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I'm aware of MCMC methods such as Metropolis Hastings and friends, but these methods assume a stable posterior distribution. Is there a way to draw samples from a multidimensional timeseries? For example, the purchases on a debit card. There is correlation and seasonality in purchases: buy coffee in the morning, lunch in the afternoon, dinner at night. These purchases are heavily correlated - when you buy coffee (\$2) you buy it at a coffee shop, when you buy groceries (\$100) you buy it in a supermarket, and so on.

Edit for specificity:

For example, given a dataset of N observations, like debit card transactions:

| time | merchant | purchase | cost |
-------------------------------------
|17:00 |  shop A  |   food   | $10  |
|09:30 |  shop B  |  coffee  | $2   |
| ...  |   ...    |    ...   | ...  | 

Time is a continuous variable, merchant and purchase are categorical and cost is discrete.

Is there a method for drawing samples from these sorts of distributions, with or without making model assumptions (I'm sure there is)? I mention MCMC because it's the closest thing I know to this, but I don't know if it is the solution!

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    $\begingroup$ Metropolis-Hastings can sample from any target distribution. $\endgroup$
    – jaradniemi
    May 4 '17 at 13:52
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    $\begingroup$ @Dan If you have a specific model for the process, it's generally possible (indeed often simple) to simulate a sequence from the process-model. If you don't have a model, it's rather difficult. $\endgroup$
    – Glen_b
    May 4 '17 at 13:59
  • $\begingroup$ @Glen_b how about resampling methods designed for time series? (e.g. bootstrap) $\endgroup$
    – GeoMatt22
    May 4 '17 at 14:58
  • $\begingroup$ @Glen_b well let's say for example we have a dataset of N debit card transactions with variables time, merchant, purchase and cost. We don't know/assume the underlying distribution. How would we sample from it? $\endgroup$
    – Dan
    May 4 '17 at 15:24
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    $\begingroup$ @GeoMatt22 I specifically had those in mind when I wrote that; none of the ones I have used have typically been very satisfactory except on relatively simple kinds of problems I can already identify reasonable models for; more sophisticated versions exist than I have tried (and still more may exist than I know about) which may do better but then I believe we're already well into "difficult" for most people that would need to ask this question. When you add that the question mentions (and so presumably relates to) MCMC, the direct suitability of the bootstrap option becomes less clear. $\endgroup$
    – Glen_b
    May 4 '17 at 23:26
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You can calculate some statistic of a given multi-variate time series using boot::tsboot. According to the documentation for boot:

The replicate time series can be generated using fixed or random block lengths or can be model-based replicates.

library(tidyverse)
library(magrittr)
library(mvtnorm)
library(boot)

set.seed(123)
sigma <- diag(c(2,3))    
sigma[1,2] <- sigma[2,1] <- 0.8
n <- 200
eps <- mvtnorm::rmvnorm(n = n, mean = rep(0, 2), sigma = sigma)

## Simulate two correlated AR(1) process
xy <- data.frame(x = arima.sim(n = n, model = list(ar = c(0.7)), innov = 
  eps[,1]), y = arima.sim(n = n, model = list(ar = c(-0.4)), innov = eps[,2]))

## Examine the correlation
cor_xy <- function(xy){cor(xy$x, xy$y)}
xy %>% cor_xy()

## Examine the correlation using bootstrap
(b = boot::tsboot(xy, cor_xy, R = 1000, l = 20, sim = "fixed"))
plot(b)
BLOCK BOOTSTRAP FOR TIME SERIES

Fixed Block Length of 20 

Call:
boot::tsboot(tseries = xy, statistic = cor_xy, R = 1000, l = 20, 
    sim = "fixed")


Bootstrap Statistics :
      original       bias    std. error
t1* 0.07804724 -0.003418753  0.05543288

The bootstrap results can be visualized immediately using plot().

enter image description here

Discussion about the general bootstrap method can be found in How do you do bootstrapping with time-series data?. See Understanding the output of a bootstrap performed in R for detailed interpretation of output in boot::tsboot.

The way I simulate the multivariate time series is from How do you simulate two correlated AR(p) time series?.

For now, I don't know how to extract the samples from the bootstrap, which might be important in some cases. I am not sure if the algorithm is similar to that for univariate time series. Also, I don't know if the method applies to a time series and a categorical series.

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