WhiteNoiseFactor

Below is partly generated with ChatGPT 4o, needs to be verified.

Overview¶

The WhiteNoiseFactor in GTSAM is a binary nonlinear factor designed to estimate the parameters of zero-mean Gaussian white noise. It uses a mean-precision parameterization, where the mean $\mu$ and precision $\tau = 1/\sigma^2$ are treated as variables to be optimized.

Parameterization¶

The factor models the negative log-likelihood of a zero-mean Gaussian distribution as follows,

where:

• $z$: Measurement value (observed data).
• $\mu$: Mean of the Gaussian distribution (to be estimated).
• $\tau$: Precision of the Gaussian distribution $\tau = 1/\sigma^2$, also to be estimated).

This formulation allows the factor to optimize both the mean and precision of the noise model simultaneously.

Use Case: Estimating IMU Noise Characteristics¶

The WhiteNoiseFactor can be used in system identification tasks, such as estimating the noise characteristics of an IMU. Here’s how it would work:

1. Define the Measurement:

   • Collect a series of IMU measurements (e.g., accelerometer or gyroscope readings) under known conditions (e.g., stationary or constant velocity).

2. Set Up the Factor Graph:

   • Add WhiteNoiseFactor instances to the factor graph for each measurement, linking the observed value $z$ to the mean and precision variables.

3. Optimize the Graph:

   • Use GTSAM’s nonlinear optimization tools to solve for the mean $\mu$ and precision $\tau$ that best explain the observed measurements.

4. Extract Noise Characteristics:

   • The optimized mean $\mu$ represents the bias in the sensor measurements.
   • The optimized precision $\tau$ can be inverted to compute the standard deviation $\sigma = 1/\sqrt{\tau}$, which represents the noise level.