Descubre el Tenis W50 Taizhou China con Nuestros Partidos Actualizados y Predicciones de Apuestas
¿Eres un entusiasta del tenis buscando seguir de cerca los emocionantes encuentros del W50 Taizhou China? ¡Estás en el lugar correcto! Nuestra plataforma está especialmente diseñada para brindarte una experiencia inigualable con actualizaciones diarias de partidos, garantizando que nunca te pierdas un solo punto. Además, contamos con expertos que te ofrecen predicciones de apuestas precisas para cada partido, ayudándote a tomar decisiones informadas y maximizar tus chances de éxito.
Por qué Te Interesa el W50 Taizhou China
El W50 Taizhou China es uno de los torneos más emocionantes del circuito profesional femenino, ubicado en la vibrante ciudad china de Taizhou. Este torneo es parte del circuito WTA, ofreciendo una oportunidad única para ver a las mejores jugadoras del mundo en acción en un ambiente dinámico y acogedor. Además, este evento es ideal para quienes buscan oportunidades de apuestas deportivas en tenis, ya que nuestras predicciones están basadas en un análisis detallado de las estadísticas de las jugadoras, su desempeño reciente, y otros factores claves.
Actualizaciones Diarias de Partidos
Mantente al día con los últimos resultados y próximos encuentros gracias a nuestras actualizaciones diarias. No importa si estás en casa o fuera, podrás seguir cada set y cada punto en tiempo real. Nuestra plataforma está diseñada para ofrecerte un acceso rápido y fácil a toda la información relevante de los partidos, asegurando que nunca te pierdas un solo detalle.
- Resultados en Vivo: Sigue cada partido en directo con resultados actualizados al instante.
- Análisis de Partidos: Profundizamos en cada encuentro para brindarte el contexto necesario.
- Horarios de Partidos: Nunca te pierdas un partido gracias a nuestro calendario detallado y actualizado.
Predicciones de Apuestas con Expertos
Las apuestas deportivas pueden ser una aventura emocionante, pero también requieren una preparación adecuada. Aquí es donde nuestras experticias en predicciones de apuestas entran en juego. Nuestros expertos en tenis analizan cada aspecto del juego para ofrecerte predicciones precisas que te ayudarán a tomar decisiones informadas y aumentar tus probabilidades de ganar.
- Análisis Estadístico: Utilizamos datos históricos y estadísticas detalladas para entender el rendimiento de las jugadoras.
- Evaluación de Condiciones: Consideramos factores como la superficie de juego, el clima y las condiciones locales.
- Perfil de Jugadora: Analizamos el estilo de juego, la forma física y la psicología de cada participante.
¿Cómo Funcionan Las Apuestas en Tenis?
Si eres nuevo en el mundo de las apuestas deportivas, especialmente en tenis, te invitamos a explorar más sobre cómo funcionan. Las apuestas en tenis pueden ser tan simples como apostar por la ganadora del partido, o tan complejas como predecir el número exacto de juegos ganados por una jugadora. Aquí te explicamos las opciones más populares:
- Ganadora del Partido: Apuesta por quién crees que ganará el encuentro.
- Ganador de Set: Predice quién ganará cada set del partido.
- Total de Juegos: Averigua si el total de juegos jugados será mayor o menor que un número estimado.
- Tie-Break Set: Apuesta sobre si habrá o no tie-break en un set específico.
Tips para Mejorar Tu Estrategia de Apuestas
Más allá de las predicciones, es importante tener una estrategia sólida para maximizar tus posibilidades de éxito en las apuestas. Aquí te compartimos algunos consejos:
- Infórmate Bien: Conoce a las jugadoras, su forma actual, y cualquier factor relevante que pueda impactar su desempeño.
- Gestiona Tu Banca: Establece límites de apuestas y asegúrate de no arriesgar más de lo que estás dispuesto a perder.
- Varía Tus Apuestas: Distribuye tus apuestas en diferentes tipos de juegos y encuentros para diversificar el riesgo.
- Analiza Minuto a Minuto: Sigue la acción y adapta tus decisiones de apuestas según avance el partido.
Conoce las Mejores Locaciones para Ver en Vivo el Tennis W50 Taizhou
Si prefieres vivir la emoción desde Taizhou, aquí tienes información sobre dónde puedes disfrutar de los partidos en vivo. El W50 es conocido por su ambiente acogedor y sus instalaciones modernas que ofrecen una experiencia inolvidable tanto para jugadores como aficionados.
- Taizhou Tennis Stadium: El recinto principal donde se realizan la mayoría de los encuentros. Con capacidad para miles de espectadores, ofrece la mejor experiencia visual posibles.
- Zon<|repo_name|>eshayper/thesis<|file_sep|>/code/latex/old/paper.tex
documentclass{article}
usepackage{graphicx}
usepackage{amsmath}
usepackage{amssymb} % for mathbb
usepackage{hyperref}
newcommand{R}{mathbb R} % real field
newcommand{N}{mathbb N} % natural numbers
begin{document}
title{LARGE Chattering for Multi-Target Tracking}
author{Elliott Shayper, Jeremy Estepp, Carlell Nelson}
maketitle
begin{abstract}
The exercise of tracking an airborne target is often complicated by the possibility of presence of multiple similar targets within the local region of the observer (e.g., oncoming airliners on a runway). A relatively simple model for an observer tracking an unknown number of targets in its local region is represented by the chattering model. This paper presents a suite of data-driven methods to detect and separate multiple targets from within the filter noise of a chattering model and applies these methods to simulated data from an airborne target with a Lagrangian Airborne Surveillance Radar (LASR).
end{abstract}
section{Introduction}
The problem of tracking multiple targets in close proximity is not only difficult from a theoretical perspective but also requires significant attention when implementational aspects are considered. In this paper, we consider an observer tracking an unknown number of similar targets in its local region.
A relatively simple model for multi-target tracking is to augment the single target chattering model with two key changes. Typical tracking models consider a target whose position is given by an observation drawn from a probability distribution centered around the target's current position. However, when considering multiple targets, there is potential for the single observation to be drawn from a distribution centered around some other target.
Let $mathbf{z}_t$ be the observation at time $t$. If we let $i_t$ be the index of the target that $mathbf{z}_t$ centers on at time $t$, then for each time $t$:
begin{align}
mathbf{x}_t &= mathbf{x}_{t-1} + boldsymbol{eta}_t \
mathbf{z}_t &= mathbf{x}_{i_t} + boldsymbol{epsilon}_t.
end{align}
Note that both $boldsymbol{eta}_t$ and $boldsymbol{epsilon}_t$ are normally distributed around zero and that $i_t$ is an integer between $1$ and $N$, where $N$ is the number of targets.
This model is known as the chattering model because of the phenomenon that as the targets get closer together, their movement becomes more similar and sometimes indistinguishable via current observations. The chattering model has been extensively studied by Salomond (cite{salomond1989hidden}, cite{salomond1989discrete}) and revisited much more recently by Lindqvist and Niranjan (cite{lindqvist2019chattering}, cite{lindqvist2019posterior}).
The chattering model presented by Salomond (cite{salomond1989hidden}) and Lindqvist and Niranjana (cite{lindqvist2019posterior}) are illustrated below in Figure ref{fig:1}.
It may be obvious that the curves in Figure ref{fig:1a} converge for $t = 10$, but not as obvious for Figure ref{fig:1b}; yet we can still recover the truth when using the methods presented in this paper. However, if backward recursion is performed on the likelihood and density functions, recovery will converge on an incorrect result. This makes the chattering problem a fitting case for particle filtering where Markov Chain Monte Carlo (MCMC) methods can be used to resample the particles, thus providing approximations to the densities involved.
In addition to the complication of multiple nearby targets, there is ancillary complicating factor involving measurement noise that accompanies the addition of any new sensor. This paper explores predicting outliers in a large amount of data by exploring which predictions of particle filtering are incorrect over many samples.
% DIFFERENCE IN FILTERING APPROACHES
% Error Propagation based Filtering
% Likelihood Ratio
% Information based
% Matching Filtering
% Bayesian Filtering
% Particle Filtering
The most common method of particle filtering consists of alternating a prediction step with an update step.
The prediction step draws a sample from the proposed distribution $q$ and then weights them using the importance ratio $w^{(t)}=p_s(x_t|x_{t-1})/q(x_t|x_{t-1})$.
This paper focuses more on revisiting and modifying this approach to include heuristics that can recover the truth and correct for chattering when particle filtering with the chattering model.
The problems we are attempting to solve include:
begin{itemize}
item Predicting outliers during the observation step of particle filtering
item Adjusting weights of particles based on these predictions
item Calculating weights of matched particles based on the degree to which they are matched
item Deciding when a particle has moved so far away that it should be discarded, rather than propagated
item How we should handle resampling
end{itemize}
A challenge in implementing these methods comes from the fact that particles should be drawn from distributions based on state equations where contemporaneous measurements are not conditioned on the observations yet to come. This means that when drawing values for the particle for $x_t$, we need to select a value for $boldsymbol{eta}$ without knowledge of $mathbf{z}_t$. We take care of this by drawing a value for $boldsymbol{eta}$ independently from from $i_t$, then drawing a value for $boldsymbol{epsilon}$ based off of what we have already drawn in $mathbf{x}_{t-1} + boldsymbol{eta}$.
For notation, in this paper we will refer to state variables as $mathbf{x}_t$, observations as $mathbf{z}_t$, belief as $pi_t$, dynamic or state transition equation as $mathbf{x}_t = B(mathbf{x}_{t-1}, boldsymbol{eta}_t)$, observation equation as $mathbf{z}_t = O(mathbf{x}_{i_t}, boldsymbol{epsilon}_t)$, importance sampling proposal distribution as $q$, and importance weighting function as $w$.
% section{Particle Filtering}
% TODO: Reference: https://github.com/IRIS-HEP/particle/blob/master/notes/particlelearning.pdf
% Draw $N$ particles from the transition density for each particle at time $t$. This step provides values for each particle candidate at time $t-1$. These particles can now be updated based on their likelihood of arriving there given the observations.
% $$x^{(j)}_t sim B(mathbf{x}_{t-1}^{(j)}, q(cdot|x_{t-1}^{(j)}))$$
% $$ w^{(j)}_t = p_s(x_t|x_{t-1})/q(x_t|x_{t-1}) $$
% $$ w^{(j)}_t propto w^{(j)}_{t-1}L(mathbf{z}_t|mathbf{x}_t^{(j)}) = L(mathbf{z}_t|mathbf{x}_t^{(j)}) / q(mathbf{x}_t^{(j)}|mathbf{x}_{t-1}^{(j)}) $$
% Resampling should occur if and only if the effective sample size is less than half the variance in predictions. If resampling occurs, then $N$ particles are drawn with replacement based on their weights from the time step right before resampling. This results in a time step where all particles have equal weights again.
% Using our proposal distribution $mathcal{N}$ and importance weight function $w$,
% begin{align*}
% & x^{(j)}_t sim mathcal{N}(mathbf{x}_{t-1}^{(j)}, sigma)\
% & w_t^{(j)} = L(mathbf{z}_t|mathbf{x}_t^{(j)}) / mathcal{N}(x_t^{(j)}|mathbf{x}_{t-1}^{(j)}, sigma)
% end{align*}
% To normalize:
% $$ sum_{j=1}^N w_t^{(j)} = 1 $$
% where
% $$ w_t^{(j)} = w^{(j)}_t / (sum_{k=1}^N w_t^{(k)}) $$
% The only tuning parameter of this algorithm is $sigma$, which determines how "wiggly" our distribution is around the previous state value.
section{Methods for Detection and Separation}
The standard filtering approach begins by predicting from prior density and update based on observation process. One modification that allows us to better handle the chattering problem occurs in the prediction step. We can modify the type of distribution we draw from during this step based on prior information. The process involves drawing excess particles from a uniform distribution and throwing away some particles based on looking at some cut-off percentile of these excess particles in prediction. We refer to this method as sampling with concentration (SWC).
subsection{Sampling with Concentration}
Let $w_0$ represent the desired probability of keeping any given particle. To propose particles using SWC, we sample $N_0 = frac{N}{w_0}$ from some proposal distribution and weigh these particles proportional to their likelihood. We then thin them down to just $N$ particles by discarding some percent of them based on decile distribution within the likelihoods.
If we choose SWC with $w_0 = 0.1$, then for each particle set we would throw away $90%$ based on likelihoods to ensure that our weights reflected our desired level of concentration on these areas for more accurate estimates for the density.
Note that probabilities for each sub-set of particles must be normalized based on all particles currently considered so that they sum to unity. To do this we calculate new normalized particle weights as
$$ w_t^{prime(j)} = frac{w_t^{(j)}}{sum_{k=1}^N w_t^{(k)}}. $$
This method results in creating more particles than necessary, where we lose information over time by throwing away particles and then resampling them. Perhaps a better method would be to keep track of these discarded particles and add them back in later if they have potential movement that interests us.
subsection{Incorporating Passed Particles}
Passing particles consist of preserving particles from one time-slice to another, but not considering them when computing likelihoods or determining weights (Kastner et al., cite{kastner2013p}). We can modify this method so that we only incorporate passed particles when updating likelihoods.
Suppose we have $N$ total particles at time $t=0$. At each subsequent time step, we predict from $N_p$ passed particles and $N_0$ new particles, such that $N=N_p+N_0$. The algorithm for incorporating passed particles is as follows:
begin{algorithm}
We should choose $N_p propto Var(p(mathbf{x}_{i_t}))$ if we are interested in small outlier detection and $N_p propto Var(q(mathbf{x}_{i_t}))$ if we are interested in larger outlier detection.
begin{algorithmic}
State Predict from past particles:
For{$j=1,...,N_p$}
State $boldsymbol{eta}_t^{(j)} sim q(cdot)$
State $mathbf{x}_t^{(j)} gets B(mathbf{x}_{t-1}^{(j)}, boldsymbol{eta}_t^{(j)})$
State $pi_t^p
