[Supervised ML] Regression/ Cost function - 2

1. Linear Regression

 

Purpose: To predict the output based on the given examples or dataset

1. 1 Terminology

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Univariate linear regression: Linear regression with one(single feature x) variable

fw,b(x) = wx + b

1.2 Code

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Necessary Libraries

#Numpy, a popular library for scientific computing
import numpy as np 
#Matplotlib, a popular library for plotting data
import matplotlib.pyplot as plt

x_train = np.array([1.0 , 2.0])
y_train = np.array([300, 500])

#Number of training examples
m = len(x_train)

Plotting the data

# Plot the data points
plt.scatter(x_train, y_train, marker = "x", c = "r")
# Set the title
plt.title("Housing Prices")
# Set the y-axis label
plt.ylabel("Price")
#Set the x-axis label"
plt.xlabel("Size")
plt.show()

Calculating Model output

def calculate_model_output(w, b, x):
	#Find the dimension of x
	m = x.shape
    #Fill in the array with the dimension of x
    f_wb = np.zeros(m)
    for i in range(len(x)):
    	f_wb[i] = w * x[i] + b
        
    return f_wb

Implementation

tmp_f_wb = calculate_model_output(w, b, x_train)

# Plotting our model prediction
# plt.plot = contiguous line / c == "color
plt.plot(x_train, tmp_f_wb, c= "b", label = "Our prediction")

#Plot the data points
#plt.scatter = scatterplot
plt.scatter(x_train, y_train, marker = "x", c = "r', label = "Actual Values)

#Set the title and axis
plt.title("Housing Prices")
plt.xlabel(size)
plt.ylabel(Price)
# Shows the label
plt.legend()
plt.show()

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2. Cost function

 

Purpose: To check how well the model is doing and how can we make it better

 

In f(x) = wx + b, w and b are parameters

 

We can find w,b by finding y-hat(i) close y(i) for all (x(i), y(i))

 

2.1 Cost Function Formula

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Q:How do we measure the performance of the model?

A: By measuring the difference between the estimate output of the model and the actual output of the dataset

 

 

Squared error cost function

1/(2m) meant to make later calculation better

 

Q: What is the purpose of using Squared Error Cost function?

A: Squaring the cost emphasizes relatively large cost and also eliminates the negative cost values.

2.2 Cost Function Intuition

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Goal of Linear Regression: 

 

Example: 

2.3 Visualization Examples

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2.4 Cost function code

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def compute_cost(x, y, w, b):
	
    m = len(x)
    cost = 0
    
    for i in range(m):
    
        #calculate the model predication
        f_wb = w * x[i] + b

        #calculate the cost
        cost = cost + (f_wb - y[i])**2
    
    total_cost = 1/ (2 * m) * cost
    
    return total_cost