# Time derivative of a gaussian process

I am currently working on biomass. I am trying to quantify how much the level of uncertainties in biomass estimations will affect the level of uncertainty in biomass fluxes.

For example, I know the above ground biomass AGB of a forest $F$ at the years $y_1$, $y_2$, ..., and I know that the variance is $s^2$.

Biomass fluxes between the forest and the atmosphere are important because it allows us to qualify the forest as a sink or a source of carbon, whether the flux is positive or not.

So I want to quantify the uncertainties on the fluxes in a simple way to have rough estimates.

I am assuming I have a Gaussian Process, at the times $y_1, y_2,...$ So my process is $AGB(y, s)$. For each $y$, I have $b_y = AGB(y,s)$ which is a random variable with a normal distribution. Let's assume that I have the same variance for each $y$.

If I look at the flux $dAGB/dy$, with a covariance function which is squared exponential, I have: $$\frac{dAGB}{dy} = \frac{\mathcal N\left(b_{y+dy} - b{y},~~ 2s^2 + 2*s^2*\exp\left(\frac{-(b_{y+dy} - b{y})^2}{2*l^2}\right)\right)}{dy}$$ Let's say that $l = 100$ metric tons / ha. So between 2 consecutive years ($dy = 1$), basically if the biomass density is not changing much the variance will be about $4s^2$...

However, my measurement error in the biomass may be large, but I am pretty sure that the trend between different measures of biomass estimates is good. So I think the assumption of a Gaussian process is fundamentally wrong.

Would you have any suggestions to model something like that?

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 Welcome to the site, @ThomasC. I took the liberty of editing your question to use the $\LaTeX$ markup that CV allows. Please make sure it still says what you want. – gung Nov 24 '12 at 3:22