Artificial Intelligence (AI) refers to the simulation of human intelligence in machines that are programmed to think and learn like humans. The term can also be applied to any machine that exhibits traits associated with a human mind, such as learning and problem-solving.

Key Aspects of AI:

• Learning: AI systems can learn from data and improve over time. Machine learning, a subset of AI, focuses on the development of algorithms that can learn from and make predictions or decisions based on data.
• Reasoning: AI systems can solve problems through logical deduction. They can be designed to provide approximate or definite solutions to complex problems.
• Problem Solving: AI can be used to solve specific problems by considering possible actions, predicting their outcomes, and implementing the best solution.
• Perception: Many AI systems can interpret the world around them by recognizing objects, speech, and text. Computer vision and natural language processing are examples of AI in this area.
• Language Understanding: AI can interpret and respond to human language. This is evident in chatbots, translation services, and voice assistants.

AI is a broad field that includes many subfields and technologies, like neural networks, deep learning, robotics, and expert systems. It has applications in a wide range of areas, from simple tasks like filtering spam emails to complex ones like autonomous driving or personalized medicine.

Creating a high-level, minimal example of how AI works can be best illustrated using a simple machine learning model in Python. Here's an example using the scikit-learn library to create a basic linear regression model. This model learns from data to predict a continuous value:

from sklearn.linear_model import LinearRegression

import numpy as np

# Sample data: 

# X - feature (e.g., hours studied), 

# y - target (e.g., exam score)

X = np.array([[1], [2], [3], [4], [5]])  # hours studied

y = np.array([2, 4, 6, 8, 10])           # exam score

# Create and train the model

model = LinearRegression()

model.fit(X, y)

# Predicting with the model

prediction = model.predict([[6]])

print("Predicted exam score for 6 hours of study:", prediction[0])

In this example, LinearRegression is a simple AI model that learns the relationship between hours studied (X) and exam scores (y). After training, the model can predict the exam score for a given number of study hours. This demonstrates the fundamental concept of AI: learning from data to make predictions or decisions.

The term "linear regression" is used because this method models the relationship between a dependent variable and one or more independent variables using a linear equation. The key components of this term are:

1. Linear: This refers to the form of the relationship being modeled. The equation of a straight line, typically written as \(y = mx + b\) or \(y = \beta_0 + \beta_1x\), is used to describe this relationship. Here, \(y\) is the dependent variable, \(x\) is the independent variable, \(m\) or \(\beta_1\) is the slope of the line (indicating how much \(y\) changes for a unit change in \(x\)), and \(b\) or \(\beta_0\) is the \(y\)-intercept (the value of \(y\) when \(x\) is \(0\)).
2. Regression: This term originates from "regression to the mean," a concept introduced by Sir Francis Galton in the context of genetic traits. In statistics, it has come to mean any approach to modeling the relationship between variables. Specifically, in the context of linear regression, it refers to the process of finding the best-fitting line through the data points that minimizes the differences (errors) between the predicted and actual values.

In the linear regression example provided earlier, the model learns a linear relationship between the input variable \(X\) (hours studied) and the output variable \(y\) (exam score). This relationship is represented by a linear equation: \[y = mx + b\]

Here, \(m\) is the slope of the line (which represents the weight), and \(b\) is the \(y\)-intercept (which represents the bias).

To show the equation obtained by the LinearRegression model's fit method, we can extract the slope and intercept from the trained model. The equation obtained by fitting the linear regression model to the given data is: \[y = 2.0x + 0.0\]

This means for each additional hour studied, the exam score increases by 2.0 points. The y-intercept is 0.0, indicating that if no time is spent studying, the predicted exam score would be 0.0. This model perfectly fits the provided data, as it is a simple linear relationship.

The equation in linear regression is obtained through a process called "least squares fitting." The goal is to find the line that minimizes the sum of the squared differences (errors) between the observed values and the values predicted by the line. Here's how it works:

1. Define the Equation: The linear equation is typically of the form \(y = mx + b\), where \(y\) is the dependent variable, \(x\) is the independent variable, \(m\) is the slope of the line, and \(b\) is the \(y\)-intercept.
2. Calculate the Best-Fit Line: The best-fit line is calculated by minimizing the sum of the squares of the vertical distances (residuals) of the points from the line. The residuals are the differences between the observed values and the values predicted by the line.
3. Use Least Squares Method: The least squares method provides a way to calculate the best values for \(m\) and \(b\) that minimize the sum of the squared residuals. The formulas for \(m\) and \(b\) in simple linear regression are derived from calculus and are as follows: \[ \begin{aligned} m &= \frac{N\sum(xy) - \sum x \sum y}{N\sum x^2 - (\sum x)^2}\\ b &= \frac{\sum y - m\sum x}{N} \end{aligned} \] Here, \(N\) is the number of data points, \(\sum\) denotes summation, \(x\) and \(y\) are the individual data points.
4. Fit the Model:: In practice, when using a software library like scikit-learn, these calculations are done automatically when you call a method like fit(). The library uses efficient algorithms to compute the coefficients \(m\) and \(b\) that best fit the data.

Objective: Minimize the Sum of Squared Residuals

1. Residuals: For each data point, the residual is the difference between the observed value (actual \(y\)-value) and the predicted value (the \(y\)-value on the line). Mathematically, for a data point \((x_i, y_i)\), the residual is \(r_i = y_i - (mx_i + b)\), where \(m\) is the slope and \(b\) is the \(y\)-intercept of the line.
2. Squared Residuals: Squaring each residual helps to treat both positive and negative deviations equally, as well as to emphasize larger errors more than smaller ones.
3. Sum of Squared Residuals (SSR): The sum of squared residuals is calculated as: \[SSR = \sum_{i = 1}^n r_i^2 = \sum_{i = 1}^n (y_i - (mx_i+b))^2\] Here, \(n\) is the total number of data points.

Least Square Method

• The "least squares" criterion seeks to find the values of \(m\) and \(b\) that minimize the SSR. This is essentially a problem of finding the minimum of a quadratic function, which can be solved using calculus.
• By differentiating SSR with respect to \(m\) and \(b\) and setting these derivatives to zero, we can find the values of \(m\) and \(b\) that minimize SSR. This results in a set of linear equations, known as the normal equations, which can be solved to find the optimal \(m\) and \(b\).

Result

• The line defined by the calculated values of \(m\) and \(b\) is considered the best-fit line because it has the smallest possible sum of squared residuals compared to any other line. This means that, on average, the line is as close as possible to all the data points.
• The process doesn't guarantee that the line passes through all the points (which it generally won't), but it ensures that the overall deviation of points from the line is minimized.