# Quadratic Discriminant Analysis

Examples & Code

Quadratic Discriminant Analysis (QDA) is a dimensionality reduction algorithm used for classification tasks in supervised learning. QDA generates a quadratic decision boundary by fitting class conditional densities to the data and using Bayes’ rule. As a result, QDA is a useful tool for solving classification problems with non-linear decision boundaries.

{% embed url="https://youtu.be/_rmZk_-4mq8?si=PR9rnpJM9p7D_QcN" %}

## Quadratic Discriminant Analysis: Introduction

| Domains          | Learning Methods | Type                     |
| ---------------- | ---------------- | ------------------------ |
| Machine Learning | Supervised       | Dimensionality Reduction |

Quadratic Discriminant Analysis, commonly referred to as QDA, is a classification algorithm that falls under the category of dimensionality reduction. As a supervised learning method, QDA works by fitting class conditional densities to the data and using Bayes' rule to generate a quadratic decision boundary.

QDA is a useful tool for modeling complex relationships between variables and has been applied in various fields, including image and speech recognition, finance, and biology. This algorithm is often used when the classes have different covariances matrices, making it more flexible than linear discriminant analysis (LDA).

Understanding the fundamentals of QDA is crucial for anyone interested in artificial intelligence and machine learning, as it is a powerful technique that has proven to be effective in many real-world applications.

In this introduction, we will delve deeper into the working principles of QDA and explore its advantages over other classification algorithms.

## Quadratic Discriminant Analysis: Use Cases & Examples

Quadratic Discriminant Analysis (QDA) is a type of dimensionality reduction algorithm used in supervised learning. It generates a classifier with a quadratic decision boundary by fitting class conditional densities to the data and using Bayes' rule.

One use case of QDA is in medical diagnosis. By analyzing patient data, QDA can accurately classify whether a patient has a certain disease or not based on their symptoms and medical history.

Another use case is in image recognition. QDA can be trained to classify images into different categories based on their features, such as color, texture, and shape.

QDA can also be used in financial analysis to predict stock market trends. By analyzing historical data, QDA can classify whether the stock market is likely to go up or down in the future.

## Getting Started

Quadratic Discriminant Analysis (QDA) is a supervised learning algorithm used for classification tasks. It is a dimensionality reduction technique that fits class conditional densities to the data and uses Bayes’ rule to classify new data points. QDA assumes that each class has its own covariance matrix, unlike Linear Discriminant Analysis (LDA) which assumes that all classes share the same covariance matrix.

Getting started with QDA in Python is easy with the help of popular machine learning libraries like NumPy, PyTorch, and scikit-learn. Here is a code example using scikit-learn:

```
import numpy as np
from sklearn.discriminant_analysis import QuadraticDiscriminantAnalysis

# create sample data
X = np.array([[-1, -1], [-2, -1], [-3, -2], [1, 1], [2, 1], [3, 2]])
y = np.array([1, 1, 1, 2, 2, 2])

# fit QDA model
qda = QuadraticDiscriminantAnalysis()
qda.fit(X, y)

# predict new data points
print(qda.predict([[-0.8, -1]]))
print(qda.predict([[2.5, 1.5]]))

```

In this example, we first create some sample data with two features and two classes. We then fit a QDA model to the data using scikit-learn's QuadraticDiscriminantAnalysis class. Finally, we use the model to predict the class of two new data points.

## FAQs

### What is Quadratic Discriminant Analysis (QDA)?

Quadratic Discriminant Analysis (QDA) is a supervised learning algorithm used for classification tasks. It assumes that the data for each class comes from a Gaussian distribution, and generates a quadratic decision boundary by fitting class conditional densities to the data and using Bayes’ rule.

### What is the difference between QDA and Linear Discriminant Analysis (LDA)?

The main difference between QDA and Linear Discriminant Analysis (LDA) is that LDA assumes that the covariance matrix of the predictor variables is the same for each class, while QDA does not make this assumption. As a result, QDA can model more complex decision boundaries than LDA.

### What are the advantages of using QDA?

QDA can model more complex decision boundaries than LDA and is often more accurate when the assumptions of LDA are not met. It also works well with small training datasets and is less sensitive to imbalanced class distributions.

### What are the disadvantages of using QDA?

QDA can be computationally expensive when the number of predictors is large, as it requires estimating a separate covariance matrix for each class. It is also more prone to overfitting than LDA when the number of predictors is small relative to the number of training observations.

### What are some applications of QDA?

QDA has been used in a variety of applications, including image and speech recognition, medical diagnosis, and fraud detection.

## Quadratic Discriminant Analysis: ELI5

Quadratic Discriminant Analysis, or QDA for short, is a fancy tool that helps us group things together based on similarities. Imagine you are a teacher and you have a bunch of students with different grades in math and science. You want to divide them into two groups-- those who are good at math and those who are good at science. QDA can give you a decision boundary that separates the two groups by looking at the individual grades of each student, fitting them to math and science models, and using Bayes' rule to make the best classification decision.

More technically speaking, QDA is a type of dimensionality reduction algorithm that uses supervised learning methods to classify data points into different groups based on the shape of the data distribution. It is called 'Quadratic' because it uses a quadratic decision boundary to separate the data, meaning it can capture more complex relationships between the input features and target variables compared to a linear decision boundary.

By using QDA, we can get a better understanding of our data and group them in a way that makes sense for our specific problem. This can be useful for a variety of real world applications, from identifying different types of cancers based on medical data to predicting whether someone will default on a loan.

So, in a nutshell, QDA is a powerful tool for supervised learning that helps us make better decisions about our data and how to group them together.

Remember, though, that like any other tool, it is not perfect-- it can sometimes overfit the data or create false positives. It's always important to evaluate the results of any algorithm and make sure it's a good fit for our specific problem.

\*\[MCTS]: Monte Carlo Tree Search [Quadratic Discriminant Analysis](https://serp.ai/quadratic-discriminant-analysis/)
