Question: are multiple observations at the same time allowed?

[leavesandlemmas]

If one has multiple observations of the same quantity in biocro, e.g., having mulitple leaf N or leaf C values, do I need to average those into a single observation for that time point ?

[justinmcgrath]

Is this in the context of fitting parameters to observed values? There can be more than one observation for each time point. The function to minimize can only return a single value though, so the multiple observations need to be handled there. So if you use sums of squared residuals for example, it's the sum of deviations of each observation from the predicted value at that time point.

If this is an input to the model, then it needs to be a single value. For example, you can't have two values of vcmax for the same time point. You would need to run the model multiple times to do something like that.

[eloch216]

In my understanding, you're asking about a scenario like the following: Say I have a predicted value of $Y$ ($Y_{pred}$) and two values of $Y$ observed at the same time ($Y_1$ and $Y_2$). For example, maybe biomass was measured from two different plots on the same day. Should my least-squares error term be $E_a = (Y_{pred} - Y_{mean})^2$ (where $Y_{mean} = (Y_1 + Y_2) / 2$) or $E_b = (Y_{pred} - Y_1)^2 + (Y_{pred} - Y_2)^2$? I think these are not proportional to each other... according to some hasty napkin math, if you write them out, $E_a$ has a $Y_1 Y_2$ term but $E_b$ doesn't.

I haven't tested, but I don't think BioCroValidation::objective_function would have any issue with two observed values of something at the same time point. But my guess is that the $E_a$ option is most often the better choice, so it would be better to first calculate the mean (and use the standard deviation to weight the observations).

[justinmcgrath]

I haven't tested, but I don't think BioCroValidation::objective_function would have any issue with two observed values of something at the same time point. But my guess is that the E a option is most often the better choice, so it would be better to first calculate the mean (and use the standard deviation to weight the observations).

You would want to use the two values, $E_b$, so that the fitting can use the knowledge of uncertainty. Variance likely differs at each time point, and you want time points with the most certainly to most strongly influence the fitting. If you use the mean at each time point, each time point has equal weight. Using the mean isn't "incorrect", but it has less statistical power.

Similarly, say you collected more data at one time point than another. There is more information at that point, and it makes sense for it to more strongly influence the model, but using the mean would weight each time point equally.

If we ever did things with parameter uncertainty, it would affect that as well. But it's not really a consideration at this point.

[leavesandlemmas]

In my understanding, you're asking about a scenario like the following: Say I have a predicted value of Y ( Y p r e d ) and two values of Y observed at the same time ( Y 1 and Y 2 ).

Yes, that's what I mean. Mostly out of laziness for not wanting to compute the Y_mean

For example, maybe biomass was measured from two different plots on the same day. Should my least-squares error term be E a = ( Y p r e d − Y m e a n ) 2 (where Y m e a n = ( Y 1 + Y 2 ) / 2 ) or E b = ( Y p r e d − Y 1 ) 2 + ( Y p r e d − Y 2 ) 2 ? I think these are not proportional to each other... according to some hasty napkin math, if you write them out, E a has a Y 1 Y 2 term but E b doesn't.

They aren't the same objective function value, but the minimizer is the same. Differentiating E_b and setting it equal to zero gives

Y_pred  - Y_1 +  Y_pred - Y_2  = 0 

which is true when Y_pred = (Y_1 +Y_2)/2. It's not necessarily equivalent if one uses a standard error (or other weight).

I haven't tested, but I don't think BioCroValidation::objective_function would have any issue with two observed values of something at the same time point. But my guess is that the E a option is most often the better choice, so it would be better to first calculate the mean (and use the standard deviation to weight the observations).

I guess I'll test it first to avoid any weird bugs. I think theoretically theE_b is more general, since least squares is a special of maximizing the log-likelihoods where the likelihood is a gaussian distribution. So if the likelihood was something else (like log-normal, gamma or weibull) then calculating the mean first wouldn't make much sense. However, I don't think it really matters here since we probably won't ever do anything like that.