Scaling ODEModel initial value parameters #284
Description
Since 0.5.2 (and this merge) symfit can now estimate the initial values of ODEModels. However, there's no clear way to provide the scale of those initial values. For example, the following code fits a logistic curve to some data. Important information is the growth rate r and the initial signal s1_0. This works fine if I give a good guess for s1_0, but I know that the local minima in the model are evenly distributed in the "log" space of the initial values. If I brute force the residuals, I find the global minima occurs at r = 0.8 and log10(s1_0) = -7.5, but there are other local minima at log10(s1_0) ~ -5.5, -6.5, -8.5, -9.5 with r = 0.6, 0.7, 0.9, and 1.0, respectively.
Is there a way around this? Perhaps sympy could be used more explicitly, setting s1_0 as a relation to an intermediate value that lives in log-space?
import numpy as np
import symfit as sf
signal1 = np.array([ 5.86039495e-03, 3.61153194e-03, 2.24330124e-03, -2.92966086e-03,
-1.34148231e-03, 7.86443626e-04, 1.37743090e-03, 1.63747582e-05,
-2.96159949e-03, -4.91545334e-03, 2.25744104e-03, 8.02806834e-03,
6.86979013e-03, -3.99727245e-03, -5.43338189e-03, -4.08880449e-03,
2.11456048e-03, 1.97120626e-02, 5.00087356e-02, 1.14839342e-01,
2.26256041e-01, 3.75486015e-01, 5.36502495e-01, 6.94936368e-01,
8.08767694e-01, 8.73566838e-01, 9.04374323e-01, 9.18962105e-01,
9.33002509e-01, 9.26539485e-01, 9.22790406e-01, 9.27219537e-01,
9.30125574e-01, 9.37782682e-01, 9.31860668e-01, 9.25248598e-01,
9.27760548e-01, 9.21753351e-01, 9.28642494e-01, 9.33063441e-01,
9.26756334e-01, 9.24954549e-01, 9.24742853e-01, 9.21482340e-01,
9.26562524e-01, 9.23340814e-01, 9.32431892e-01, 9.27025424e-01,
9.18785776e-01, 9.17351300e-01, 9.22062558e-01, 9.27517326e-01,
9.25602973e-01, 9.27516517e-01, 9.26961663e-01, 9.31088965e-01,
9.24303093e-01, 9.17609376e-01, 9.23493194e-01, 9.12655604e-01])
cycles = np.arange(len(signal1))
s1,c = sf.variables('s1,c')
r,K1,s1_0 = sf.parameters('r,K1,s1_0')
lg2_e = np.log2(np.exp(1))
s1_0.min = 1e-8
s1_0.value = 10**-7.5
s1_0.max = 1e-7
K1.min = 0.1
K1.value = 1
K1.max = 2
r.min = 0.1/lg2_e
r.value = 1/lg2_e
r.max = 2/lg2_e
logistic_eqn = {
sf.D(s1,c): r*s1*(1-s1/K1)
}
logistic_model = sf.ODEModel(logistic_eqn, initial = {c:0.0,s1:s1_0})
logistic_fit = sf.Fit(logistic_model,c=cycles,s1=signal1)
fit_result = logistic_fit.execute()
#print(fit_result)
fit = logistic_model(c=cycles, **fit_result.params)
plt.plot(cycles, signal1, 'oC0')
plt.plot(cycles, fit.s1,'C0')
try:
print(fit_result)
except:
for param in [r,K1,s1_0]:
print(f'{param}={fit_result.value(param):.2f}')Here's the code for brute-forcing the residuals. Obviously, this is a very hard problem, particularly for a local minimizer...
k = 0.9190785714863928
logistic_eqn = {
sf.D(s1,c): r*s1*(1-s1/K1)
}
inits = np.arange(-12,-2,0.1)
rates = np.arange(0.1,2,0.1)
rss = np.zeros([inits.size,rates.size])
for (i,j),_ in np.ndenumerate(rss):
logistic_model = sf.ODEModel(logistic_eqn, initial = {c:0.0,s1:10**inits[i]})
fit = logistic_model(c=cycles, K1=k,r=rates[j])
residual = signal1-fit.s1
rss[i,j] = np.sum(residual**2)
And plotting the result:
plt.subplots(1,1,figsize = [12,8])
plt.contourf(rates,inits,np.log10(rss),cmap='Blues_r')
plt.ylabel('log10 Initial')
plt.xlabel('Growth Rate (Base $e$)')
cbar = plt.colorbar()
cbar.ax.set_ylabel('Log10 RSS')