Data Modelling using Linear regression .ipynb

Kernel: Python 3 (ipykernel)

Machine Learning

Training data 


Predict a Trend , Classification  (ML -Supervsied)

SALES= 20m $
y = 100k$

tRAIN:
    
    y=2026,
    
    y=MX+C 
    Profit = m1(Sales)+ m2(Profit)+M3(Season)+C
    
    
    
    
## Classification

Apple and Orange 

1. Color: red, Yellow
2. Weight:
   
    1. Default Modelling , Binary Modelling
       (healthy or Not)
        1,0


## Data Points (Unsupervised) 

Salary vs Spend

10000  20
30000  30
20000  30

Linear Regression Overview

Linear Regression is a statistical method used to model the relationship between a dependent variable 𝑌(output) and one or more independent variables X (input). The simplest form is Simple Linear Regression, which deals with a single input feature.

Assumptions of Linear Regression

• Linearity: The relationship between independent and dependent variables is linear.

• Independence: Observations are independent.

• Homoscedasticity: Constant variance of residuals across all levels of 𝑋

• Normality of Residuals: Residuals should be approximately normally distributed.

• No Multicollinearity (for multiple regression): Independent variables should not be highly correlated.

Implementation Using NumPy

import numpy as np
import matplotlib.pyplot as plt

# Generate sample data
np.random.seed(42)
heights = np.random.normal(180, 20, 30)  # Heights in cm
weights = 0.5 * heights + np.random.normal(0, 5, 30)  # Generate weights with some noise

heights

array([189.93428306, 177.23471398, 192.95377076, 210.46059713,
       175.31693251, 175.31726086, 211.58425631, 195.34869458,
       170.61051228, 190.85120087, 170.73164614, 170.68540493,
       184.83924543, 141.73439511, 145.50164335, 168.75424942,
       159.74337759, 186.28494665, 161.83951849, 151.75392597,
       209.31297538, 175.48447399, 181.35056409, 151.50503628,
       169.11234551, 182.21845179, 156.98012845, 187.51396037,
       167.9872262 , 174.166125  ])

weights

array([ 91.95860847,  97.87874791,  96.40939926,  99.94174392,
        91.77119081,  81.55441218, 106.83644613,  87.87599667,
        78.6643259 ,  96.40990662,  89.05815597,  86.19954387,
        91.8413813 ,  69.36167908,  65.35821172,  80.77790367,
        77.56849494,  98.42808446,  82.63785069,  67.06176221,
       106.27690754,  85.81682559,  87.29067205,  78.81089958,
        89.71117037,  95.76562649,  74.29397661,  92.2109183 ,
        85.64993026,  91.96078814])

# Implementing Linear Regression using Numpy
def linear_regression(x, y):
    # Calculate coefficients
    n = len(x)
    x_mean, y_mean = np.mean(x), np.mean(y)
    beta_1 = np.sum((x - x_mean) * (y - y_mean)) / np.sum((x - x_mean)**2)
    beta_0 = y_mean - beta_1 * x_mean
    return beta_0, beta_1

# Calculate coefficients
intercept, slope = linear_regression(heights, weights)

# Predicted values
predicted_weights = intercept + slope * heights

# Visualizing the results
plt.scatter(heights, weights, label="Actual Data", color="blue")
plt.plot(heights, predicted_weights, label=f"Fitted Line (y = {intercept:.2f} + {slope:.2f}x)", color="red")
plt.xlabel("Height (cm)")
plt.ylabel("Weight (kg)")
plt.title("Linear Regression - From Scratch")
plt.legend()
plt.show()

print(f"Intercept (Beta 0): {intercept:.2f}")
print(f"Slope (Beta 1): {slope:.2f}")

Intercept (Beta 0): -5.11
Slope (Beta 1): 0.53

## Given ()

-5.11+0.53*(150)

74.39

Implementation Using sklearn

To implement Linear Regression with sklearn, we use the LinearRegression class from sklearn.linear_model.

from sklearn.linear_model import LinearRegression
import numpy as np
import matplotlib.pyplot as plt

# Data Preparation
heights = heights.reshape(-1, 1)  # Reshape for sklearn (2D input required)
weights = weights

# Create and train the model
weight_pred = LinearRegression()
weight_pred.fit(heights, weights)

# Predictions
predicted_weights = weight_pred.predict(heights)

# Visualizing the results
plt.scatter(heights, weights, label="Actual Data", color="green")
plt.plot(heights, predicted_weights, label=f"Fitted Line (y = {weight_pred.intercept_:.2f} + {weight_pred.coef_[0]:.2f}x)", color="red")
plt.xlabel("Height (cm)")
plt.ylabel("Weight (kg)")
plt.title("Linear Regression - sklearn")
plt.legend()
plt.show()

print(f"Intercept: {weight_pred.intercept_:.2f}")
print(f"Slope (Coefficient): {weight_pred.coef_[0]:.2f}")

Intercept: -5.11
Slope (Coefficient): 0.53

Examples of Linear Regression

Linear regression is widely used in various fields. Here are some examples:

• Predicting House Prices Input: Square footage, number of bedrooms. Output: Price of the house.

• Salary Prediction Input: Years of experience. Output: Annual salary.

• Health Analysis Input: Body Mass Index (BMI). Output: Risk of certain diseases.

• Advertising Effectiveness Input: Money spent on advertising. Output: Sales revenue.

Quick Practice Predicting Attrition Rate of a Company

Year Attrition Rate (%) 2004 12.5 2005 13.2 2006 11.8 2007 12.1 2008 14.0 2009 15.5 2010 14.8 2011 13.3 2012 13.0 2013 12.7 2014 11.5 2015 11.0 2016 12.2 2017 12.9 2018 13.5 2019 14.0 2020 16.5 2021 18.0 2022 17.5 2023 16.0

You need to predict Attrition equation and alos Attrition rate for 2025 and 2026.

import numpy as np
import pandas as pd

# Creating the dataset
years = np.arange(2004, 2024)  # 20 years of data
attrition_rates = [12.5, 13.2, 11.8, 12.1, 14.0, 15.5, 14.8, 13.3, 13.0, 12.7, 
                   11.5, 11.0, 12.2, 12.9, 13.5, 14.0, 16.5, 18.0, 17.5, 16.0]

data = pd.DataFrame({"Year": years, "AttritionRate": attrition_rates})
print(data.head())

   Year  AttritionRate
0  2004           12.5
1  2005           13.2
2  2006           11.8
3  2007           12.1
4  2008           14.0

# Linear Regression Functions
def linear_regression(x, y):
    x_mean, y_mean = np.mean(x), np.mean(y)
    beta_1 = np.sum((x - x_mean) * (y - y_mean)) / np.sum((x - x_mean)**2)
    beta_0 = y_mean - beta_1 * x_mean
    return beta_0, beta_1

def calculate_r2(y, y_pred):
    ss_total = np.sum((y - np.mean(y))**2)
    ss_residual = np.sum((y - y_pred)**2)
    return 1 - (ss_residual / ss_total)

def calculate_mse(y, y_pred):
    return np.mean((y - y_pred)**2)

# Prepare data
years_normalized = years - years.min()  # Normalize years for better numerical stability
intercept, slope = linear_regression(years_normalized, attrition_rates)

# Predictions
predicted_attrition = intercept + slope * years_normalized

# Metrics
r2 = calculate_r2(attrition_rates, predicted_attrition)
mse = calculate_mse(attrition_rates, predicted_attrition)

print(f"Intercept (β₀): {intercept:.2f}")
print(f"Slope (β₁): {slope:.2f}")
print(f"R² Score: {r2:.2f}")
print(f"MSE: {mse:.2f}")

from sklearn.linear_model import LinearRegression
from sklearn.metrics import r2_score, mean_squared_error

# Reshape data for sklearn
years_reshaped = years.reshape(-1, 1)

# Create and train the model
attrition_rate = LinearRegression()
attrition_rate.fit(years_reshaped, attrition_rates)

# Predictions
predicted_attrition = attrition_rate.predict(years_reshaped)

# Metrics
r2_sklearn = r2_score(attrition_rates, predicted_attrition_sklearn)
mse_sklearn = mean_squared_error(attrition_rates, predicted_attrition_sklearn)

print(f"Sklearn Intercept: {attrition_rate.intercept_:.2f}")
print(f"Sklearn Slope: {attrition_rate.coef_[0]:.2f}")
print(f"Sklearn R² Score: {r2_sklearn:.2f}")
print(f"Sklearn MSE: {mse_sklearn:.2f}")

Sklearn Intercept: -358.32
Sklearn Slope: 0.18
Sklearn R² Score: 0.30
Sklearn MSE: 2.62

import matplotlib.pyplot as plt

plt.scatter(years, attrition_rates, label="Actual Data", color="blue")
plt.plot(years, predicted_attrition, label="Fitted Line (Manual)", color="red")
plt.plot(years, predicted_attrition_sklearn, label="Fitted Line (Sklearn)", color="green", linestyle="dashed")
plt.xlabel("Year")
plt.ylabel("Attrition Rate (%)")
plt.title("Attrition Rate Prediction Using Linear Regression")
plt.legend()
plt.show()

Attrition_Rate = .18(Year)+ (-358) +2.62