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1D Correlation in Python

Learn how to perform 1 dimensional correlation between two signals in Python.

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Imports

The tutorial below imports NumPy, Pandas, SciPy and Plotly.

In [1]:
import plotly.plotly as py
import plotly.graph_objs as go

import numpy as np
import pandas as pd
import scipy

from scipy import signal

Correlation Between Saw and Square Wave

Similar to convolution, the 1D Cross-Correlation between two functions $f$ and $g$ is a measure of their similarity in terms of the lag of one to another (source). Since we are dealing with arrays of data rather than continuous functions, the cross-correlation is mathematically defined as:

$$ \begin{align*} (f*g)[n] = \sum_{m=-\infty}^{\infty} f^{*}[m]g[n+m] \end{align*} $$

where $f^*$ is the complex conjugate of $f$.

To give us an intuitive glance at what this does, let us look at the cross-correlation between a saw wave and a square wave.

In [2]:
x = np.linspace(0, 20, 50)
saw_y = signal.sawtooth(t=x)
square_y = signal.square(t=x)
convolve_y = signal.convolve(saw_y, square_y)

trace_saw = go.Scatter(
    x = range(len(x)),
    y = saw_y,
    mode = 'lines',
    name = 'Saw',
    marker=dict(
        color='#57D1C9'
    )
)

trace_square = go.Scatter(
    x = range(len(x)),
    y = square_y,
    mode = 'lines',
    name = 'Square',
    marker=dict(
        color='#ED5485'
    )
)

trace_convolution = go.Scatter(
    x = range(len(x)),
    y = convolve_y,
    mode = 'lines',
    name = 'Convolution',
    marker=dict(
        color='#FFE869'
    )
)

data = [trace_saw, trace_square, trace_convolution]
py.iplot(data, filename='1d-convolution-of-saw-and-square')
Out[2]:

Correlation Between Saw and Shifted Saw Wave

To compare with the plot above, we can plot a saw wave, a phase shifted saw wave and the convolution between the two to see how they correlate along the axis.

In [3]:
x = np.linspace(0, 20, 50)

saw_y = signal.sawtooth(t=x)
square_y = signal.square(t=x)
shifted_saw_y = signal.sawtooth(t=np.linspace(10, 30, 50))
convolve_y = signal.convolve(saw_y, shifted_saw_y)

trace_saw = go.Scatter(
    x = range(len(x)),
    y = saw_y,
    mode = 'lines',
    name = 'Saw',
    marker = dict(
        color='#FF7844'
    ),
    opacity = 0.8
)

trace_shifted_saw = go.Scatter(
    x = range(len(x)),
    y = shifted_saw_y,
    mode = 'lines',
    name = 'Shifted Saw',
    marker = dict(
        color='#A64942'
    ),
    opacity = 0.8
)

trace_convolution = go.Scatter(
    x = range(len(x)),
    y = convolve_y,
    mode = 'lines',
    name = 'Convolution',
    marker = dict(
        color='#53354A'
    )
)

data = [trace_saw, trace_shifted_saw, trace_convolution]
py.iplot(data, filename='1d-convolution-of-saw-and-shifted-saw')
Out[3]:
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