这里我的目标是估计由
给出的阻尼谐振子的参数（gama和omega）DX ^ 2 / dt的^ 2 +伽马* DX / DT +（2 * PI *欧米加）^ 2 * X = 0。 （我们可以在系统中添加白高斯噪声。）
import pymc
import numpy as np
import scipy.io as sio
import matplotlib.pyplot as plt;
from scipy.integrate import odeint
#import data
xdata = sio.loadmat('T.mat')['T'][0] #time
ydata1 = sio.loadmat('V1.mat')['V1'][0] # V2=dV1/dt, (X=V1),
ydata2 = sio.loadmat('V2.mat')['V2'][0] # dV2/dt=-(2pi*omega)^2*V1-gama*V2
#time span for solving the equations
npts= 500
dt=0.01
Tspan=5.0
time = np.linspace(0,Tspan,npts+1)
#initial condition
V0 = [1.0, 1.0]
# Priors for unknown model parameters
sigma = pymc.Uniform('sigma', 0.0, 100.0)
gama= pymc.Uniform('gama', 0.0, 20.0)
omega=pymc.Uniform('omega',0.0, 20.0)
#Solve the equations
@pymc.deterministic
def DHOS(gama=gama, omega=omega):
V1= np.zeros(npts+1)
V2= np.zeros(npts+1)
V1[0] = V0[0]
V2[0] = V0[1]
for i in range(1,npts+1):
V1[i]= V1[i-1] + dt*V2[i-1];
V2[i] = V2[i-1] + dt*(-((2*np.pi*omega)**2)*V1[i-1]-gama*V2[i-1]);
return [V1, V2]
#or we can use odeint
#@pymc.deterministic
#def DHS( gama=gama, omega=omega):
# def DOS_func(y, time):
# V1, V2 = y[0], y[1]
# dV1dt = V2
# dV2dt = -((2*np.pi*omega)**2)* V1 -gama*V2
# dydt = [dV1dt, dV2dt]
# return dydt
# soln = odeint(DOS_func,V0, time)
# V1, V2 = soln[:,0], soln[:,1]
# return V1, V2
# value of outcome (observations)
V1 = pymc.Lambda('V1', lambda DHOS=DHOS: DHOS[0])
V2 = pymc.Lambda('V2', lambda DHOS=DHOS: DHOS[1])
# liklihood of observations
Yobs1 = pymc.Normal('Yobs1', mu=V1, tau=1.0/sigma**2, value=ydata1, observed=True)
Yobs2 = pymc.Normal('Yobs2', mu=V2, tau=1.0/sigma**2, value=ydata2, observed=True)
通过将上述代码保存为DampedOscil_model.py，我们可以按如下方式运行PYMC
import pymc
import DampedOscil_model
MDL = pymc.MCMC(DampedOscil_model, db='pickle')
MDL.sample(iter=1e4, burn=1e2, thin=2)
gama_trace=MDL.trace('gama')[- 1000:]
omega_trace=MDL.trace('omega')[-1000:]
gama=MDL.gama.value
omega=MDL.omega.value
效果很好（见下文）。
现在我将此代码转换为PYMC3以使用NUTS和ADVI。
import matplotlib.pyplot as plt
import scipy.io as sio
import pandas as pd
import numpy as np
import pymc3 as pm
import theano.tensor as tt
import theano
from pymc3 import Model, Normal, HalfNormal, Uniform
from pymc3 import NUTS, find_MAP, sample, Slice, traceplot, summary
from pymc3 import Deterministic
from scipy.optimize import fmin_powell
#import data
xdata = sio.loadmat('T.mat')['T'][0] #time
ydata1 = sio.loadmat('V1.mat')['V1'][0] # V2=dV1/dt, (X=V1),
ydata2 = sio.loadmat('V2.mat')['V2'][0] # dV2/dt=-(2pi*omega)^2*V1-gama*V2
#time span for solving the equations
npts= 500
dt=0.01
Tspan=5.0
time = np.linspace(0,Tspan,npts+1)
niter=10000
burn=niter//2;
with pm.Model() as model:
#Priors for unknown model parameters
sigma = pm.HalfNormal('sigma', sd=1)
gama= pm.Uniform('gama', 0.0, 20.0)
omega=pm.Uniform('omega',0.0, 20.0)
#initial condition
V0 = [1.0, 1.0]
#Solve the equations
# do I need to use theano.tensor here?!
@theano.compile.ops.as_op(itypes=[tt.dscalar, tt.dscalar],otypes=[tt.dvector])
def DHOS(gama=gama, omega=omega):
V1= np.zeros(npts+1)
V2= np.zeros(npts+1)
V1[0] = V0[0]
V2[0] = V0[1]
for i in range(1,npts+1):
V1[i]= V1[i-1] + dt*V2[i-1];
V2[i] = V2[i-1] + dt*(-((2*np.pi*1)**2)*V1[i-1]-gama*V2[i-1]);
return V1,V2
V1 = pm.Deterministic('V1', DHOS[0])
V2 = pm.Deterministic('V2', DHOS[1])
start = pm.find_MAP(fmin=fmin_powell, disp=True)
step=pm.NUTS()
trace=pm.sample(niter, step, start=start, progressbar=False)
traceplot(trace);
Summary=pm.df_summary(trace[-1000:])
gama_trace = trace.get_values('gama', burn)
omega_trace = trace.get_values('omega', burn)
对于此代码，我收到以下错误： V1 = pm.Deterministic（＆＃39; V1＆＃39;，DHOS [0]）
TypeError: 'FromFunctionOp' object does not support indexing
简单地说，我想知道如何将PYMC代码的以下部分转换为PYMC3。
@pymc.deterministic
def DOS(gama=gama, omega=omega):
V1= np.zeros(npts+1)
V2= np.zeros(npts+1)
V1[0] = V0[0]
V2[0] = V0[1]
for i in range(1,npts+1):
V1[i]= V1[i-1] + dt*V2[i-1];
V2[i] = V2[i-1] + dt*(-((2*np.pi*omega)**2)*V1[i-1]-gama*V2[i-1]);
return [V1, V2]
V1 = pymc.Lambda('V1', lambda DOS=DOS: DOS[0])
V2 = pymc.Lambda('V2', lambda DOS=DOS: DOS[1])
问题是，首先，确定性函数的论证在PYMC3中与PYMC不同，其次，在PYMC3中没有Lambda函数。
感谢您帮助解决PYMC3中的ODE，解决生物系统中的参数估计任务（从数据中估算方程参数）。
非常感谢您的帮助。
亲切的问候，
Meysam
答案 0 :(得分：0)
我建议并已成功使用“黑匣子”方法与PYMC3进行接口。在这种情况下，这意味着您自己计算对数可能性，然后使用PYMC3对其进行采样。这需要以Theano和PYMC3可以与它们交互的方式编写函数。
在PYMC3页面上的notebook中对此进行了概述，该页面以cython为例。
这里是需要做的事情的简短示例。
首先，您可以加载数据并设置所需的任何参数，例如时间步长等。
import pymc3 as pm
import numpy as np
import theano
import theano.tensor as tt
#import data
xdata = sio.loadmat('T.mat')['T'][0] #time
ydata1 = sio.loadmat('V1.mat')['V1'][0] # V2=dV1/dt, (X=V1),
ydata2 = sio.loadmat('V2.mat')['V2'][0] # dV2/dt=-(2pi*omega)^2*V1-gama*V2
#time span for solving the equations
npts= 500
dt=0.01
Tspan=5.0
time = np.linspace(0,Tspan,npts+1)
#initial condition
V0 = [1.0, 1.0]
然后，您像以前一样定义数据生成功能，但是您不需要为此使用PYMC的任何装饰器。该函数的输出应该是您需要与数据进行比较以计算似然性的结果。
def DHOS(theta):
gama,omega=theta
V1= np.zeros(npts+1)
V2= np.zeros(npts+1)
V1[0] = V0[0]
V2[0] = V0[1]
for i in range(1,npts+1):
V1[i]= V1[i-1] + dt*V2[i-1];
V2[i] = V2[i-1] + dt*(-((2*np.pi*omega)**2)*V1[i-1]-gama*V2[i-1]);
return [V1, V2]
接下来，您编写一个函数，该函数调用前一个函数，并使用所需的任何分布（在这种分布中为正态分布）来计算似然性。
def my_loglike(theta,data,sigma):
"""
A Gaussian log-likelihood function for a model with parameters given in theta
"""
model = DHOS(theta) #V1 and V2 from the DHOS function
#Here data = [ydata1,ydata2] to compare with model
#sigma is either the same shape as model or a scalar
#which corresponds to the uncertainty on the data.
return -(0.5)*sum((data - model)**2/sigma**2)
现在您必须在这里定义Theano类，以便它可以与PYMC3接口。
# define a theano Op for our likelihood function
class LogLike(tt.Op):
"""
Specify what type of object will be passed and returned to the Op when it is
called. In our case we will be passing it a vector of values (the parameters
that define our model) and returning a single "scalar" value (the
log-likelihood)
"""
itypes = [tt.dvector] # expects a vector of parameter values when called
otypes = [tt.dscalar] # outputs a single scalar value (the log likelihood)
def __init__(self, loglike, data, sigma):
"""
Initialise the Op with various things that our log-likelihood function
requires. Below are the things that are needed in this particular
example.
Parameters
----------
loglike:
The log-likelihood (or whatever) function we've defined
data:
The "observed" data that our log-likelihood function takes in
x:
The dependent variable (aka 'x') that our model requires
sigma:
The noise standard deviation that our function requires.
"""
# add inputs as class attributes
self.likelihood = loglike
self.data = data
self.sigma = sigma
def perform(self, node, inputs, outputs):
# the method that is used when calling the Op
theta, = inputs # this will contain my variables
# call the log-likelihood function
logl = self.likelihood(theta, self.data, self.sigma)
outputs[0][0] = array(logl) # output the log-likelihood
最后，您可以使用PYMC3建立模型并进行相应采样。
ndraws = 10000 # number of draws from the distribution
nburn = 1000 # number of "burn-in points" (which we'll discard)
# create our Op
logl = LogLike(my_loglike, rdat_sim, 10)
# use PyMC3 to sampler from log-likelihood
with pm.Model():
gama= pm.Uniform('gama', 0.0, 20.0)
omega=pm.Uniform('omega',0.0, 20.0)
# convert m and c to a tensor vector
theta = tt.as_tensor_variable([gama,omega])
# use a DensityDist (use a lamdba function to "call" the Op)
pm.DensityDist('likelihood', lambda v: logl(v), observed={'v': theta})
trace = pm.sample(ndraws, tune=nburn, discard_tuned_samples=True)
您可以使用内部绘图查看采样结果
_ = pm.traceplot(trace)
这仅是从示例笔记本中的链接改编而成的，并且如前所述，如果您想使用NUTS，则需要渐变信息，而您没有为其提供自定义功能。在链接中，它讨论了如何对渐变进行采样并构建渐变，以便您可以将其传递到采样器中，但此处没有显示。
此外，如果您要使用solve_ivp（或odeint或其他求解器），您所要做的就是更改DHOS函数，就像平时调用求解器一样。其余代码应可移植到您或任何其他人需要的任何问题上。