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Covariance

ROS / ros world / navigation and localization / robot localization

  • Covariance: is the tendency for two variables to vary together, which is a way of being correlated!
  • Variance: measures how much a single variable (e.g., sensor noise in one axis) fluctuates around its mean.
  • Bias: a systematic error, usually handled separately (e.g., calibration, bias states in the filter).

Reference:

- The Covariance Matrix : Data Science Basics

Demo: Calculate the covariance for optical velocity estimator

build the covariance matrix Σ that describes the uncertainty of (vx,vy)

Step1: Collect samples

Collect estimate velocity for N frames

Step2: Compute the mean velocity

\[ \mu_v = \begin{bmatrix} \mu_{v_x} \\ \mu_{v_y} \end{bmatrix}=\frac{1}{N}\sum_{i=1}^{N} v_i \]

Step3: Compute the deviations

For each measurement: \(\(d_i = v_i - \mu_v\)\)

This gives you how much each velocity reading deviates from the mean.

Step4: Build the covariance matrix

\[\Sigma = \begin{bmatrix} \text{Var}(v_x) & \text{Cov}(v_x,v_y) \\ \text{Cov}(v_y,v_x) & \text{Var}(v_y) \end{bmatrix}\]
\(\(\text{Var}(v_x) = \frac{1}{N-1}\sum (v_{x,i} - \mu_{v_x})^2\)\)
\(\(\text{Var}(v_y) = \frac{1}{N-1}\sum (v_{y,i} - \mu_{v_y})^2\)\)
\(\(\text{Cov}(v_x,v_y) = \frac{1}{N-1}\sum (v_{x,i} - \mu_{v_x})(v_{y,i} - \mu_{v_y})\)\)

Code example

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import numpy as np

velocities = np.array([
    [1.0, 0.5],
    [1.1, 0.4],
    [0.9, 0.6],
    [1.2, 0.5]
])

# covariance (rows=variables, cols=samples)
cov_matrix = np.cov(velocities.T, bias=False)
print(cov_matrix)