This article shares a Filerpy documentation and code sample documentation about Kalman and Bayesian filters in Python. This method can be applied to fields such as meteorological remote sensing.
Documentation: https://filterpy.readthedocs.io/en/latest/kalman/KalmanFilter.html
Reference code link: https://nbviewer.org/github/rlabbe/Kalman-and-Bayesian-Filters-in-Python/ blob/master/table_of_contents.ipynb
A simple example is used to illustrate the application of Kalman filter and Bayesian filter in remote sensing time series data.
Suppose we have a remote sensing sensor that observes the water level of a lake every day. Every observation can have some degree of noise due to errors in the weather and the sensors themselves. We hope to use these observations to estimate the actual water level of the lake.
- Kalman filter application:
- First, we use a Kalman filter to estimate the water level of the lake. We assume that the water level of the lake is a time-varying state, and assume that the change in state is a linear model. The Kalman filter can continuously update the current state estimate based on current observations and previous state estimates.
Assume that on day t, we get an observation z_t representing the water level observation of the lake. Our state variable is x_t, representing the actual water level of the lake.
The process of Kalman filter is as follows:
-
Prediction step (prediction state and covariance):
x_t^- = A * x_t-1 (state prediction)
P_t^- = A * P_t-1 * A^T + Q (covariance prediction)
where A is the state transition matrix , P_t^- is the predicted covariance matrix, and Q is the covariance matrix of the process noise. -
Update step (update state and covariance from observations):
K_t = P_t^- * H^T * (H * P_t^- * H^T + R)^-1 (Kalman Gain) x_t
= x_t^- + K_t * (z_t - H * x_t^-) (status update)
P_t = (I - K_t * H) * P_t^- (covariance update)
where, H is the observation matrix, R is the covariance matrix of the observation noise, K_t is the Kalman gain. -
Through the Kalman filter, we can dynamically estimate the actual water level of the lake based on the previous state estimation value and the current observation value, and reduce the influence of observation noise on the estimation result.
- Bayesian filter application:
Next, we consider the use of Bayesian filters to fuse observations from multiple sensors. In addition to the previous sensor, we now introduce another sensor, which also observes the water level of the lake every day.
Let's say the first sensor has observations of z_t 1 and the second sensor has observations of z_t 2. We wish to estimate the actual water level x_t of the lake based on the observations from these two sensors.
-
The process of Bayesian filter is as follows:
According to Bayesian theorem, we can get the posterior probability distribution P(x_t | z_t^1, z_t^2) ∝ P(z_t^1, z_t^2 | x_t) * P( x_t) -
where P(z_t^1, z_t^2 | x_t) is the joint probability distribution of two sensor observations given the actual water level x_t, and P(x_t) is the prior probability distribution representing our previous estimate of the actual water level.
-
We can use various methods to estimate the posterior probability distribution, such as particle filter, extended Kalman filter, etc. By fusing data from multiple sensors, we can get a more accurate estimate of the actual water level of the lake.
- Summary:
Through Kalman filter and Bayesian filter, we can perform state estimation, remove noise, fill in missing values and fuse data from multiple sensors on remote sensing time series data, thereby improving data quality and providing more accurate estimation results . In practical applications, the specific selection and parameter setting of filters will depend on specific data and application scenarios.