The stochastic-processes tag has no wiki summary.
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Importance sampling of finite path of stochastic difference equation
Before passing to question, let me briefly recap what's importance sampling of random variables is about. Suppose $\xi$ is a real-valued random variable with density $f$, and let $g:\Bbb R\to \Bbb R$ ...
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26 views
Gillespie Stochastic Simulation in Discrete Time using R [migrated]
I'm simulating a Stochastic Simulation for Epidemiology. How do I simulate it in a discrete time? I managed to obtain for continuous time using the coding below.
...
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16 views
How to approximate a Stochastic volatility process with Markov Chain
It is easy to use a Markov Chain to approximate an AR(1) process --- Tauchen (1986), Tauchen and Hussey (1991).
For a simple stochastic volatility process (discrete), which in it's very basic form is ...
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0answers
27 views
Learning parameters of non-parametric Bayesian models
I have a sample of Chinese restaurant process which I want to model as Pitman–Yor process. How do I determine parameters of Pitman-Yor model from given sample?
For Dirichlet process I would just use ...
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2answers
54 views
Are all Levy processes memoryless?
We know that the two canonical Levy processes, namely the Wiener process and Poisson process, are both memoryless, so I wonder if there are any Levy process that is not memoryless. Specifically, are ...
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1answer
39 views
Ripley's K Function and L Function for Point Patterns
The following is a spatial point pattern:
and these are the corresponding Ripley's K function and L function for this data:
How are these functions interpreted?
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1answer
42 views
Tests for spatial stationarity (homogeneity)
There are many models for spatial point patterns and spatial marked point patterns that assume spatial homogeneity or stationarity.
i) Is there a statistical test for determining this, where the ...
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1answer
57 views
Degrees of freedom for Gaussian Process
I am reading this paper on Generalised Wishart Process (GWP). It is about modelling covariance matrix of D - dimensional gaussian processes (GP) as GWP. I fail to understand interpretation of "degrees ...
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33 views
Closed form Karhunen-Loeve/PCA expansion for gaussian/squared-exponential covariance
The Gaussian, or squared exponential covariance is $k_{SE}(s,t) = \exp \left\{ -\frac{1}{2l} (s - t)^2 \right\}$. It is a common covariance function used in Gaussian processes. The Karhunen-Loeve ...
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15 views
Find the distribution of the increments between two consecutive steps of the walk from a Langevin equation?
Given a Langevin equation of a stochastic process: $X_{I+1}=X_I-F(X_I)+W_I$ - where $F(X_I)$ is a position dependent force, and $W_I$ is the Wiener process term (i.e., Gaussian / white-noise).
How do ...
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44 views
Question about infinte Markov Chains
Do 2 Markov chains $\left\{X_n\right\}^\inf_{n=0} $ and $\left\{Y_n\right\}^\inf_{n=0} $ with all of the following properties exist so that the probability for infinite n values to maintain ...
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20 views
Interpreting variance, conditional variance and variance of residuals in a stationary time series
I'm trying to properly understand variance (in its various guises) in a stationary process.
As I understand the constant mean and variance is the mean and variance of the unconditional (or marginal) ...
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1answer
85 views
what is meant by stochastic drift?
could anybody define what is meant by stochastic drift?
I think I have a rough idea, say on a random walk X(n)= a + X(n-1) + Z(n) where Z(n) are iid zero mean and constant variance, then E[X(n)] = ...
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39 views
Renewal Process Hypothesis Test
I have $n$ realisations $s_1,\, \dots , s_n$ of random variables $S_1,\, \dots, S_n$ which are assumed to be i.i.d. with unknown distribution. These measure the time between events.
I want to ...
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0answers
24 views
Test for martingale
Given a time series, I am trying to determine whether the underlying stochastic process is a martingale or not. Until now, I have deduced that since I know that the process is always bounded after a ...
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1answer
111 views
Using a Markov Chain to find the limiting probability?
Let's say a website makes available only one of three online quizzes A, B and C, daily.
If the majority of visitors pass the quiz then the next day the website will randomly publish either quiz A, B, ...
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56 views
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1answer
51 views
Name for a function mapping time slices to probabilities
Is there a commonly used name for a function that maps time slices in a day to probability of a specific event happening in that slice? I was using probability distribution but I guess since the total ...
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0answers
50 views
What is meant by a “stochastic constant”?
I've seen it in a few pieces of econometric literature, and googling it turns up lots of papers using it, almost always in reference to state-space models and other dynamic linear regressions.
No ...
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4answers
332 views
Is winning a soccer match independent of previous wins\losses?
$\quad$ I have a friend of mine who is a bit of a gambler ask me this question. He is of poor mathematical background, but has a sense of logic and will probably accept a logical answer in the natural ...
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1answer
55 views
description of a Wiener Process assuming a Laplace Distribution
Is there a description of the Wiener Process when a Laplace distribution is assumed rather than a normal one?
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26 views
How to estimate a stochastic time delay
Say I have irregular samples of two random processes that are both based on an unobserved process, $Z(t)$. Their SDEs are
$\mathrm{d}X(t) = \mathrm{d}Z(t-u(t)) + \epsilon(t)$
$\mathrm{d}Y(t) = ...
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1answer
177 views
Intuitive understanding covariance, cross-covariance, auto-/cross-correliation 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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38 views
Ergodicity, mixing and stationarity
When considering sequences, I know that mixing and stationarity implies ergodicity. This is a stationary mixing process is ergodic. But, if I have a stationary and non-ergodic sequence, can I conclude ...
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1answer
115 views
Estimate the second moment of a latent variable using a conditionally unbiased proxy
The Setup: Let $X_t$ denote an unobservable stochastic sequence where the first two unconditional moments are finite constants; ie $\mathbb{E} X_t = \mu < \infty$ and $\mathbb{E} X_t^2 = \gamma ...
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General definition of stochastic processes
I'm trying to understand the basic concept of random processes.I already understood that a continuous-time random process is defined by X(¥,t) where ¥ is each element in the Sample Space and t is the ...
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1answer
161 views
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$\{n \in \mathbb{N} | p_{ii}^{(n)} > 0\} =1$
...
3
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1answer
97 views
Stationary matrix given a transition matrix
I am given the following transition matrix
$$P= \pmatrix{ 1-\alpha & \alpha \\ \beta & 1-\beta}, \ \alpha,\beta \in (0,1)$$
with the states $S=\{1,2\}$.
I want to determine the stationary ...
3
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1answer
101 views
Proof of theorem on recurrent states and its equivalence class
A theorem states the following:
Theorem
if $i \in S$ is a state which is recurrent, then every state in the equivalence class of $i$ $(\ K(i) \ )$ is recurrent.
Additional information on the ...
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2answers
149 views
Values for integral of square of standard Brownian process
I am trying to generate values in a table for the following function:
$$
W = \int_0^1 [B(t)]^2 dt
$$
Where $B(t)$ is a standard Brownian motion.
Example: $W_{0.05} = 1.656$, $W_{0.025} = 2.135$.
...
2
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1answer
93 views
Recurrence definition for a Markov chain
We define a state i to be recurrent if $\sum\limits_{n=0}^\infty P(X_n=i,X_k \neq i$ for $1\leq k < n | X_0=i)$=1.
Why do we take infinite series over the probability? Why don't we define ...
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24 views
Ito equation generalization?
Ito's equations
$$dx = a(x,t) dt + b(x,t) \delta W$$
describes processes for which
$$ \frac{<x-x_0>}{\Delta t} = a(x_0, t_0)\quad \Delta t = t-t_0 \to 0$$
$$ \frac{<(x-x_0)^2>}{\Delta ...
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87 views
Test for independent but not identically distributed time-series?
So far, runs test or bds test are doing OK with i.i.d data but not so for independent but not identically distributed data set based on my practical experiences with them. Are there any other tests ...
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0answers
56 views
Distribution/expected length of the shortest path in infinite random geometric graphs
Consider an infinite random geometric graph $G(\rho,d)$ in which vertices are uniformly and independently scattered over the 2D plane with density $\rho$ and edges connect the vertices that are closer ...
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1answer
126 views
Density of robots doing random walk in an infinite random geometric graph
Consider an infinite random geometric graph in which the node locations follow a Poisson point process with density $\rho$ and edges are placed between the nodes that are closer than $d$. Therefore, ...
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Can we reconstruct the hidden (latent) variables after executing EM?
The question is in the title. I know that EM algorithm could do maximum likelihood estimation for models that have latent variables. I would like to know can we get the (estimated) value of these ...
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2answers
203 views
Random walk with drift are differences white noise?
If I have a random walk without drift the differences form a white noise process.
But what happens if I incorporate a drift $d$? Does this still hold true? I'm not sure because with the drift term d:
...
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112 views
Expected Value of Integral of Stochastic Process
Suppose we have a continuous-time stochastic process $X(t)$, which consists of a sequence of delta functions that, at each time $t$, have a probability $p(t)$ of taking a non-zero value. $p(t)$ lies ...
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20 views
the approach for checking whether a process is stable?
I have two scenarios for time series data.
1) I have a uni-variate variable spanning across the time axis, are there any approaches or statistic to check whether this process is stable?
2) I have a ...
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74 views
How does integrating the Kolmogorov forward equation give $P = \exp (Qt)$?
If $Q$ is a generator matrix of a continuous time Markov chain (CTMC), and I need to use this matrix to solve the Kolmogorov forward equation, I would need to start by integrating it. But I haven't ...
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44 views
Comparing two different leagues of similar but not equal distributions around a standard deviation of error of a prediction from a rating system
This query ties a lot of my interests in rating sports teams together, because as I’ve mentioned before I do a version of the Kenneth Massey method (as per his 1997 thesis ...
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227 views
is there any package / simulator for discrete time markov chain simulation
is there any package or simulator for discrete time markov chain simulation.
Matlab/ Python solutions ?
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1answer
590 views
Solving the Kolmogorov forward equation for transition probabilities
Let $\lambda \mu > 0$ and let $X$ be a Markov chain on $\{1,2\}$ with generators
$$
Q = \begin{pmatrix} -\mu & \mu \\ \lambda & -\lambda \end{pmatrix}$$
Write down the forward equations ...
2
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1answer
136 views
How can I measure the effects certain events have on the frequency of other events over time?
EDIT: added more details following @kjetil comments
I have the following problem:
I monitor one stream of events of type A - those events can be considered instantaneous.
I also monitor additional ...
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1answer
68 views
What is the distribution that can properly describe the PE fluctuation of a stock
I have observed the historical PE (price / profit) value of a stock and realized that it roughly follows a log normal distribution. However, even when the next earning data point is easily ...
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2answers
305 views
Calculating probabilites of an nth step transition matrix for discrete time markov chains
"Let $\{X_n, n \geq 0\}$ be a DTMC with state space $S = \{1, 2, 3, 4, 5\}$ and the following transition probability matrix:
$$
P = \begin{pmatrix} 0.1 & 0.0 & 0.2 & 0.3 & 0.4 \\ 0.0 ...
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1answer
160 views
Distribution of arrival times to server for an M/M/1 queue (what the server experiences)
In an M/M/1 queue, we know that inter-arrival times are exponentially distributed, and that service times are the same. What is the distribution of to-server inter-arrival times (aka service start ...
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0answers
23 views
Is a sequence of random variables indexed by a homogeneous Poisson process process strictly stationary?
I'm revising for an exam and have no idea how to approach this question:
Let $\{N_t\}_{t\geq 0}$ be a homogeneous Poisson process of parameter $\lambda > 0$. Let $\{X_k\}_{k\geq 0}$ be a sequence ...
2
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1answer
55 views
Predicting dichotomous outcome of temporal data set with covariates
I have a set of data, with outcome and time-varying variables, for patients during the course of their respective stays in the hospital. There is a dichotomous outcome on the last day. The length of ...
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44 views
About Galton watson process
My question is about a homework question that I found interesting. It gives another proof (without using martingales) for that the critical Galton Watson tree dies out eventually. But it has given a ...
