Assignment $4+5$: Predicting $10-$Day Velocity Components $\backslash ($ u $\backslash )$ and $\backslash ($ v $\backslash )$ Using EOF Analysis $($SVD$),$ Transformers, and Diffusion Models

Objective

The objective of this assignment is to use Empirical Orthogonal Function $($EOF$)$ analysis, performed through Singular Value Decomposition $($SVD$),$ and advanced deep learning models to reduce the dimensionality of the $\backslash ($ u $\backslash )$ and $\backslash ($ v $\backslash )$ velocity data. These reduced representations will be used to forecast future states over a $10-$day period and reconstruct velocity data across the entire region of interest.

EOF Analysis Using Singular Value Decomposition $($SVD$)$

Empirical Orthogonal Function $($EOF$)$ analysis simplifies high$-$dimensional data and captures the most significant patterns through Singular Value Decomposition $($SVD$).$

Step$-$by$-$Step SVD$-$Based EOF Analysis

$1.$ Construct the Data Matrix $\backslash ($ D $\backslash )$ : Arrange the velocity data into a matrix $\backslash ($ D $\backslash )$ of size $\backslash ($ n $\backslash$times d $\backslash ),$ where $\backslash ($ n $\backslash )$ represents time steps and $\backslash ($ d $\backslash )$ represents spatial points.

$2.$ Apply SVD: Decompose $\backslash ($ D $\backslash )$ as $\backslash ($ D$=$U $\backslash$Sigma V$^\{$T$\}\backslash ),$ identifying spatial patterns $($EOFs$)$ and temporal dynamics $($PCs$).$

$3.$ Interpret EOFs and PCs: Use the matrices from SVD to identify principal spatial and temporal patterns.

$4.$ Dimensionality Reduction: Select the first $\backslash ($ k $\backslash )$ components that capture the majority of the variance for further analysis.

$5.$ Reconstruct the Data $($Optional$)$: Use the significant modes to approximate the data matrix $\backslash ($ D $\backslash )$ with $\backslash ($ D $\backslash$approx U$\_\{$k$\}\backslash$Sigma$\_\{$k$\}$ V$\_\{$k$\}^\{$T$\}\backslash ).$ Forecasting with Transformers

Using EOF Analysis

$-$ Model Setup: Utilize a transformer model to forecast future velocity components using the Principal Components $($PCs$)$ as input, which are derived from the EOF analysis.

$-$ Data Preparation: Format the reduced data from EOF analysis for sequential processing in the transformer. Ensure that the input sequence to the transformer appropriately encapsulates temporal dynamics as represented by the PCs$.$

$-$ Implementation Details: 'Train the transformer to predict the future state of the PCs over a $10-$day period. The training should focus on capturing the temporal dependencies and dynamics encapsulated by the EOF$-$derived PCs$.$

Without Using EOF Analysis

$-$ Model Setup: In this approach, use the transformer to directly forecast future velocity components based on the original, non$-$reduced data.

$-$ Data Preparation: Prepare the original velocity data by structuring it into an appropriate sequence format for the transformer, maintaining the full complexity of the spatial$-$temporal patterns.

$-$ Implementation Details: Train the transformer model on the full dataset to predict the future velocity components over the same $10-$day period. This approach tests the transformer's ability to handle high$-$dimensional data without the preprocessing step of dimensionality reduction.

Comparison Task: After completing both forecasting approaches, students are required to compare the performance, efficiency, and accuracy of the forecasts generated by the transformers with and without the use of EOF analysis. Discuss the advantages and potential limitations of each approach.

Reconstruction Using Diffusion Models

$-$ Objective: Use a diffusion model to reconstruct the velocities from partial observations of the map to the entire region of interest.

$-$ Data Provision: Different data sets will be provided specifically for this task.

$-$ Methodology: Implement the diffusion process to generate complete velocity fields from sparse or incomplete data. Task Implementation

$1.$ EOFs and PCs Extraction: Identify EOFs and PCs through SVD on the data matrix to be used in subsequent forecasting tasks.

$2.$ Machine Learning Forecasting:

$-$ Using EOF$-$derived PCs:

$-$ Train LSTM$,$ RNN$,$ and MLP models on the PCs to predict future values, focusing on capturing the essential dynamics.

$-$ Train a Transformer model to forecast the future state of the PCs over a $10-$day period, using the reduced representation from EOF analysis.

$-$ Using original, non$-$reduced data:

$-$ Train a Transformer model directly on the original, nonreduced data to forecast future velocity components, testing the ability to process complex high$-$dimensional data.

$3.$ Comparison of Forecasting Methods: Students are required to compare the performance of LSTM$,$ RNN$,$ MLP$,$ and Transformer models when using EOF$-$derived PCs versus using the original dataset in the Transformer model. For this comparison, students should use the following metrics:

$-$ Correlation Coefficient: Measure the linear correlation between the actual and forecasted values to evaluate the accuracy and predictive performance of each model.

$-$ Mean Squared Error $($MSE$)$: Calculate the MSE to assess the average squared difference between the estimated values and what is estimated. This metric will help evaluate the precision of the forecast.

$-$ Computational Efficiency: Analyze the time and resources required for training and predicting with each model, assessing how well each method handles the complexity and size of the data.

Discuss the advantages and potential limita $-$ Evaluate Reconstruction Quality: Assess the quali