Hi, I am working on an ODE system:
dx/dt = ay + bz - cx
dy/dt = -dx + ry
dz/dt = -ex + kz
Im trying to infer the values of a, b, c, d, r, e, k using 3 time series of x(t), y(t), and z(t). The problem I encounter is that values of x is between 0 and 5, but the values of y(t) and z(t) are between 10^4 to 10^7. this makes the values of a,b,r,k very small compared to the other parameters. Do you have any suggestion on how I can reparameterize the system to overcome this issue? Tia
You can change the units of things.
So do:
\hat{y} = \frac{y}{1000}
That implies:
\frac{d \hat{y}}{dt} = \frac{d y}{dt} \frac{1}{1000} = (-d x + r y) \frac{1}{1000}
\frac{d \hat{y}}{dt} = -d x \frac{1}{1000} + r \hat{y}
So now you can work with \hat{y} instead of y.
If you look at dx/dt = -cx, c is in the unit of \text{time}^{-1}, so in general you want to normalize x, y, z and t. Following @bbbales2 's notation, you’ll need \hat{t}=t/t_0 for some t_0. The rest is just chain-rule.
I did try to do what @bbbales2 suggested but I forgot to change my priors according. Now it works out very well, thanks so much :)