human
I am working on the file "Kalman Filter/Kalman-first.ipynb".
The first task is:
Kalman Filter
You might be wondering where is the control in all this. So now that's easily introduced in the actual formulation of Kalman filter which we will look at now. In the actual Kalman Filter we deal with multi-variable setting unlike that of the voltmeter example. So let me describe the inputs and the outputs and the parameters
Inputs
* $\textbf{Z}_m : $ The measurements vector at each instant m
* $\textbf{U}_m : $ This is new!, it denotes the controls provided to the system at instance m (this is just like force being the control for velocity of the car)
Outputs
* $\textbf{X}_m : $ Newest estimate of the current state (state can be thought of as a parameter vector)
* $P_m :$ The newest estimate for the average error of the state
Parameters
* A : State transition matrix, basically the constant matrix multiplied to the previous estimate in the process equation
* B : Control matrix, one to be multiplied to the control vector in the process equation
* H : Observation matrix, the proportionality factor for state to be equal to the measurements
* Q : Covariance matrix for the process error (this is again assumed to be known)
* R : Covariance matrix for the measurements error (again known)
Now we are ready to write the basic equations for the KF, don't worry much of the above will be cleared as you look at the equation.
Lets start with the two basic equations, first is the process equation
$$\textbf{X}m = A\textbf{X}{m-1} + B\textbf{U}_{m-1} + \pmb{\omega}_m \qquad (8)$$
and then we have the measurements equation
$$\textbf{Z}_m = H\textbf{X}_m + \pmb{\nu}_m \qquad (9)$$
Here, we model the two error terms $\pmb{\omega}_m$ and $\pmb{\nu}_m$ as multi-variate Gaussian distributions with zero mean and covariance matrices Q and P respectively, i.e. $\pmb{\omega}_m \textrm{~} \mathcal{N}(0,Q)$
and $\pmb{\nu}_m \textrm{~} \mathcal{N}(0,P)$
So now following the exact same philosophy of that we followed for the derivation in voltmeter example we get the update equations for the general Kalman Filter. I will just...
Can you write Python code for it?

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Can you provide a the Python code you requested?
assistant
#Here is the Python code you requested.

class kalmanFilter:
    def __init__(self, X0, P0, A, B, H, Q, R):
        self.A = A                       # State Transition Matrix
        self.B = B                       # Control Matrix
        self.H = H                       # Observation Matrix
        self.Q = Q                       # Covariance for the process error
        self.R = R                       # Covariance for the measurements error
        self.current_estimate = X0       # this is the initial guess of the state
        self.current_error_estimate = P0 # initial guess for the state estimate error
        
    def getEstimate(self):
        # returns the current state estimate
        return self.current_estimate
    
    def getErrorEstimate(self):
        # returns the current state error estimate
        return self.current_error_estimate
    
    def iteration(self, U, Z):
        # here is where the updates happen
        # U = control vector
        # Z = measurements vector
        
        # prior prediction step
        prior_estimate = self.A * self.current_estimate + self.B * U
        prior_error_estimate = (self.A * self.current_error_estimate) * np.transpose(self.A) + self.Q
        
        # intermediate observation
        y = Z - self.H * prior_estimate
        y_covariance = self.H * prior_error_estimate * np.transpose(self.H) + self.R
        
        # Correction Step
        K = prior_error_estimate * np.transpose(self.H) * np.linalg.inv(y_covariance)
        self.current_estimate = prior_estimate + K * y
        # We need the size of the matrix so we can make an identity matrix.
        size = self.current_error_estimate.shape[0]
        # eye(n) = nxn identity matrix.
        self.current_error_estimate = (np.eye(size) - K * self.H) * prior_error_estimate