Regression:

Regression algorithms are used to predict continuous numerical values based on input variables. The main goal is to establish a relationship between the independent variables and the dependent variable or output. Some key points about regression algorithms include:

Mathematical Concepts: Regression models fit a mathematical function to the data that can predict the output value as a function of the input features. This is typically done by minimizing a loss function (e.g., mean squared error) that represents the error between the predicted values and the actual values.

Working Domain and Applications: Regression algorithms are commonly used in areas such as finance (predicting stock prices), healthcare (predicting patient outcomes based on medical history), and marketing (forecasting sales based on advertising expenditure).

Based on your request, the example has been updated to predict the housing price based on the number of rooms in a house instead of the year. Here is the revised explanation:

Mathematics of Regression for Housing Price Prediction:

Regression analysis is a statistical method used to model the relationship between a dependent variable, which in this case is the housing price \(y\), and one or more independent variables, such as the number of rooms \(x\). In a simple linear regression model, the relationship is defined by the equation:

\[ y = \beta_0 + \beta_1 x + \varepsilon \]

In this context: - \(y\) represents the dependent variable (housing price). - \(x\) represents the independent variable (number of rooms). - \(\beta_0\) is the intercept term, indicating the base price of a house. - \(\beta_1\) is the coefficient that captures the effect of the number of rooms on the housing price. - \(\varepsilon\) represents the error term, accounting for variability not explained by the model.

Example: Housing Price and Number of Rooms

Consider a scenario where we aim to predict the housing price based on the number of rooms in a house. We can formulate a regression equation for predicting housing prices using the number of rooms:

\[ \text{Housing Price} = \beta_0 + \beta_1 \times \text{Number of Rooms} + \varepsilon \]

By fitting a regression model to real estate data with information on house prices and the number of rooms, we can estimate the model coefficients (\(\beta_0\) and \(\beta_1\)), make price predictions based on the room count, and gain insights into how the number of rooms influences housing prices.

Regression analysis enables us to quantify the relationship between variables, make informed predictions, and aid decision-making in the housing market or real estate industry.

# Create a dataset with housing cost, number of rooms, and house color
set.seed(123)
num_rooms <- sample(3:6, 50, replace = TRUE)
house_color <- sample(c("Red", "Blue", "Green", "White"), 50, replace = TRUE)
housing_cost <- 100000 + 20000 * num_rooms + rnorm(50, mean = 0, sd = 5000)

house_data <- data.frame(Num_Rooms = num_rooms, House_Color = house_color, Cost = housing_cost)
# Build a regression model to predict housing cost based on number of rooms
lm_model <- lm(Cost ~ Num_Rooms, data = house_data)
summary(lm_model)
## 
## Call:
## lm(formula = Cost ~ Num_Rooms, data = house_data)
## 
## Residuals:
##      Min       1Q   Median       3Q      Max 
## -13231.8  -2805.8    125.1   2229.8  10995.6 
## 
## Coefficients:
##             Estimate Std. Error t value Pr(>|t|)    
## (Intercept)  98196.1     2827.9   34.72   <2e-16 ***
## Num_Rooms    20581.6      631.7   32.58   <2e-16 ***
## ---
## Signif. codes:  0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1
## 
## Residual standard error: 4534 on 48 degrees of freedom
## Multiple R-squared:  0.9567, Adjusted R-squared:  0.9558 
## F-statistic:  1061 on 1 and 48 DF,  p-value: < 2.2e-16
# Create a scatter plot of Number of Rooms vs. Housing Price
plot(house_data$Num_Rooms, house_data$Cost, main = "Number of Rooms vs. Housing Price", 
     xlab = "Number of Rooms", ylab = "Housing Price", col = "blue")

# Add the regression line to the plot
abline(lm_model, col = "red")

Mathematics of Classification:

Classification is a machine learning technique that involves predicting the category or class of a data point based on its features. In binary classification, there are two classes - typically labeled as 0 and 1. The goal is to determine the decision boundary that separates the classes in the feature space.

Let’s define the mathematical concepts involved in classification:

  1. Binary Classification:

In binary classification, the decision boundary is represented by a linear equation of the form:

\[ \beta_0 + \beta_1 x_1 + \beta_2 x_2 + \ldots + \beta_n x_n = 0 \]

Here: - \(\beta_0\) is the intercept term, - \(\beta_1, \beta_2, \ldots, \beta_n\) are the coefficients of the features \(x_1, x_2, \ldots, x_n\), - \(x_1, x_2, \ldots, x_n\) are the input features, - The decision boundary separates the classes when the equation equals 0.

  1. Predicting Class Probability:

The logistic function (sigmoid function) is commonly used in binary classification to predict the probability of a data point belonging to a particular class. It’s defined as:

\[ P(y=1|x) = \dfrac{1}{1 + e^{-(\beta_0 + \beta_1 x_1 + \beta_2 x_2 + \ldots + \beta_n x_n)}} \]

Where: - \(P(y=1|x)\) is the probability of the data point belonging to class 1, - \(e\) is Euler’s number, approximately equal to 2.71828.

  1. Decision Rule:

To make the final classification, a decision rule is applied. For instance, if the predicted probability is greater than a threshold (e.g., 0.5), the data point is assigned to class 1; otherwise, it’s assigned to class 0.

Classification involves finding the optimal parameters (coefficients) that best separate the classes and using these parameters to classify new data points based on their features. It’s a fundamental technique used in various fields such as healthcare, finance, and marketing for tasks like disease diagnosis, fraud detection, and customer segmentation.

# Load the rpart package
library(rpart)
library(rpart.plot)

# Convert house color to a factor for classification
house_data$House_Color <- as.factor(house_data$House_Color)

# Build a classification model to determine cost based on house color
class_model <- rpart(Cost ~ House_Color, data = house_data, method = "class")

# Print the complexity parameter table to display the model performance
printcp(class_model)
## 
## Classification tree:
## rpart(formula = Cost ~ House_Color, data = house_data, method = "class")
## 
## Variables actually used in tree construction:
## [1] House_Color
## 
## Root node error: 49/50 = 0.98
## 
## n= 50 
## 
##         CP nsplit rel error xerror xstd
## 1 0.020408      0   1.00000 1.0204    0
## 2 0.010000      3   0.93878 1.0204    0
rpart.plot(class_model)

# Load required libraries
library(ggplot2)

# Create a scatter plot with color representing house color and scattered data points
ggplot(house_data, aes(x = House_Color, y = Cost, color = House_Color)) + 
  geom_jitter(width = 0.3, height = 0) +
  labs(x = "House Color", y = "Housing Price", title = "Housing Price vs. House Color") +
  theme_minimal()