I want to check if a time series is independent or not. The time series are nonlinear, generated from a nonlinear dynamical system.

For this I am using the autocorrelation (AC). Autocorrelation refers to the correlation of a time series with its own past and future values.

A time series is a collection of ordered random variables. I have two different time series that are obtained from a non-linear dynamical system denoted by variable phi and phit respectively.

The sample length of each time series is 5000.

Statistical test for independence :

Question 1:

In general, for checking if the samples of a time series are independent and identically distributed do we check for auto correlation (ACF) or cross-correlation (CCF)?

What is the value that tells us whether the sequence is independent?

Question 2: An example

The first plot is

CST = xcorr(xphit,xphi); AST = xcorr(xphi,xphi);

The second graph is the Auto and cross-correlation for the bit stream obtained from the same nonlinear dyanamical system.

I am not sure how to interpret the two graphs, the spike at lag zero and whether the samples are correlated or independent. Please help

UPDATE: Here is the code to generate the plot using a nonlinear dynamical system.

A = .5;  B = 1.99;  phin = .5;  xphi(1) = A - (B*phin);

for it = 2:1:5000    
 xphi(it) = A - (B*abs(xphi(it-1)));    
end
phi = 2*(xphi>=0.5)-1; 
A = .51;  B = 1.90;  phin = .5;  xphit(1) = A - (B*phin);

for it = 2:1:5000   
 xphit(it) = A - (B*abs(xphit(it-1)));    
end
phit = 2*(xphit>=0.5)-1; 


CST = xcorr(phit,phi);
AST = xcorr(phi,phi);

Time = 0;
for nn = 2:1:length(CST)
Time(nn) = Time(nn-1) + 1; 
end

plot(Time,AST,'r',Time,CST,'g');       title('\bf AUTO & CROSS correlation');    
xlabel('\bf Time');    ylabel('\bf Auto & Cross correlation'); legend('red-AC','green-CC');

I am trying to plot the auto and cross correlation of the bitstream that is obtained from a nonlinear dynamical system. 
The graph is the Auto and cross-correlation for the bitstream obtained from the same nonlinear dyanamical system.

I am not sure how to interpret the graph. Whether the program is incorrect or not. I wanted to obtain the plot of lags vs ACF but what I got is different than the actual plot given in a book. Please help.

A = .5;  B = 1.99;  phin = .5;  xphi(1) = A - (B*phin);

for it = 2:1:5000    
 xphi(it) = A - (B*abs(xphi(it-1)));    
end
phi = 2*(xphi>=0.5)-1; 
A = .51;  B = 1.90;  phin = .5;  xphit(1) = A - (B*phin);

for it = 2:1:5000   
 xphit(it) = A - (B*abs(xphit(it-1)));    
end
phit = 2*(xphit>=0.5)-1; 


CST = xcorr(phit,phi);
AST = xcorr(phi,phi);

Time = 0;
for nn = 2:1:length(CST)
Time(nn) = Time(nn-1) + 1; 
end

plot(Time,AST,'r',Time,CST,'g');       title('\bf AUTO & CROSS correlation');    
xlabel('\bf Time');    ylabel('\bf Auto & Cross correlation'); legend('red-AC','green-CC');

The plot should resemble

where the figure on top is the cross-correlation and the bottom is the auto-correlation

I want to check if a time series is independent or not. The time series are nonlinear, generated from a nonlinear dynamical system. Following the paper titled : A New Pseudo-Random Number Generator Based on Two Chaotic Maps

I am learning what techniques are there to show if a system such as the one above, known as chaotic nonlinear dynamical system from which the time series is obtained can be a good generator of PRBS. I am following the tests given in paper :

For this I am using the autocorrelation (AC). Autocorrelation refers to the correlation of a time series with its own past and future values.

A sequence of samples and I treat this as a time series -denoted by variables phi and phit respectively.

The sample length of each time series is 5000.

Statistical test for independence :

Question 1:

In general, for checking if the samples of a time series are independent and identically distributed do we check for auto correlation or cross-correlation?

Considering one time series only, if I want to check for independence then auto correlation should be zero.

Is this wrong?

Question 2:

The first plot is

CST = xcorr(xphit,xphi); AST = xcorr(xphi,xphi);

The second graph is the Auto and cross-correlation for the bitstream obtained from the same nonlinear dyanamical system.

I am not sure how to interpret the two graphs.

UPDATE: Here is the code to generate the plot. As I cannot upload the data set, I am using a simulated data set using a nonlinear dynamical system.

A = .5;  B = 1.99;  phin = .5;  xphi(1) = A - (B*phin);

for it = 2:1:5000    
 xphi(it) = A - (B*abs(xphi(it-1)));    
end
phi = 2*(xphi>=0.5)-1; 
A = .51;  B = 1.90;  phin = .5;  xphit(1) = A - (B*phin);

for it = 2:1:5000   
 xphit(it) = A - (B*abs(xphit(it-1)));    
end
phit = 2*(xphit>=0.5)-1; 


CST = xcorr(phit,phi);
AST = xcorr(phi,phi);

Time = 0;
for nn = 2:1:length(CST)
Time(nn) = Time(nn-1) + 1; 
end

plot(Time,AST,'r',Time,CST,'g');       title('\bf AUTO & CROSS correlation');    
xlabel('\bf Time');    ylabel('\bf Auto & Cross correlation'); legend('red-AC','green-CC');

I want to check if a time series is independent or not. The time series are nonlinear, generated from a nonlinear dynamical system.

For this I am using the autocorrelation (AC). Autocorrelation refers to the correlation of a time series with its own past and future values.

A time series is a collection of ordered random variables. I have two different time series that are obtained from a non-linear dynamical system denoted by variable phi and phit respectively.

The sample length of each time series is 5000.

Statistical test for independence :

Question 1:

In general, for checking if the samples of a time series are independent and identically distributed do we check for auto correlation (ACF) or cross-correlation (CCF)?

What is the value that tells us whether the sequence is independent?

Question 2: An example

The first plot is

CST = xcorr(xphit,xphi); AST = xcorr(xphi,xphi);

The second graph is the Auto and cross-correlation for the bit stream obtained from the same nonlinear dyanamical system.

I am not sure how to interpret the two graphs, the spike at lag zero and whether the samples are correlated or independent. Please help

UPDATE: Here is the code to generate the plot using a nonlinear dynamical system.

A = .5;  B = 1.99;  phin = .5;  xphi(1) = A - (B*phin);

for it = 2:1:5000    
 xphi(it) = A - (B*abs(xphi(it-1)));    
end
phi = 2*(xphi>=0.5)-1; 
A = .51;  B = 1.90;  phin = .5;  xphit(1) = A - (B*phin);

for it = 2:1:5000   
 xphit(it) = A - (B*abs(xphit(it-1)));    
end
phit = 2*(xphit>=0.5)-1; 


CST = xcorr(phit,phi);
AST = xcorr(phi,phi);

Time = 0;
for nn = 2:1:length(CST)
Time(nn) = Time(nn-1) + 1; 
end

plot(Time,AST,'r',Time,CST,'g');       title('\bf AUTO & CROSS correlation');    
xlabel('\bf Time');    ylabel('\bf Auto & Cross correlation'); legend('red-AC','green-CC');

I want to check if a time series is (a) random or not (b) independent. The time series are nonlinear, generated from a nonlinear dynamical system. Following the paper titled : A New Pseudo-Random Number Generator Based on Two Chaotic Maps

For these I am using the autocorrelation (AC). Autocorrelation refers to the correlation of a time series with its own past and future values.

If random variables are independent then there should be no correlation between them. The reverse is not true.

The sample length of each time series is 5000. I have plotted the AC for time series phi. I cannot interpret the graph, what the spike indicates and if the graph provides information about randomness and independence.

I am not sure what statistical tests I should be performing to check for randomness and independence.

A = .5;  B = 1.99;  phin = .5;  xphi(1) = A - (B*phin);

for it = 2:1:5000    
 xphi(it) = A - (B*abs(xphi(it-1)));    
end
phi = 2*(xphi>=0.5)-1; 
A = .51;  B = 1.90;  phin = .5;  xphit(1) = A - (B*phin);

for it = 2:1:5000   
 xphit(it) = A - (B*abs(xphit(it-1)));    
end
phit = 2*(xphit>=0.5)-1; 


CST = xcorr(phit,xphi);
AST = xcorr(xphi,xphi);

Time = 0;
for nn = 2:1:length(CST)
Time(nn) = Time(nn-1) + 1; 
end

plot(Time,AST,'r',Time,CST,'g');       title('\bf AUTO & CROSS correlation');    
xlabel('\bf Time');    ylabel('\bf Auto & Cross correlation'); legend('red-AC','green-CC');

I want to check if a time series is independent or not. The time series are nonlinear, generated from a nonlinear dynamical system. Following the paper titled : A New Pseudo-Random Number Generator Based on Two Chaotic Maps

For this I am using the autocorrelation (AC). Autocorrelation refers to the correlation of a time series with its own past and future values.

The sample length of each time series is 5000.

Statistical test for independence :

Question 1:

In general, for checking if the samples of a time series are independent and identically distributed do we check for auto correlation or cross-correlation?

Considering one time series only, if I want to check for independence then auto correlation should be zero.

Is this wrong?

Question 2:

The first plot is

CST = xcorr(xphit,xphi); AST = xcorr(xphi,xphi);

The second graph is the Auto and cross-correlation for the bitstream obtained from the same nonlinear dyanamical system.

I am not sure how to interpret the two graphs.

A = .5;  B = 1.99;  phin = .5;  xphi(1) = A - (B*phin);

for it = 2:1:5000    
 xphi(it) = A - (B*abs(xphi(it-1)));    
end
phi = 2*(xphi>=0.5)-1; 
A = .51;  B = 1.90;  phin = .5;  xphit(1) = A - (B*phin);

for it = 2:1:5000   
 xphit(it) = A - (B*abs(xphit(it-1)));    
end
phit = 2*(xphit>=0.5)-1; 


CST = xcorr(phit,phi);
AST = xcorr(phi,phi);

Time = 0;
for nn = 2:1:length(CST)
Time(nn) = Time(nn-1) + 1; 
end

plot(Time,AST,'r',Time,CST,'g');       title('\bf AUTO & CROSS correlation');    
xlabel('\bf Time');    ylabel('\bf Auto & Cross correlation'); legend('red-AC','green-CC');