# Generate Time Series with specific ACF and multiple LAG

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In summary, the conversation discusses the use of non-parametric methods in generating time series data with different lag values and ACF values. The goal is to compare the effectiveness of these methods by calculating autocorrelation and residues. The speaker is seeking help in generating time series with higher lag values and specific ACF patterns.
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Hello Everyone,

I will try to explain what am I doing here and I hope someone will understand.
ACF - autocorrelation function

I'm doing a research about non-parametric methods utility. How they fit and are useful in a different environment. I'm generating time series with different sizes of data (N=128, N=256, N=512 and so on, N - represents how many values are in time series.). For the results to be more accurate I generate 10 000 samples for each N=128, N=256... I'm using different ACF values (0,1 , 0,3 , 0,5 , 0,7 and 0,9) to generate those time series. And I also use 4 different distributions (Normal, Exponential, Cauchy and Uniform).

I've managed to generate time series with LAG=1 using autoregressive process X(t) = aX(t-1) + e(t), where a = ACF value, e(t) are successively chosen at random using random number distribution (normal,exp,uni or cauchy).

After having time series generated with specific ACF, I calculate autocorrelation from those generated time series. Having those two different values (ACF and autocorrelation from time series) I can calculate different residues and see where one or another method is more effective.

I couldn't figure out how to generate time series with LAG>1.

So, what I need to do now is to:

A) Generate time series with higher LAG than 1. And while the LAG is increasing, I need ACF value to go down slowly and reach 0 (lets say in 10-20 LAGS)

B) Generate time series with higher LAG than 1. And while the LAG is increasing, I need ACF value to go down FAST and reach 0 (lets say in 2-4 LAGS). ACF = 0 value should repeat itself until LAG=100 and then it should go up significantly

C) Same thing. Generate time series by increasing it's LAG, but the ACF value should be cyclic. It should repeat itself every 4 or 8 or 12 LAG.

I hope you understood what I was trying to say. Can you help me?

For lag > 1, just try something like X(t) = aX(t-lag) + e(t)

Is this what you are looking for?
A) X(t) = 1/5.5* [X(t-1) + 0.9*X(t-2)+ 0.8*X(t-3)+ 0.7*X(t-4)+ 0.6*X(t-5) + ... ] + e(t)

(I divided by 5.5 = 1+.9+.8+.7+.6+.5+.4+.3+.2+.1. I think that will make it stationary. Not sure. Would have to think about that. You can always divide by more.)

## What is a time series?

A time series is a set of data points measured at consistent intervals over time. It is used to track the change in a variable or phenomenon over time and can be used to make predictions about future values.

## What is an ACF?

An ACF (Autocorrelation Function) is a measure of the correlation between a time series and a lagged version of itself. It shows how the current value of a time series is related to previous values.

## How can I generate a time series with a specific ACF?

To generate a time series with a specific ACF, you can use statistical software such as R or Python. These programs have functions or packages that allow you to specify the desired ACF and generate a time series that follows it.

## What is the significance of multiple lags in a time series?

Multiple lags in a time series can help to identify patterns and trends in the data. By looking at the ACF of a time series with multiple lags, you can see how the current value is related to previous values at different time intervals. This can provide useful insights for forecasting and analysis.

## Can I manipulate the ACF and lags in a time series?

Yes, you can manipulate the ACF and lags in a time series by changing the parameters or input data. This can be done in statistical software or by using different models and techniques to generate a desired ACF and multiple lags.

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