Solving simple chemical network odes in pymc3 with theano - python-3.x

Question

Im trying to solve a simple chemical network A->B(reaction rate k1) and A1->B(reaction rate k2) with Bayesian inference. My hopes are to get sensitivity analysis of k1 and k2. If A, A1 and B are my constant variables only logical thing would be that if for example k1 decreases k2 should increase for some proportional amount and vice versa. But I am having some troubles with ODE's in pymc3. So here is my attempt:
import numpy as np
import matplotlib.pyplot as plt
from scipy.integrate import odeint, solve_ivp
import seaborn
import pymc3 as pm
import theano.tensor as T
from theano.compile.ops import as_op
from sys import exit
time = 10
Nt = 11
tt = np.linspace(0,time, Nt)
y0 = [1,2,0]
k1, k2 = 1, 1
#Actual Solution of the Differential Equation(Used to generate data)
def real(t,c):
da_dt = -k1*c[0]
da1_dt = -k2*c[1]
db_dt = k1*c[0] + k2*c[1]
return da_dt, da1_dt, db_dt
c_est = solve_ivp(real, t_span = [0,time], t_eval = tt, y0 = y0)
#Method For Solving the ODE
def lv(xdata, k1=1, k2=1):
def equat(c,t):
da_dt = -k1*c[0]
da1_dt = -k2*c[1]
db_dt = k1*c[0] + k2*c[1]
return da_dt, da1_dt, db_dt
Y, dict = odeint(equat,y0,xdata,full_output=True)
return Y
#Generating Data for Bayesian Inference
k1, k2 = 1, 1
ydata = c_est.y
# Adding some error to the ydata points
yerror = 10*np.random.rand(Nt)
ydata += np.random.normal(0.0, np.sqrt(yerror))
ydata = np.ravel(ydata)
#as_op(itypes=[T.dscalar, T.dscalar], otypes=[T.dvector])
def func(al,be):
Q = lv(tt, k1=al, k2=be)
return np.ravel(Q)
# Number of Samples and Initial Conditions
nsample = 5000
y0 = 1.0
sd = 0.2
# Model for Bayesian Inference
model = pm.Model()
with model:
# Priors for unknown model parameters
k1 = pm.HalfNormal('k1', sd = sd)
k2 = pm.HalfNormal('k2', sd = sd)
# Expected value of outcome
mu = func(k1,k2)
# Likelihood (sampling distribution) of observations
Y_obs = pm.Normal('Y_obs', mu=mu, sd=yerror, observed=y_data)
trace = pm.sample(nsample, nchains=1)
pm.traceplot(trace)
plt.show()
But it doesn't "loop" through equat function. Output error:
Traceback (most recent call last):
File "<ipython-input-16-14ca425a8735>", line 1, in <module>
runfile('/folder/code.py', wdir='/folder')
File "/anaconda3/lib/python3.7/site-packages/spyder_kernels/customize/spydercustomize.py", line 786, in runfile
execfile(filename, namespace)
File "/anaconda3/lib/python3.7/site-packages/spyder_kernels/customize/spydercustomize.py", line 110, in execfile
exec(compile(f.read(), filename, 'exec'), namespace)
File "/code.py", line 77, in <module>
mu = func(k1,k2)
File "/anaconda3/lib/python3.7/site-packages/theano/gof/op.py", line 674, in __call__
required = thunk()
File "/anaconda3/lib/python3.7/site-packages/theano/gof/op.py", line 892, in rval
r = p(n, [x[0] for x in i], o)
File "/anaconda3/lib/python3.7/site-packages/theano/compile/ops.py", line 555, in perform
outs = self.__fn(*inputs)
File "/code.py", line 60, in func
Q = lv(tt, k1=al, k2=be)
File "/code.py", line 42, in lv
Y, dict = odeint(equat,y0,xdata,full_output=True)
File "/anaconda3/lib/python3.7/site-packages/scipy/integrate/odepack.py", line 233, in odeint
int(bool(tfirst)))
File "/code.py", line 39, in equat
da1_dt = -k2*c[1]
IndexError: index 1 is out of bounds for axis 0 with size 1
I'm going nuts here. :( I don't even know if I am on the right path.
Edit, corrected that but now it shows another error.

Answer 1

If anyone else has difficulty here I solved it!
from scipy.integrate import odeint, solve_ivp
import numpy as np
import matplotlib.pyplot as plt
from theano.compile.ops import as_op
import theano.tensor as T
import pymc3 as pm
import copy
from sys import exit
time = 10
Nt = 11
tt = np.linspace(0,time, Nt+1)
y0 = [1,2,0]
k1, k2 = 1, 1
def real_equat(t,c):
da_dt = -k1*c[0]
da1_dt = -k2*c[1]
db_dt = k1*c[0] + k2*c[1]
return da_dt, da1_dt, db_dt
z = solve_ivp(real_equat, t_span=[0,time], t_eval= tt, y0 = y0)
def lv(xdata, k1=k1, k2=k2):
def equat(c,t):
da_dt = -k1*c[0]
da1_dt = -k2*c[1]
db_dt = k1*c[0] + k2*c[1]
return da_dt, da1_dt, db_dt
Y, dict = odeint(equat,y0,tt,full_output=True)
return Y
a = z.y
ydata = copy.copy(a)
yerror = 10*np.random.rand(Nt+1)
ydata += np.random.normal(0.0, np.sqrt(yerror))
ydata = np.ravel(ydata)
#as_op(itypes=[T.dscalar, T.dscalar], otypes=[T.dvector])
def func(al,be):
Q = lv(tt, k1 = al, k2 = be)
return np.ravel(Q)
niter = 10
model = pm.Model()
with model:
# Priors for unknown model parameters
k1 = pm.Uniform('k1', upper = 1.2, lower = 0.8)
k2 = pm.Uniform('k2', upper = 1.2, lower = 0.8)
# Expected value of outcome
mu = func(k1,k2)
# Likelihood (sampling distribution) of observations
Y_obs = pm.Normal('Y_obs', mu=mu, sd=0.2, observed=ydata)
trace = pm.sample(niter = niter, nchains=4)