CSCS 520 Program Manual

Center for the Study of Complex Systems
University of Michigan
March 11, 1999

This manual is designed to document the CSCS 520 suite of programs. The programs are:

• dimension - a dimension calculator.
• lyapunov- a program to approximate the lyapunov exponent of a time series.
• deltacalc- a program to calculate the predictability and deltas of a time series.

These programs are designed to perform simple analyses on a time series. They all reside on the CSCS machines, which are located in the CSCS Computer Lab, 4477 Randall. They can be found in the directory

/s1/courses/pscs520/bin

Using command line options to specify program parameters is generally faster than using the user interface. If a specific option is not specified, the default value for that option will be used. The defaults are listed in the tables below.

Common command-line options

How to convert binary datasets into ascii

The datasets are stored in binary format as big endian double precision floating point numbers. The best way to import one of the datasets into Mathematica is to export it first into an intermediate file. This can be done by using the utility program /s1/courses/pscs520/bin/convert, which takes a dataset and converts it into an ASCII file.

Usage: convert [-p precision] [-n num_pts] [-s data_set] -o
where:
precision = number of decimal places in each output file 'myfile'.
num_pts = number of points to convert (default = 1000)
data_set = which data set to convert (a letter between 'a' and 'i')

Once you have exported the dataset, Mathematica should be able to read the ASCII file.

Dimension Calculator

General Description

In the following discussion, x is a time series. The dimension calculator divides the range, r into P equally sized segments, where P is the specified number of initial partitions. It then constructs d-tuples from the time series by taking time lags, where d is a dimension chosen for the reconstruction of the time series. Thus, the division of the range into P segments produces a set of boxes in the d-dimensional space, each of which has a linear size . The program then determines the box in which the d-tuple is contained, and increments the count of the number of d-tuples that are within that box. When all d-tuples in the time series have been exhausted, the program computes the quantities in expressions (1) and (2) below.

Next, the program multiplies the number of segments by m, a multiplicative factor, and divides each axis of the d-dimensional space into mP segments, thus forming d-dimensional boxes of linear dimension   in the d-dimensional reconstruction space. The number of segments along each axis is guaranteed to increase by at least one for each iteration of this process. The program then repeats the calculations described above for this new set of boxes. After that, the program again multiplies the number of segments by m, thus forming d-dimensional boxes of linear size  , and repeating the calculations. This process continues for the number of specified iterations.

The index k labels which dimension we are computing. For a given k, and for each iteration (i.e. each segmentation of the range) the program calculates the following quantities:

k not 0

k=0

Here V is the total number of d-dimensional vectors in the time series. If T is the length of the time series, then V=T-d. N is the total number of boxes into which the d-dimensional reconstruction space is divided. (This is just the number of segments into which each axis of the space is divided, raised to the d power.) n_i is the number of d-tuples contained in box i, and epsilon is the linear size of the box, as defined above.

Although the dimension D_k+1 are, strictly speaking, defined in the epsilon -> 0 limit, when considering finite data sets, or data sets in which there may be noise added to some underlying deterministic structure, one cannot use this definition practically. The reason is that for small epsilon, statistical limitations due to finite data and noise effects become dominant. Rather, a more reasonable strategy is to look for a range of epsilon over which the ratio

or

is sensibly constant. Thus, in practice, one plots (1) or (2) vs. log(epsilon) and looks for a region where the resulting curve is approximately a straight line. The slope of that line will then be an estimate for D_k.

Dimension Program Output File

The dimension program produces an output file containing the following information: For each value of k that was specified, the output file has a series of epsilons and dimension calculations. For example, the output below details five iterations (the five different epsilons) and three dimensions.

Results for k = -1
                              Dimension
log10(epsilon)    1             2               3
0.249975        0.000400        0.000800        0.001700
0.199980        0.000500        0.001100        0.002200
0.166650        0.000600        0.001300        0.002600
0.142843        0.000700        0.001600        0.003200
0.124987        0.000800        0.001600        0.003500

Examples

The following examples demonstrate a few ways of using the dimension calculator.
 

dimension -zA	Uses all of the default values on data set A.
dimension -z A -o mydim.out	Same as above except output is written to the file "mydim.out"
dimension -d3:5 -n20000 -f infile.dat	Calculates 20,000 points of the input file "infile.dat" for dimensions 3 to 5 inclusive.
dimension -d4 -zA -c30000:40000	Uses the 10,000 points of data set A from index 30,000 to 40,000 and calculates dimension 4.
dimension -zA -k"-2:2"	Calculates values of k equal to -2, -1, 0, 1, and 2 for data set A, using all other default values.

Lyapunov Exponent Calculator

General Description

The lyapunov exponent calculator works as follows:

For every n-tuple ( x_i, x_i+1, x_i+2, ... x_i+n-1), the program finds all n-tuples ( x_j, x_j+1, x_j+2, ... x_j+n-1) such that

and d < e , where e is the specified value of epsilon. For each pair of n-tuples that are within epsilon, the value

is calculated, where T is the specified time offset. From these two values, d and D, the following estimation of the lyapunov exponent is calculated:

A histogram of the lambdas is generated by the program.

Lyapunov Program Output File

The program produces an output file containing histogram data, which can be used to obtain the lyapunov exponent.

Examples

lyapunov -zC -n50000 -d3:5 -T3	Performs the calculation on 50,000 points of dataset C for dimensions 3, 4, and 5 with a time offset of 3.
lyapunov -c40000:50000 -d2 -zb	Analyzes 10,001 points of dataset b from index 40,000 to 50,000 inclusive for dimension 2.

Delta Calculator

General Description

The delta calculator computes the probability that two vectors are within e , where e is a calculated value of epsilon. It works as follows:

Initially, the value of e is equal to s mu _init , where s is the standard deviation, and mu_init is a user specified parameter. For every d-tuple ( x_i, x_i-1, x_i-2, ... x_i-d+1), the program counts all d-tuples ( x_j, x_j-1, x_j-2, ... x_j-d+1) such that

For each pair of n-tuples that are within e , the count is incremented. The probability

is calculated, where n is the count of the number of pairs within e , and N is the total number of d-tuples in the time series.

From these calculated values of C, the deltas and predictabilities are then calculated.

In each successive iteration of the program, e = e mu , where mu is a user-specified parameter. The loop repeats through i iterations, with i being specified by the user as well.

Delta Program Output File

The program produces an output file containing the following information:

CSCS 520 Delta Calculator Results - Thu Mar 18 16:08:50 1998

Program parameters:
        Input file:   /s1/courses/pscs520/data/dataA.bin
        Time Series Length: 10000 points, Range: 0 - 10000
        Starting Dimension: 1     Ending Dimension:  3
        Mu:  0.5000000    Number of Mu's: 4
        Sigma: 0.2903502
        Lag Values :    0, 1, 2

*************************************************************************

Epsilon            C01             C02             C03
0.1451751       0.1500574       0.0843727       0.0472669
0.0725875       0.0739823       0.0394495       0.0210504
0.0362938       0.0368568       0.0191454       0.0099546
0.0181469       0.0183442       0.0093495       0.0047707


*************************************************************************


Epsilon         Delta  01       Delta  02       Delta  03
0.1451751       0.7331217       0.0036645       0.0000000
0.0725875       0.8612563       0.0007028       0.0000000
0.0362938       0.9290472       0.0009426       0.0000000
0.0181469       0.9640077       0.0011719       0.0000000

*************************************************************************


Epsilon            S01             S02             S03
0.0725875       0.0739823       0.5332295       0.5336045
0.0362938       0.0368568       0.5194545       0.5199446
0.0181469       0.0183442       0.5096689       0.5102669
0.0090734       0.0091349       0.5041898       0.5054090   
  

Examples

deltacalc -za -c "1000:2000" -d "3:5"	Analyze 1001 points between 1000 and 2000 inclusive for dimensions 3 to 5 of data set A
deltacalc -M 0.33 -i 15 -f myfile.inp	Analyze points from myfile.inp with initial mu of 0.33 and perform 15 iterations.
deltacalc -zf -l "0,1,5:9"	Analyze data set F and set the lag map to 0, 1, 5, 6, 7, 8, 9