Cross correlation in python

How to Calculate Cross Correlation in Python

Cross-correlation is a measure of similarity of two series as a function of the displacement of one relative to the other. It’s a standard method of estimating the degree to which two variables or datasets move in relation to each other.

Cross-correlation is a powerful tool in many fields, including statistics, probability theory, and signal processing. It can help identify lagged dependencies between two signals or datasets, which is often useful in time series analysis, image processing, and many other areas.

For instance, in time series analysis, the cross-correlation between a variable and a lagged version of another variable is often used to find patterns of the time delay between these two variables. If a peak in the cross-correlation function appears at a positive lag, it means that the change in the first variable might be causing a change in the second variable after some time delay.

It’s important to note that cross-correlation does not imply causation: just because two variables or datasets have a high cross-correlation, it doesn’t mean that one causes changes in the other.

Also, it’s worth noting that cross-correlation is sensitive to time shifts, as it measures how much two signals “look alike” when one is shifted in time. If you’re not interested in these shifts, then you might want to use another method, such as calculating the correlation coefficient, to measure the relationship between your variables or datasets.

How to Calculate Cross-Correlation in Python?

Calculating cross-correlation in Python can be done with the numpy library using the correlate function. This function computes the correlation as generally defined in signal processing texts.

Here’s an example using two lists of numbers:

import numpy as np # Here are two lists of numbers: a = [1, 2, 3, 4, 5] b = [2, 3, 4, 5, 6] # You can calculate the cross-correlation with numpy's correlate() function: cross_corr = np.correlate(a, b, mode='valid') print('Cross-correlation: ', cross_corr) 

In this example, np.correlate(a, b, mode=’valid’) calculates the cross-correlation of the two lists. The mode parameter determines the size of the output:

• ‘valid’ mode returns output of max(M, N) length.
• ‘full’ mode returns output of size N+M-1.
• ‘same’ mode returns output of max(M, N) length centered with respect to the ‘full’ output.

Please note that this is a simple cross-correlation calculation. Depending on your specific use case, you might need to normalize your data before calculating the cross-correlation or use more advanced techniques to take into account the specifics of your data or problem.

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Функция np.correlate() в Python — что делает

Python numpy.correlate(v1,v2, mode) выполняет свертку массива v1 с обращением массива v2 и дает результат, обрезанный с использованием одного из трех указанных режимов.

Что такое функция np.correlate() в Python?

Метод numpy.correlate() в Python используется для поиска взаимной перекрестной корреляции между двумя одномерными векторами. Функция np.correlate(), которая вычисляет корреляцию, как обычно определено в тексте с одиночной обработкой, задается как: c_ [k] = sum_n v1[n+k] * conj(v2[n]) с последовательностями v1 и v2, при необходимости дополняется нулями, а conj является сопряженным.

Синтаксис

Параметры

Функция numpy.correlate() принимает не более четырех параметров:

• v1: array_like, первый одномерный входной массив. Предположим, он имеет форму(M,)
• v2: array_like, второй одномерный входной массив. Предположим, он имеет форму(N,)
• mode: . Это необязательный параметр, который имеет три различных режима, которые поясняются ниже:

1. «valid»: это режим по умолчанию. Режим «действителен» возвращает вывод длины max(M, N) – min(M, N) + 1. Продукт свертки дается только тогда, когда v1 и v2 полностью перекрывают друг друга. Значения вне границы сигнала не влияют.
2. «same»: возвращает выходные данные длины min(M, N). Пограничные эффекты все еще видны.
3. «full»: это возвращает свертку в каждой точке перекрытия с выходной формой(M + N-1). В конечных точках свертки векторы v1 и v2 не перекрываются полностью, и могут наблюдаться граничные эффекты.

• ‘old_behavior: bool, это логический параметр, который может принимать значения true или false.

В случае, если old_behavoiur принимает значение true из числового значения(correlate(v1, v2) == correct(v2, v1), сопряжение не берется для сложных массивов. В противном случае, если old_behavoiur принимает значение false из числового значения, тогда используется обычное определение обработки сигналов.

Возвращаемое значение

Метод numpy.correlate() возвращает взаимную корреляцию одномерных векторов v1 и v2.

Примеры программ с функцией numpy.correlate()

Пример 1

Программа для демонстрации работы метода numpy.correlate():

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How to Do Cross-Correlation in Python: 4 Different Methods

Cross-correlation is a basic signal processing method, which is used to analyze the similarity between two signals with different lags. Not only can you get an idea of how well the two signals match with each other, but you also get the point of time or an index, where they are the most similar.

Whenever you need to find similarities between two signals, datasets, or functions, cross-correlation is one of the tools that you should try.

Below you can see an illustration of the cross-correlation between sine and cosine functions. Unsurprisingly, the maximum is when the phase of the functions (lag) is off by \(\frac<3\pi>\), which is the delay that makes the two signals overlap

Note that autocorrelation can be viewed as a special case of cross-correlation, where the cross-correlation is taken with respect to the signal itself.

If you are looking for efficient packages to compute autocorrelation, check our post for 4 Ways of Calculating Autocorrelation in Python.

Data set and number of lags to calculate

Before going into the methods of calculating cross-correlation, we need to have some data. You can find below the data set that we are considering in our examples. The data set consists of two sinusoidal functions with \(\frac<\pi>\) phase difference.

Cross-correlation: 3 essential package + pure python implementation

Our brief introduction to cross-correlation is done and we are ready for the code. Here are three essential packages from math, signal processing, and statistics disciplines to calculate cross-correlation. As a bonus, we’ve thrown in a pure Python implementation without any external dependencies.

Python only

This is a Python-only method without any external dependencies for calculating the cross-correlation.

Output with our test data set

[-0.471998494510103, -0.24686753498102817, -0.019269956645980538, 0.20852016072607304, 0.4342268135797527, 0.6555948156484444, 0.8704123310300105, 1.076532974119988, 1.271897255587048, 1.4545531601096169, 1.62267565026772]

NumPy

NumPy is the defacto numerical computation package for Python. It comes as no surprise that NumPy comes with a built-in method for cross-correlation.

''' Numpy implementation ''' import numpy as np corr = np.correlate(a=sig1, v=sig2) print(corr)

Output with our test data set

[-0.47199849 -0.24686753 -0.01926996 0.20852016 0.43422681 0.65559482 0.87041233 1.07653297 1.27189726 1.45455316 1.62267565]

SciPy

When NumPy falls short, SciPy is most of the package to look at. It contains helpful methods for varying fields of science and engineering. When it comes to cross-correlation, we need to import the signal processing package. SciPy cross-correlation automatically pads the signal at the beginning and end, which is why it returns a longer signal response for cross-correlation than our pure Python implementation and the NumPy package. In our test case, we remove these padded components, to make the result comparable.

Output with our test data set

[-0.47199849 -0.24686753 -0.01926996 0.20852016 0.43422681 0.65559482 0.87041233 1.07653297 1.27189726 1.45455316 1.62267565]

Statsmodels

Statsmodels is a really helpful package for those working with statistics. Here, it must be kept in mind that in statistics cross-correlation always includes normalization, which ensures that the correlation is within \([-1,1]\).

To this end, we first show you how to do the normalization for the NumPy example and then compare the results.

Basically, the normalization involves moving the signal mean to 0 and dividing by the standard deviation and signal length.

Output with our test data set

Now, let’s have a look at what the Statsmodels package provides

Output with our test data set

Notice that, similarly to the SciPy implementation, we needed to remove the padding. Also, Statsmodels provides the cross-correlation response in reversed order with respect to the other schemes, which is why we needed to flip the result.

Summary

As usual, it is up to you, which implementation is the best suited for you. If the performance is not an issue, go with the package that you are using anyways. For performance, NumPy is usually quite a safe bet. However, we have not done any performance comparison here. For the least dependencies go with NumPy or Python-only implementation.

Further reading

3 Techniques for Calculating Percentiles in R

Percentiles, a pivotal tool in the world of statistics, represent a measure that tells us what proportion of a dataset falls below a particular value. In statistical analysis, percentiles are used to understand and interpret data by providing a means to compare individual data points to the rest of the

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When it comes to programming, iteration or looping is a fundamental building block. Just like many other languages, Rust offers a variety of ways to do loops. In this article, we will dive deep into the various methods of looping in Rust, exploring for loops, while loops, and the loop

Three Ways Compute Cross Product in Python

Python, as a high-level and general-purpose programming language, offers a variety of libraries and built-in functionality that streamline complex computations, such as calculating the cross product of two vectors. This article will delve into three distinct methods to compute the cross product in Python using * NumPy * SymPy * Pure Python implementation.

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Cross Correlation in Python

Cross-correlation is an essential signal processing method to analyze the similarity between two signals with different lags. Not only can you get an idea of how well the two signals match, but you also get the point of time or an index where they are the most similar.

This article will discuss multiple ways to process cross-correlation in Python.

Cross-Correlation in Python

We can use Python alone to compute the cross-correlation of the two signals. We can use the formula below and translate it into a Python script.

sig1 = [1,2,3,2,1,2,3] sig2 = [1,2,3]  # Pre-allocate correlation array corr = (len(sig1) - len(sig2) + 1) * [0]  # Go through lag components one-by-one for l in range(len(corr)):  corr[l] = sum([sig1[i+l] * sig2[i] for i in range(len(sig2))]) print(corr) 

Now, let’s go through multiple Python packages that use cross-correlation as a function.

Use NumPy Module

The standard Python module for numerical computing is called NumPy . It is not surprising that NumPy has a built-in cross-correlation technique. If we don’t have NumPy installed, we can install it with the command below:

import numpy as np  sig1 = [1,2,3,2,1,2,3] sig2 = [1,2,3]  corr = np.correlate(a=sig1, v=sig2)  print(corr) 

Use SciPy Module

When NumPy fails, SciPy is the main package to consider. It includes practical techniques for numerous engineering and scientific disciplines.

But first, we must import the cross-correlation-related signal processing software. Then, the signal is automatically padded at the start and finish by the SciPy cross-correlation.

As a result, compared to our pure Python code and the NumPy module, it provides a more extensive signal response for cross-correlation. Therefore, we deleted these padding components to make the outcome equivalent in our test case.

If we don’t have SciPy installed, we can install it with the command below:

import scipy.signal  sig1 = [1,2,3,2,1,2,3] sig2 = [1,2,3]  corr = scipy.signal.correlate(sig1, sig2)  # Removes padded Correlations corr = corr[(len(sig1)-len(sig2)-1):len(corr)-((len(sig1)-len(sig2)-1))]  print(corr) 

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