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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What does a "closed-form solution" mean?
I have come across the term "closed-form solution" quite often. What does a closed-form solution mean? How does one determine if a close-form solution exists for a given problem? Searching online, I ...
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Is a time series the same as a stochastic process?
A stochastic process is a process that evolves over time, so is it really a fancier way of saying "time series"?
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Why does the variance of the Random walk increase?
The random walk that is defined as $Y_{t} = Y_{t-1} + e_t$, where $e_t$ is white noise. Denotes that the current position is the sum of the previous position + an unpredicted term.
You can prove that ...
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The "Amazing Hidden Power" of Random Search?
I have the following question that compares random search optimization with gradient descent optimization:
Based on the (amazing) answer provided over here Optimization when Cost Function Slow to ...
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Time taken to hit a pattern of heads and tails in a series of coin-tosses
Inspired by Peter Donnelly's talk at TED, in which he discusses how long it would take for a certain pattern to appear in a series of coin tosses, I created the following script in R. Given two ...
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Intuitive explanation for periodicity in Markov chains
Can someone explain me in a intuitive way what the periodicity of a Markov chain is?
It is defined as follows:
For all states $i$ in $S$
$d_i$=gcd$\{n \in \mathbb{N} | p_{ii}^{(n)} > 0\} =1$
...
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What are examples of statistical experiments that allow the calculation of the golden ratio?
There are some very simple experiences that can be done by a kid at home, whose result allows one to statistically approach famous numbers such as $\pi$ or $e$.
An example where $\pi$ shows up is ...
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How do you see a Markov chain is irreducible?
I have some trouble understanding the Markov chain property irreducible.
Irreducible is said to mean that the stochastic process can "go from any state to any state".
But what defines whether it can ...
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What are the main differences between Granger's and Pearl's causality frameworks?
Recently, I ran across several papers and online resources that mention Granger causality. Brief browsing through the corresponding Wikipedia article left me with the impression that this term refers ...
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How likely am I to be descended from a particular person born in the year 1300?
In other words, based on the following, what is p?
In order to make this a math problem rather than anthropology or social science, and to simplify the problem, assume that mates are selected with ...
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How do betting houses determine betting odds for sports?
Let's take football (soccer) for example. There are 3 possible outcomes, home win, draw, away win. I took a random game from bet365
...
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How to create a multivariate Brownian Bridge?
It is known, that a standard multivariate Brownian bridge $ y(\mathbf u) $ is a centered Gaussian process with covariance function
$$ \mathbb E(y(\mathbf u) y(\mathbf v)) = \prod_{j=1}^d (u_j \wedge ...
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Wavelet-domain Gaussian processes: what is the covariance?
I've been reading Maraun et al, "Nonstationary Gaussian processes in wavelet domain: Synthesis, estimation, and significant testing" (2007) which defines a class of non-stationary GPs that ...
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What are some techniques for sampling two correlated random variables?
What are some techniques for sampling two correlated random variables:
if their probability
distributions are parameterized
(e.g., log-normal)
if they have non-parametric
distributions.
The data are ...
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Derivative of a Gaussian Process
I believe that the derivative of a Gaussian process (GP) is a another GP, and so I would like to know if there are closed form equations for the prediction equations of the derivative of a GP? In ...
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When can a Gaussian Process solve an SDE?
Considering an SDE of the form
$$dX_t = \mu(X_t, t)dt + \sigma(X_t, t)dW_t ,$$
where $W_t$ is a Wiener process, is there a set of necessary and sufficient conditions on the structure of the functions $...
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Whether a AR(P) process is stationary or not?
In practice, how to evaluate whether a AR(P) process is stationary or not?
How to determine the order for the AR and MA model?
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stochastic vs deterministic trend/seasonality in time series forecasting
I have moderate background in time series forecasting. I have looked at several forecasting books, and I don't see the following questions addressed in any of them.
I have two questions:
How would I ...
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What is the difference between Markov chains and Markov processes?
What is the difference between Markov chains and Markov processes?
I'm reading conflicting information: sometimes the definition is based on whether the state space is discrete or continuous, and ...
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How will studying "stochastic processes" help me as a statistician?
I wish to decide if I should take a course called "INTRODUCTION TO STOCHASTIC PROCESSES" which will be held next semester in my University.
I asked the lecturer how studying such a course would help ...
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Why does the supremum of the Brownian bridge have the Kolmogorov–Smirnov distribution?
The Kolmogorov–Smirnov distribution is known from the Kolmogorov–Smirnov test. However, it is also the distribution of the supremum of the Brownian bridge.
Since this is far from obvious (to me), I ...
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Finding the MLE for a univariate exponential Hawkes process
The univariate exponential Hawkes process is a self-exciting point process with an event arrival rate of:
$ \lambda(t) = \mu + \sum\limits_{t_i<t}{\alpha e^{-\beta(t-t_i)}}$
where $ t_1,..t_n $ ...
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Random walk with momentum
Consider an integer random walk starting at 0 with the following conditions:
The first step is plus or minus 1, with equal probability.
Every future step is: 60% likely to be in the same direction ...
user1566
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What does it mean to say that an event "happens eventually"?
Consider a 1 dimensional random walk on the integers $\mathbb{Z}$ with initial state $x\in\mathbb{Z}$:
\begin{equation}
S_n=x+\sum^n_{i=1}\xi_i
\end{equation}
where the increments $\xi_i$ are I.I.D ...
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Closed form expression for the quantiles of $\alpha_1\sin(x)+\alpha_2\cos(x)$
I have two random variables, $\alpha_i\sim \text{iid }U(0,1),\;\;i=1,2$ where $U(0,1)$ is the
uniform 0-1 distribution.
Then, these yield a process, say:
$$P(x)=\alpha_1\sin(x)+\alpha_2\cos(x), \;\...
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What is the difference between a distribution and a process (Poisson)?
I'm doing my PhD in geomechanics. I thought we use a Poisson-Weibull distribution (for the variability of a parameter at the rock), but reading more about the subject I think maybe is a Poisson-...
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What are the differences between stochastic and fixed regressors in linear regression model?
If we have stochastic regressors, we are drawing random pairs $(y_i,\vec{x}_i)$ for a bunch of $i$, the so-called random sample, from a fixed but unknown probabilistic distribution $(y,\vec{x})$. ...
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Numeric solvers for stochastic differential equations in R: are there any?
I'm looking for a general, clean and fast (i.e. using C++ routines) R package for simulating paths from a non-homogeneous nonlinear diffusion like (1) using the Euler-Maruyama scheme, the Milstein ...
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A question related to Borel-Cantelli Lemma
Note:
Borel-Cantelli Lemma says that
$$\sum_{n=1}^\infty P(A_n) \lt \infty \Rightarrow P(\lim\sup A_n)=0$$
$$\sum_{n=1}^\infty P(A_n) =\infty \textrm{ and } A_n\textrm{'s ...
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Is stationarity preserved under a linear combination?
Imagine we have two time-series processes, which are stationary, producing: $x_t,y_t$.
Is $z_t=\alpha x_t +\beta y_t$, $\forall \alpha, \beta \in \mathbb{R}$ also stationary?
Any help would be ...
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What is the difference between deterministic and stochastic model?
Simple Linear Model:
$x=\alpha t + \epsilon_t$ where $\epsilon_t$ ~iid $N(0,\sigma^2)$
with $E(x) = \alpha t$ and $Var(x)=\sigma^2$
AR(1):
$X_t =\alpha X_{t-1} + \epsilon_t$ where $\epsilon_t$ ~...
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Ergodicity explained in layman terms
I've been told that Ergodicity gives us a practical vision of processes WSS (Wide-sense stationary) and a bunch of integrals. For me, it is not enough to fully understand it.
Could someone explain me ...
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Exploratory analysis of spatio-temporal forecast errors
The data: I have worked recently on analysing the stochastic properties of a spatio-temporal field of wind power production forecast errors. Formally, it can be said to be a process $$ \left (\epsilon^...
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Intuitive understanding covariance, cross-covariance, auto-/cross-correlation and power spectrum density
I'm currently studying for my finals in basic statistics for my ECE bachelor.
While I think I have the math mostly down, I lack intuitive understanding what the numbers actually mean.(Preamble: I'll ...
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Plain English explanation of Ito's integral?
I'm looking for a plain English explanation of Ito's integral. I don't need an exhaustive proof, derivation, etc. Just a simple ~this is effectively what it does and why it's better than a Riemann sum ...
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Is anything inherently random?
Is anything inherently random? Or is all randomness observed in data either "errors in measurement" or "lack of understanding"? Assume we could measure everything with infinite ...
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Why is an unbiased random walk non-ergodic?
Wikipedia says "An unbiased random walk is non-ergodic."
Let's look at a simple random walk. It's defined as: take independent random variables $Z_{1},Z_{2}$, where each variable is either $1$ or $−1,...
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Prove/Disprove probability of 0 or 1 (almost surely) will never change and has never been different
Prove/Disprove $E[1_A | \mathscr{F_t}] = 0 \ \text{or} \ 1 \ \text{a.s.} \ \Rightarrow E[1_A | \mathscr{F_{s}}] = E[1_A | \mathscr{F_t}] \ \text{a.s.}$
Given a filtered probability space $(\Omega, \...
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Special probability distribution
If $p(x)$ is a probability distribution with non-zero values on $[0,+\infty)$, for what type(s) of $p(x)$ does there exist a constant $c\gt 0$ such that $\int_0^{\infty}p(x)\log{\frac{ p(x)}{(1+\...
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Optimization of stochastic computer models
This is a tough topic for myself to google since having the words optimization and stochastic in a search almost automatically defaults to searches for stochastic optimization. But what I really want ...
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Expected number of coin tosses to get N consecutive, given M consecutive
Interviewstreet had their second CodeSprint in January that included the question below. The programmatic answer is posted but doesn't include a statistical explanation.
(You can see the original ...
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If a time series is second order stationary, does this imply it is strictly stationary?
A process $X_t$ is strictly stationary if the joint distribution of $X_{t_1},X_{t_2},...,X_{t_m}$is the same as the joint distribution of $X_{t_1+k},X_{t_2+k},...,X_{t_m+k}$ for all $m$, for all $k$ ...
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Example of a process that is 2nd order stationary but not strictly stationary
Does anybody have a nice example of a stochastic process that is 2nd-order stationary, but is not strictly stationary?
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The Fishing Problem
Suppose you want to go fishing at the nearby lake from 8AM-8PM. Due to overfishing, a law has been instated that says you may only catch one fish per day. When you catch a fish, you can choose to ...
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How to test if "previous state" has influence on "subsequent state" in R
Imagine a situation:
We have historical records (20 years) of three mines. Does the presence of silver increases the probability of finding gold in next year? How to test such question?
Here is ...
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Will there ever be an unhappy Tribble in Oz?
Here's an amusing problem brought to me by a student. Although it was originally phrased in terms of mutually annihilating bullets fired at regular intervals by a gun, I thought you might enjoy a ...
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What's the name for a time series with constant mean?
Consider a random process $\{X_t\}$ for which the mean $\mathbb{E}(X_t)$ exists, and is constant, for all times $t$, i.e. $\mathbb{E}(X_t)=\mathbb{E}(X_{t+\tau})$ for all times $t$ and time shifts (or ...
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Modifying linear ballistic accumulator (LBA) simulation in R
The "Linear Ballistic Accumulator" model (LBA) is a rather successful model for human behaviour in speeded simple decision tasks. Donkin et al (2009, PDF) provide code that permits estimating the ...
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How can I show that a random walk is not covariance stationary?
How can I show that a random walk ($y$ follows a random walk) is not covariance stationary? I tried to work on the formula below (with no results) Could you give me just a hint on how to proceed ...
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Dealing with different definitions of the Ornstein-Uhlenbeck process
I've run up against a wall in reconciling two different definitions of the Ornstein-Uhlenbeck process, and would appreciate some help.
On the one hand, as discussed here, we can define an Ornstein-...
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