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# Copyright (c) DataLab Platform Developers, BSD 3-Clause license, see LICENSE file.
"""
.. Stability Analysis (see parent package :mod:`sigima.algorithms.signal`)
"""
from __future__ import annotations
import numpy as np
from sigima.tools.checks import check_1d_arrays
@check_1d_arrays(x_evenly_spaced=True)
def allan_variance(x: np.ndarray, y: np.ndarray, tau_values: np.ndarray) -> np.ndarray:
"""
Calculate the Allan variance for given time and measurement values at specified
tau values.
Args:
x: Time array
y: Measured values array
tau_values: Allan deviation time values
Returns:
Allan variance values
"""
if len(x) != len(y):
raise ValueError(
"Time array (x) and measured values array (y) must have the same length."
)
dt = np.mean(np.diff(x)) # Time step size
if not np.allclose(np.diff(x), dt):
raise ValueError("Time values (x) must be equally spaced.")
allan_var = []
for tau in tau_values:
m = int(round(tau / dt)) # Number of time steps in a tau
if m < 1:
raise ValueError(
f"Tau value {tau} is smaller than the sampling interval {dt}"
)
if m > len(y) // 2:
# Tau too large for reliable statistics
allan_var.append(np.nan)
continue
# Calculate the clusters/bins
clusters = y[: len(y) - (len(y) % m)].reshape(-1, m)
bin_means = clusters.mean(axis=1)
# Calculate Allan variance using the definition
# σ²(τ) = 1/(2(N-1)) Σ(y_(i+1) - y_i)²
# where y_i are the bin means
squared_diff = np.sum(np.diff(bin_means) ** 2)
n = len(bin_means) - 1
if n > 0:
var = squared_diff / (2.0 * n)
allan_var.append(var)
else:
allan_var.append(np.nan)
return np.array(allan_var)
@check_1d_arrays(x_evenly_spaced=True)
def allan_deviation(x: np.ndarray, y: np.ndarray, tau_values: np.ndarray) -> np.ndarray:
"""
Calculate the Allan deviation for given time and measurement values at specified
tau values.
Args:
x: Time array
y: Measured values array
tau_values: Allan deviation time values
Returns:
Allan deviation values
"""
return np.sqrt(allan_variance(x, y, tau_values))
@check_1d_arrays(x_evenly_spaced=True)
def overlapping_allan_variance(
x: np.ndarray, y: np.ndarray, tau_values: np.ndarray
) -> np.ndarray:
"""
Calculate the Overlapping Allan variance for given time and measurement values.
Args:
x: Time array
y: Measured values array
tau_values: Allan deviation time values
Returns:
Overlapping Allan variance values
"""
if len(x) != len(y):
raise ValueError(
"Time array (x) and measured values array (y) must have the same length."
)
dt = np.mean(np.diff(x)) # Time step size
if not np.allclose(np.diff(x), dt):
raise ValueError("Time values (x) must be equally spaced.")
overlapping_var = []
for tau in tau_values:
tau_bins = int(tau / dt)
if tau_bins <= 1 or tau_bins > len(y) / 2:
overlapping_var.append(np.nan)
continue
m = len(y) - tau_bins # Number of overlapping segments
avg_values = [np.mean(y[i : i + tau_bins]) for i in range(m)]
diff = np.diff(avg_values)
overlapping_var.append(0.5 * np.mean(np.array(diff) ** 2))
return np.array(overlapping_var)
@check_1d_arrays(x_evenly_spaced=True)
def modified_allan_variance(
x: np.ndarray, y: np.ndarray, tau_values: np.ndarray
) -> np.ndarray:
"""
Calculate the Modified Allan variance for given time and measurement values.
Args:
x: Time array
y: Measured values array
tau_values: Modified Allan deviation time values
Returns:
Modified Allan variance values
"""
if len(x) != len(y):
raise ValueError(
"Time array (x) and measured values array (y) must have the same length."
)
dt = np.mean(np.diff(x))
if not np.allclose(np.diff(x), dt):
raise ValueError("Time values (x) must be equally spaced.")
mod_allan_var = []
for tau in tau_values:
tau_bins = int(tau / dt)
if tau_bins <= 1 or tau_bins > len(y) / 2:
mod_allan_var.append(np.nan)
continue
m = int(len(y) / tau_bins)
reshaped = y[: m * tau_bins].reshape(m, tau_bins)
avg_values = reshaped.mean(axis=1)
squared_diff = (np.diff(avg_values)) ** 2
mod_allan_var.append(np.mean(squared_diff) / (2 * (tau_bins**2)))
return np.array(mod_allan_var)
@check_1d_arrays(x_evenly_spaced=True)
def hadamard_variance(
x: np.ndarray, y: np.ndarray, tau_values: np.ndarray
) -> np.ndarray:
"""
Calculate the Hadamard variance for given time and measurement values.
Args:
x: Time array
y: Measured values array
tau_values: Hadamard deviation time values
Returns:
Hadamard variance values
"""
if len(x) != len(y):
raise ValueError(
"Time array (x) and measured values array (y) must have the same length."
)
dt = np.mean(np.diff(x))
if not np.allclose(np.diff(x), dt):
raise ValueError("Time values (x) must be equally spaced.")
hadamard_var = []
for tau in tau_values:
tau_bins = int(tau / dt)
if tau_bins <= 1 or tau_bins > len(y) / 3:
hadamard_var.append(np.nan)
continue
m = len(y) - 2 * tau_bins
avg_values = [np.mean(y[i : i + tau_bins]) for i in range(m)]
diff = np.diff(avg_values, n=2) # Second differences
hadamard_var.append(np.mean(diff**2) / 6)
return np.array(hadamard_var)
@check_1d_arrays(x_evenly_spaced=True)
def total_variance(x: np.ndarray, y: np.ndarray, tau_values: np.ndarray) -> np.ndarray:
"""
Calculate the Total variance for given time and measurement values.
Args:
x: Time array
y: Measured values array
tau_values: Total variance time values
Returns:
Total variance values
"""
if len(x) != len(y):
raise ValueError(
"Time array (x) and measured values array (y) must have the same length."
)
dt = np.mean(np.diff(x))
if not np.allclose(np.diff(x), dt):
raise ValueError("Time values (x) must be equally spaced.")
total_var = []
for tau in tau_values:
tau_bins = int(tau / dt)
if tau_bins <= 1 or tau_bins > len(y) / 2:
total_var.append(np.nan)
continue
m = int(len(y) / tau_bins)
reshaped = y[: m * tau_bins].reshape(m, tau_bins)
avg_values = reshaped.mean(axis=1)
squared_diff = np.diff(avg_values) ** 2
total_var.append(np.mean(squared_diff))
return np.array(total_var)
@check_1d_arrays(x_evenly_spaced=True)
def time_deviation(x: np.ndarray, y: np.ndarray, tau_values: np.ndarray) -> np.ndarray:
"""
Calculate the Time Deviation (TDEV) for given time and measurement values.
Args:
x: Time array
y: Measured values array
tau_values: Time deviation time values
Returns:
Time deviation values
"""
allan_var = allan_variance(x, y, tau_values)
return np.sqrt(allan_var) * tau_values
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