Classify each point in the testing dataset based on the fitted cluster centers. Report the numbers of points classified into each center and its corresponding MSE. Use two pie charts to convey the same information $($training and testing$).$ Does the training data fit the test data well? Draw conclusions about model performance. In python refering to code below:

import sys

import numpy as np

from pyspark import SparkConf, SparkContext

from pyspark.mllib.clustering import KMeans

# Helpers

def parse$\_$vector$($line$,$ sep$=\text{'},\text{'})$:

$"""$Parses a line.

Returns: numpy array of the latitude and longitude

$"""$

fields $=$ line.strip$().$split$($sep$)$

latitude $=$ float$($fields$[1])$

longitude $=$ float$($fields$[2])$

return np$.$array$([$latitude$,$ longitude$])$

# Main

if $\_\_$name$\_\_=="\_\_$main$\_\_"$:

if len$($sys$.$argv$)!=3$:

print $>>$ sys$.$stderr, "Usage: kmeans $"$

exit$(-1)$

# Configure Spark

conf $=$ SparkConf$().$setMaster$("$local$")\backslash$

$.$setAppName$("$Earthquake Clustering"$)\backslash$

$.$set$("$spark$.$executor.memory", $"2$g$")$

sc $=$ SparkContext$($conf$=$conf$)$

# Create training RDD of $($lat$,$ long$)$ vectors

earthquakes$\_$file $=$ sys$.$argv$[1]$

training $=$ sc$.$textFile$($earthquakes$\_$file$).$map$($parse$\_$vector$)$

# Train KMeans models for different values of k

k$\_$values $=$ range$(2,11)$

mse$\_$values $=[]$

for k in k$\_$values:

model $=$ KMeans.train$($training$,$ k$,$ maxIterations$=10,$ initializationMode$=$"random"$)$

mse $=$ model.computeCost$($training$)$

mse$\_$values.append$($mse$)$

# Find the optimal k using the elbow method

import matplotlib.pyplot as plt

plt$.$plot$($k$\_$values, mse$\_$values$)$

plt$.$xlabel$(\text{'}$Number of Clusters $($k$)\text{'})$

plt$.$ylabel$(\text{'}$Mean Squared Error $($MSE$)\text{'})$

plt$.$title$(\text{'}$Elbow Method for Optimal k$\text{'})$

plt$.$show$()$

# Train the model with the optimal k

optimal$\_$k $=3$ # You can change this based on the elbow plot

model $=$ KMeans.train$($training$,$ optimal$\_$k$,$ maxIterations$=10,$ initializationMode$=$"random"$)$

# Print the cluster centers

print$("$Earthquake cluster centers:"$)$

print$($model$.$clusterCenters$)$

# Plot the cluster centers on a map $($using a library like Folium$)$

# This part requires an additional library $($Folium$)$ and code for map visualization

sc$.$stop$()$