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My question assumes the following set up for continuous time discrete space: Consider $n$ individuals indexed by $i = 1, \ldots, n$. For each individual $i$: $Y_i(t) \in \{1, 2, \ldots, k\}$ denotes ...
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Model Consider a mechanistic / quasi-compartmental model for binary variables $d:$ "depression" and $h:$ "hazardous drinking" (Figure). Individuals enter the model not depressed or ...
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In many books about Bayesian inference based on Gaussian process, it is assumed that one can only observe a set of data/signals at discrete points. This is a very realistic assumption. However, in ...
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So, I am doing a meta-analysis of odds ratios to find out the risk factors for a certain medical outcome. One of those risk factors would be (simplified) the time spent in the operation room. My ...
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Lets say we want to predict a single target variable and we have 10 regressors/features. Assume we would like to predict 30 days ahead (daily predictions up to 30 days ahead) and our data is a daily ...
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Recently, I've been questioning myself on the possibility of combining a discrete update and a continuous update on a single MCMC. Stephens (2000) in Algorithm 3.2 runs the process for a fixed amount ...
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if a data set is set up such that it has 5 independent variables and an array of 201 elements of "smooth continuous" data as the dependent variable. I was wondering if there was a model ...
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In Econometrics, majority of the models are in discrete-time setting, whereas when you move on to quantitative finance, continuous-time models are most prevalent. I get the theory and idea behind ...
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Lets say I simulate a immigration-death process: $P(X(t + \delta t) = x+ 1 | X(t) = x) = \lambda \delta t$ $P(X(t + \delta t) = x-1 | X(t) =x) = v x \delta t$ using a Gillespie simulation - I pick a ...
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It is often said that a AR(1) process can be viewed as a discretized version of the continuous-time Ornstein-Uhlenbeck process. Can we really claim this to be valid considering that the Ornstein-...
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How can I handle discrete events in a continuous time stream in the context of an F1 metric? To give an example, let's say the Earthquake Forecasting Bureau would report the following for their ...
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Let's say we have a two dimensional MC defined on the state space $\mathbb{N}\times \mathbb{N}$ evolving as below: $(i,j) \rightarrow (i,j+1)$ with rate $\lambda$ for all $i,j$. $(i,j) \rightarrow (i-...
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First I find the stationary distribution of this problem by solving $\pi G=0$ to get $\pi \approx (0.20675105, 0.29535865, 0.05907173, 0.23628692, 0.1350211, 0.06751055)$. Using $\pi$, we can ...
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If a discrete-time autogressive AR(p) model is fit to data x at t=1,2,..., what is the probability distributiom of x at time n+h, denoted x(n+h), where 0 < h < 1 and x(n) and x(n+1) are known?
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I am considering an example where a person flips his (unfair) coin to examine what is the probability of getting head. I could find some posts saying that the posterior distribution follows Beta ...
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Most classical epidemic models such as SIR and variants are formulated as differential equations. However, to me discrete-time models feel more natural to measure the evolution of a disease on a day-...
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Consider a stationary Continuous-time AutoRegressive (CAR) process on a bounded time-interval $(a, \, b)$. This article by Emmanuel Parzen describes the corresponding Reproducing Kernel Hilbert Space (...
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I have been studying continuous time markov chains through Dobrow's book. Everything went fine until the author introduced the concept of infinitesimal generator, which he refers to as $\textbf{Q}$. ...
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In this question, I asked about validating the assumption of geometric Brownian motion in a analytic model using ARIMA. Here, I want to generalise this idea. If I'm building a decision model that ...
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I have read that Brownian motion, or more precisely, a Wiener process, is a scaling limit of a random walk. Hence, when attempting to model a real time-series of energy prices, if I discover that an $...
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Is there a way to calculate the riskiest places to be infected by COVID-19? My friends and I are having an argument of whether being in a "high traffic-short contact time" situation (public transport) ...
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I'm trying to model data with a time-homogenous CTMC with a number of states with corresponding constant transition rates $\lambda_{i}$ when I notice that much of the transition times from one state ...
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I am interested in an overview over the connection and correspondence between time series models in continuous vs. discrete time in finance. E.g. take ARMA(p,q) or GARCH(s,r) or ARMA(p,q)-GARCH(s,r) ...
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I have learned that the Fourier transform of a continuous-time unit-periodic stochastic process is: $$x(t) = \sum\limits_{k=-\infty}^{\infty} a_k e^{i2\pi kt} \quad \quad \text{ where } \quad \quad ...
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Let X be a Cox process (doubly-stochastic Poisson process) driven by a Poisson process with fixed intensity(rate) $\lambda=50$ , and choose some small time interval $dt=0.01$ . Is the proper way to ...
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Consider a continuous-time stochastic process $y(t)$ having the following linear (Gaussian) state-space representation for $t \geq 0$ $$ \left\{ \begin{array}{c c l} \text{d}{\boldsymbol{\...
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I have an irreducible continuous-time Markov chain (CTMC) with a finite state space. The CTMC also does not have any one-step transitions from any state to itself. I have the transition rate matrix $Q$...
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The negative binomial distribution with parameters $p\in(0,1)$ and $t>0$ is sometimes defined as the distribution of the number of failures before the $t$th success. This is supported on the set $\{...
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Given a logistic function of the form. \begin{align*} f(t) = \frac{\alpha}{1 + \beta e^{\gamma t}} \end{align*} Harvey (1984) differentiates this and takes logs to yield: \begin{align*} \ln f' = 2 ...
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I am analysing a dataset from a randomised controlled trial (2 treatment groups) with measurements at 3 time points (weeks 0, 1 and 8). I am struggling with whether to analyse this with the three time ...
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I have two continuous time series of discrete events. The data from them are timestamps of occurences, e.g. ...
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