Fit it with a cluster-weighted model using a local linear function in the clusters, $y = a + bx$. Plot the data and the resulting forecast $⟨y|x⟩$, uncertainty $⟨σ2 y |x⟩$, and cluster probabilities $p(⃗x |cm)p(cm)$ as the number of clusters is increased (starting with one), and animate their evolution during the EM iterations.¶
For this problem set, corresponding equations are in chapter 16.4 (page 202). In the book, EM is explained: 'Calculating an expected distribution given parameters, then finding the most likely parameters given a distribution, is called Expectation-Maximization (EM), and it converges to the maximum likelihood distribution starting from the initial guess [Dempster et al., 1977].' To get a better understanding of how to translate this part of the equation into code, I found Eyal's documentation helpful.
Then I asked ChatGPT the following: Do you know how to make a cluster-weighted model using a local linear function? Sharing the results here for future reference (and fun).
Yes, I can explain how to make a cluster-weighted model using a local linear function. A cluster-weighted model (CWM) is a type of mixture model where each mixture component is a locally weighted linear regression model. The idea is to partition the input space into clusters and fit a linear regression model to each cluster. The weights of the local regression models are determined by a probability distribution over the clusters, which is learned from the data.
Here's the basic algorithm to construct a CWM using a local linear function:
1. Partition the input space into clusters using a clustering algorithm such as k-means or Gaussian mixture model. The number of clusters and their locations are typically chosen based on prior knowledge or through cross-validation.
2. For each cluster, fit a locally weighted linear regression model to the data points in that cluster. The weights are determined by a kernel function that assigns higher weights to nearby points and lower weights to distant points.
3. Compute the probability of each data point belonging to each cluster, using the learned clustering probabilities. These probabilities can be computed using Bayes' rule and the Gaussian kernel density estimator.
4. Predict the output for a new input point by taking a weighted average of the predictions of the local regression models, where the weights are the probabilities of the point belonging to each cluster.
5. Tune the hyperparameters of the model, such as the kernel bandwidth and the number of clusters, using cross-validation.
In practice, the algorithm can be implemented using existing machine learning libraries such as scikit-learn in Python. I hope this helps!
Next, I asked, Do you know what an Expectation-Maximization (EM) iteration is?
Yes, I can explain what an Expectation-Maximization (EM) iteration is. The EM algorithm is a widely used iterative method for finding maximum likelihood estimates of parameters in statistical models where some of the variables are not observed. The basic idea of EM is to iteratively estimate the missing variables, while maximizing the likelihood of the observed data.
The algorithm consists of two steps: the Expectation (E) step and the Maximization (M) step. In the E step, the missing data is estimated based on the current estimate of the parameters. In the M step, the parameters are updated based on the estimated missing data. These two steps are repeated until convergence.
Here is the EM algorithm in more detail:
1. Initialization: Choose an initial estimate for the parameters of the model.
2. E step: Compute the expected value of the complete data log-likelihood given the observed data and the current estimate of the parameters. This involves estimating the missing data by computing the posterior distribution of the missing data given the observed data and the current estimate of the parameters.
3. M step: Maximize the expected value of the complete data log-likelihood with respect to the parameters. This involves updating the parameters to increase the likelihood of the observed data, using the estimated missing data from the E step.
4. Repeat steps 2-3 until convergence: Check for convergence, and if the convergence criterion is not met, return to step 2.
The EM algorithm is used in a wide range of applications, including image processing, natural language processing, and bioinformatics, among others. It is particularly useful in cases where some of the data is missing, or where the likelihood function is difficult to optimize directly.
With that information, I'm attempting to create the cluster weighted model.
