Questions tagged [random-walk]

A stochastic process that describes a path arising from a succession of random steps.

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2k 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 ...
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1answer
427 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
148 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 ...
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357 views

Is a random walk necessarily a martingale?

I read in the following notes (http://www2.econ.iastate.edu/classes/econ672/Falk/_notes/lecture_4_martingales.pdf, p.2) that "a random walk is a martingale." Although it seems logical with the used ...
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1answer
92 views

Estimating error from an autocorrelated random walk

Hello Cross Validated, I am trying to simulate autocorrelated, lognormal values such that their SD matches that of an existing time series. I plan to generate these values using the following ...
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2answers
559 views

Error term in Random walk with drift

In random walk model the error term is generally modelled using normal distribution. I have two questions. Q1. How to calculate the parameters of error term in case of Random walk with drift. Q2. ...
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66 views

the length of a random walk in two dimensions

This is a matlab code that could generate a random walk in two dimensions of length 1000: xy = cumsum (-1+2*rand(1000,2),1). The 'stepsize' is between $[-1,1]$. In this case, we are talking about ...
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599 views

Estimating the white noise of a Random Walk

I want to simulate a random walk without drift to predict a time serie. The random walk model is $X_t = X_{t-1} + \epsilon_t$ where $\epsilon_t$ a white noise with a Normal distribution. It seems ...
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66 views

How to draw a representative gaussian random walk?

I would like to draw (as in, with a pen on paper) a line chart that is representative of a Guassian random walk defined as the sequence of random variables $(Y_t)$ where $t = 0, 1, 2, \dots$, such ...
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1answer
2k views

Lag between predicted output and real output in time series prediction (directional prediction)

I modeled a directional prediction of a time series. In every step, I predict next direction of that series (up or down). Currently I have a lag in predicted outputs compared to real outputs. For ...
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1answer
814 views

Estimating Covariance Matrix of Innovations of Multivariate Random Walk

Suppose that I have a multivariate random walk: $X_{t+1} = X_t + \epsilon_t$ where $\epsilon_t \sim N(0,\Sigma)$ Estimating the covariance matrix $\Sigma$ is straightforward from first differences $...
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1answer
638 views

Problem Sampling with Metropolis Hastings

I'm trying to use the Metropolis-Hastings (MH) algorithm to obtain an approximation of the distribution of the parameters of the following model: $$ y_{t} = \alpha x_{t}^\beta + w_{t} $$ $$ w_{t} \ \...
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1answer
114 views

How does the random walk know the height of the posterior in Bayesian inference?

How does the sampling procedure work if the posterior that is being sampled from is unknown? If the proposed jump has a higher probability than the current one the random walk jumps to the proposed ...
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28 views

Is a first differenced Markov process still Markovian?

If a discrete random variable forms a Markov process, then you obtain the time series from it. If you first difference that time series (i.e. Markov process). Do you still get the Markov property or ...
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1answer
217 views

Metropolis-Hastings acceptance ratio

I'm using Metropolis-Hastings to sample from an Inv-Gamma$(a,b)$ posterior distribution. My jumping distribution $J_t(\theta_*|\theta_{t-1})$ is N$(\theta_{t-1},0.5^2)$. After I sample a $\theta_*$ ...
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1answer
290 views

Can someone explain Importance Sampling to me? [duplicate]

So even the combined efforts of my professor, my book and the internet have not been able to make me understand the concept of Importance Sampling. I know that it is a way to help with estimating the ...
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1answer
3k views

Hamiltonian Monte Carlo vs. Sequential Monte Carlo

I am trying to get a feel for the relative merits and drawbacks, as well as different application domains of these two MCMC schemes. When would you use which and why? When might one fail but the ...
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Tuning MALA (Metropolis-adjusted Langevin) proposal

I'd like to implement a version of Metropolis-adjusted Langevin sampling, but I'm unsure how to go about tuning the parameters of the proposal density. My understanding is that in MALA, a proposal ...
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1answer
2k views

Random Walk with Restart vs. Personalized Pagerank

Are Personalized Pagerank and Random Walk with Restart really the same thing? From this source, it seems to be: http://web.eecs.umich.edu/~dkoutra/papers/fabp_pkdd2011.pdf I've used the RANKS RWR ...
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2answers
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Why are random walks intercorrelated?

I have observed that, on average, the absolute value of Pearson correlation coefficient is a constant close to 0.560.42 for any ...
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1answer
380 views

Variance of an ARIMA(0,1,0) model

The question goes like Given a model $$X_t=X_{t-1}+Y_t$$ where $Y_t$ are iid random variables with $\mathbb{E}(Y_t)=0$ and $\text{Var}(Y_t)=4$, evaluate $\text{Var}(X_1+X_2+X_3+...+X_8)$. How ...
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Why is a random walk not a stationary process? [duplicate]

In the book Analysis of Financial Time Series by Rue Tsay, I read: A time series $\{p_t\}$ is a random walk if it satisfies $p_t = p_{t−1} + a_t$ where $p_0$ is a real number denoting the ...
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Estimate next observation in a noisy random walk with known variances

Suppose we have an infinite sequence of observations $x_n, x_{n-1}, ...$ (going back in time) of a "noisy" random walk: For all $k \leq n$, $x_k \sim N(\mu_k, 1)$ $\mu_k \sim N(\mu_{k-1}, \sigma^...
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185 views

Variance of geometric random walk

I am trying to calculate the mean and variance of the following simple random walk: suppose we start from 1. With probability $p$ it can increase to $a$, and with probability $q(=1−p)$ decrease to $b$....
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1answer
52 views

1D random walk with variable probability

How can I model the following problem by a random walk: Consider that at time $t=0$ Mike has $M$ dollars. He receives a dollar at each of the times $R_1, R_1+ R_2, R_1+R_2+R_3,\cdots$, where $...
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2answers
228 views

How can A and B have equal chances to be visited last if A has higher chances to be visited before B?

The classical problem considered by Ross is a random particle visiting all chairs under a circular table. That is, there is a circular buffer of size m+1, were ...
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1answer
237 views

Dirichlet distribution and its applications in data science [closed]

What is the Dirichlet distribution and what are its applications?
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1answer
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Explain what is meant by a deterministic and stochastic trend in relation to the following time series process? [closed]

Explain what is meant by a deterministic and stochastic trend in relation to the following time series process? $y_t = c + y_{t-1} + \varepsilon_t$ where $\varepsilon_t\sim iid(0, \sigma^2)$ this ...
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1answer
219 views

Probability that simple random walk is bounded within $[-M,M]$

Suppose $\{ X_n \}$ is a simple random walk that moves to left with probability $p$ and right with probability $1-p$. How can I argue that the probability that the simple random walk stays within ...
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440 views

Check if time series has drift or drift and trend

Let's say you have some time series data that you think exhibits a drift, but you are unsure. Is it better to test first a regression with a drift and a trend to see if that is stationary? Secondly, ...
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1answer
1k views

LSTM Fitting Random Walk

I've got a question about an LSTM neural net fitting a random walk. I've made the LSTM [network shape: 1, 50, 100, 200, 50, 1] and out of interest made a completely random walk (by using a normal ...
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79 views

Show that $ f(p) \approx \sum_{k=0}^n f (\tfrac{k}{n}) \binom{n}{k} p^k (1-p)^{n-k}$

I have a probability question. We have a function $f:[0,1] \to \mathbb{R}$ that we would like to estimate. So we use the Bernstein polynomials and write: $$ f(p) \approx \sum_{k=0}^n f (\tfrac{k}{...
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151 views

Connection between optimal exponential smoothing and autocorrelation

Say we have a Bernoulli random walk with time evolving parameter $\theta_t$ which varies pretty smoothly in small increments over time. We have $n+1$ observations of $x_i$. Intuitively, if we wanted ...
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68 views

RW Metropolis and ARMS fail

I've been trying to estimate a series of simulated Gamma-distributed random variables and its structural parameters with MCMC for a stochastic volatility model. However, both the random walk ...
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1answer
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Random Walk with Drift: Why is the change in a trending variable also a function of a random variable when $E(\epsilon_t) = 0$?

I came across Pearson’s companion site of Murray, M. P. (2005). Econometrics: A modern introduction. Pearson Higher Education. While skimming through the related lecture slides here http://wps.aw.com/...
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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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Computing properties of non-uniform random walk/diffusion

I have a lot of numerical data which I'm looking to characterise as a (possibly continuous) random walk with variable (in space) step size, for example, along $x$ between $-1$ and $1$ with a step size ...
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119 views

Random Walk: Right p-value method?

Based on a stochastic matrix/graph $G$, I want to conduct a series of random walks to get an idea which are frequent interaction partners for a starting node (not only connected by the first edge). I ...
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1answer
81 views

Sequential parameter estimation of multiple discrete random walks

We have $M$ Bernoulli random walks, $X_1$ has parameter $p< 1/2$ and all the rest have parameter $1/2$. The steps are $\pm 1$. I want to identify the parameter $p$ random walk with probability of ...
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247 views

2 Dimensional Random Walk Simulation

I am trying to simulate random diffusion of particles using a random walk diffusion model. I have used probabilities of movement of particles in a 2D area, to be 1/4 in all 4 directions. The confusion ...
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1answer
4k views

Difference between random walk and martingale

I am trying to understand the diffrence between random walk and martingale. According to my understanding, a random walk without drift is $$ y_{t} = y_{t-1} + u_{t} $$ where $u_{t}$ is $i.i.d.(0, \...
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1answer
11k views

How to create a random walk model using {forecast} R package

I have a good understanding of ARIMA models but I've always found significant spikes in ACFs and PACFs that gave me the appropriate AR and MA parameters. Now I'm dealing with a series that is more ...
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1answer
332 views

Predicting point forecast using Random walk model coefficients

I have created a random walk model ARIMA(0,1,0) in R. The coefficients and R output is as shown below: ...
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81 views

Learn a random walk process with RTS smoother

I'm trying to learn a random walk process as described at section 7 of http://www.cs.cmu.edu/~epxing/papers/SDM08_Ahmed.pdf I have a set of points over N epochs. Given a set of clusters $K$, every ...
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1answer
45 views

probability that noone will have to wait for change [closed]

This is the exact question text: $2n$ children are waiting in a queue for movie ticket. Tickets are priced at a quarter each. Each child pays for the ticket either with a quarter or with half dollar ...
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322 views

continuous vs discrete random walk

For 1D random walk in discrete case the probability of finding walker at position X after N steps(P_N(X)) is binomial distribution(http://mathworld.wolfram.com/RandomWalk1-Dimensional.html), moreover ...
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1answer
85 views

Expected Number of steps

At each stage a person either moves one step to the right with probability 0.6 or one step to the left with probability 0.4.Assuming the direction of each step is independent I want to find the ...
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Expand unbounded walk

For an unbounded walk that goes up or down by one unit with probability p and q=p-1. I have that is the probability of being at position -1 at step n where $H(s)=\sum\limits_{n=0}^\infty h_ns^n$ $...
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0answers
76 views

Can my data be described as a random walk or not?

I'm trying to figure out whether the observed "time series" can be described as a random walk or not. Unfortunately a major problem regarding my data is in time intervals: Problem: Time ...
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0answers
83 views

Random walk with continuously distributed steps on $[-1,1]$

A simple random walk $S_n = X_1 +\cdots +X_n$, where $P(X_i = 1) = p \not = 0.5$ and $P(X_i=-1)= q \triangleq 1-p$, admits the following probability $$P(S_n \textrm{ reaches } a \textrm{ before} -b) =...