Delta Var Preprocessor¶

The delta var preprocessor function shifts the input signal by the var of the signal. The is defined as:

\[ x_{shifted_{i}} = x_{i} - \sigma^{2}_{x}, \quad \forall i \in \{1, \dots, N\} \]

For shifting signals by a custom \(\delta\), see the delta preprocessor function. For more on how we compute the var of a signal, check out var function.

Preprocess the signal x shifting each element of x by the variance of x.

Parameters:

Name	Type	Description	Default
`x`	`ndarray`	The array to shift by its variance.	required
`ddof`	`Union[int, int_]`	The delta degrees of freedom. Default is `0`. See `numpy.var` for more information.	`0`
`where`	`Callable[[Union[int, float, int_, float_]], Union[bool, bool_]]`	A function that takes a value and returns `True` or `False`. Default is `lambda x: not np.isnan(x)` i.e. a measurement is valid if it is not a `NaN` value.	`lambda : not numpy.isnan(x)`

Returns:

Type	Description
`ndarray`	The shifted signal.

Examples¶

Here we look at an example where we shift two signals by their var to demonstrate the effect of the delta var preprocessor function.

Transform Signal¶

First, we define the signals as two normal distributions with different means and variances. Then, we apply the delta var preprocessor function to both signals.

A univariate normal distribution with mean \(\mu\) and var \(\sigma\) is defined as:

\[ \mathcal{N}(\mu, \sigma) = \frac{1}{\sigma \sqrt{2 \pi}} e^{-\frac{(x - \mu)^{2}}{2 \sigma^{2}}} \]

import numpy as np
import autonfeat.preprocess.functional as PF

# Number of samples
n_samples = 100

# Generate sample data
x1 = np.random.normal(0, 1, n_samples)
x2 = np.random.normal(5, 5, n_samples)

shifted_x1 = PF.delta_var_tf(x1)
shifted_x2 = PF.delta_var_tf(x2)

Visualize Transform¶

Next, we visualize the effect of the transform on the signals.

import matplotlib.pyplot as plt

# Plot original data
plt.figure(figsize=(8, 4))
plt.subplot(1, 2, 1)
plt.plot(x1, '.', color='green', label='x1')
plt.plot(x2, '.', color='orange', label='x2')
plt.legend()
plt.title('Original Data')

# Plot shifted data
plt.subplot(1, 2, 2)
plt.plot(shifted_x1, '.', color='green', label='x1 shifted')
plt.plot(shifted_x2, '.', color='orange', label='x2 shifted')
plt.legend()
plt.title('Shifted Data')

plt.tight_layout()
plt.show()

This can be seen in the figure below.

DeltaVar

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