“Non-identifiability”, by the way, is the search term that does the trick and finds something useful. Please see: Daly et al. [1], section 3. They study indentifiability characteristics of logistic sigmoid (that has rate r and goes from zero to carrying capacity K at t=0..30) via Fisher information matrix (FIM). Quote:
When measurements are taken at times t ≤ 10, the singular vector (which is also the eigenvector corresponding to the single non-zero eigenvalue of the FIM) is oriented in the direction of the growth rate r in parameter space. For t ≤ 10, the system is therefore sensitive to changes in the growth rate r, but largely insensitive to changes in the carrying capacity K. Conversely, for measurements taken at times t ≥ 20, the singular vector of the sensitivity matrix is oriented in the direction of the growth rate K[sic], and the system is sensitive to changes in the carrying capacity K but largely insensitive to changes in the growth rate r. Both these conclusions are physically intuitive.
Then Daly et al. proceed with MCMC scheme to numerically show that samples at different parts of time domain result in different identifiability of rate and carrying capacity parameters (Figure 3.)
[1] Daly, Aidan C., David Gavaghan, Jonathan Cooper, and Simon Tavener. “Inference-Based Assessment of Parameter Identifiability in Nonlinear Biological Models.” Journal of The Royal Society Interface 15, no. 144 (July 31, 2018): 20180318. https://doi.org/10.1098/rsif.2018.0318
EDIT.
To clarify, because someone might miss it: this is not only a reply to shminux. Daly et al 2018 is (to some extent) the paper Stuart and others are looking for, at least if you are satisfied with their approach by looking what happens to effective Fisher information of logistic dynamics before and after inflection, supported by numerical inference methods showing that identifiability is difficult. (Their reference list also contains a couple of interesting articles about optimal design for logistic, harmonic models etc.)
Only thing missing that one might want AFAIK is a general analytical quantification of the amount of uncertainty, and comparison to specifically exponential (maybe along the lines Adam wrote there), and maybe writing it up in easy to digest format.
I had been looking at Fisher information myself during the weekend, noting that it might be a way of estimating uncertainty in the estimation using the Cramer-Rao bound (but quickly finding that the algebra got the better of me; it *might* be analytically solvable, but messy work).
“Non-identifiability”, by the way, is the search term that does the trick and finds something useful. Please see: Daly et al. [1], section 3. They study indentifiability characteristics of logistic sigmoid (that has rate r and goes from zero to carrying capacity K at t=0..30) via Fisher information matrix (FIM). Quote:
Then Daly et al. proceed with MCMC scheme to numerically show that samples at different parts of time domain result in different identifiability of rate and carrying capacity parameters (Figure 3.)
[1] Daly, Aidan C., David Gavaghan, Jonathan Cooper, and Simon Tavener. “Inference-Based Assessment of Parameter Identifiability in Nonlinear Biological Models.” Journal of The Royal Society Interface 15, no. 144 (July 31, 2018): 20180318. https://doi.org/10.1098/rsif.2018.0318
EDIT.
To clarify, because someone might miss it: this is not only a reply to shminux. Daly et al 2018 is (to some extent) the paper Stuart and others are looking for, at least if you are satisfied with their approach by looking what happens to effective Fisher information of logistic dynamics before and after inflection, supported by numerical inference methods showing that identifiability is difficult. (Their reference list also contains a couple of interesting articles about optimal design for logistic, harmonic models etc.)
Only thing missing that one might want AFAIK is a general analytical quantification of the amount of uncertainty, and comparison to specifically exponential (maybe along the lines Adam wrote there), and maybe writing it up in easy to digest format.
Awesome find! I really like the paper.
I had been looking at Fisher information myself during the weekend, noting that it might be a way of estimating uncertainty in the estimation using the Cramer-Rao bound (but quickly finding that the algebra got the better of me; it *might* be analytically solvable, but messy work).