Part 1

As we discussed in class, a common analytical optimization technique is the first derivative test. In this test, you take the first derivative of a function and solve for its zeroes. Because of the way slope works, we know then that these points will be either a minimum or a maximum.

1. Are these points global or local minima/maxima

First Derivative provides us LOCAL minimum or maximum values

2. Apply the First Derivative test to the function. How many minima/maxima does this function have? Can you identify which zeroes are a minimum and which are a maximum without graphing the function?

$f(x) =3 x+ 10*cos(x) $

Answer:
$f’(x)= 3- 10*sin(x) $

Let’s set f’(x)=0, then we have sin(x)=0.3. Therefore, there are infinite many minima/maxima for this function. Since Sin(x) is an alternating function, therefore x=17.5+2n*Pi are local Max, and x=17.5+ (2n+1)*Pi is local Min, where n is in Z.

3. Apply the fsolve() function, as discussed in class, to write a simple program to find minima and maxima for the above function.

from scipy.optimize import fsolve
import math
import numpy as np

def system(coeff):
    b0= coeff
    f0=3-10*math.sin(math.radians(b0))
    return(f0)

b_guess=(0)
b0=fsolve(system,b_guess)
print("Beta0= {}".format(b0))

Beta0= [ 17.45760312]

Part 2

1. Least-Squares Regression

Using the least-squares regression module discussed in class, perform a regression analysis upon the data provided in the assignment_6.csv file on Moodle. Split the data provided into training and testing data. Use whatever means you like to determine the appropriate form of the model, $f(x)$, to use.

from scipy.optimize import leastsq
import pandas as pd
import matplotlib.pyplot as plt

#Read in the data
data=pd.read_csv("C:/Users/fatan/Downloads/assignment6.csv",names=["X","Y"]) 

#Show Data structure
print(data[0:10])

#Split Training set and testing set 
#Traning set should be around 70% to 80%
train=data[0:78]
test=data[79:100]

def residual(b,x,y):
    return b[1]*x + b[0]-y

b_guess=[0,0]
line= 12.465*train.X -1.53

#calculate the optimized parameters for training set
b,_ =leastsq(residual,b_guess,args=(test.X, test.Y))
print(b,_)

#data visulization
plt.scatter(test.X, test.Y)
plt.plot(train.X, line,"r", label= "Training Set Linear Reg Line")
plt.xlabel("Test Set X value")
plt.ylabel("Test Set Y value")
plt.title("Linear Model Example")
plt.legend()
plt.show()

      X          Y

0 0.916092 10.973234
1 4.610461 63.649082
2 0.164516 8.143623
3 1.089609 13.759627
4 1.589659 15.190665
5 2.264226 23.217127
6 2.656766 27.918476
7 2.665267 28.458073
8 4.358936 56.519672
9 2.882788 26.703205
[ -0.83044318 12.66276417]

Part 3¶

Naive Bayes Classifier¶

In Python, implement a Naïve Bayes Classifier, as discussed in class.

from sklearn.naive_bayes import GaussianNB

x= np.array([[-3,7],[1,8], [1,1], [-1,0], [2,3], [-4,4], [-2,3], [1,2], [-1,4], [3,6], [-3,0], [-2,5]])
Y = np.array(["Y", "N", "Y", "Y", "Y", "N", "N", "Y", "Y", "Y", "N", "N"])
model = GaussianNB()
model.fit(x, Y)
predicted= model.predict([[1,1]])
print (predicted)

[‘Y’]