How do I compute the bounds of random walk for a given ellapsed time?

Question

I am trying to model random walk of a gyro, given some manufacturer specifications of maximum random walk in units of degrees per root-hour.

My first step was to generate white noise with a standard deviation derived from the specifications, like so.

imu_rate  = 200   # hz
stop_time = 3600  # sec
n_samples = imu_rate * stop_time
time = np.arange(n_samples) / imu_rate

# monte carlo
runs = 5
arw = 0.05                           # deg/rt-hour (MAX
arw /= 60                            # deg/rt-sec  
sigma = arw * np.sqrt(imu_rate) / 3  # deg - standard deviation of noise

white_noise = np.random.normal(0, sigma, size=(n_samples, runs))
random_walk = white_noise.cumsum(0) / imu_rate

plt.plot(time, random_walk);

Now, to the point of the question, I want to verify that that this method of modelling the sensor noise required to generate random walk within these bounds is correct. I want to do so by generating random walk and plotting it against some sort of bounds.

My first inclination was just to take the square root of time do a cumulative sum at each time step. However, that gives me really large bounds, completely dwarfing the scale of the random walk.

bounds = arw * np.sqrt(t)
plt.plot(time, random_walk);
plt.plot(time, bounds.cumsum())

So as the title says, how do I compute the bounds of random walk vs time?

Answer 1

It is more appropriate for this site to use equations rather than code.

You are basically constructing a random walk: $$ Y=\sum_{n=1}^N X_n $$ with $(X_n)$ iid centered gaussian variables of variance $\sigma^2$. You need to be careful of what you mean by bound. If you are talking about the standard variation of $Y$: $\Delta Y$, then it is just (you don’t even need the fact that it is gaussian): $$ \Delta Y^2=n\sigma^2\\ \Delta Y=\sqrt n\sigma $$ If in your first time step, your were computing the cumulative sum of the square root of time, then you are pretty off: $$ \sum_{n=1}^N\sqrt n\sim \sqrt n^3 $$ which dwarfs the $\sqrt n$ growth of the variance.

Another definition of the bound could be the quantiles. Since $Y$ is also a centered gaussian of variance $ n\sigma^2$, then these also grow as $\sqrt n$.

Hope this helps.