Questions tagged [random-walk]
A stochastic process that describes a path arising from a succession of random steps.
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18 views
Integrated Random Walk is invertible and/or stationary?
I am wondering if the integrated Random Walk $y_t$ is Stationary and/or invertible $ y_t = 2y_{t-1} - y_{t-2} + e_t $ where $ e_t$ is a White Noise.
To prove non stationarity I tried to say: If, by ...
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13 views
Analyzing animal occurrence data to detect a movement bias in R
I have occurrence data for animal locations over several months. These UTM coordinates are coupled with a date and time of occurrence, but they differ from tracking data in that it is not over ...
6
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1answer
71 views
How can we verify the intuition that in the RW-Metropolis-Hastings algorithm with Gaussian proposal too small and too large variances are bad choices
Let $d\in\mathbb N$ and consider the Random Walk Metropolis-Hastings algorithm with a Gaussian proposal kernel $Q$ such that $Q(x,\;\cdot\;)=\mathcal N_d(x,\sigma^2_dI_d)$ for all $x\in\mathbb R^d$.
...
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1answer
73 views
Average time for a random walk on the edges of a cube
In a interview i had something similar to this question Random walk on the edges of a cube.
But this time there is only a ant, it takes one minute to go through a single edge. The ant can use the ...
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0answers
15 views
Random walks-heavy tailed case
Let $\beta > 0$ and $S_{0}=0$, and let $S_{n}=\xi_{1}+\dots+\xi_{n}$,$n \geq 1$, be a random walk with i.i.d. increments $\{\xi_{n}\}$ having a common distribution
$P(\xi_{1}=-1)=1-C_{\beta}$ and $...
2
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1answer
29 views
User-Specific Activity Level Along Weeks
I am assigning Low | Medium | High activity levels to users once a week.
At the end of an entire period,a user has been assign n weekly activity levels among {Low, Medium, High}.
Let k be the total ...
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1answer
76 views
Probability you end up at the origin after taking $2n$ steps?
Starting at the origin on the line we take a step of unit to the left or to the right
with probability $\frac12$. We do this repeatedly with independent steps. If we take $2n$ steps, what is the
...
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42 views
Expected number of steps in Gambler's ruin game with two players
Let's say we have two players A and B. Player A has 3 coins and player B has 5 coins. If player wins the other player gives one coin. During game second player probability of loosing is $2/3$, while ...
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99 views
Expectation of a random walk that can't go below zero
Suppose we have a random walk $S_n$ that is constrained to be positive or zero, that is:
$$S_0 > 0$$
$$S_{i+1} = \max(S_i+x_i,\space 0)$$
$$x_i \sim N[\mu,\sigma^2]$$
Can we analytically ...
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1answer
61 views
Why is the probability of a random walk reaching 1 (in n steps) squared greater than the probability of it reaching 2 (in n steps)?
Let $S_n$ be a simple random walk. i.e.
$$
S_n = \sum_{t=1}^n X_t,
$$
where ${X_t}$ are i.i.d random variables with
$$
X_t =
\begin{cases}
+1, & \textrm{w/ probability } p \\
-1, & \...
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1answer
47 views
Applying Bayesian Gaussian movement question
I have a question from my stats class that I am confused about how to proceed with. I have a general idea of what I am to do but I am not sure how to start.
The question is about a car that is moving ...
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1answer
70 views
Random walk with “negative coefficient”
I was playing around in R with simulating random walks. At some point I tried this model:
x = NULL
x[1] = 0
for (i in 2:2000) {
x[i] = -x[i-1] + rnorm(1)
}
...
4
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0answers
53 views
Survival probability of a random walk with renewal timings
A random walker starting at time $t=0$ and location $x=0$ moves to the right ($x+1$) or the left ($x-1$). The $k^{\mathrm{th}}$ moves to the right and left occure at the times $\sum_{i=1}^{k} R_i$ and ...
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0answers
25 views
Probability of never returning to the origin until time $2n$ in asymmetric Bernoulli random walk
I have the following asymmetric random walk problem.
$X_1, \cdots, X_{2n} \overset{iid}{\sim} F(p)$, where $F(p) : \begin{cases} P(X = 1) = p \\ P(X = -1) = q=1-p\end{cases}$
So what I need to ...
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12 views
What does a Drift Diffusion Model tell you about choice and reaction time?
I'm having trouble understanding what exactly the drift diffusion model, LBA, LCA, etc. models tell you about a set of 2 (or multi) alternative forced choice tasks. I know these models are supposed to ...
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0answers
29 views
How to measure “cyclicity” of a directed weighted graph?
Say you have a weighted directed graph with (potentially) some cycles in it. You want to have some sort of a measure of how "cyclical" this graph is. The requirements are:
This measure C=0 on an ...
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66 views
The Hessian of multinomial Probit model
I wanted to implement multinomial probit in Bayesian with random-walk Metropolis Hasting. To achieve the best numerical efficiency when drawing $\beta$, I need to use the hessian matrix of $\beta$. ...
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18 views
Modelling product purchase history as a random walk in n-space
I have a large dataset of customers making monthly purchases of multiple products. Customers usually purchase between 3 and 10 products, from a large product list (1000s). I'm interested in clustering ...
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1answer
20 views
Difference between $2^{nd}$ order random walk and personalized pagerank
I've been recently working with graph sampling, and I can't seem to find useful explanation of the following two aspects. On one side there are pagerank-based algorithms, which converge to a ...
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1answer
86 views
Proving that a random walk that diverges to infinity may not become negative
Consider a random walk $S_n= \sum_{k=1}^n X_k$, where $\{X_k\}_{k=1}^\infty$ are independent and identically distributed random variables. Assume that $S_n \rightarrow \infty$ almost surely as $n \...
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1answer
139 views
Proving that a random walk using a maximum likelihood estimator can diverge to infinity
Consider a sequence of continuous random variables $\{X_n\}^\infty_{n=1}$ that are independent and identically distributed under the probability density function $f_\theta (x)$, where $\theta \in [\...
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18 views
How does the range of a random walk vary with sample size?
I want to compare the ranges of 2 sets of data, but one set has N times the number of steps as the other.
I'm willing to treat the underlying process as a random walk.
Is sqrt(N) an appropriate ...
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2answers
396 views
Are S&P 500 monthly (or annual) returns a random walk?
I'm using financial software that assumes that yearly market returns are random and independent in their Monte Carlo analysis. Its not clear to me that this is the case. Is there an easy way for a "...
2
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1answer
140 views
Expectation of the absolute value in a sequence of Bernoulli trials
On this tweet:
Can I get some help in understanding the proposed solution by N. Taleb:
It is not clear how he describes success, i.e. $n-x$ to come up with $\binom{n}{n-x}.$ It almost seems as ...
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1answer
287 views
random walk and covariance stationary
I was preparing for CFA and encountered this question, which is quite puzzling.
To use autoregressive model, it has to be covariance stationary (same mean, covariance). If a model's residual is not ...
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0answers
14 views
Why switching edges definition changes the result of random walk search for communities? (walktrap)
If a graph object is a not directed graph, then the following set of operations should yield the same result:
...
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1answer
344 views
Showing that R-squared might not be useful in time series data
I understand that using $R^2$ in time series models may not be the best as $R^2$ is non-decreasing.
I also read this post: What is the problem with using R-squared in time series models? on the ...
6
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1answer
103 views
Principal Components of Random Walk
In this blog figure 4 shows that the principal components of a random walk are sinusoidal with increasing frequency for decreasing eigenvalue. Is there an intuitive way of understanding this?
If I ...
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40 views
What is the “distance” of a random walk on a graph?
What is the definition of the distance of a random walk? In the Statistical significance of a cluster of this paper, and this ...
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694 views
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49 views
Brownian bridge to unknown via extremum
Suppose, I know what's the minimum $\min$ of a random walk $w_t$ in period $[0,\Delta t]$. I also know $w_0$ and $\sigma$. How to construct the Brownian bridge for the latter period?
I guess it's not ...
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0answers
140 views
Estimate standard deviation of random-walk using Kalman filter
I'm new to Kalman filters so this might be a stupid question.
I created a Kalman filter that takes in time series observations and estimates the mean of that time series. This is simply modeling a ...
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1answer
106 views
Random Walk Process - Time Series
Is it true that the mean of a random walk process does not depend on time and the sequence can be considered mean stationary?
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1answer
160 views
Power Spectral Density of Random Walk
The Brownian motion has a power spectral density (PSD) dependency on frequency like $\frac{1}{f^2}$. As far as I understand, power spectral density is defined only for wide sense stationary processes ...
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139 views
Variance sum of two independent random walks
I have two random walks, which represent fishing mortality in season 1 and season 2 of year $t$ ($X_{t,\mathrm{summer}}$ and $X_{t,\mathrm{winter}}$). If I add up both series to obtain annual fishing ...
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1answer
166 views
Does it make sense to have the dependent variable in growth rates or rather in levels?
I am currently investigating the impact of uncertainty on investment dynamics. I have an unbalanced panel data set (approximately 100,000 observations). To study the relation between investment and ...
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2answers
58 views
Forecasting average values with varying number of observations
Every day $t$ we observe several (independent) realizations of a variable $X_t$. This variable is the sum of a time dependent mean value and white noise:
$$X_t=\mu_t+\epsilon_t$$
You can assume $\...
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2answers
295 views
How to prove that the probability of spurious correlation increases with random walk length?
Define a simple random walk $y_{t}$ as:
$$y_{t} = y_{t-1} + 2\times Bernoulli\left(0.5\right)-1,$$
so that at time $t$ the value of $y$ equals its previous value plus a perturbation from the "flip-a-...
3
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1answer
233 views
Interpretation of an I(2) process?
I know that an ARIMA(0,0,0) process is white noise and ARIMA(0,1,0) is a random walk,
Is there an interpretation of what an ARIMA(0,2,0) process is?
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1answer
153 views
Predictor for averaged Brownian motion
The best forecast (predictor) for a Brownian motion at time $t+h$ is the present value at time $t$ since it's a martingale. The same holds for random walks with independent steps and without drift.
...
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2answers
4k views
How to interpret ARIMA(0,1,0)?
I ran the auto.arima()command in R on a set of data and it chose the appropriate model to be ARIMA(0,1,0).
I know ARIMA(0,0,0) is just white noise, but what does ...
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0answers
91 views
Random walk touching or exceeding thresholds
What is the formula to estimate the probability of a random walk touching or exceeding a particular threshold? The threshold starts and stops at particular times. (Without starting and stopping times ...
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0answers
327 views
Stochastic or Deterministic Trend: Supported by the Augmented Dickey-Fuller Test
Below are the sequential steps/question regarding my problem:
I am attempting to specify a VAR model in order to analyze impulse response functions. In plotting my first variable (Figure 1) I ...
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1answer
75 views
How to add random walk in rstanarm [closed]
I have used rstanarm GLM model without the intercept like below in R
...
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1answer
271 views
Variance of random walk, mean reverting and trending series
I am reading Ernie Chan's blog post "Mean reversion, momentum, and volatility term structure". It says that
To be precise, if $z$ is the log price, then volatility, sampled at intervals of $\tau$, ...
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82 views
Ito's integral formula for non-standard Brownian motion
Concerning Ito's integral formula,
$$\int_0^t B(s)dB(s) = \frac{1}{2}B^2(t)-\frac{1}{2}t,$$
the MIT lecture notes give a proof that "the standard Brownian motion has a.s. finite quadratic variation ...
4
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1answer
265 views
How to implement a uniform random walk on a simplex?
I am looking for an uniform random walk algorithm on a simplex for MCMC purposes. Hence the process should on average spent the same time in any given area. I want this to be my proposal algorithm. ...
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2answers
1k views
Moving Average, Exponential Smoothing, and Random Walk for Forecasting
I would like to confirm my understanding. Is it true that a (simple) exponential smoothing model with alpha (smoothing constant) = 1 is the same as MA(1), which is in turn the same as a random walk ...
2
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1answer
291 views
Runs test and Durbin-Watson test yield different outcomes
I have analysed the market return using the runs test and the Durbin-Watson test to determine whether the return series follow the random walk or not.
The problem I have found is that some return ...
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1answer
71 views
Proof the increment variance of the Scaled Random Walk
Start by defining the Symmetric Random Walk:
$$ M_t = \sum_{i=1}^{t}X_i, ~~ \text{with}~X_0=0 $$
where $X_i$ is equal to 1 or -1 with $p=(1-p)=0.5$.
Consider $t > s$, the variance of its ...
