Questions tagged [stochastic-processes]

A stochastic process describes 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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28 views

What does it mean “a conditional expectation given a stochastic process”?

Let $(\Omega,\mathscr{F},P)$ be a probability space. Let $X,Y$ be random variables on $\Omega$. Then, we say $Z\sim X|Y$ iff (i) $\int_{Y^{-1}(A)} X dP = \int_{Y^{-1}(A)} Z dP$ and (ii) $Z$ is $\...
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What is the extension of fractional Brownian motion to describe statistical multiscaling?

A random variable $X(t)$ is said to be monoscaling if $$ X(t) = a^{-H}X(at).$$ $H$ is called the Hurst exponent, and $a$ is a scaling factor. A key model of statistical monoscaling is the fractional ...
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When is it appropriate to model data as a Stochastic Process? [on hold]

I understand the question may be poorly worded, but please bear with me. I have a background in computer science and mathematics, but never took any upper-level statistics courses. I currently work ...
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Give an example of an IID sequence of random variables (different from dice throwing) where one can observe weak LLN [closed]

Give an example of an IID sequence of random variables (different from dice throwing) where one can observe weak LLN.
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27 views

Weak stationarity of the stochastic process and the impact of the lagged white noise

I am struggling with following exerises: Consider following discrete-time stochastic processes $Y_t$: $$Y_t=\frac {5}{3} Y_{t-1}-\frac{2} {3} Y_{t-2}+\epsilon_t$$ $$Y_t=1.3 Y_{t-1}-0.4 Y_{t-2}+\...
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Quantiles of the Q values of an MDP

Cross-posted from Math StackExchange: Consider an MDP with $n$ states, $k$ actions, and discount factor $\gamma \in [0,1)$. We are uncertain of its reward function $R \in \mathbb{R}^{n \times k}$ and ...
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27 views

mean time to return random walk continuous whit drift to origen

Let $S_t$ a random walk with rate $a>0$ on $Z$ that such has a drift in direction of $0$, this is, if $(0,0)\in Z\times[0,\infty)$, and defined a sequence $(S_n, T_n)_{n\in N}$ such that $(S_0, T_0)...
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40 views

What is the probability that maximum likelihood algorithm is right in Bernoulli trials?

Let $\varepsilon\in(0,1)$ and $p:=\frac{1+\varepsilon}{2}$. Suppose that we have a sequence of independent Bernoulli random variables of parameter $p$, say $(X_k)_{k\in\mathbb{N}}$ defined on a ...
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expected value of a time series/ stochastic process

I am new to time-series analysis and I found myself getting confused by the most fundamental concepts. A stochastic process is a collection of random variables $X_t$ for $t=1,\dots,n$ where each $X_t$...
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Stationary distribution of AR(1) process with AR(1) shocks

I am trying to find the stationary distribution of an AR(1) process, where the shock terms themselves are an AR(1) process. That is, the process moves subject to the following 2 equations: \begin{...
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Is there a name for this approach to find a class probability via density estimation?

I am studying a stochastic process that produces event 1 with a probability $p \ll 1$ and event 0 with probability $1 - p$. I have large amounts of data consisting of about $n=20$ features and ...
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Questions about the stability (and stationarity) of a system and state space representations

I'm pretty new to the topic and I'm trying to understand how to determine the stability of a process. I'm giving this discrete-time stochastic system: $$ \cases{ s_t = 2s_{t-2} + 3w_{t-2} \\ y_t = ...
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Geometric Brownian Motion with two-state diffusion/volatility

let's assume a GBM process S(t) with dynamics: dS(t) = a S(t) dt + b S(t) dB(t) where B(t) is a Brownian motion, a and b are constants, and S(0)>0. For any time s>t, we have that E_t[S(s)^k] = S(t)...
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Supremum of infinite-horizon gaussian process that converges to 0 at infinity

Suppose $X(t),t\in[0,\infty)$ is a centered gaussian process with covariance function $\Gamma(t,s)$, such that $\Gamma(t,t)$ is uniformly bounded over $t\in[0,\infty)$, and $\Gamma(t,t)\rightarrow 0$ ...
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probability of getting to state a before state b starting from state a

If $X_n$ is a Markov Chain. P(0,0) = 0.5, P(0,1) = 0.5. For all states x > 0, P(x, x) = 0.5, P(x, x+1) = P(x, x-1) = 0.25. My goal is to find $P_0$ ($T_0$ < $T_5$), which is the probability of ...
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What is the volatility surface of a parameter that depends on a Stochastic process?

Hi this question takes a few steps to setup so please bear with me :) Short version: Does a parameter that depends on the output of a stochastic model inherently have variance? If so why? Long ...
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When developing the Yule-Walker equations, why can we assume $E(X_{t-k} \cdot Z_t) = 0$

I have a AR(1) process $X_t = 0.4X_{t-1} + Z_t$. I want to derive the Yule-Walker equation. I multiplied by $X_{t-k}$, took the expectation of both sides, and got: $$E(X_{t-k}X_t) = 0.4E(X_{t-k}X_{t-...
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Is it possible to perform a regression where you have an unknown / unknowable feature variable?

Is it possible to perform a regression where you have an unknown / unknowable feature variable? Say I have $y_n = a_0 + a_1 x_1 + a_2 x_2 + a_3 x_3$ but I do not / cannot measure the value of the ...
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30 views

What are the differences between probabilistic models, stochastic models, and statistical models? [closed]

What are the differences between probabilistic models, stochastic models, and statistical models? Do statistical models deal with data sets, and model them mathematically to capture the summary ...
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56 views

Probability of a stochastic process hitting a certain level?

Suppose I have observations of a stochastic process $\{X_t\}$. I.e., time series data. Say presently $X_t < u$ for a level $u$, for every $t=1,2,...,N$. From these observations, how can I model ...
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Correlations between two sequences of irrational numbers

If $x$ is an irrational number and $b$ an integer, let's define $g(x,k) = \mbox{Correl}(\{nx\},\{nb^kx\})$. Here $k=1,2,\cdots$ is an integer. The brackets represent the fractional part function. ...
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60 views

Calculating the AIC based on histograms for selection of stochastic models

I am modelling a nonlinear stochastic process and have data to compare model output against. My aim is to obtain an evolution equation of the form, $$\frac{du}{dt} = f(u,\theta_f)+\alpha(u,\theta_\...
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47 views

Why do we treat a value at each location as a random variable?

Apology if my question is very simple to you (I am very new to geostatistic). Suppose that X is a random variable (e.g., the concentration of the Zinc at a specific ...
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Which distribution or process should be used for wearout reliability modeling?

When modeling the reliability of a system, it is usual to use exponential distribution to model errors that occur randomly throughout the system's useful lifetime (the middle part of the well-known ...
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44 views

Ornstein-Uhlenbeck process

Given a multivariate Ornstein-Uhlenbeck process that is a stochastic process, is it correct that each component of this process is a univariate Ornstein-Uhlenbeck process?
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25 views

Stochastic processes whose current state depends on a distant past state

I am new to the field of stochastic processes. I want to ask if there is any specific term for a multi-dimensional random process $\overrightarrow{Z_n} = \left \langle X_n, Y_n \right \rangle$ whose ...
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Is the joint distribution the conditional distribution of part of a Bernoulli process

Note: all the discussion below is in the context of the Bernoulli process section 6.1 of Bertsekas et. al's book "Introduction to Probability, 2nd Edition" says Generalizing from the case of a ...
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What is the minimum ingredients to construct a stochastic process in discrete time?

This post gives A stochastic process in discrete time n ∈ $N$ = {0, 1, 2, . . .} is a sequence of random variables (rvs) $X_0, X_1, X_2$, . . . denoted by $X = \{X_n : n ≥ 0\}$. ... what is ...
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Definition of the dirichlet process: what is the sequence of random variables

Reference material by Dr. Teh Definition Given a measureable set S, a base probability distribution H and a positive real number $\alpha$, the Dirichlet process $DP(H, \alpha)$ is a stochastic ...
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Is there a name for this type of transition probability diagram which seems not to be a transition probability graph?

This is a "transition probability graph" coming from the book "Introduction to Probability, 2nd Edition by Dimitri P. Bertsekas and John N. Tsitsiklis". That book also gives this figure, which seems ...
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What are the differences between Bayesian filters and adaptive filters?

I am learning about state estimation and I am having difficulty understanding the difference between Bayesian filters such as Kalman filter and particle filters compared to adaptive filters. According ...
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Given a transition matrix P for weather conditions (modeled as either rainy or sunny), is $P^n$ the n-Step Transition Probabilities for day n+1?

wiki uses this example to illustrate Markov chains. The probabilities of weather conditions (modeled as either rainy or sunny), given the weather on the preceding day, can be represented by a ...
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How a dataset can be modeled as Stochastic Process

I have a question about time series and stochastic processes representation in python Recently, I read this web page for modeling the temperature dataset and it considers a time series as a ...
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22 views

The dynamics of a normal distribuition in stochastic processes (food court example)

Suppose I want model a huge food court. Let's assume that the number of people who start having a lunch is a function of time $f(t)$. Also, let's consider that the time people spend having a lunch ...
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Model to Bootstrap/ simulate hourly shape from daily data

How do I simulate/boot strap hourly shape from daily data ? $\mathbf {Data set:}$ My first data has hourly granularity, its hourly temperature, $T_1$ through $T_{24}$ and but I have no means to ...
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Understanding the stochastic part of rank-based inverse normal transformation

I'm checking out some methods for rank-based inverse normal transformation in Python and found this: https://github.com/edm1/rank-based-INT/blob/master/rank_based_inverse_normal_transformation.py ...
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Gradient of the variance of a Gaussian process

I'm trying to compute the spatial derivatives of the expected improvement acquisition function in Gaussian-process optimisation, and doing so requires the spatial derivatives of the predictive ...
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9 views

Difference between Autocorrelation of a function and stochastic average of the same function squared

What is the difference between < F(t)F(t') > - computed for all |t-t'| and <|F(t)|^2> - computed for all 't', conceptually, given F(t) is real and random? For simplicity, let us say Auto Corr of ...
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39 views

Correct definition of the probability space of a $ \mathcal{F}_{t} $-measurable random variable

Let the triple $ \left ( \Omega, \mathcal{F}, \mathbb{P} \right ) $ designates the probability space of a stochastic event and let us consider the filtration $ \left ( \mathcal{F}_{t} \right )_{t \in \...
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How to use the Phillips-Perron and Dickey-Fuller unit root test to test for AR(1) or Ornstein-Uhlenbeck process

My question comes from this paper (p. 10), where the authors say: The $\ln(\hat{p_t}) - \ln(p_t) \sim AR(1)$ condition expresses that the LPPLS fitting residuals can be modeled by a mean-reversal ...
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How do I model the chaotic behaviour(like the sequence from Lorenz attractor) in a stochastic sense?

Recently, I encountered a difficulty of prediction Lorenz attractor by using a GRU. (See the code from here.) I think that it's inevitable since the original system, i.e. Lorenz equation, is too ...
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How do you predict the sequence from Lorenz attractor?

Now, I'm verifying the capability of the GRU(Gated-Recurrent-Unit) by checking the prediction performance of Lorenz Attractor. (See the code from here.) It can predict an initial part of sequence, ...
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Difficulty in understanding the proof of the Wold decomposition theorem

The proof of the Wold Decomposition [1] of $x_t$ involves the definition of the process $$w_t = x_t - P_{\mathcal{M}_{t-1}^x} x_t,$$ where $x_t$ is a stationary zero-mean process, $\mathcal{M}_n^x = ...
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Questions about calculate the first k-th principal components of VR-PCA?

Variance-reduced PCA VR-PCA, focus on the first component calculation, what about the k-th PCs? Any idea? Thank you!
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Time distibution in Markov chain

Let $E=\{A,B\}$ be a set and $X_{1,t}, X_{2,t}, X_{3,t}$ three independent Markov chains on the set $E$ with respective transition probability $P^{(1)}, P^{(2)}, P^{(3)}$ where $$P^{(i)}=\begin{...
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23 views

Expectation of $dX_t$ for $X_t$ being an Ito process

Let $X_t$ be an Ito Process: $$dX_t = f(t, X_t)dt + g(t, X_t)dW_t$$ What is $E_t[dX_t]$? How can we compute it and importantly what is the intuitive explanation of $E_t[dX_t]$?
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How to use Gamma distributions to estimate the number of failures?

I need to calculate the expected number of failures of a product within 6 years. The time until failure is said to be gamma distributed with $\alpha=2$ and $\beta=0.5$ for a mean time between ...
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34 views

Variance of linear combination of AR(1) process

Let $ \{X_t\}$ ~ AR(1): $$ X_t=2.62-0.84X_{t-1}+\epsilon_t, \ \ \ \epsilon_t\sim WN(0,2.27)$$ Compute the variance of $$ \overline{X}= \frac{1}{3}\sum_{t=1}^{3} X_t $$ The solution is: Var($\...
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68 views

First difference of AR(1) process

Given AR(1): $$X_t - \mu = \phi(X_{t-1}-\mu) + \epsilon_t$$ where $$ \mu = 0.85 \\ \phi=0.59 $$ and $$ W_t = X_t - X_{t-1} $$ Compute $$ Corr(W_t,W_{t-1})=-0.205 \\ Cov(W_t,W_{t-4})=-0.43 \\ Corr(...
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Continuous-time Kalman filter with no observation/measurement noise

The continuous-time (linear) state space model can be written \begin{align*} \text{d}\mathbf{x}_t &= \mathbf{F} \,\mathbf{x}_t \, \text{d}t + \mathbf{G} \,\text{d} \boldsymbol{\...