn does not use a

template-wise approach when estimating conditional probabilities. To be

defined, SampEn requires only that one template find a match of length m +

1.

Grassberger and Procaccia ,defined

Cm(r) = (N - m + 1)

- 1

1

1

mN

i

Cm

i(r),

(6.1)

the average of the C

mi(r) defined above. This

differs from

m(r) only in that

m(r) is the average of the natural logarithms of the C

mi(r).

They suggest

approximating the Kolmogorov entropy of a process represented by a time

series by

limr 0 limn limN - ln [C

m + 1(r)/ C

m(r)]

(6.2)

In this form, however, the limits render it unsuitable for the analysis of finite

time series with noise. Therefore two alterations are made to adapt it to this

purpose. First, it is followed their later practice in calculating correlation

integrals and did not consider self-matches when computing Cm(r). Second,

114

only the first N - m vectors of length m are considered, ensuring that, for

1 i N - m, xm(i) and xm + 1(i) are

defined.

It is defined Bm

i(r) as (N - m - 1)- 1

times the number of vectors xm( j) within r

of xm(i), where j ranges from 1 to N - m, and j

i to exclude self-matches. It is

then defined

Bm(r) = (N - m)

- 1

mN

i 1

Bm

i(r). (6.3)

Similarly, Am

i(r) may be defined as (N - m - 1)- 1

times the number of vectors

xm + 1( j) within r of xm + 1(i),

where j ranges from 1 to N - m ( j i), and set

Am(r) = (N - m)

- 1

mN

i 1

Am

i(r). (6.4)

Bm(r) is then the probability that two sequences will match for

m points,

whereas Am(r) is the probability that two sequences will match for

m + 1 points.

We then defined the parameter

SampEn(m, r) = limN {-ln [A

m(r)/ B

m(r)]},

(6.5)

which is estimated by the statistics

SampEn(m, r, N ) = -ln [A

m(r)/B

m(r)]. (6.6)

Where there is no confusion about the parameter r and the length m of

the template vector, we set

B = {[(N - m - 1)(N - m)]/2} B

m(r) (6.7)

115

and

A = {[(N - m - 1)(N - m)]/2} Am(r), (6.8)

so that B is the total number of template matches of length m and A is the total

number of forward matches of length m + 1. It noted that A/B = [A

m(r)]/B

m(r)],

so SampEn(m, r, N ) can be expressed as -ln (A/B).

The quantity A/B is precisely the conditional probability that two

sequences within a tolerance r for m points remain within r of each other at the

next point. In contrast to ApEn(m, r, N ), which calculates probabilities in a

template-wise fashion, SampEn(m, r, N ) calculates the negative logarithm of a

probability associated with the time series as a whole. SampEn(m, r, N ) is

defined except when B = 0, in which case no regularity has been detected, or

when A = 0, which corresponds to a conditional probability of 0 and an infinite

value of SampEn(m, r, N ). The lowest nonzero conditional probability that

this algorithm can

report is 2[(N - m - 1)(N - m)]-1

. Thus, the statistic

SampEn(m, r, N ) has ln (N - m) + ln (N - m - 1) - ln (2) as an upper bound,

nearly doubling ln (N - m), the dynamic range of ApEn(m, r, N).

SampEn is not defined unless template and forward matches occur and

is not necessarily reliable for small numbers of matches. The calculation of

SampEn(m, r, N ) has been reviewed as a process of sampling information

about regularity in the time series and used sample statistics to inform about

the reliability of the calculated result. For operational purposes,

it is assumed

that the sample averages follow a Student's td distribution, where d is the

number of degrees of freedom. It is obvious that the "true" average conditional

probability of the process is within SDt(B 1,0.975)/ of the sample average,

where SD is the sample standard deviation and tB 1,0.975 is the upper 2.5th

percentile of a t distribution with B - 1 degrees of freedom. The

size of the

116

confidence intervals depends on the number B and the number of forward

matches. Informally, large confidence intervals around SampEn(m, r, N )

indicate that there are insufficient data to estimate the conditional probability

with confidence for that choice of m and r. In addition, confidence intervals

allow standard statistical tests of the significance of differences between data

sets.

Because of the logarithms inside the summation, m(r)(v u) will not

generally be equal to m(r)(u v). Thus cross-ApEn(m, r, N )(v u) and its

direction conjugate cross-ApEn(m, r, N )(u v) are unequal in most

cases.

In defining cross-SampEn, it is convenient to set Bm

i(r)(v u) as (N- m)-1

times the number of vectors ym( j) within r of xm(i), where j ranges from 1 to

N - m.

It is then defined

Bm(r)(v u) = (N - m)

- 1

mN

i 1

B

mi(r)(v u). (6.9)

Similarly, it is set A

mi(r)(v u) as (N

- m)

- 1 times the number of vectors ym + 1(

j) within r of xm + 1(i), where j ranges from 1 to N - m.

It is then defined

Am(r)(v u) = (N - m)

-1

mN

i 1

Am

i(r)(v u). (6.10)

Finally, SampEn can be expressed in the form

SampEn(m, r, N )(v u) = -ln {[Am(r)(v u)]/ [B

m(r)(v u)]}.

(6.11)

117

Examining this definition for direction dependence, it is found that (N -

m) Bm

i(r)(v u) is the number of vectors from v within r of the ith template of

the series u. Summing over the templates, it is found that

mN

i 1

(N - m) Bm

i(r)(v

u) simply counts the number of pairs of vectors from the two series that

match within r. The number of pairs that match is clearly independent of which

series is the template and which is the target. Because the last summation is

equal to (N - m)2 B

m(r)(v u), it follows that B

m(r)(v u) is also direction

independent, implying that cross-SampEn(m,

r, N )(v u) = cross-

SampEn(m, r, N )(u v). It should be noted that cross-SampEn will be defined

provided that Am(r)(v u) 0. Cross-SampEn, on the other hand, requires only

that one pair of vectors in the two series match for m + 1

points.

It is obvious that SampEn statistics appear to be relatively consistent

over the family of processes, whereas ApEn statistics are not. Although we

believe that relative consistency should be preserved for processes for which

probabilistic character is understood, we see no general reason why ApEn or

SampEn statistics should remain relatively consistent for all time series and all

choices of parameters.

It is proposed as a general, but by no means exhaustive, explanation for

this phenomenon. SampEn is, in essence, an event-counting statistic, where the

events are instances of vectors being similar to one another. When these events

are sparse, the statistics are expected to be unstable, which might lead to a lack

of relative consistency. The value of SampEn(m, r, N ) is less than or equal

to

ln (B), the natural logarithm of the number of template matches.

If

SampEn(m, r, N )(S) < SampEn(m, r, N ) (T ) and the number of T's template

matches, BT, is less than the number of S's template matches, BS, which would

be consistent with T displaying less order than S. Provided that AT

and AS, the

118

number of forward matches, are relatively large, both SampEn statistics will be

considerably lower than their upper bounds. As r decreases, BT and AT

are

expected to decrease more rapidly than BS and AS. Thus, as BT becomes very

small, SampEn(m, r, N )(T ) will begin to decrease, approaching the value

ln (BT), and could cross over a graph of SampEn(m, r, N )(S), where or

while

BS is still relatively large. Furthermore, as the number of template matches

decreases, small changes in the number of forward matches can have a large

effect on the observed conditional probability. Thus the discrete nature of the

SampEn probability estimation could lead to small degrees of crossover and

intermittent failure of relative consistency, and it cannot be said that SampEn

will always be relatively consistent. It is obvious, however, that SampEn is

relatively consistent for conditions where ApEn is not, and it is not observed

any circumstance where ApEn maintains relative consistency and SampEn

does not.

5.2 Method for the calculation of SampEn

Consider a time series with many matching points (data) of length N.

Two data points are considered to be in matching if they are within the

tolerance window „r‟. Now the sequences of data points are matched. The

sequence length is „m‟. For m=2, the match of two data points is considered.

The total number of matches for a particular pair is counted & the process is

repeated for all the possible pairs.

Now the total number of two component matches from the series is added.

The first data point from the series is omitted i.e. now number of data points

are (N-m+1) where m=2. Then the above procedure is repeated. At last add the

total number of pairs of two components to the previous. Say it is „B‟.

119

Similarly pair of 3-components (m=3) by using above procedure is obtained.

Then all the possible 3-component pairs are added. Say it is „A‟.

Sample entropy is defined as the negative natural logarithm of the ratio of A &

B. So mathematically SampEn can be written as

SampEn (m , r , N) = -ln(A/B) (6.12)

5.3 Multi-Scale Entropy Analysis

MSE method is dependent of coarse-graining procedure. It incorporates two

steps.

Consider a given time series nix,........,x,x,xx

321

The length of the series is N. Then we construct consecutive coarse-grained

time series by averaging a successively increasing number of data points in

non-overlapping windows. Figure 5.1 shows a schematic illustration of the

coarse-graining procedure for scale 2 and 3. Each element of the coarse-

grained time series, is calculated accordingly to the equation

j

1)1j(i

i

)(

jx

1y

(6.13)

Where, τ represents the scale the factor and i ≤ j ≤ The length of each

coarse-grained time-series is N/τ. For scale one, the time series { y(1)

} is

simply the original time-series.

Finally, sample entropy (Samp En) is calculated for each coarse-grained time-

series, and then SampEn is plotted as a function of the scale-factor. Let

1mi1iimx...,,.........x,x)i(u 1 ≤i N-m

(6.14)

120

be vectors of length m. Let nim(r) represent the number of vectors Um(i) .

Within distance r of Um(i) where j range from 1 to (N-m) and j≠ 1 to exclude

self matches.

1mN

)r(n)r(c

imm

i

(6.15)

Is probability that any vector Um (i) is within tolerance range r of Um (i). We

then define

)r(clnmN

1)r(u

mN

1i

m

ii

m

(6.16)

The parameter Sample entropy (SampEn) is defined as

)(

)(ln),(

1

limru

rurmSampEn

m

m

N

(6.17)

For a Time Series of finite length (N), the sample entropy is estimated by

statistics,

)r(u

)r(uln)N,r,m(SampEn

m

1m

(6.18)

121

Sample entropy is the natural logarithm of the ratio of the total number of two

components templates matches to the total number of three components

templates matches.

For scale one, the value of entropy is higher for the white noise time series in

comparison to the 1/f noise.

This result explains the facts that the 1/f noise contains complex structures

across multiple scales in contrast to the white noise.

122

Scale 1: x1 x2 x3 x4 x5 x6 …..xi

xi+1

Scale 2: x1 x2 x3 x4 x5 x6 ……..xi xi+1

Y1 Y2 Y3 Yj =

Scale 3: x1 x2 x3 x4 x5 x6 ………… ..… xi xi+1 xi+2

Y1 Y2 Yj =

[Fig 5.1 Coarse graining procedure for time series data]

123

[Fig 5.2 Procedure for calculating sample entropy (SampEn) for the case in which the

pattern length, m, is 2, and the similarity criterion, r, is 20]

124

5.4 Data Analysis and Results

The data base of various sectoral indices of National Stock Exchange of India

from tt5 (Advance) of India Infoline Securities pvt. Limited. Daily closing

values of indices are taken from 17th

june 2003 to 9th feb 2010. Total no of data

points are taken to be 1623.

Figure 5.3(a) and 5.3(b) show the Graph of variation inVarious Sectoral

Indices of Nifty. Figure 5.4(a) and 5.4(b) show corresponding MSE Profile of

same Sectoral Indices. Fig 5.5 shows the variation of tick value of NIFTY for

pre budget hours, just after the budget announcement and at later hours of the

market. Figures 5.6 and 5.7 show the corresponding MSE profiles for m=2 and

m=3 values respectively. Figure 5.8 shows the variation in the tick value of

NIFTY for 24th

feb 2010 to 2 march 2010, in which there are two days of pre-

budget days, budget day, and one post- budget day. Fig 5.9(a) and 5.9(b) are

MSE Profiles of Pre Budget, Budget and post budget hours & Later hours of

the Market.

125

[Fig 5.3a Graph of Various Sectoral Indices of Nifty]

[Fig 5.3b Graph of Various Sectoral Indices of Nifty]

0

5000

10000

15000

20000

25000

0 500 1000 1500 2000 2500

Tick Numbers

Tick

Valu

es

BSE Auto BSE Cap BSE Tech

0

2000

4000

6000

8000

10000

12000

14000

16000

18000

20000

0 500 1000 1500 2000 2500 Tick Numbers

Tick Values

BSEMetal BSE Bank BSE IT

126

[Fig 5.4a MSE Profile of Sectoral Indices of BSE]

[Fig 5.4b MSE Profile of Sectoral Indices of BSE]

m=2 , r=0

6

8

10

12

14

16

0 10 20 30 40 50Scale Factor

Sam

ple

entro

py

BSE M BANKEX BSEIT

m=2, r=0

7

9

11

13

15

0 10 20 30 40 50Scale Factor

Sam

ple

Ent

ropy

BseAuto BseTech BsePsu BseCap.Goods

127

[Fig 5.5 Tick Value of NIFTY for Pre Budget, Budget and post budget hours

& Later hours of the Market]

[Fig 5.6 MSE Profile of Pre Budget, budget and post budget hours & Later

hours of the Market ( m=2)]

4840

4880

4920

4960

5000

0 1000 2000 3000 4000 5000Tick Values

Rate

sBudget Hr Pre Budget Hr Later Hr

MSE Plot Of 26FEB( m=3,r=.001)

0

2

4

6

8

10

12

14

16

18

0 5 10 15 20 Scale Factor

Samle Entropy

9-11.59am 12-12.59pm 1-3 pm

128

[Fig 5.7 MSE Profile of Pre Budget, Budget and post budget hours & Later

hours of the Market ( m=3)]

[Fig 5.8 Variation in the Tick value of NIFTY for 24th feb 2010 to 2 march

2010]

MSE Plot of 26 FEB (m=2,r=.001)

0

2

4

6

8

10

12

14

16

0 5 10 15 20Scale Factor

Sam

ple

Ent

ropy

9-11.59am 12-12.59pm 1-3 pm

4800

4850

4900

4950

5000

5050

0 2000 4000 6000 8000 10000 12000Tick Values

Rat

es

24-Feb 25-Feb 26-Feb 2-Mar

129

[Fig 5.9a MSE Profile of Tick value of NIFTY from 24th

feb 2010 to 2 march

2010]

[Fig 5.9b MSE Profile of Tick value of NIFTY from 24th

feb 2010 to 2

march 2010]

m=2,r= .001

0

2

4

6

8

10

12

14

16

18

0 5 10 15 20Scale Factor

Sam

ple

Ent

ropy

24-Feb 25-Feb 26-Feb 2-Mar

m=3, r= .001

0

2

4

6

8

10

12

14

16

18

0 5 10 15 20Scale Factor

Sam

ple

Entr

py

24-Feb 25-Feb 26-Feb 2-Mar

130

5.4 Discussion

Studies of the MSE profile of daily variations of different indices show

that the BSE Metal Index exhibit lower MSE pattern as compared to other

sectoral indices. This is due to the cyclical nature of the constituents of the

index. Thus a pattern emerging out of the cyclical nature gives rise to a lower

MSE pattern at all scales. All other indices show identical MSE pattern

indicating equivalent complexity levels of daily data. Similarly NIFTY tick

values [34, 35] are studied for pre budget hours, just after budget, and later

hours of market timing on the budget day. The MSE profile shows that the

entropy of market is the maximum at pre budge hours. The higher entropy

profile indicates higher entropy of the data. The index value of the budget and

post budget hours shows lower MSE profile showing low complexity, i.e.

higher degree of order. As the provisions of the budget were made public

market started interacting with the information showing lower MSE profile

indicating higher degree of order (low complexity).

Data of later hours again shows higher entropy profile [32] but still

having lower values of entropies at different scales than the pre budget hours.

However the same study performed on the NIFTY tick values for the period

from 24th

feb 2010 to 2nd

march 2010 in which there are two days of pre

budget, budget day and one post budget day, indicates that the difference

among their MSE profile disappear and their profiles converge showing the

identical behavior. This indicates that the market responds to receive

information with a higher degree of order and adjusts itself interacting with the

information. As the information has been received, the market behaves like an

isolated system with higher entropy. Multiscale entropy measurements could

be an effective alternative nonlinear approach for analyzing the stock market

data [33].

131

Limitations

The long-standing problem of deriving useful measures of time series

complexity is important for the analysis of financial, physical and biological

systems. MSE is based on the observation that the outputs of complex systems

are far from the extrema of perfect regularity and complete randomness.

Instead, they generally reveal structures with long-range correlations on

multiple spatial and temporal scales. These multiscale features, ignored by

conventional entropy calculations, are explicitly addressed by the MSE

method.

The complexity [31] is associated with the ability of financial systems to

adjust to an ever-changing environment, which requires integrative multiscale

functionality. In contrast, under free-running conditions, a sustained decrease

in complexity reflects a reduced ability of system to function in certain

dynamical regimes possibly due to decoupling or degradation of control

mechanisms. The MSE method requires an adequate length of data to provide

reliable statistics for the entropy measure on each scale. The minimum number

of data points required to apply the MSE method depends on the level of

accepted uncertainty. Another important consideration is related to

nonstationarity. To calculate SampEn , one has to fix the value of a parameter

that depends on the time series SD. Therefore, the results may be significantly

affected by nonstationarities , outliers , and artifacts. In contrast, attempts to

remove nonlocal nonstationarities , e.g.,trends , will most likely modify the

structure of the time series over multiple time scales. Thus, given the temporal

complexity of data on multiple scales, a novel technique, multiscale entropy,is

a robust measure of complexity [17].

132

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