1. KNN_LOOCV_wrapper function:

   • Purpose: To perform KNN classification on a given dataset with LOOCV, returning the classification accuracy and a matrix of distances between instances.

   KNN_LOOCV_wrapper() Function: This function implements the KNN classifier with LOOCV. Given a dataframe df, a list of feature indices feat, an array prior_distances, and a hyperparameter k (number of neighbors):

   • It first initializes a distances matrix to zero where each element (i, j) will eventually represent the distance between sample i and sample j.

   • It then loops through each row in the dataframe. For each row j, it treats that row as the "test" sample and calculates the distance between this test sample and all other samples in the dataframe (the "training" samples).

   • The distance calculations are made using the Euclidean distance formula.

   • After calculating the distances, the function identifies the k nearest neighbors to the test sample and their associated class labels.

   • The mode of those labels is then computed to determine which class to assign to the test sample. If there's a majority of 1s, the mode is set to 1; otherwise, it's 0.

   • If the predicted label matches the actual label, a 1 is entered into the diagonal of the distances matrix; otherwise, it remains 0.

   • Lastly, the accuracy is computed as the number of correct predictions divided by the total number of samples, multiplied by 100 to convert it to a percentage.

   • The function returns the computed accuracy and the updated distances matrix.

   • Process:

     • Calculates the distances between each test instance and the other training instances using Euclidean distance.
     • Determines the k nearest neighbors and calculates the mode of their labels to perform classification.
     • Updates a matrix of distances with the calculated distance values, using diagonal entries to store classification correctness (1 if correct, 0 otherwise).
     • Computes accuracy as the ratio of correct predictions over all instances in the dataset.

   • Returns: The classification accuracy percentage and the updated distances matrix.

2. sfs function (Sequential Forward Selection):

   • Purpose:** To sequentially select the best-performing feature set for the KNN classification.

   This function implements the Sequential Forward Selection algorithm:

   • It starts with an empty model and iteratively adds features that contribute to the highest increase in accuracy, as determined by the KNN_LOOCV_wrapper() function.

   • It maintains a list of best_feat (initially empty) to keep track of the features selected so far.

   • In each iteration, it tries adding each of the remaining features (those not already selected) to the list of best_feat and observes the change in accuracy.

   • It sorts the obtained accuracy values in descending order and updates the best_feat list with the feature that brings the highest accuracy improvement.

   • If adding a new feature does not improve the accuracy, the algorithm terminates; otherwise, it adds the feature to the best_feat list and continues.

   • The function returns the final list of selected features and their corresponding accuracies.

Timing and running the SFS:

• The code sets up a timer to measure how long it takes to run the SFS algorithm.

• The timer starts, sfs() is called with the dataframe x (which is assumed to be previously defined), and the timer ends after SFS completes.

• The duration of running SFS is printed out.

Printing the selected features and accuracies:

• It creates a new dataframe df with the selected features and their accuracies from the SFS run.

• Prints the LOOCV classification accuracies.

3. Final LOOCV Accuracy:

   • Finally, the script calls KNN_LOOCV() (which is not defined in the provided snippet but is assumed to be another function for KNN with LOOCV) with the dataframe x and the list of selected features test_feat, outputs the accuracy rounded to three decimal places.