# Physical meaning of autocorrelation

• jollage
In summary: This low-frequency drift accumulates over time, and the autocorrelation function will show it. In summary, the autocorrelation function indicates how well the system can predict future values based on past values. The shorter the time delay, the more rapid the decay of the autocorrelation. This suggests that the low-frequency component in the data is due to drift, and that it will decrease over time.
jollage
Hi All,

I was in a process of processing my vibration-test data. I now generated a plot of the autocorrelation function of the object acceleration. Please see the attachements (the second attachment is the close-up for small tau's).

The x-axis in the plot is the time delay tau. You can see from the second attachment that, at tau=0,the autocorrelation reaches its maximum.

My question is, as for the signal like this, in the long time delay (tau=60s, 70s), the autocorrelation still does not decay too much. What could this behavior imply? In terms of the noise frequency, what can we say?

Mz

#### Attachments

• autocor.jpg
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• autocor2.jpg
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The autocorrelation function gives the amount of correlation of values of the original function at times delayed by ##\tau##. If the autocorrelation function vanishes quickly, it means that the value of the function at some time cannot be predicted with success from neighboring values.

If it does not vanish quickly, it means that the behavior of the function is less random; from the value at time 0 one can estimate possible values at future times with better success. For example, if the function is periodic, after one wave one can make reasonable estimate what the next values will be, and the autocorrelation function comes out also periodic. But if the function is position of Brownian particle, the next values are hard to predict successfully, and also the autocorrelation function decays very fast (exponentially).

If the autocorrelation does not die away then there must be some significant periodic elements in the signalbecause the delayed signal correlates well with an earlier version of itself.

jollage said:
Hi All,

I was in a process of processing my vibration-test data. I now generated a plot of the autocorrelation function of the object acceleration. Please see the attachements (the second attachment is the close-up for small tau's).

Mz

I think of the autocorrelation as a measure of how deterministic the system is- a perfectly deterministic system will always have a normalized autocrrelation of '1', while a perfectly random system will have have an autocorrelation that looks like a delta function. For real systems with dissipation present, the autocorrelation function will generally follow a decaying exponential, and the width of the exponential is a measure of the (de-)correlation time (or length, if we extend the analysis to cover spatially random systems such as reflection from a rough surface). Multiple dissipation processes result in multi-exponential autocorrelation functions- the dynamic light scattering field is a good place to get familiar with interpreting autocorrelation curves.

Without knowing any details about what generated the data (what is vibrating? How was it excited?) it's tough to parse your data. However, when we see a low-frequency component like your first graph has, it's due to drift in our detector.

Hello Mz,

The autocorrelation function is a measure of the similarity between a signal and a delayed version of itself. In other words, it tells us how much a signal is correlated with itself at different time intervals. The maximum value of the autocorrelation function at tau=0 indicates that the signal has a high degree of self-similarity.

In terms of the noise frequency, the behavior of the autocorrelation function at long time delays (tau=60s, 70s) can provide information about the underlying frequency components of the signal. If the autocorrelation function does not decay too much at these longer time delays, it suggests that there may be low frequency components present in the signal. This could indicate that the vibration-test data contains low frequency noise or that the object being tested has low frequency vibrations.

It is important to note that the autocorrelation function is just one tool for analyzing signals and should be used in conjunction with other techniques to fully understand the characteristics of the signal. I hope this helps to clarify the physical meaning of autocorrelation in your data analysis. Best of luck with your research.

Sincerely,

## 1. What is autocorrelation and why is it important in scientific research?

Autocorrelation is a statistical tool used to measure the linear relationship between a variable and its past values. It is important in scientific research because it helps identify patterns and trends in data, and can be used to make predictions and test hypotheses.

## 2. How is autocorrelation calculated?

Autocorrelation is typically calculated using the Pearson correlation coefficient, which measures the strength and direction of the linear relationship between two variables. It can also be calculated using other methods such as the Spearman correlation coefficient or the cross-correlation function.

## 3. What is the difference between autocorrelation and cross-correlation?

Autocorrelation measures the relationship between a variable and its own past values, while cross-correlation measures the relationship between two different variables. Autocorrelation is used to analyze time series data, while cross-correlation is used to analyze the relationship between two different signals or datasets.

## 4. How is autocorrelation interpreted?

The autocorrelation coefficient ranges from -1 to 1, with a value of 0 indicating no correlation and values closer to 1 or -1 indicating a stronger linear relationship. A positive autocorrelation indicates that the variable and its past values are positively related, while a negative autocorrelation indicates a negative relationship.

## 5. What are some common applications of autocorrelation in scientific research?

Autocorrelation is commonly used in fields such as economics, finance, and meteorology to analyze time series data. It is also used in signal processing to analyze the relationship between signals and in geology to study the relationship between geological events and their effects on the Earth's surface.

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