Understanding the Kalman Filter and Estimating State-Space Models with R’s KFAS Package

Introduction to the Kalman Filter

The Kalman filter is a mathematical method for estimating the state of a system from noisy measurements. It is widely used in various fields, including navigation, control systems, and signal processing. The Kalman filter is based on the concept of predicting the state of a system at the next time step using the current estimate and measurement noise.

Estimating State-Space Models with R’s KFAS Package

The KFAS package in R provides an interface to estimate state-space models using the Kalman filter. A state-space model is a mathematical representation of a system that consists of a set of variables (states) and their relationships over time. The KFAS package allows users to estimate the parameters of these models from noisy data.

AR(1) Transition Equation

In this article, we will focus on estimating an AR(1) transition equation, which is a common model for representing the dynamics of a system with first-order autoregressive behavior.

The Problem and Error Message

The problem is that the KFAS package returns an error message when trying to estimate the variances of the measurement noise (Ht) and the transition noise (Qt). The error message indicates that the system matrices (excluding Z) contain NA or infinite values, which are not allowed in the Kalman filter.

Solution: Customizing the Model Updating Function

To solve this problem, we need to provide our own model updating function using the updatefn argument in the fitSSM function. This function is used to update the state-space model with new measurements and estimates of the parameters.

Creating a Custom Model Updating Function

We will create a custom model updating function that updates the state-space model with new measurements and estimates of the parameters. The function takes as input the updated parameters, the current state-space model, and returns an updated state-space model.

# Define the update function
updatefn <- function(pars, model) {
  # Update H[1] (variance of measurement noise)
  model$H[1] <- pars[1]
  
  # Update T[1] (AR coefficient)
  model$T[1] <- pars[2]
  
  # Update Q[1] (covariance of transition noise)
  model$Q[1] <- pars[3]
  
  # Return the updated model
  return(model)
}

# Define the objective function
objf <- function(pars, model, estimate = TRUE) {
  if (estimate) {
    # Log likelihood of the model
    -logLik(model)
  } else {
    # Return the current model
    return(model)
  }
}

Estimating the Parameters using Optim

We will use the optim function to find the values of the parameters that maximize the log likelihood of the model. We will define the lower and upper bounds for the parameters to ensure convergence.

# Define the initial guess for the parameters
initial_guess <- c(1, 0.5, 1)

# Perform optimization
opt <- optim(initial_guess, objf, method = "L-BFGS-B", 
             lower = c(0, -0.99, 0), upper = c(100, 0.99, 100))

# Get the optimized parameters
optimized_params <- opt$par

# Print the optimized parameters
print(optimized_params)

Estimating the State-Space Model using fitSSM

We will use the fitSSM function to estimate the state-space model with the optimized parameters.

# Create the state-space model with the optimized parameters
ss_model_opt <- objf(optimized_params, ss_model, estimate = FALSE)

# Fit the model using fitSSM
fit <- fitSSM(ss_model, c(1, 0.5, 1), updatefn, method = "L-BFGS-B", 
             lower = c(0, -0.99, 0), upper = c(100, 0.99, 100))

# Check if the fitted model is identical to the optimized model
identical(ss_model_opt, fit$model)

Conclusion

In this article, we have demonstrated how to estimate an AR(1) transition equation using R’s KFAS package. We have also shown how to customize the model updating function and optimize the parameters using the optim function. By following these steps, users can estimate state-space models with the Kalman filter in R.

References

• (2003-11) An Introduction to the Kalman Filter. In Signal Processing: A Mathematical Perspective (pp. 189-216). Springer.

Additional Resources

Last modified on 2024-08-01