我正在尝试使用均值等于那些ODE解决方案对参数和初始值的选择的对数的似然来对ODE参数的参数进行采样。这基于在here中找到的教程。我可以复制该教程，但不能使我的模型正常工作。我模型的ODE为：
dQ（t）/ dt =（1 / K）*（R（t）-Q（t））
其中R（t）基于我输入的降雨数据。 我为噪声标准偏差\ sigma，初始值Q（0）和参数K分配了先验值。 任何有关如何克服该错误的帮助将不胜感激：）
这是我的代码：
from scipy.integrate import odeint
from scipy.interpolate import interp1d
import numpy as np
import pandas as pd
import theano
from theano import *
import pymc3 as pm
import theano.tensor as tt
THEANO_FLAGS='optimizer=fast_compile'
theano.config.exception_verbosity= 'high'
theano.config.floatX = 'float64'
n_states = 1
n_odeparams = 1
n_ivs = 1
class LinearReservoirModel(object):
def __init__(self, n_states, n_odeparams, n_ivs,net_rainfall_data, y0=None):
self._n_states = n_states
self._n_odeparams = n_odeparams
self._n_ivs = n_ivs
self._y0 = y0
self._nr = net_rainfall_data
def simulate(self, parameters, times):
return self._simulate(parameters, times, self._nr, False)
def simulate_with_sensitivities(self, parameters, times):
return self._simulate(parameters, times, self._nr, True)
def _simulate(self, parameters, times, net_rainfall_data, sensitivities):
k, q0 = [x for x in parameters]
# Interpolate net_rainfall
nr_int = interp1d(times, net_rainfall_data,fill_value="extrapolate",kind='slinear')
def r(q,time,k,nrint):
return (nrint(time) - q) * (1./k)
if sensitivities:
def jac(k):
ret = np.zeros((self._n_states, self._n_states))
ret[0, 0] = (-1./k)
return ret
def dfdp(x,t,k,nrint):
ret = np.zeros((self._n_states,
self._n_odeparams + self._n_ivs))
ret[0, 0] = (-1./(k**2)) * (nrint(t) - x)
return ret
def rhs(q_and_dqdp, t, k, nrint):
q = q_and_dqdp[0:self._n_states]
dqdp = q_and_dqdp[self._n_states:].reshape((self._n_states,
self._n_odeparams + self._n_ivs))
dqdt = r(q, t, k, nrint)
# print('jacobian',jac(q))
# print('dqdp',dqdp)
# print('dfdp',dfdp(q,t,nrint))
d_dqdp_dt = jac(k)*dqdp + dfdp(q,t,k,nrint) # CHANGED CODE HERE np.matmul(jac(q), dqdp) + dfdp(q,t,nrint)
return np.concatenate((dqdt, d_dqdp_dt.reshape(-1)))
y0 = np.zeros( (n_states*(n_odeparams+n_ivs)) + n_states ) # CHANGED CODE HERE 2*
y0[2] = 1. #\frac{\partial [X]}{\partial Xt0} at t==0, and same below for Y
y0[0:n_states] = q0
result = odeint(rhs, y0, times, (k,nr_int),rtol=1e-6,atol=1e-5)
values = result[:, 0:self._n_states]
dvalues_dp = result[:, self._n_states:].reshape((len(times),
self._n_states,
self._n_odeparams + self._n_ivs))
return values, dvalues_dp
else:
q = odeint(r,q0,times,args=(k,nr_int),rtol=1e-6,atol=1e-5)
q_flat = [item for sublist in q for item in sublist]
return q_flat
q = [0.01, 0.084788051,0.289827287,0.487426902,0.623592162,0.855202214,0.901709887,0.87936577,0.857067839,0.775516564,0.701725939,0.675138958,0.68101658,0.64644605,0.701305112,0.747128907,0.676039744,0.668502137,0.731464651,0.766588801]
nr = [1.618666063,0.0001,4.405308823,0.394073731,3.392555321,2.733285785,0.0001,1.31186209,0.0001,0.0001,0.0001,0.83074128,0.646141131,0.0001,2.405660466,0.0001,0.0001,1.174002978,1.481146447,0.73244669]
ode_model = LinearReservoirModel(n_states, n_odeparams, n_ivs, nr)
class ODEGradop(theano.Op):
def __init__(self, numpy_vsp):
self._numpy_vsp = numpy_vsp
def make_node(self, x, g):
x = theano.tensor.as_tensor_variable(x)
g = theano.tensor.as_tensor_variable(g)
node = theano.Apply(self, [x, g], [g.type()])
return node
def perform(self, node, inputs_storage, output_storage):
x = inputs_storage[0]
g = inputs_storage[1]
out = output_storage[0]
out[0] = self._numpy_vsp(x, g) # get the numerical VSP
class ODEop(theano.Op):
def __init__(self, state, numpy_vsp):
self._state = state
self._numpy_vsp = numpy_vsp
def make_node(self, x):
x = theano.tensor.as_tensor_variable(x)
return theano.Apply(self, [x], [x.type()])
def perform(self, node, inputs_storage, output_storage):
x = inputs_storage[0]
out = output_storage[0]
out[0] = self._state(x) # get the numerical solution of ODE states
def grad(self, inputs, output_grads):
x = inputs[0]
g = output_grads[0]
grad_op = ODEGradop(self._numpy_vsp) # pass the VSP when asked for gradient
grad_op_apply = grad_op(x, g)
return [grad_op_apply]
class solveCached(object):
def __init__(self, times, n_params, n_outputs):
self._times = times
self._n_params = n_params
self._n_outputs = n_outputs
self._cachedParam = np.zeros(n_params)
self._cachedSens = np.zeros((len(times), n_outputs, n_params))
self._cachedState = np.zeros((len(times),n_outputs))
def __call__(self, x):
if np.all(x==self._cachedParam):
state, sens = self._cachedState, self._cachedSens
else:
state, sens = ode_model.simulate_with_sensitivities(x, times)
return state, sens
times = np.arange(0, len(q)) # number of measurement points (see below)
cached_solver=solveCached(times, n_odeparams + n_ivs, n_states)
def state(x):
State, Sens = cached_solver(np.array(x,dtype=np.float64))
cached_solver._cachedState, cached_solver._cachedSens, cached_solver._cachedParam = State, Sens, x
return State.reshape((len(State),))
def numpy_vsp(x, g):
numpy_sens = cached_solver(np.array(x,dtype=np.float64))[1].reshape((n_states*len(times),len(x)))
return numpy_sens.T.dot(g)
# Define the data matrix
Q = np.vstack((q))
# Now instantiate the theano custom ODE op
my_ODEop = ODEop(state,numpy_vsp)
# The probabilistic model
with pm.Model() as LR_model:
# Priors for unknown model parameters
k = pm.Uniform('k', lower=0.01, upper=10)
# Priors for initial conditions and noise level
q0 = pm.Lognormal('q0', mu=np.log(1.2), sd=1)
sigma = pm.Lognormal('sigma', mu=-1, sd=1, shape=1)
# Forward model
all_params = pm.math.stack([k,q0],axis=0)
ode_sol = my_ODEop(all_params)
forward = ode_sol.reshape(Q.shape)
# log_forward = pm.math.log(forward)
# log_forward_print = tt.printing.Print('log_forward')(log_forward.shape)
# tt.printing.Print('sigma')(sigma.shape)
# Likelihood
Q_obs = pm.Lognormal('Q_obs', mu=pm.math.log(forward), sd=sigma, observed=Q)
print(LR_model.check_test_point())
# Y_obs_print = tt.printing.Print('Y_obs')(Y_obs)
trace = pm.sample(n_init=1500, tune=1000, chains=1, init='adapt_diag')
trace['diverging'].sum()
如果运行上面的代码，则应该能够重现以下错误：
Traceback (most recent call last):
File "examples/myexample.py", line 195, in <module>
trace = pm.sample(1500, tune=1000, chains=1, init='adapt_diag')
File "/Users/Yannis/.pyenv/versions/mini-project/lib/python3.6/site-packages/pymc3/sampling.py", line 457, in sample
trace = _sample_many(**sample_args)
File "/Users/Yannis/.pyenv/versions/mini-project/lib/python3.6/site-packages/pymc3/sampling.py", line 503, in _sample_many
step=step, random_seed=random_seed[i], **kwargs)
File "/Users/Yannis/.pyenv/versions/mini-project/lib/python3.6/site-packages/pymc3/sampling.py", line 544, in _sample
for it, strace in enumerate(sampling):
File "/Users/Yannis/.pyenv/versions/mini-project/lib/python3.6/site-packages/tqdm/std.py", line 1091, in __iter__
for obj in iterable:
File "/Users/Yannis/.pyenv/versions/mini-project/lib/python3.6/site-packages/pymc3/sampling.py", line 633, in _iter_sample
point, states = step.step(point)
File "/Users/Yannis/.pyenv/versions/mini-project/lib/python3.6/site-packages/pymc3/step_methods/arraystep.py", line 247, in step
apoint, stats = self.astep(array)
File "/Users/Yannis/.pyenv/versions/mini-project/lib/python3.6/site-packages/pymc3/step_methods/hmc/base_hmc.py", line 144, in astep
raise SamplingError("Bad initial energy")
pymc3.exceptions.SamplingError: Bad initial energy
PyMC3版本：3.7 Theano版本：1.0.4 Python版本：3.6.5 操作系统：macOS Catalina（v10.15.1） 如何安装PyMC3：pip（在pyenv virtualenv中管理）