diff --git a/Task_1.R b/Task_1.R new file mode 100644 index 0000000..5df4a3c --- /dev/null +++ b/Task_1.R @@ -0,0 +1,66 @@ +#time series plots +plot(Dataframe$time, Dataframe$x1, type = "n", xlim = c(30, 50), ylim = c(-5, 4), xlab="Time", ylab="EEG Signals") +lines(Dataframe$time, Dataframe$x1, col="blue", lwd="1") +lines(Dataframe$time, Dataframe$x2, col="red", lwd="1") + +plot(Dataframe$time, Dataframe$x1, type = "n", xlim = c(50, 54), ylim = c(-5, 3.5), xlab="Time", ylab="EEG Signals") +lines(Dataframe$time, Dataframe$x1, col="blue", lwd="2") +lines(Dataframe$time, Dataframe$x2, col="red", lwd="2") +lines(Dataframe$time, Dataframe$y, col="purple", lwd="1") + + +plot(Dataframe$time, Dataframe$y, type = "l", xlim = c(30, 50), ylim = c(-50, 50), col="purple", xlab="Time", ylab="Audio input") + +#distribution of the signals +hist(Dataframe$x1, col="blue", main = ("Distribution of X1"), xlab="", ylab="Frequency") +hist(Dataframe$x2, col="red", main = ("Distribution of X2"), xlab="", ylab="Frequency") +hist(Dataframe$y, col="purple", main = ("Distribution of Y"), xlab="", ylab="Frequency") + + +#variance +print(var(Dataframe$x1)) +print(var(Dataframe$x2)) +print(var(Dataframe$y)) +#mean +print(mean(Dataframe$x1)) +print(mean(Dataframe$x2)) +print(mean(Dataframe$y)) +#standard deviation +print(sd(Dataframe$x1)) +print(sd(Dataframe$x2)) +print(sd(Dataframe$y)) +#correlation +print(cor(Dataframe$x1, Dataframe$y)) +print(cor(Dataframe$x2, Dataframe$y)) + +#scatter plot with linear model +plot(Dataframe$x1, Dataframe$y, xlab = "X1", ylab = "Y")#positive correlation +abline(lm(Dataframe$y ~ Dataframe$x1), col = "red") + +plot(Dataframe$x2, Dataframe$y, xlab = "X2", ylab = "Y") #no correlation +abline(lm(Dataframe$y ~ Dataframe$x2), col = "blue") + +df=data.frame(Dataframe$x1,Dataframe$x2, Dataframe$y) + +#box plots +library(ggplot2) +df=data.frame(X, Dataframe$y) + +# Plot the box plot +ggplot(data = df, aes(x = "x1", y = x1)) + + geom_boxplot() + + xlab("EEG Signal") + + ylab("Value") + + ggtitle("Boxplot of EEG Signal x1") + +ggplot(data = df, aes(x = "x2", y = x2)) + + geom_boxplot() + + xlab("EEG Signal") + + ylab("Value") + + ggtitle("Boxplot of EEG Signal x2") + +ggplot(data = df, aes(x = "y", y = Dataframe.y)) + + geom_boxplot() + + xlab("EEG Signal") + + ylab("Value") + + ggtitle("Boxplot of Sound y") diff --git a/Task_2.7.R b/Task_2.7.R new file mode 100644 index 0000000..1951c0c --- /dev/null +++ b/Task_2.7.R @@ -0,0 +1,53 @@ +set.seed(123) +#split the matrix of data 70% training data and 30% testing data +data = sort(sample(nrow(X), nrow(X)*.7)) +train_data<-Dataframe[data,] +test_data<-Dataframe[-data,] + +#task 2.7.1 +#create the theta bias of length X1 in training data. +const = rep(1, times=length(train_data$x1)) + +#create the theta bias of length X1 in training data. +const_test = rep(1, times=length(test_data$x1)) + +#create the model +model3=cbind((train_data$x1^3), train_data$x2, train_data$x1, const) +model3_test_data=cbind((test_data$x1^3), test_data$x2, test_data$x1, const_test) + +#estimate model parameters using least squares (training data) +thetahat=solve(t(model3) %*% model3) %*% t(model3) %*% train_data$y + +#compute model prediction using test data +y_pred=model3_test_data %*% thetahat + +#plot 95% confidence interval +residuals = test_data$y - y_pred +rss = sum(residuals^2) +n = length(test_data$y) +#calculate variance of the residuals +sigma_squared = rss/(n-1) +#compute the estimated covariance matrix of the model parameters +cov_thetaHat = sigma_squared * (solve(t(model3_test_data) %*% model3_test_data)) + +var_y_hat = matrix(0 , n , 1) + +#compute the variance for of y_pred +for( i in 1:n){ + X_i = matrix( model3_test_data[i,] , 1 , 4 ) # X[i,] creates a vector. Convert it to matrix + var_y_hat[i,1] = X_i %*% cov_thetaHat %*% t(X_i) +} + +#get all three variables to order +CI = 2 * sqrt(var_y_hat) # Confidence interval +y=test_data$y +time=test_data$time + +#create dataframe and order it to plot +data = data.frame(y, y_pred, CI, time) +data = data[order(data$time), ] + +#plot prediction, testing data, and confidence intervals +plot(data$time, data$y_pred, type = "o", xlim = c(41, 47), ylim = c(-1, 1), col="steelblue1", xlab="Time", ylab="Y", lwd=2) +points(data$time, data$y, col="forestgreen", lwd=2) +segments(data$time, data$y_pred - data$CI, data$time, data$y_pred + data$CI, col="red1", lwd=3) diff --git a/Task_2.R b/Task_2.R new file mode 100644 index 0000000..ff23235 --- /dev/null +++ b/Task_2.R @@ -0,0 +1,63 @@ +set.seed(123) + +#create the theta bias +constant = rep(1, times = length(y)) + +#split X into X1 and X2 +X1=Dataframe[,1] +X2=Dataframe[,2] + +#create the models +model1=cbind((X1^3), (X2^5), constant) +model2=cbind((X1^4), (X2^2), constant) +model3=cbind((X1^3), X2, X1, constant) +model4=cbind(X1, (X1^2),(X1^3), (X2^3), constant) +model5=cbind((X1^3), (X1^4), X2, constant) + +for (i in 1:5) { + #get the model for every index: (model + i) + var_name=paste0("model", i) + my_model=get(var_name) + + #estimate model parameters using least squares (task 2.1) + thetahat=solve(t(my_model) %*% my_model) %*% t(my_model) %*% y + #print(thetahat) #print thetahat to include in the report + + #compute the models prediction + y_pred=my_model %*% thetahat + + #compute model residual (error) and calculate the sum of squared errors (task 2.2) + residuals = y - y_pred + rss = sum(residuals^2) + print(paste0("--------------------- Model ", i," -----------------------")) + print(paste0("RSS for model ",i, ": ", rss)) + + #compute the log-likelihood function (task 2.3) + n = length(y) + sigma_squared = rss/(n-1) + loglikelihood = -(n/2) * log(2*pi) - (n/2) * log(sigma_squared) - (1/(2*sigma_squared)) * rss + print(paste0("log likelihood for model ",i, ": ", loglikelihood)) + + #define k for AIC and BIC this is the number of estimated parameters in each model + #K is 4 in model 1 and 2, K is 5 in model 3 and 5, K is 6 in model 4 + if (i == 1 || i == 2) { + k= 4 + } else if (i == 4){ + k= 6 + } else { + k= 5 + } + #compute AIC (task 2.4) + AIC = 2 * k - 2 * loglikelihood + print(paste0("AIC for model ",i, ": ", AIC, " info: K is equal to: ", k)) + + #compute BIC + BIC = log(n) * k - 2 * loglikelihood + print(paste0("BIC for model ",i, ": ", BIC)) + + #Check if the residuals are close to normal/Gaussian (task 2.5) + qqnorm(residuals, main=paste("Q-Q Plot for Model", i)) + qqline(residuals) + hist(residuals, col="limegreen", main=paste("Distribution of the residuals for model", i), xlab="Model prediction errors", ylab="Frequency") + print(shapiro.test(residuals)) +} \ No newline at end of file diff --git a/Task_3.R b/Task_3.R new file mode 100644 index 0000000..50c6d66 --- /dev/null +++ b/Task_3.R @@ -0,0 +1,58 @@ +set.seed(123) +constant = rep(1, times = length(y)) +n=length(Dataframe$y) +X1 = Dataframe$x1 +X2 = Dataframe$x2 + +model_matrix = cbind((X1^3), X2, X1, constant) + +# Use a Uniform distribution as prior, around the estimated parameter values for those 2 parameters +prior1 = runif(10000, 4.181390 - 1, 4.181390 + 1)#0.078 because the when set to 1, the max and min accepted ranges +-0.57 +prior2 = runif(10000, -6.651400 -1, -6.651400 + 1)#0.1 + +# Create empty array to store parameters +accepted_params = matrix(nrow = 0, ncol = 3) + +num_iter = 100000 +tolerance = 0.018 + +for (i in 1:num_iter) { + # Draw parameters from prior + theta3 = sample(prior1, 1) + thetaBias = sample(prior2, 1) + + # Fix other parameters as constant + thetahat = c(2.715713, -3.151350, theta3, thetaBias) + theta=matrix(thetahat) + + # Compute simulated data using the model + y_sim = model_matrix %*% theta + + # Compute distance between simulated and observed data + dist = (sqrt(sum((y_sim - y)^2)))/n + + print(dist) + # Accept or reject parameter values based on the distance + if (dist < tolerance) { + accepted_params = rbind(accepted_params, c(theta3, thetaBias, dist)) + } +} + +# Plot joint posterior distribution +library(ggplot2) + +# Extract accepted parameters +theta3_posterior = accepted_params[, 1] +thetaBias_posterior = accepted_params[, 2] + +# Plot joint posterior distribution +#plot(accepted_params[,1], accepted_params[,2], type = "p", +# xlab = "Theta 3", ylab = "Theta Bias",col="blue", main = "Joint Posterior Distribution", lwd=1) + +ggplot(data.frame(theta3_posterior, thetaBias_posterior), aes(x=theta3_posterior, y=thetaBias_posterior)) + + geom_bin2d(bins = 15, colour="green") + + labs(x = expression(theta[3]), y = expression(theta[Bias])) + +#plot the marginal distribution +hist(theta3_posterior, col="orangered1", main = ("Posterior distribution for Theta 3"), xlab="Value", ylab="Frequency") +hist(thetaBias_posterior, col="limegreen", main = ("Posterior distribution for Theta Bias"), xlab="Value", ylab="Frequency") \ No newline at end of file diff --git a/X.csv b/X.csv new file mode 100644 index 0000000..cdbd212 --- /dev/null +++ b/X.csv @@ -0,0 +1,2401 @@ +x1,x2 +-0.5623197529376424,-2.2744913096467174 +0.505550488720138,-1.4515861652820807 +-0.857811647528123,-0.27121493056582446 +-0.7352616768582236,-3.40250053308577 +-1.244368749652711,-2.199161883401363 +1.0339634760650147,-2.6430473499459035 +-1.1078683945266046,-1.941599373726427 +0.9771568233661176,-1.5294837750835688 +-1.4893489406363887,-2.0272966913128085 +0.32626646361956413,-2.5025678591235074 +-0.7054832931223808,-1.3631820734888334 +-1.0656968680008638,-1.6850036520313552 +1.736795074066959,-0.5546874753380111 +-0.0616941224555599,-0.7116271121553213 +-1.2130858207854165,-3.0981101626515506 +-2.5372227628531316,-2.004896768361141 +1.336317339992,-3.0451189932749436 +-1.1381622379485599,-1.219410899904661 +-0.17943805505022464,-0.8858627351038395 +0.9226989537689183,-1.3349715546523326 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