Hierarchical Clustering Algorithm - Vector of Posterior Probabilities - IT/Computer Science Assignment Help

Assignment Task:

Task:

Exercise A.
For the following questions, write R functions that complete the required task and execute each function on some random inputs. Each function needs to match the input and output arguments indicated in the question. Each set of random inputs must correspond to an appropriate scenario for testing each function and should be thoroughly justified. The code in each function must be explained mathematically and fully justified as well.
 

1. Task: Creation of a basis of step functions for nonparametric supervised learning.
Inputs: Data vector x with n observations; number of cut-off points K
Output: Matrix of basis functions.
2. Task: First iteration in a hierarchical clustering algorithm.
Inputs: Matrix X of n observations in p variables.
Output: List where each element contains the indices of the n-1 clusters.
3. Task: Linear aggregation of M classifiers.
Inputs: Logical vector of size M containing the binary predictions from the M classifiers; Numeric vector of size M containing the weights.
Output: Logical variable containing the prediction from the aggregate classifier.
4. Task: Computation of posterior probabilities for quadratic discriminant analysis in a classification problem with one input variable.

Inputs: A point x corresponding to the value taken by the input variable; vector π of size K containing the prior probabilities; vector μ of size K containing the sample averages for each class; vector σ of size K containing the sample standard deviations for each class.
Output: Vector of posterior probabilities.

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