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.
1,059
questions
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4 views
Bernoulli process with nonstationary probability
Say we have a process $X_t\vert P_t\sim \mathrm{Bin}(n,P_t)$ where $X_t$ is observable but $P_t$ is not. Also, the success probability $P_t$ might vary over time and I don't assume some ...
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2answers
69 views
If $X(t)=A\sin(\omega t + \theta)$ with $\theta\sim U[0,2\pi]$ then is $f(X(t))$ jointly WSS with $X(t)$ for a function $f(z)$?
Suppose a continuous time process $X(t)=A\sin(\omega t + \theta)$ with $A$, $\omega$ fixed and $\theta\sim uniform[0,2\pi]$.
It is easy to see that this is a strict sense stationary process.
Now, let ...
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0answers
10 views
Car Driving Behavior: using non-instant transition with a Markov Transition Matrix
I have a dataset on driving behavior/trips (workable example below), with information on the departure location, arrival location, duration of the trip and length of the trip. The states of the ...
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13 views
artificial neural network
ann.compile(optimizer='adam', loss='binary_crossentropy', metrics=['accuracy'])
ann.fit(x_train,y_train, batch_size=32, epochs=100)
this is few lines of my code my questions is
adam optimizer works ...
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11 views
Hurst estimation in small samples
I'm trying to estimate the Hurst exponent of a time series which I believe behaves as a fractional Brownian motion. My problem is that all the estimation methods I have found so far (r/s, Whittle, etc....
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1answer
18 views
Struggling to understand the difference between a DTMC and a CTMC
My textbook, Modeling and Analysis of Stochastic Systems, third edition, by Kulkarni, introduces continuous-time Markov chains (CTMCs) as follows:
In Chapters 2, 3, and 4 we studied DTMCs. They ...
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13 views
How to simulate the supremum of a Gaussian Process
I have a problem where I need to estimate the quantiles of the supremum of a Gaussian Process certain point $t_0$ in time. This should be achieved by simulations.
I have a centered Gaussian Process $...
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8 views
Poisson counting process subinterval distribution
Suppose $N(t)$ is a homogeneous Poisson counting process with a constant parameter $\lambda$. Given positive real numbers $T$ and $\tau$, and non-negative integer $n$, what is the probability that $N(...
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11 views
probabilty , events , stochastic process [closed]
Problem 1
Suppose that the universal set S is defined as S = {1, 2,āÆ, 10} and A = {1, 2, 3},
B = {X ā S : 2 ā¤ X ā¤ 7}, and C = {7, 8, 9, 10}.
a. Find A āŖ B.
b. Find (A āŖ C) ā B.
c. Find AĀÆ āŖ (B ā C).
d....
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6 views
Minimum variance hedge ratio - returns or differences for Monte Carlo
So If I want to simulate for two stocks a monte carlo simulation since I want to show that in the end a portfolio with the minimum variance hedge ratio has the lowest standard deviation do I use as ...
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5 views
Minimum Variance Hedge Ratio for Prices and Returns
So from my understanding Hull (2012) f.e. shows that the optimal hedge ratio minimizes the variance of the returns. But what happens to the variance of the prices? Is the Minimum variance hedge ...
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0answers
5 views
Number of Susceptible individuals in stochastic SIR ICM
I am using SIR-ICM to model an outbreak in large population (around 10 million). I want to know the size of susceptible individuals to use in this model. I am in a doubt to use the actual population ...
2
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1answer
23 views
Variance of the autocovariance function at a particular lag
For a uniform random number(u) having mean $\mu$, the autocovariance at a lag $\tau$ is given by
$$C(\tau)=\frac{1}{N-\tau} \sum_{i=0}^{n-\tau} (u_i-\mu)(u_{i+\tau}-\mu)$$
For the uniformly ...
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19 views
How do I find the common factor from a stochastic process ARMA model
How would you factorise this ?
There should be a common factor between the Lag polynomial of the auto regressing moving average A(L) and the for the lag polynomial moving average B(L).
It should be ...
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0answers
27 views
How does polynomial trend get 'detrended' by the d-differencing?
Suppose $X_t$ is an ARIMA(p, d, q) process, then so is $X_t + m(t)$ where
$$
m(t) =a_0 + a_1t +...+ a_{d-1}t^{d-1}
$$
is some polynomial of degree $d-1$.
How does such a polynomial trend get '...
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0answers
15 views
Impact of correlation bounds for Monte Carlo simulations
As the lognormal distribution imposes bounds of attainable correlations as discussed in Attainable correlations for lognormal random variables my question would be what happens if say we want to do a ...
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0answers
16 views
Monte Carlo Error for Minimum Variance Hedge Simulation
So I was running a monte carlo simulation for two assets and a portfolio consisting of 1 quantity of the first asset and short a fraction x of the second asset to hedge, where the fraction is ...
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0answers
5 views
Equal Limiting Distributions in a Markov Chain Implications
I am studying Stochastic Models, and I came across the concept of limiting distributions, and I just wanted to make sure that I have understood it correctly.
If the limiting probabilities are equal ...
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0answers
19 views
Comparing Two Poisson Processes
I'm trying to solve this problem:
Regional and international planes arrive at an airport following independent Poisson processes with rates $\lambda$ and $\mu$, respectively. Each regional plane has ...
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0answers
10 views
Does the unconditional mean of a non stationary ARMA process exists?
Assume that we are dealing with an $ARMA(1,1)$ model:
$$
y_{t} = \theta y_{t-1} + \epsilon_{t} + \alpha \epsilon_{t-1}
$$
where $$ \epsilon_{t} \sim WN(0, \sigma^{2})
$$
Then, we can rewrite the model ...
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0answers
26 views
Show that this expression is a Poisson Process
How can I show that a Poisson process can be represented by $N(t) = \sum_{i=1}^{\infty} \mathbb{1}(T_{i} \leq t)$ .
Note that $T_{i} = \sum_{j=1}^{i}X_{j}$ where $X_{j}$ is the inter-arrival time of ...
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1answer
21 views
how to model random changes in line segments
I have a discrete time stochastic process where an interval from 0 to L consists of smaller sub-segments, where the boundaries are always at integers, and L is some large integer. At each iteration, a ...
3
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2answers
180 views
why independence assumption not applicable on time series analysis?
I just started my course in time series analysis. I saw a statement there: "Statistical methods that depend on independence assumption are no longer applicable in time series analysis". Why it so?
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15 views
Autocovariance lag dependence imply variance finite?
Suppose {$X_t,t \in T$} Gaussian Process, such that:
$E(X_t)=\mu$, where $\mu$ constant $\forall t \in T$ (constant mean)
$\gamma_X(X_s,X_t)=\gamma_X(X_{s+h},X_{t+h})$ $\forall t,s \in T$ and $h \in \...
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0answers
13 views
Is *conditional* stationarity (of a stochastic process) a useful concept?
Typical stationarity is defined in terms of some notion of invariance of a stochastic process w.r.t. a time shift operator.
I think you could talk about conditional invariance:
$
P(X_{t_1}, ... X_{...
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13 views
Where is the “Markov property” in Markov Random Fields? Are Markov Random Fields stochastic processes at all?
I am learning about Markom Random Fields. As I understand them they are constituted of some Random variables $X_{i}$ and some potential functions $\phi(X_s)$ where $X_s$ is a set of some random ...
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1answer
12 views
Is it possible to combine SGD/ Minibatch GD and Batch Gradient Descent to find the global optimum quickly?
Say the model is bouncing around the optimum with SGD. Is there a way to know that it's near the minimum, pause the model, and continue with an extremely small learning rate
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10 views
Average of the product of two autocorrelated variables
I am not expert in statistics, so I hope this question doesn't sound too trivial.
I have a discrete random variable that evolves in discrete time, . The stochastic process behind this variable has ...
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1answer
14 views
Don't understand one particular normal distribution notation from opaper Stochastic Variational Deep Kernel Learning, help needed
I'm in progress working with paper "Stochastic Variational Deep Kernel Learning" NIPS 2016
and I have the problem with understanding the meaning of this normal distribution notation from part 2 ...
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1answer
12 views
How to estimate model coefficients for option-like response: ie. response is related to variables in the form y = max( formula, 0)
Rewrite
Hopefully someone can point me to a resource on how to estimate the parameters I'm trying to model. I've had trouble giving a title to my question and googling for resources.
Suppose $y$ ...
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0answers
42 views
Intuitive explanation of Gaussian Process Regression
How would you intuitively explain the idea behind Gaussian Process Regression to someone unfamiliar with stochastic processes? Especially the point where you discuss modeling covariance, choice of ...
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0answers
14 views
Method to generate 2D arrays with specified autocorrelation?
Suppose I have a specified value of autocorrelation (say 0.9) at lag 1. Is there a way to generate a random 2D (NxN) array which will have this particular autocorrelation value at the same lag?
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1answer
20 views
Renewal processes, interarrival time distributions
I am dealing with renewal processes recently and I have some questions and I hope you can help me :).
Why interarrival time distributions need to be independent in a renewal process?
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28 views
Renewal processes simulation using Monte Carlo
I am dealing with renewal processes recently and I have some questions and I hope you can help me :). I'll post each question separately.
Is this a valid way to simulate a renewal process using Monte ...
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19 views
Poisson process, combined waiting time
Passengers arrive at a railway station according to a homogeneous Poisson process
with rate Ī». At the beginning (time 0), there are no passengers at the station.
The train departs at time T. Denote by ...
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55 views
Renewal processes, relationship between distribution of the events amplitudes and the interarrival times
One of the basic assumptions of a renewal process seems to be that the distributions of the interarrival times are iid. However, this makes me think if there must exist a relationship between the ...
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6 views
k-nearest neighbor kernel density estimation for a an unknown stochastic process?
Suppose we have a sequence of random variables $\{X_t:t=1,\cdots,T\}$ following an unknown stochastic process (possibly stationary or non-stationary). Now I have two questions:
1- Would it be ...
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1answer
23 views
Can a random variable be expressed as a sum of deterministic and random variable?
Say we have a sequence of random variables $\{X_t:t\geq 0\}$ following an unknown stochastic process with distribution $X_t\sim N(\mu_X,\sigma_X^2)$. This idea came to me from the additive noise model....
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7 views
Linking Correlated Dependent Variable with Independent Variable
I have a Monte Carlo model that generates a distribution of possibilities $X_i$ for the non-normal stochastic process $Z$ it describes. The distribution of $X$ and $Z$ is fat tailed but for the most ...
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14 views
Parameter estimation for time-varying autoregressive processes in R
I want to estimate the parameters of an autoregressive process with time-dependent coefficients. For example TVAR(1) model with 1 lag: $$ X_t = \phi_t X_{t-1} + \sigma_tW_t $$ where $\phi_t$ and $\...
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15 views
Extending normalized probability generating function (pgf) in branching process
My question is in the context of branching processes and simply how to extend a normalized negative binomial probability generating function (pgf) from $\frac{1}{y}[G(s)]^y$ to $\frac{n}{y}[G(s)]^y$ ...
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1answer
22 views
Can we express the following unconditional probability as follows?
Some of you may be aware that I have been asking a nagging question for quite a while on this forum, in different shapes and forms. Although I may have been a nuisance, may I thank you as this has ...
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53 views
How to estimate a single AR model for multiple time series?
I have ACF of a time series data, I want to interpolate the ACF using AR model for which I need to calculate the coefficient of the AR model The problem is I have (6000x6000) matrix of ACF and I cant ...
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2answers
90 views
AR(1) with known initial and terminal condition: how to draw the innovations?
Suppose I have the following stationary $AR(1)$ process:
$$ y_{t}=\alpha_{0}+\alpha_{1}y_{t-1} + u_{t} $$
where $u_{t} \sim \mathbb{N}(0,\sigma^{2})$, with $\sigma^{2}$ known. Suppose I have an ...
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1answer
33 views
The covariance function of a stochastic process is positive semidefinite
Let $\{X_t, t \in \mathbb{Z}\}$ a real-valued stochastic process and $\gamma : \mathbb{Z} \times \mathbb{Z} \to \mathbb{R}$ the autocovariance function. I would like to show it is a positive ...
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1answer
52 views
The question is related to stochastic process and the Markov chains
This is the complete question.
Jim is currently living in Scranton. Each year that he lives in Scranton, he has a
probability of 1/2 of staying in Scranton the next year. Otherwise, he has an equally
...
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20 views
In predictive regressions, can we assume that the predictor follows the Ornstein-Uhlenbeck process?
A predictive regression is a regression of the form
\begin{equation}
y_t=\beta x_{t-1}+\varepsilon_t
\end{equation}
where $x_{t-1}$ is generally assumed to be a highly persistent stochastic variable, ...
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0answers
18 views
Is a grid-like network more efficient than a random network?
Imagine two different types of street networks with the same number of nodes and where edges are weighted according to their length. Both network types cover the same geographic area (say 1kmĀ²) and ...
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8 views
Can you find the dynamic unconditional mean and variance of a process without assuming a model for the DGP?
Let us say $\{X_t:t \geq 0\}$ is a weakly stationary stochastic process with initial condition $X_0$. Suppose we know that the process is normally distributed, but we wish to estimate the dynamic mean ...
3
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1answer
93 views
Probability distributions associated to the logarithm numeration system
The most elementary logarithmic numeration system is defined as follow. Any random number $X \in [0, 1]$ can be represented uniquely as
$$X=\log_3(A_1 + \log_3(A_2+\log_3 (A_3 + \cdots)))$$
with $A_k \...
