Questions tagged [discrete-time]

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Training discrete time hazard model with xgboost on right censored loan data

I am currently developing a loan default risk model using a discrete time hazard approach with xgboost. The goal is to generate a series of predicted monthly default probabilities using a new ...
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1 vote
1 answer
24 views

Censoring in a discrete-time hazard model of time-to-drop-out of a clinical trial

I am conducting a discrete-time hazard model where the outcome being analysed is time to drop out of treatment during a 12-week clinical trial examining the effect of an agonist drug on days of ...
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Is is acceptable to use cox proportional hazard regression when time-to-event is a discrete, numeric variable?

I recently submitted a paper where I performed a cox proportional hazards regression model modelling the effect of group allocation in a randomised controlled trial on treatment retention. The event ...
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19 views

When are continuous-time models important?

In Econometrics, majority of the models are in discrete-time setting, whereas when you move on to quantitative finance, continuous-time models are most prevalent. I get the theory and idea behind ...
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Determine order of events happening in discrete timespace

At $t=0$ I have a certain (active) population, lets say 1,000 customers. In each time step I expect a certain amount I have a probability of death of 1% and a prepayment rate of 5%. At the end of each ...
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Probability $X_n=k$ for a Markov Chain over the integers

Let $\{X_n\}$ be a Markov chain with state space $\mathcal{S}=\mathbb{Z}$, and $X_0=0,$ and $p_{0,1}=p_{0,-1}=1/2,$ and $p_{i,i+1}=1$ for all $i \ge 1,$ and $p_{i,i-1}=1$ for all $i\le -1.$ The ...
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62 views

Proving positive recurrence for a finite state space DTMC

Let $$P = \begin{pmatrix} 0.3 & 0.5 & 0 & 0 & 0 & 0.2 \\ 0 & 0.5 & 0 & 0.5 & 0 & 0 \\ 0 & 0 & 1 & 0 & 0 & 0 \\ 0 & 0.3 & 0 & 0 &...
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4 votes
1 answer
226 views

Calculating the expected number of visits to a state by a DTMC

Suppose we have a DTMC $X$ : $\{X_n : n = 0, 1,2,\dots\}$, a transition probability matrix $P$, and state space $S = \{1,2,3\}$. Suppose I want to calculate the expected amount of times we visit state ...
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  • 427
3 votes
1 answer
92 views

Conditional probabilities with Discrete Time Markov Chain non-perishable inventory

Suppose $\{X_n : n =0,1,2,\dots \}$ is a DTMC that represents the inventory level at the end of day $n$. We have inventory policy $(2,4)$, i.e., if $X_n < 2$, we order enough units to have ...
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What statistical tests allow for a comparison of the "roughness" of discrete functions?

I have a data set of the revenues of 10 different large companies from the years 2000 through 2019. Here they are all plotted in one graph. The y-axis has a unit of billions of Euros:                  ...
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1 answer
35 views

Using test hypothesis to prove I ran a sufficient number of simulations

I simulated a system using discrete-events simulation, more particularly next-event simulation. The system is M/M/1 (infinite FIFO waiting room). Whenever the system is jobless, the server shutdowns. ...
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0 answers
21 views

Discrete Time Logistic

I'm trying to use panel survey data to determine married event in life using discrete logistic regression. However, some observations enter the survey not the same time as other i.e. some enter in ...
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0 answers
62 views

Generating correlated discrete random variables

Suppose that we have $q_t \in \{-1, 1\}$ where $\mathbb{P}(q_t = -1) = \mathbb{P}(q_t = 1) = \frac{1}{2}$. Further, assume that \begin{align} Cor\left( q_t, q_{t-k} \right) = \begin{cases} ...
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3 votes
1 answer
334 views

Z transform of $ 2^{-|n|} $

Dears, I'm trying to compute the Z-transform of $$ x(n) = 2^{-|n|} $$. My procedure is as follows: Using definition of Z transform: $$ X(z) = \sum_{n=-\infty}^{\infty}2^{-|n|} z^{-n} = \sum_{n=-\infty}...
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2 votes
1 answer
27 views

Variable data measured discretely with poor resolution

Is there an appropriate method for estimating variation when variable data is measured discretely where the resolution is poor. Here's an example: You are looking at how long it takes for a solid to ...
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2 votes
0 answers
46 views

Negative Binomial vs. Discrete Survival

I am trying to model a discrete duration variable (number of weeks) and I am not sure which model to use. Many would argue that durations are modeled using survival models. However, in J. Hilbes ...
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25 views

Any methods for clustering short time series data

I have around 200 different sets of time series data with a semi-annual periodicity. The length of each series varies from 1 to 6 years. I essentially want to build unsupervised clusters by grouping ...
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2 votes
0 answers
183 views

Proving/showing that the Markov property holds in discrete time Markov chain example

I am currently studying the textbook Introduction to Modeling and Analysis of Stochastic Systems, Second Edition, by V. G. Kulkarni. In a section on discrete-time Markov chains, the author presents ...
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1 vote
1 answer
653 views

Rows and columns of the one-step transition probability matrix

I am currently studying the textbook Introduction to Modeling and Analysis of Stochastic Systems, Second Edition, by V. G. Kulkarni. In a section on discrete-time Markov chains, the author introduces ...
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  • 1,126
0 votes
1 answer
700 views

Regression analysis with small data set

We've performed a regression analysis with 14 data points (Y's) that originate from 9 X's (plant batches were split up). I'm well aware that this is a small data set and there are caveats. However, ...
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2 votes
1 answer
90 views

Statistical modeling multidimensional discrete system

I have a system which its state is described by a vector $v=(a, b, c)$, where $a$, $b$ and $c$ can take any value between $0$ and $100$ and where $a+b+c <= 100$. I have observations of the state ...
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0 votes
1 answer
49 views

Detecting Significant Changes in Time Series

I’m hoping someone might be able to point me in the right direction for this problem. I’ve anonymised the problem by describing it in terms of supermarket purchase data rather than the real context so ...
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1 vote
1 answer
205 views

Posterior distribution from piecewise likelihood

Consider a hierarchical Bayesian model for analysing data from an inhomogeneous Poisson process that we observe in discrete time. Let $Y_i, i = 1,...,n$, be the number of events occurring in the time ...
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1 vote
0 answers
74 views

Modeling Time in the Discrete Time Analysis using Left-truncated Data

I am doing multiple-spell, discrete-time analysis to examine factors shaping time to exiting a renter spell, with time-varying covariates at monthly interval. My data has left-truncated spells, ...
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-1 votes
1 answer
457 views

INAR(1) simulation in R [closed]

How do we simulate values of $Y_t$ for a maximum value of $t=60$ when we have an INAR(1) process as follows: $Y_t=ρ^*Y_{(t-1)}+R_t$ where $t$ takes values from 1 to 60, $ρ=0.3,0.8$, $ρ^*$ is ...
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  • 1
0 votes
1 answer
291 views

Simulating a (discretized) Cox process via binomial sampling

Let X be a Cox process (doubly-stochastic Poisson process) driven by a Poisson process with fixed intensity(rate) $\lambda=50$ , and choose some small time interval $dt=0.01$ . Is the proper way to ...
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0 votes
1 answer
111 views

How to recognize an ARMA process?

By looking at the autocovariance, how could you recognise what discrete model (MA(q), AR(p), or ARMA(p,q)) is more appropriate to describe your data?
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4 votes
2 answers
426 views

Simulation -Discrete Time Hazard model

How we can simulate the survival times for fitting discrete time hazard model?
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5 votes
1 answer
2k views

How to forecast integer time series in R?

For a while now I used to forecast integer/count time series as I would do for any other continuous time series, meaning : I use models like ARIMA, ETS, THETA, TBATS ... And then I simply round the ...
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3 votes
0 answers
77 views

best statistical approach to study the time evolution of clustering in a data set

I am using a stochastic method for the clustering of a data set. The number of clusters that this approach returns, can differ in each iteration. On the other hand, I would like to study the evolution ...
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7 votes
0 answers
92 views

Is average stopping time a continuous function of Bernoulli parameter?

Consider an infinite sequence $X = (X_i)_{i \in \mathbb N}$ of i.i.d Bernoulli random variables with (unknown) parameter $p \in (0,1)$, and let $N$ be a stopping time on $X$. Is it always the case ...
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2 votes
0 answers
29 views

Gaussian random variable over multiple periods

Let's assume that : $$X_0=X_0$$ $$X_1=(1+R_1)*X_0$$ $$...$$ $$X_N=(1+R_N)*X_{N-1}$$ where $R_i \sim \mathcal{N}(0,u^2)$. I am trying to compute the probability $P(X_N < a)$. How accurate is it to ...
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0 votes
1 answer
2k views

Linear mixed model: Time as continuous or discrete variable?

I am analysing a dataset from a randomised controlled trial (2 treatment groups) with measurements at 3 time points (weeks 0, 1 and 8). I am struggling with whether to analyse this with the three time ...
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  • 1
0 votes
1 answer
132 views

Probability distribution for day of the year [closed]

"Day of the year" is an integer from 1 to 365, indicating the day within the year. I'd like to model the day of the year that a particular event occurs. The event must occur between day 100 and day ...
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  • 1,051
4 votes
1 answer
3k views

discrete time survival analysis

I would greatly appreciate if you could let me know how to do discrete time survival analysis with time varying covariates. Some part of my data set is as follows: ...
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0 votes
1 answer
1k views

parametric survival regression and discrete time survival regression

I would greatly appreciate if you could let me know how to choose among different parametric distributions including gama, Weibull, lognormal, loglogistic and etc for panel (time series cross ...
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  • 227
0 votes
0 answers
28 views

Looking for suitable method to update predictions

consider an e-commerce website which daily updates their transaction data. I take interest in predicting the revenue an account generates over a certain time range, say until the fifth year after the ...
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2 votes
0 answers
160 views

Difference between count time and discrete time survival analysis

Is there a practical or theoretical difference between count time survival analysis (e.g. ctset/cttost in Stata), where counts ...
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3 votes
0 answers
426 views

Interpreting a (integer valued) discrete predictor as a continuous variable?

I am currently working on a multiple linear regression in order to evaluate the impact of some variables on a contract's length, which can only take integer values (years). As I am only interested in ...
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5 votes
0 answers
366 views

Describe AR process with additive white noise using ARMA process

Disclaimer: This is a homework problem This is a problem from "Adaptive Filter Theory" by Haykin. Problem 2.10 (2nd edition). Problem A discrete-time stochastic process $\{x(n)\}$ that is real-...
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  • 333
1 vote
1 answer
220 views

Algorithms for extracting moving average intervals

I have a series of discrete events and a global clock. Whenever an event happens, I record a timestamp and do some processing. Specifically, I am trying to compute "moving average" (not really) ...
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0 votes
0 answers
203 views

How to check assumptions for a discrete time survival analysis

What tests are required before and after estimating a discrete time survival model? In other words, how the assumptions should be tested and how the generalization of results should be investigated ...
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  • 227
1 vote
1 answer
118 views

Coefficients in discrete proportional hazards model

I am aware of the rationale behind taking the exponentiated coefficients in a proportional hazards model as representing change in hazard per unit change in the corresponding predictor. This also ...
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1 vote
1 answer
251 views

I have a discrete-valued time series and would like to analyse it, but dont know where to start

Due to some health issues, each day at about the same time, I give myself a score which represents the state of my health: specifically, fatigue, with 1 being the worst and 10 the best. In practice, ...
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0 votes
1 answer
340 views

How to calculate the probability of death between two discrete time periods using survival curves

I was hoping somebody could help, I am trying to work out the probability of death between two time points on a survival curve. i.e. I have my survival curve as follows (this is an example Kaplan-...
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0 votes
2 answers
2k views

When is time treated as a discrete variable?

Time is usually treated as a continuous variable but in some cases it is discrete. An example would be with a drug study and measurements are taken at 1, 2 and 3 hours. Am I right to think an ...
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1 vote
2 answers
229 views

Resource request : How to prove the output of a process is random variables?

I am reading through articles which present the spectral properties of chaotic systems such that they can be candidates for generating pseudo random binary sequences. One such article, is http://...
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1 vote
0 answers
94 views

Difference between Cox Proportional Harzards Model and interval-based logistic model for survival regressions

I have a dataset of timestamped trader transactions (to the second). I want to conduct a survival analysis on the trades where a trade is started at startTime and ...
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0 votes
2 answers
134 views

Discrete survivor function expressed in terms of hazard

Let $T$ can take on values $t_1,t_2,\ldots,$ with $0\le t_1\le t_2,\ldots,$ and let the probability function be $$f(t_j)=Pr(T=t_j),\quad j=1,2,\ldots$$ The survivor function is then $$S(t)=Pr(T\ge ...
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3 votes
0 answers
242 views

Discrete time generator of stochastic process

While looking at one paper about Metropolic Hasting optimal convergence rates, I came accross a discrete time generator of Markov chain. It is defined as follows: $$G V(x)=nE\left [ \left( V(y)-V(x)\...
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