
Task 1  Preliminary data analysis
 Analysis of time series plots (of audio and EEG signals)
 Analysis of the distribution for each signal
 Outlier detection
 Analysis of correlation and scatter plots (between the audio and brain signals)

Task 2  Regression  Modelling the relationship between audio and EEG signals
Five candidate nonlinear polynomial regression models have been proposed, and the objective is to identify the model that can truly describe this relationship through a series of tasks. The five candidate models are:
Model 1:
$y=θ_1 Χ_1^3+θ_2 Χ_2^5+θ_b+ϵ$
Model 2:$y=θ_1 Χ_1^4+θ_2 Χ_2^2+θ_b+ϵ$
Model 3:$y=θ_1 Χ_1^3+θ_2 Χ_2+θ_3 Χ_1+θ_b+ϵ$
Model 4:$y=θ_1 Χ_1+θ_2 Χ_1^2+θ_3 Χ_1^3+θ_4 Χ_2^3+θ_b+ϵ$
Model 5:$y=θ_1 Χ_1^3+θ_2 Χ_1^4+θ_3 Χ_2+θ_b+ϵ$  Task 2.1  Estimate model parameters using least squares method
 Tasks 2.2, 2.3, 2.4  Compute RSS, Loglikelihood function, AIC, and BIC
 Task 2.5  Evaluate the distribution of the candidate models residuals
 Task 2.6  Select the best model based on the destribution of the residuals, AIC, BIC and RSS
 Task 2.7  Compute the model’s prediction on the testing data and plot the 95% confidence intervals
 Split the data into training and test sets
 Estimate model parameters using only the training dataset
 Compute the model’s output/prediction on the testing data
 Compute and plot the 95% confidence intervals together with the model prediction, and the testing data samples

Task 3  Approximate Bayesian Computation (ABC)

Perform rejection ABC to compute the posterior distributions of the selected regression model parameters
The focus was on the two parameters with the largest absolute values in the least squares estimation of the selected model, while all the other parameters were fixed as constant using the estimated values from Task 2.1. A Uniform distribution was used as a prior around the estimated parameter values, and samples were drawn from this prior distribution to perform the rejection ABC algorithm. In each iteration, the model parameters were updated, and the new parameter matrix was multiplied by the model to generate predictions, the distance was computed and the pairs of parameters were accepted if the distance was smaller than the tolerance.

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