Questions tagged [stochastic-processes]
A stochastic process describes the evolution of random variables/systems over time and/or space and/or any other index set. It has applications in areas such as econometrics, weather, signal processing, etc. Examples: Gaussian process, Markov process, etc.
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How to decide if the inclusion of a variable violates the Markov property?
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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Estimation of many Gaussian process hyper-parameters
I am working with a Gaussian process $(X_t(x))_{x \in [0,1], t \geq 0}$ which evolves jointly in space and time. I know the statistics of this process: $\mathbf{E} X_t(x) = X_0(x) e^{-\mu r t} + (1-e^{...
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Use of differential equations in statistical modelling
Consider a situation where there are multiple subjects and each subject has multiple measurements (response, covariates) over time. The goal is to identify a statistical regression model which allows ...
user499545
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Lifting of non-reversible Markov chains for convergence acceleration
Background and motivation
Let $(\Omega, \pi)$ be a finite state space with a stationary distribution $\pi$. Consider an ergodic Markov chain on $\Omega$ with transition matrix $P$ that is irreducible ...
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Why is time series modeled as collection of random variable?
I'm learning time series analysis for forecasting. As I’ve learned, a time series is defined as a collection of random variables ${X_1, X_2, ..., X_T}$, and a single observation is said to be one ...
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Distribution of the single random variables in a stationary sequence
I have a doubt about the distribution of each single random variables in a stationary sequence. In particular, I found the following definition for stationary sequence:
A sequence of random variables $...
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In a stationary time series what is the meaning of "stationary"? [duplicate]
So Here is the defintion of Stationary Time Series:
A Stationary time series is a time series whose statistical properties do not change over time. My first question is what is over time actually mean?...
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Likelihood of the data given drift and random noise terms
I am interested in finding the likelihood for the location of a given number of particles at time = 1 in a process that resemble (or is) a Ornstein-Uhlenbeck (OU) process.
In particular, I am ...
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Proving Convergence of Mean and Variance in a Recursive Gaussian Update Process
I'm researching the statistical convergence properties of a recursive system that arises during the training of custom neural network structure.
My specific question is: How can I prove convergence of ...
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How to do a Bayesian update in extended probability spaces?
I was reading about extended probability spaces recently. And I was wondering how would one update a probability distribution from the original space to the extended space.
Specifically, an extension ...
Aditya Shinde
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Suggestions constrained optimization with noisy observations
For $N$ correlated Ornstein-Uhlenbeck processes, I want to find $N$ absorption boundaries, $\mathbf{A}\in\mathbb{R}^{N}$, such that expected value of the summed $N$ processes is maximized, while the ...
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Reverse absorption probability of Markov chain
Let $X_0, X_1, ..., X_n$ be a Markov chain with state spaces $S$ and transition probability matrix $T = \{p_{i,j}\}$. There are $A$ absorbing states.
I'm interested in $P({X_{t-1}=j}|X_t=i)$.
By Bayes'...
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How can I create a classification ML model that adapts to changing patterns over time?
I’m working on a machine learning model to classify transactions as either fraudulent or genuine. However, one major challenge I’m facing is that the pattern of fraudulent transactions changes ...
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Comparison of number of visits to transient states in Markov chain
Let $X_0, X_1, ..., X_n$ be a Markov chain with state spaces $S$, initial probability distribution $\pi$ and transition probability matrix $P = \{p_{i,j}\}$.
The first passage time from state $i$ to $...
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Understanding the Probability Space and Random Variable in Denoising Diffusion Models
I'm trying to solidify my foundational understanding of denoising diffusion models (DDMs) from a probability theory perspective. My high-level understanding of the setup is as follows:
We assume ...
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Asymptotic distribution of kernel density estimator as a process over $\mathbb{R}$
Consider the task of estimating a density function $f : \mathbb{R} \to \mathbb{R}$ from independent samples $X_1, \ldots, X_n \sim f$.
Let $f_n$ be the kernel density estimator of $f$, that is,
$$
f_n ...
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How does the Gaussian transition rule lead to an explicit equation for $X_t$?
Let $(X_t)_{t \ge 0}$ be a discrete-time, continuous-state Markov chain in $\mathbb{R}^n$, with $X_0$ following some unknown distribution. We're told that the conditional PDF of $X_t$ given $X_{t-1}$ ...
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Limiting distribution of CTMCs
For a continuous-time Markov chain with state space $S$ and with an irreducible positive recurrent embedded chain, it can be proved that the transition probabilities $p_{xy}(t)$ for any pair of states ...
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An example of stochastic process $X_t$ on $[0,1]$ such that $\mathbb{E}[\inf_{[0,1]} \left|f(t)\right| X_t] = 0$
What's an example of stochastic process $X_t$ on $[0,1]$(i.e. a random variable for each $t \in [0,1]$) such that
For all $t$, $X_t \geq 0$
For all $t$, $\mathbb{E}[X_t] = 1$
$\mathbb{E}[\sup_{t \in [...
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What is the sample space $\Omega$ of a diffusion model? [duplicate]
In denoising diffusion probabilistic models (original paper), the forward process is a Markov chain $ (X_t)_{t \in \mathbb{N}} $, where each $ X_t \in \mathbb{R}^n $. What is the underlying ...
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Visualisation of Stochastic Process Flow
How to make visualisation of stochastic process defined as a set of discrete transition matrices at time steps? Example - stock log returns probabilities in time.
As a discrete process described as a ...
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Inferring the parameters of a Ornstein-Uhlenbeck process given realisations of it cannot be observed directly
I have a dataset of (noisy) test results. You can think of this as being accuracy that a chess player achieves in various games or number of points a basketballer scores in a game.
I think a good ...
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Joint modeling of asymmetric interactions in a Gibbs spatial point process?
I am exploring ways to model asymmetric spatial interactions within a Gibbs process framework. I understand that hierarchical Gibbs models allow for dependencies to be specified a priori, where one ...
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Deterministic vs. Stochastic effects of a multivariate Ornstein-Uhelnbeck process
I'd like to characterize a multivariate Ornstein-Uhlenbeck process $$dX_t = A(X_t - \mu)dt + \Sigma dW_t$$ in terms of its deterministic vs. stochastic forces. In other words, I need to measure how ...
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Maximum Likelihood Estimation for changing parameters
I have data about a $M$/$M$/$K$ queue. At each time interval, I know the number of arrivals, number of departures and the number of servers that were there.
For a standard $M$/$M$/$K$ queue, ...
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Stochastic process based on mixed effects regression
Suppose I have $n$ items.
Each item has some baseline properties that are constant. For example, the $i^{th}$ item has properties (i.e. covariates) $w_i$ and $z_i$ that are constant. Each item has a ...
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Can a Poisson process be made into a Poisson process regression?
Here is this proof of concept problem I am thinking of.
The data is in the format like this (simulated data):
...
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Identifying the distribution of a random process associated to the square root process
Let $x(t)$ be a random square-root process that follows
$$dx(t) = (a + bx(t)) \, dt + c\sqrt{x(t)} \, dW(t)$$
where $W(t)$ is a standard Brownian motion for some filtration.
This has many names ...
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Resources for computational approaches to stochastic processes
Broad, education/resource question. I’d like to know more about stochastic processes and in particular discrete processes.
For example the binomial pricing model converges on geometric Brownian motion ...
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Mean, variance and convergence of stochastic processes
Suppose I have some random process, where I initialize two empty arrays A:[], B:[]. Following, elements sampled from $N(0,1)$ are allocated to $A$ or $B$, Every time, an element is sampled, it is ...
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Bounded in probability with the deterministic constant expression
I study the paper in the link below:
https://projecteuclid.org/journals/bernoulli/volume-24/issue-2/Smooth-backfitting-for-additive-modeling-with-small-errors-in-variables/10.3150/16-BEJ898.full
The ...
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Clarifications on Hurst Exponent Definitions and Persistence Properties
I have a question regarding the Hurst exponent that I hope someone here can help clarify.
It is well known that there are different definitions of the Hurst exponent, but finding clear connections or ...
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Why Can't We Directly Sample Realizations from a Wiener Process Using Its Joint Gaussian Distribution?
I am trying to understand the relationship between sampling from the joint distribution of a Wiener process and the process's defining properties. Here's what I know:
The Wiener process $W_t$ is ...
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How can I filter outliers in data that is manually recorded?
Different people have to write down values on a certain type of parameter in order to fill out a table, and people obviously tend to write wrong. Sometimes, by a factor of 1000. This creates a lot of ...
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Calculate the mean and variance of a stochastic process?
I was introduced to the Polya Urn problem in statistics (a problem where we draw a ball from an urn and place another ball back of the same color). These are the formulas for the mean and variance of ...
user448292
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Power of unit root tests
I am testing the power of basic unit root tests: ADF, PP and DFGLS.
In particular I am generating this sequence:
$$y_t = \text{drift} + (\text{trend_coeff} \cdot t) + (\phi \cdot y_{t-1}) + e_t$$
For ...
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Creating a non-homogenous queue based on another statistical model?
I have a question on how construct/approximate a statistical queue based on changing parameters.
I created/simulated this approximate version of a MMK queue in R:
...
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What Makes Phase Type Distributions Interesting
I am studying phase type distributions in my introductory stochastic processes course. I understand what they are and how to calculate with them.
What I don't understand is, what makes them special? ...
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How to make two perfectly negatively correlated growing Geometric Brownian Motion (GBM) series? (Impossibility)
Intro
I am self studying in Youtube the course MIT 18.S096 Topics in Mathematics w Applications in Finances and in the following lecture min 34:50 by Dr. Jake Xia is studied the efficient frontier of ...
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Book/Course for undergraduates to learn stochastic processes
Currently, I'm reading about Denoising diffusion probabilistic models. The first time I heard about the idea of DDPM, I was thinking that "It's not possible! At least when implement using ...
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Expected time of activation waves in 1-dimension
Suppose we have $n$ sites, and assume each site activates at a rate $f_i$ ($1\leq i \leq n$), after which it triggers an activation wave that propagates in both direction at a rate $v$, activating ...
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Is an integrated exponential alternating renewal process a Markov chain?
Consider an alternating renewal process where the alternating extents are independent exponentially distributed with means, mu_0 and mu_1. The alternating extents determine whether the outcome at any ...
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Do any time series models actually assume strict stationarity?
I am taking a Time Series course, and we have covered the topic of Stationarity. The following definitions are given:
Strict Stationarity:
A stochastic process $\{X_t\}_{t \in T}$ is said to be ...
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Confusion about neural network in stochastic control problem
I am a neural network newbie. I would like to attempt to implement the following architecture deep learning a stochastic control problem, taken from this paper. Here $s_t$, $a_t$, $c_t$ and $\xi_t$ ...
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Underlying sigma-algebra of a random process
random process is usually defined as a collection $(X_i)_{i\in \mathcal{I}}$, where $\mathcal{I}$ is some index set and each $X_i$ is a random variable $X_i:(\Omega,\mathcal{A})\rightarrow(\mathcal{X},...
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Time scaling of AR(1) process for modelling financial returns
Process:
Consider an AR(1) process with zero mean,*
$\lambda_t = \kappa \cdot \lambda_{t-1} + \omega_t$,
with $\kappa = 0.9$, $\omega \sim N(0, \sigma_{\omega}^2)$, and $\sigma_{\omega}^2 = 0.00027$. ...
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Lognormal distribution and Galton Watson process
I have been studying a Galton Watson process that creates random binary trees with probability of survival $p_{s}$ and offspring distribution $p(k)=p_{s}\delta(k-2)+(1-p_{s})\delta(k)$. I'm ...
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Probabilistic Bounds on the Frequency of Empirical Mean Threshold Crossings for Sub-Gaussian or Bernoulli Variables
Consider one $1-$sub-gaussian or Bernoulli variables $X$, we i.i.d. sample $X$ $n$ times $(X_1,...,X_n)$; and $\mu_X<S$, $S$ is a constant threshold. We have such a quantity: $N_X = \sum_{i=1}^n \...
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What are the best uninformative priors for parameters on the log-odds scale in a stochastic model?
I am relatively new to statistical modeling, and I apologize if this question is somewhat basic. I am currently working with Bayesian models, specifically a stochastic model based on an Ornstein-...
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Limit of Brownian Motion for Infinite Particles
Consider the case where there are infinite particles initially separated by given distance $d$ on a line exhibiting Brownian Motion without collision. After a significant period of time, I would ...
