Cup stats & predictions

Economic Impact: Revenue Generation through Football Events

The Football Cup Lithuania not only entertains but also contributes economically through various channels:

Listings**: Ticket sales generate substantial revenue while local businesses benefit from increased patronage during match days.

In-Depth Analysis: Key Matchups Explained Further

Detailed Breakdown: Team A vs. Team B Strategies

This fixture is poised as one of tomorrow’s most intriguing encounters due primarily because both squads have demonstrated formidable prowess throughout this season’s campaign thus far.

Offensive Strategy (Team A)luisadefranco/MasterThesis<|file_sep|>/tex/appendix.tex chapter{Appendix} section{Extended Experimental Results} begin{figure}[H] centering includegraphics[width=0.6textwidth]{plots/extended_experiments/updated_GN.png} caption{Performance comparison between RLS-GN($mu$) (blue) using $mu = [0 ; .1 ; .25 ; .5 ; .75 ; .9]$ with RLS (red) using $mu = [0 ; .1 ; .25 ; .5 ; .75 ; .9]$.} label{fig:extended_gn} end{figure} begin{figure}[H] centering includegraphics[width=0.6textwidth]{plots/extended_experiments/updated_ours.png} caption{Performance comparison between RLS-OGN (blue) using $mu = [0 ; .1 ; .25 ; .5 ; .75 ; .9]$ with RLS (red) using $mu = [0 ; .1 ; .25 ; .5 ; .75 ; .9]$.} label{fig:extended_ours} end{figure} begin{figure}[H] centering includegraphics[width=0.6textwidth]{plots/extended_experiments/updated_NLN.png} caption{Performance comparison between RLS-NLN($mu$) (blue) using $mu = [0 ; .1]$ with RLS (red) using $mu = [0]$.} label{fig:extended_nln} end{figure} section{Extended Implementation Details} The following section provides additional information about our implementation. subsection{Matrix Inversion Lemma} In order to efficiently update $P_{t+1}$ we used matrix inversion lemma~cite{kailath2000linear}: [ P_{t+1}^{-1} = P_t^{-1} + u_t u_t^T / (lambda - u_t^T P_t u_t). ] [ P_{t+1} = P_t - P_t u_t u_t^T P_t / (lambda + u_t^T P_t u_t). ] We also used $P_0^{-1}$ instead $P_0$ since it is more efficient than computing inverse. % MATLAB code snippet % % Initialization % P = inv(P_0); % % % Update % P_inv = P + u * u' / (lambda - u' * P * u); % P = inv(P_inv); We tested three approaches: begin{itemize} item Directly compute $P_{t+1}$ using Eqn~(ref{eqn:update_P}). This approach requires matrix inversion at every step. item Compute $P_{t+1}$ using Eqn~(ref{eqn:update_P_inv}) then invert it. item Use Eqn~(ref{eqn:update_P_inv}) directly. end{itemize} We observed that using Eqn~(ref{eqn:update_P_inv}) directly was fastest because it does not require matrix inversion. The MATLAB code snippet above shows how we implemented Eqn~(ref{eqn:update_P_inv}). For matrix inversion lemma we used LU decomposition instead Cholesky decomposition because LU decomposition works even if matrix is not positive definite. % MATLAB code snippet % % Initialization % L = lu(P_0); % % % Update % L(2:end,:) = L(2:end,:) + u * (u' / (lambda - u' * L(L'*u))); % L(end,:) = L(end,:) / lambda; % P = LL'; We also tested two approaches: begin{itemize} item Directly compute $P_{t+1}$ using LU decomposition. item Update $L$ matrix from LU decomposition directly. end{itemize} We observed that updating $L$ matrix directly was fastest. In order to update $L$ matrix we used back substitution algorithm~cite{kailath2000linear}. We did not use MATLAB built-in functions because they were too slow. In order to test if our implementation was correct we used MATLAB built-in functions as ground truth. <|file_sep|>chapter{Related Work} The problem we addressed in this thesis has been studied extensively over years by many researchers from different fields such as signal processing~cite{jain2015fundamentals}, machine learning~cite{sutton2018reinforcement}, adaptive filtering~cite{souloumiac2004adaptive}, control theory~cite:kailath2000linear. Adaptive filters learn an approximation model from data which is typically unknown or difficult to model mathematically. This section provides an overview of different approaches proposed by researchers over years. The main focus here is recursive least squares algorithms since they were used as basis algorithms in this thesis. Other adaptive filters such as least mean squares algorithm or Kalman filter were briefly discussed as well. Additionally this section covers existing extensions such as variable forgetting factor or robustified adaptive filters which were used as baseline algorithms in this thesis. The goal here was not only show different approaches but also provide background information necessary for understanding our proposed method. Adaptive filters are widely used applications therefore many papers were published about them. We focused only on most popular approaches which were cited most often over years. This section provides short overview about most popular adaptive filters such as recursive least squares algorithm or least mean squares algorithm. It also covers extensions such as variable forgetting factor or robustified adaptive filters which were used as baseline algorithms in this thesis. Moreover it discusses existing work related specifically to our proposed method. %% RLS In this thesis we focused primarily on recursive least squares algorithms. Recursive least squares algorithm was introduced by Anderson~et al.citeanderson1965introduction}. It is commonly used algorithm which has many applications such as signal processing~cite:jain2015fundamentals or control systems~kailath2000linear. It belongs to adaptive filters class which learn an approximation model from data typically unknown or difficult to model mathematically. This approach is iterative where parameters $hat{theta}_t$ are updated at every time step $t$ based on new observation $(x_t,y_t)$: [ y_t = x_t^T {theta^*} + n_t , ] where ${x_t}$ is input vector, ${y_t}$ is output value, ${n_t}$ is noise, ${{theta^*}}$ is unknown parameters vector, and $hat{theta}_t$ is estimate parameters vector. %% Least Squares Least squares algorithm aims at minimizing squared error between output value ${y_i}$ predicted by model ${x_i^T {theta}}$ given parameters vector ${theta}$ and real output value ${y_i}$: [ E(theta) = ||Y-X{theta}||_2^2 , \] where ${X=[x_1^T,ldots,x_n^T]^T}$, ${Y=[y_1,ldots,y_n]^T}$, and ${||z||_2=sqrt{sum_i z_i^2}}$. %% Recursive Least Squares Recursive least squares algorithm aims at minimizing squared error: [ E(theta) = ||Y-X{theta}||_2^2 , \] where ${X=[x_1^T,ldots,x_n^T]^T}$, ${Y=[y_1,ldots,y_n]^T}$, and ${||z||_2=sqrt{sum_i z_i^2}}$. %% Least Mean Squares Least mean squares algorithm aims at minimizing squared error: [ E(theta) = ||y-x^T {theta}||_2^2 , \] where ${x}$ is input vector, ${y}$ is output value, ${||z||_2=sqrt{sum_i z_i^2}}$. %% Kalman Filter Kalman filter was introduced by Kalman~et al.citekalman1960new}. It belongs also adaptive filters class since it learns an approximation model from data typically unknown or difficult to model mathematically. Kalman filter estimates state variables ${x_k}$ given measurements ${z_k}$ assuming system follows linear dynamics: [ x_k=f(x_{k-1},u_k)+w_k , \] where ${f(x,u)}$ describes system dynamics, ${u_k}$ describes control input vector, and ${w_k}$ describes process noise vector. %% Variable Forgetting Factor Variable forgetting factor aims at improving performance of adaptive filters by adjusting forgetting factor $lambda$ dynamically instead constant value typically set between $(0.95-1)$. There are many methods proposed by researchers over years however we focused only on most popular approaches cited often over years. %% Robustified Adaptive Filters Robustified adaptive filters aim at improving performance of adaptive filters by reducing effect of outliers which cause errors in estimated parameters vector. There are many methods proposed by researchers over years however we focused only on most popular approaches cited often over years. %% Robustified Adaptive Filters Robustified recursive least squares algorithms aim at improving performance of recursive least squares algorithms by reducing effect of outliers which cause errors in estimated parameters vector. There are many methods proposed by researchers over years however we focused only on most popular approaches cited often over years. %% Robustified Recursive Least Squares Algorithms Robustified recursive least squares algorithms aim at improving performance of recursive least squares algorithms by reducing effect of outliers which cause errors in estimated parameters vector. There are many methods proposed by researchers over years however we focused only on most popular approaches cited often over years. %% Robustified Recursive Least Squares Algorithms - Projection Method Projection method was introduced by Van Trees~et al.citevan2014optimum}. This method assumes observations corrupted by outliers follow Gaussian distribution: [ p(y|x