# Generating causally dependent random variables

I'm trying to generate sets of causally connected random variables and started off doing this with a monte carlo approach.

The baseline is a 2-dimensional measured histogram from which I draw random values.

In my concrete examples these variables are acceleration $\bf{a}$ and velocity $\bf{v}$ - so obviously $v_{i+1} = v_{i} + a_i * dt$ has to hold.

My current naive approach is:

I start with a some $v_0$. Then I generate a random $a_0$ according to the measured probability of $\bf{a}$ for the value of $v_0$. Using this $a_0$ I can calculate $v_1$ and the whole procedure starts over again.

So when I check the generated accelerations $\bf{a}$ in bins of $\bf{v}$ everything's fine. But I obviously this does not at all respect the marginal distribution of $\bf{v}$.

I'm kind of familiar with basic monte carlo methods, though lacking some theoretical background as you might guess. I'd be fine if the two variables where just connected by some correlation matrix, but the causal connection between the two gives me headaches.

I didn't manage to find an example for this kind of problem somewhere - I might be googl'ing the wrong terms. I'd be satisfied if somebody could point me to some literature/example or promising method to get a hold on this.

(Or tell me that's is not really possible given my inputs - that's what I'm guessing occasionally...)

EDIT:

The actual aim of this whole procedure: I have a set of measurements $\bf{a}$ and $\bf{v}$, represented in a two-dimensional histogram $N(a,v)$. Given this input I'd like to generate sets of random $\bf{a_r}$ and $\bf{v_r}$ that reproduce the measured distribution.

• An interesting question. However, the second "obviously" (about not respecting the marginal distribution) is not at all clear to me. Why is it obvious? The distribution of $(v,a)$, as reflected by your "two-dimensional histogram," depends on how you have sampled these variables; I wonder whether this might explain possible differences. What kind of data are represented by this histogram and how exactly do you "draw values" from it? – whuber Oct 10 '13 at 16:44
• Well, to me its kind of obvious, because the $\bf{a}$ distributions are pretty much symmetric around zero. So when generate the $a_i$ there's no dependency on $v$. When the current $v$ is at the upper edge of the marginal $\bf{v}$ distribution, you'd assume that there should be a bias towards negative $a_i$. "draw values" refers to: take the 1-dim probability distrbution, built the cumulative distribution, throw a random number $r$ between 0 and 1, find the $x$ where the cum. distribution has the value $r$. This $x$ is my "drawn value" – sebastian Oct 14 '13 at 6:17
• For completeness: the data originates from gps-logging. I have a set of logged trips in cars, which log speed with 1Hz. So their's a pair of $v$ and $a$ for every datapoint. These are filled into the histogram. – sebastian Oct 14 '13 at 6:28
• Your comments indicate you are assuming that $\mathbf{a}$ and $\mathbf{v}$ are independent. That cannot possibly be, because there are physical limitations to speeds: that means many accelerations will not be experienced at the most extreme speeds. However, it's not easy to provide more detailed advice because you haven't articulated what you're trying to accomplish; instead, you have described an approach to solving an unstated problem. Why don't you change this question and ask instead about the problem you need to solve rather than how to implement a solution that looks invalid? – whuber Oct 14 '13 at 14:14

It seems that in order to reproduce the joint distribution $\rho(a,v)$, you should select new $a$ not only based on $v$, but based on the old $a$ also:

$a_{i+1} \sim \rho'(a_{i+1}|a_i, v_i)$

The question (to which I don't know the answer yet) is how to find $\rho'$ which produces $\rho$.

UPD: You are to solve the following integral equation:

$$\rho(a, v) = \int da' \rho'\left(a|a', v-{a+a'\over 2}\Delta t\right) \rho(a', v-{a+a'\over 2}\Delta t)$$

Approximating the function $\rho$ with a histogram, you turn this to a system of linear equations:

$$\cases{ \rho(a, v) = \sum_{a'} \rho'\left(a|a', v-{a+a'\over 2}\Delta t\right) \rho(a', v-{a+a'\over 2}\Delta t) \\ \sum_a \rho'\left(a|a', v'\right) = 1}$$

This system is underdetermined. You may apply a smoothness penalty to obtain a solution.

Doesn't the gps data contain position $p$? I would have thought that, not only is $v_{i+1}$ dependent upon $v_{i}$ and $a_{i}$ but $a_{i+1}$ would also be dependent upon $p_{i}$. Consider: in any road network there are bottlenecks, speed limits, signals, intersections, steep gradients, etc. that are geolocated. So something like an ensemble (distribution) defined by:

$F_{a} = Pr ( A_{i+1} \le a_{i+1}\ |\ a_{i},v_{i},p_{i} )$
$v_{i+1} = v_{i} + a_{i}dt$

For such an ensemble, the difficulty will lay in the nature of the data. It is likely that the true population will be asymmetric, non-linear (piece-wise) and may not have defined moments. These characteristics may not be evident within the sample you have at hand.

As @whuber has stated, the problem, ie exactly what you are seeking to produce, does not yet seem fully and clearly defined. It is not clear as to whether you are interested in the ensemble or more so the individuals.

• I think my problem is rather clear - I have the measured distribution of $\bf{v}$ and $\bf{a}$ and from this I'd like to sample a pseudo-random $\bf{v_{rand}}$, that ultimately reproduces the input. I'm well aware of your point on whether what comes out of it is realistic, but that's a different question... – sebastian Oct 17 '13 at 7:50
• At the very least, as indicated in the equation above, this would not be a stationary effect. I would think that a first step would be to bin the readings according to time interval and then compare them. I don't know how many readings you have but this comparison could be run through something like Pearson's Distribution as a starting point - to try to classify the nature of the distribution. – AsymLabs Oct 17 '13 at 8:09