Difference between revisions of "PhET-Simulations-for-Physics/C2/Geometric-Optics/English"

Opening Slide

Learning Objectives

1. Linear Discriminant Analysis (LDA) and its implementation.
2. Assumptions of LDA
3. Limitations of LDA
4. LDA on a subset of Raisin dataset
5. Visualization of the LDA separator and its corresponding confusion matrix.

System Specifications

• Windows 11
• R version 4.3.0
• RStudio version 2023.06.1

It is recommended to install R version 4.2.0 or higher.

Prerequisites

https://spoken-tutorial.org

• Basics of R programming.
• Basics of Machine Learning using R.

If not, please access the relevant tutorials on R on this website.

Linear Discriminant Analysis

• It is used for classification.
• It constructs a data driven line that best separates different classes.
• It is based on maximization of likelihood function to classify two or more classes.

Applications of LDA

• LDA technique is used in several applications like

• • Fraud Detection
  • Bio-Imaging classification
  • Classify patient disease state

Assumptions for LDA

• All data entries are continuous, Gaussian, with equal covariance matrix for all the classes.
• Mean vectors for each class are different.
• Data records are independent and identically distributed among each class.

Limitations of LDA

• Departure from Gaussianity may increase misclassification probability in LDA.
• LDA may perform poorly if data has unequal class covariance matrix.

Implementation Of LDA

More information on raisin data is available in the Additional Reading material on this tutorial page.

Download Files

Please download this file from the Code files link of this tutorial.

Make a copy and then use it for practicing.

Point to LDA.R and the folder LDA.

Point to the MLProject folder on the Desktop.

Point to the LDA folder.

This folder is in the MLProject folder on my Desktop.

I have also set the LDA folder as my working directory.

Let us switch to RStudio.

Point to LDA.R in RStudio.

For this, click on the script LDA.R.

Script LDA.R opens in RStudio.

Highlight the command library(MASS)

Highlight the command library(ggplot2)

Highlight the command library(caret)

Highlight the command library(caret)

Highlight all the commands.

#install.packages(“package_name”)

The MASS package contains the lda() function that we will use for our analysis.

The ggplot2 package is used to plot the results of our analysis.

The caret package contains the

confusionMatrix function.

It is used as a measure for the performance of the classifier.

Please note that in order to import these libraries, we need to install them.

Please ensure that everything is installed correctly.

You can use the command install.packages(“package_name”) to install the required packages.

As I have already installed these packages, I will directly import them.

library(readxl)

library(MASS)

library(ggplot2)

library(caret)

library(lattice)

data <- read_xlsx("Raisin.xlsx")

Highlight the command data<-data[c("minorAL","ecc","class")]

Highlight the commands.

data <- read_xlsx("Raisin.xlsx")

data<-data[c("minorAL","ecc","class")]

Run these commands to import the raisin dataset.

Point to the data variable in the Environment tab.

Click the data to load the dataset.

In the Environment tab under Data heading, you will see a data variable.

Click the data variable to load the dataset in the Source window.

Type these commands in the source window.

data$class <- factor(data$class)

data$class <- factor(data$class)

Select the commands and click the Run button.

It ensures that the categorical data is properly encoded.

Select the command and run it. them.

Type the command in the source window.

set.seed(1)

index_split=sample(1:nrow(data),size=0.7*nrow(data),replace=FALSE)

set.seed(1)

Highlight the command sample(1:nrow(data),size=0.7*nrow(data),replace=FALSE)

Highlight the command replace=FALSE

Select the commands and click the Run button.

We will create a vector of indices using sample() function.

This will be 70% for training and 30% for testing.

The training data is chosen using simple random sampling without replacement.

Select the commands and run them.

Type the command

train_data <- data [index_split, ]

test_data <- data[-c(index_split), ]

train_data <- data[index_split, ]

Highlight the command

test_data <- data[-c(index_split), ]

This creates testing data, consisting of 270 unique rows.

Point to the sets in the Environment Tab

The data sets are shown in the Environment tab.

Click on test_data and train_data to load them in the Source window.

LDA_model <- lda(class~.,data=train_data)

LDA_model

LDA_model <- lda(class~.,data=train_data)

LDA_model

Highlight the command LDA_model

Click on Save and Run buttons.

Point to the output in the console window.

1. formula
2. data on which the model should train.

Select the comands and run them.

The output is shown in the console window.

Let us go through them one at a time.

Highlight the command Group means.

Highlight the command Coefficients of linear discriminants

These display the mean values of each predictor variable for each species.

These display the linear combination of predictor variables.

The given linear combinations form the decision rule of the LDA model.

predicted_values <- predict(LDA_model, test_data)

Let us check what predicted_values contain.

Point to the table.

The predicted_values table is loaded in the Source window.

head(predicted_values$class)

head(predicted_values$posterior)

head(predicted_values$x)

The output is seen in the console window.

Highlight the command output of head(predicted_values$posterior) in the console.

Highlight the command output of head(predicted_values$x) in console

It contains the posterior probability of the observation belonging to each class.

This contains the linear discriminants for each observation.

confusion <-table(test_data$class,predicted_values$class)

fourfoldplot(confusion, color = c("red", "green"), conf.level = 0, margin=1)

Click on Save and Run buttons.

Save and run the commands.

Highlight the command

fourfoldplot(confusion, color = c("red", green"), conf.level = 0, margin=1)

The fourfoldplot() function generates a visual plot of the confusion matrix,

The output is seen in the plot window.

Given the specific seed (set.seed=1), LDA has misclassified 33 out of 270 observations.

This number may change for different sets of training data.

[RStudio]

X <- seq(min(train_data$minorAL), max(train_data$minorAL), length.out = 100)

Y <- seq(min(train_data$ecc), max(train_data$ecc), length.out = 100)

min_max <- expand.grid(minorAL = X, ecc = Y)

min_max$predicted_class <- predict(LDA_model, newdata = min_max)$class

grid <- expand.grid(minorAL = X, ecc = Y)

grid$class <- predict(LDA_model, newdata = grid)$class

grid$classnum <- as.numeric(grid$class)

Click on Save and Run buttons.

This block of code operates as a setup for visual plotting.

It consists of square grid coordinates in the range of training data and their predicted linear discriminants.

The seq function generates a sequence of evenly spaced values within a range of smallest and largest values of 'minorAL' and 'ecc' variables from the training data.

The 'grid' variable contains the generated data including the prediction of the LDA_model on it.

The as.numeric function encodes the predicted classes labels into numeric values.

Select the commands and run them.

These variables contain the data for the visualization of the linear discriminants.

Click the grid data in the Environment tab.

The grid data table is loaded in the Source window.

ggplot() +

geom_point(data = train_data, aes(x = minorAL, y = ecc, color = class), size = 3) +

geom_point(data = min_max, aes(x = minorAL, y = ecc, color = predicted_class), size = 1, alpha = 0.3) +

theme_minimal()

ggplot() +

geom_raster(data=grid, aes(x=minorAL, y=ecc, fill = class),alpha=0.3) +

geom_point(data = train_data, aes(x = minorAL, y = ecc, color = class), size = 2) +

geom_contour(data= grid, aes(x=minorAL, y=ecc, z = classnum), colour="black", linewidth = 1.2) +

scale_fill_manual(values = c("#ffff46", "#FF46e9")) +

scale_color_manual(values = c("red", "blue")) +

labs(title = "LDA Decision Boundary") +

theme_minimal()

ggplot() +

geom_point(data = train_data, aes(x = minorAL, y = ecc, color = class), size = 3) +

geom_point(data = min_max, aes(x = minorAL, y = ecc, color = predicted_class), size = 1, alpha = 0.3) +theme_minimal()

ggplot() +

geom_raster(data=grid, aes(x=minorAL, y=ecc, fill = class),alpha=0.3) +

geom_point(data = train_data, aes(x = minorAL, y = ecc, color = class), size = 2) +

geom_contour(data= grid, aes(x=minorAL, y=ecc, z = classnum), colour="black", linewidth = 1.2) +

scale_fill_manual(values = c("#ffff46", "#FF46e9")) +

scale_color_manual(values = c("red", "blue")) +

labs(title = "LDA Decision Boundary") +

theme_minimal()

Select the commands and run them.

It plots the grid points with colors indicating the predicted classes.

geom_raster creates a colour map indicating the predicted classes of the grid points

geom_contour creates the decision boundary of the LDA.

The scale_color_manual function assigns specific colors to the classes and so does scale_fill_manual function.

The overall plot provides a visual representation of the decision boundary and the distribution of training data points of the model.

Select and run these commands.

Drag boundaries to see the plot window clearly.

Let us summarize.

Summary

• Linear Discriminant Analysis (LDA) and its implementation. 
• Assumptions of LDA
• Limitations of LDA
• LDA on a subset of Raisin dataset
• Visualization of the LDA separator and its corresponding confusion matrix

Assignment

• Perform LDA on inbuilt PlantGrowthdataset
• Evaluate the model using a confusion matrix and visualize the results

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