$1.$R programming: The first task is to write the code to implement the K$-$Nearest Neighbors, or KNN$,$ model from

scratch. We will do this in steps:

$$ Write a function called euclidean$\_$distance that calculates the Euclidean distance between two vectors.

There are two input arguments for this function: vector $1($vec$1),$ and vector $2($vec$2).$ The output for

this function is a numeric, the Euclidean distance $($euclDist$).$

$$ Write a function called manhattan$\_$distance that calculates the Manhattan distance between two

vectors. There are two input arguments for this function: vector $1($vec$1),$ and vector $2($vec$2).$ The

output for this function is a numeric, the Manhattan distance $($manhDist$).$

$$ Write a function called euclidean$\_$distance$\_$all that calculates the Euclidean distance between a

vector and all the row vectors in an input data matrix. There are two input arguments for this

function: a vector $($vec$1)$ and an input data matrix $($mat$1\_$X$).$ The output for this function is a vector

$($output$\_$euclDistVec$)$ which is of the same length as the number of rows in mat$1\_$X$.$ This function

must use the function euclidean$\_$distance you previously wrote.

$$ Write a function called manhattan$\_$distance$\_$all that calculates the Manhattan distance between

a vector and all the row vectors in an input data matrix. There are two input arguments for this

function: a vector $($vec$1)$ and an input data matrix $($mat$1\_$X$).$ The output for this function is a vector

$($output$\_$manhattanDistVec$)$ which is of the same length as the number of rows in mat$1\_$X$.$ This function

must use the function manhattan$\_$distance you previously wrote.

$$ Write a function called my$\_$KNN that compares a vector to a matrix and finds its K$-$nearest neighbors.

There are five input arguments for this function: vector $1($vec$1),$ the input data matrix $($mat$1\_$X$),$ the

class labels corresponding to each row of the matrix $($mat$1\_$Y$),$ the number of nearest neighbors you are

interested in finding $($K$),$ and a Boolean argument specifying if we are using the Euclidean distance

$($euclDistUsed$).$ The argument K should be a positive integer. If the argument euclDistUsed $=$ TRUE,

then use the Euclidean distance. Otherwise, use the Manhattan distance. The output of this function

is a list of length $2($output$\_$knnMajorityVote$).$ The first element in the output list should be a vector

of length K containing the class labels of the closest neighbors. The second element in the output list

should be the majority vote of the K class labels in the first element of the list. The function must use

the functions euclidean$\_$distance and manhattan$\_$distance you previously wrote.

Apply this function to predict the label of the $123$rd observation using the first $100$ observations as your input

training data matrix. Use K $=10.$ What is the predicted label when you use Euclidean distance? What is

the predicted label when you use Manhattan distance? Are these predictions correct