make answer this problem in 3 steps. each step is at least 500 words long. steps contain detailed solution with concept and logic's. use some good headings in answer.
at the end of each step give a explanation heading, in it explain the step answer. Generate a data set with two explanatory variables and from multinomial Normal distri- bution with covariance matrix a c(l, .2, .2, 4) in two classes with mean for Class O is (3, 7) and and Class 1 is (6, 10). For the Class 0, generate 50 observations and for Class 1, 50 observations. While generating this data, use the set.seed(" 99" ). Find the linear discriminant function weights. Plot the data with two colors and draw the discriminant function for classification. Also plot thc 4 test data (3.68, 5.65), (3.28, 5.20), (3.57, 8.82), (4.64, 7.98) and predict the test data. Use the R program also to predict the test data.
Note - steps length carry according to your question. Minimum 3 steps necessary.
Step 1: Generating the Data Set and Calculating the Linear Discriminant Function Weights
To begin, let's generate a data with two explanatory variables and create 50 observations for each class (Class 0 and Class 1) from aomial normal distribution with a specified covariance matrix and means. We will also set the random seed to ensure reproducibility of results in R.
# the random seed for reproducibility set.seed(99) # Number of observations for each class n <- 50 # Mean values Class 0 and Class 1 mean_class0 <- c(3, 7) mean1 <- c(6, 10) # Covariance matrix _matrix <- matrix(c(1, 0.2, 0.2,4), nrow=2, byrow=TRUE) # Generating data for Class 0 and Class 1 class0_data <-::mvrnorm(n, mean_class0, cov_matrix) class1_data <- MASS::mvrnorm(n, mean_class, cov_matrix)
Explanation of Step 1: In this step, we use the "set.seed" function to set the random seed for reproducibility. Then, we specify the number of observations for each class as 50. After that, we define the mean values for Class 0 and Class 1 as (3, 7) and (6, 10) respectively. Next, create a covariance matrix with the specified values. Using the "mvrnorm" function the "MASS" package, we generate the data for Class 0 and Class 1 based the multinomial normal distribution with the provided mean and covariance matrix.
Step 2: Calculating the Linear Discriminant Function Weights
Next, we will calculate the linear discriminant function weights using the generated data for Class 0 and Class 1. The linear discriminant function is used for classification and finding the decision boundary the two classes.
# Combining the data for both classes all_data <- rbind(class0_data, class1_dataclass_labels <- c(rep(0, n), rep(1, n)) # Calcul the linear discriminant function weights lda_model <- MASS::lda(all_data, class_labelslda_weights <- lda_model$scaling
Explanation of Step 2: In this step we combine the data for Class 0 and Class 1 into a single data set "all_data" and create class labels. Then, we calculate the linear discriminant function using the "lda" function from the "MASS" package. The obtained from the linear discriminant analysis model will be used for and drawing the discriminant function for the data.
Step 3: Visualizing the and Making Predictions
In the final step, we will visualize the generated data with two colors representing Class 0 and Class 1. Additionally, we draw the discriminant function for classification and use the model to predict the test data.
# Visualizing the data plot(class0_data, col="blue xlim=c(0, 12), ylim=c(3, 15), xlab="X1", y="X2") points(class1_data, col="red") # Drawing the discriminant function abline(-lda_weights[2]/lda_weights[1], -lda_weights[3]/lda_weights[1], colgreen") # Test data test_data <- matrix(c(3.68, 5.65,3.28, 5.20, 3.57 8.82, 4.64, 7.98), ncol=2, byrow=TRUE) # Predicting the data predicted_test <- predict(lda_model, newdata=test_data)
Explanation of Step 3: In this step, we plot the generated data with different colors representing Class 0 and Class 1. We then draw the discriminant function for classification using the linear discriminant function calculated in the previous step. Next, we define the test data and use the "predict" function to make predictions based on the linear discriminant analysis model. The predicted test results will indicate the class labels for the test data points based on the trained model.
By following these three steps, we have successfully generated the data set, calculated the linear discriminant function weights, visualized the data, and predicted the test data using R program.
Generate a data set with two explanatory variables and from multinomial Normal distri- bution with covariance matrix a c(l, .2, .2, 4) in two classes with mean for Class O is (3, 7) and and Class 1 is (6, 10). For the Class 0, generate 50 observations and for Class 1, 50 observations. While generating this data, use the set.seed(" 99" ). Find the linear discriminant function weights. Plot the data with two colors and draw the discriminant function for classification. Also plot thc 4 test data (3.68, 5.65), (3.28, 5.20), (3.57, 8.82), (4.64, 7.98) and predict the test data. Use the R program also to predict the test data.
Step 1: Generating the Data Set and Calculating the Linear Discriminant Function Weights
First, we will generate a data set with two explanatory variables and create observations for each class (Class 0 and Class 1) from a multinomial normal distribution with a specific covariance matrix and means. We will set the random seed to ensure reproducibility of results in R.
# Set random seed for reproducibility set.seed(99) # Parameters n <- 50 # Number of observations for each class mean_class0 <- c(3, 7) # Mean for Class 0 mean_class1 <- c(6, 10) # Mean for Class 1 cov_matrix <- matrix(c(1, 0.2, 0.2, 4), nrow = 2, byrow = TRUE) # Covariance matrix # Generate data for Class 0 class0_data <- MASS::mvrnorm(n, mean_class0, cov_matrix) # Generate data for Class 1 class1_data <- MASS::mvrnorm(n, mean_class1, cov_matrix)
Explanation of Step 1: In this step, we set the random seed for reproducibility and define the parameters such as the number of observations for each class, mean values for Class 0 and Class 1, and the covariance matrix. Using the "mvrnorm" function from the "MASS" package, we generate data for Class 0 and Class 1 based on the multinomial normal distribution with the specified mean and covariance matrix.
Step 2: Calculating the Linear Discriminant Function Weights and Visualizing the Data
Next, we will calculate the linear discriminant function weights using the generated data for Class 0 and Class 1. We will also plot the data with two colors to differentiate between the two classes and draw the discriminant function for classification.
# Combine the data for both classes all_data <- rbind(data.frame(class0_data, Class = 0), data.frame(class1_data, Class = 1)) # Fit the linear discriminant model lda_model <- MASS::lda(Class ~ ., data = all_data) # Plot the data with two colors plot(class0_data, col = "blue", xlim = c(0, 12), ylim = c(3, 15), xlab = "X1", ylab = "X2") points(class1_data, col = "red") # Plot the linear discriminant function abline(lda_model$scaling[2]/lda_model$scaling[1], lda_model$scaling[3]/lda_model$scaling[1], col = "green")
Explanation of Step 2: In this step, we combine the data for Class 0 and Class 1 into a single data set and fit a linear discriminant model using the "lda" function from the "MASS" package. Subsequently, we plot the generated data with different colors representing Class 0 and Class 1, and draw the discriminant function for classification using the obtained linear discriminant function weights.
Step 3: Predicting the Test Data
In this step, we will plot the test data points and use the trained model to predict their class labels.
# Test data test_data <- matrix(c(3.68, 5.65, 3.28, 5.20, 3.57, 8.82, 4.64, 7.98), ncol = 2, byrow = TRUE) # Predict the test data predicted_test <- predict(lda_model, newdata = as.data.frame(test_data))
Explanation of Step 3: In this step, we define the test data points and use the "predict" function to predict their class labels based on the trained linear discriminant analysis model.
By following these three steps, we have successfully generated the data set, calculated the linear discriminant function weights, visualized the data, and predicted the test data using R program.