WIP: Add DifferentialEquation: API for Bayesian inference of ODEs

pymc3/ode/__init__.py

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+from . import utils
+from .ode import DifferentialEquation

pymc3/ode/ode.py

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+import numpy as np
+from pymc3.ode.utils import augment_system, ODEGradop
+import scipy 
+import theano
+import theano.tensor as tt
+THEANO_FLAG = 'compute_test_value=ignore'
+
+
+class DifferentialEquation(theano.Op):
+
+    '''
+    Specify an ordinary differential equation
+
+    .. math::
+        \dfrac{dy}{dt} = f(y,t,p) \quad y(t_0) = y_0
+
+    Parameters
+    ----------
+
+    func : callable
+        Function specifying the differential equation
+    t0 : float
+        Time corresponding to the initial condition
+    times : array
+        Array of times at which to evaluate the solution of the differential equation.
+    n_states : int
+        Dimension of the differential equation.  For scalar differential equations, n_states =1.  
+        For vector valued differential equations, n_states = number of differential equations iun the system.
+    n_odeparams : int
+        Number of parameters in the differential equation.
+
+    .. code-block:: python
+
+        def odefunc(y,t,p):
+            #Logistic differential equation
+            return p[0]*y[0]*(1-y[0])
+
+        times = np.arange(0.5, 5, 0.5)
+
+        ode_model = DifferentialEquation(func = odefunc, t0 = 0, times = times, n_states = 1, n_odeparams = 1)
+    '''
+
+    __props__ = ()
+
+    def __init__(self, func, times, n_states, n_odeparams, t0=0):
+        if not callable(func):
+            raise ValueError("Argument func must be callable.")
+        if n_states<1:
+            raise ValueError('Argument n_states must be at least 1.')
+        if n_odeparams<0:
+            raise ValueError('Argument n_states must be non-negative.')
+
+        #Public
+        self.func = func
+        self.t0 = t0
+        self.times = times
+        self.n_states = n_states
+        self.n_odeparams = n_odeparams
+
+        #Private
+        self._n = n_states
+        self._m = n_odeparams + n_states
+
+        self._augmented_times = np.insert(times, t0, 0)
+        self._augmented_func = augment_system(func, self._n, self._m)
+        self._sens_ic = self._make_sens_ic()
+
+        self._cached_y = None
+        self._cached_sens = None
+        self._cached_parameters = None
+
+    def _make_sens_ic(self):
+        # The sensitivity matrix will always have consistent form.
+        # If the first n_odeparams entries of the parameters vector in the simulate call
+        # correspond to ode paramaters, then the first n_odeparams columns in
+        # the sensitivity matrix will be 0
+        sens_matrix = np.zeros((self._n, self._m))
+
+        # If the last n_states entrues of the paramters vector in the simulate call
+        # correspond to initial conditions of the system,
+        # then the last n_states columns of the sensitivity matrix should form
+        # an identity matrix
+        sens_matrix[:, -self.n_states:] = np.eye(self.n_states)
+
+        # We need the sensitivity matrix to be a vector (see augmented_function)
+        # Ravel and return
+        dydp = sens_matrix.ravel()
+
+        return dydp
+
+    def _system(self, Y, t, p):
+        '''
+        This is the function that will be passed to odeint.
+        Solves both ODE and sensitivities
+        Args:
+            Y (vector): current state and current gradient state
+            t (scalar): current time
+            p (vector): parameters
+        Returns:
+            derivatives (vector): derivatives of state and gradient
+        '''
+
+        dydt, ddt_dydp = self._augmented_func(Y[:self._n], t, p, Y[self._n:])
+        derivatives = np.concatenate([dydt, ddt_dydp])
+        return derivatives
+
+    def _simulate(self, parameters):
+        # Initial condition comprised of state initial conditions and raveled
+        # sensitivity matrix
+        y0 = np.concatenate([ parameters[self.n_odeparams:] , self._sens_ic])
+
+        # perform the integration
+        sol = scipy.integrate.odeint(func=self._system,
+                                     y0=y0,
+                                     t=self._augmented_times,
+                                     args=tuple([parameters]))
+        # The solution
+        y = sol[1:, :self.n_states]
+
+        # The sensitivities, reshaped to be a sequence of matrices
+        sens = sol[1:, self.n_states:].reshape(len(self.times), self._n, self._m)
+
+        return y, sens
+
+    def _cached_simulate(self, parameters):
+        if np.array_equal(np.array(parameters), self._cached_parameters):
+            return self._cached_y, self._cached_sens
+        else:
+            return self._simulate(np.array(parameters))
+
+    def state(self, x):
+        y, sens = self._cached_simulate(np.array(x, dtype=np.float64))
+        self._cached_y, self._cached_sens, self._cached_parameters = y, sens, x
+        return y.ravel()
+
+    def numpy_vsp(self, x, g):
+        numpy_sens = self._cached_simulate(np.array(x, dtype=np.float64))[1].reshape((self.n_states * len(self.times), len(x)))
+        return numpy_sens.T.dot(g)
+
+    def make_node(self, odeparams, y0):
+        if len(odeparams)!=self.n_odeparams:
+            raise ValueError('odeparams has too many or too few parameters.  Expected {a} paramteres but got {b}'.format(a = self.n_odeparams, b = len(odeparams)))
+        if len(y0)!=self.n_states:
+            raise ValueError('y0 has too many or too few parameters.  Expected {a} paramteres but got {b}'.format(a = self.n_states, b = len(y0)))
+
+        if np.ndim(odeparams) > 1:
+            odeparams = np.ravel(odeparams)
+        if np.ndim(y0) > 1:
+            y0 = np.ravel(y0)
+
+        odeparams = tt.as_tensor_variable(odeparams)
+        y0 = tt.as_tensor_variable(y0)
+        x = tt.concatenate([odeparams, y0])
+
+        return theano.Apply(self, [x], [x.type()])
+
+    def perform(self, node, inputs_storage, output_storage):
+        x = inputs_storage[0]
+        out = output_storage[0]
+        # get the numerical solution of ODE states
+        out[0] = self.state(x)
+
+    def grad(self, inputs, output_grads):
+        x = inputs[0]
+        g = output_grads[0]
+        # pass the VSP when asked for gradient
+        grad_op = ODEGradop(self.numpy_vsp)
+        grad_op_apply = grad_op(x, g)
+
+        return [grad_op_apply]

pymc3/ode/utils.py

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+import theano
+import theano.tensor as tt
+
+
+def augment_system(ode_func, n, m):
+    '''Function to create augmented system.
+
+    Take a function which specifies a set of differential equations and return
+    a compiled function which allows for computation of gradients of the
+    differential equation's solition with repsect to the parameters.
+
+    Args:
+        ode_func (function): Differential equation.  Returns array-like
+        n: Number of rows of the sensitivity matrix
+        m: Number of columns of the sensitivity matrix
+
+    Returns:
+        system (function): Augemted system of differential equations.
+
+    '''
+
+    # Present state of the system
+    t_y = tt.vector('y', dtype=theano.config.floatX)
+
+    # Parameter(s).  Should be vector to allow for generaliztion to multiparameter
+    # systems of ODEs
+    t_p = tt.vector('p', dtype=theano.config.floatX)
+
+    # Time.  Allow for non-automonous systems of ODEs to be analyzed
+    t_t = tt.scalar('t', dtype=theano.config.floatX)
+
+    # Present state of the gradients:
+    # Will always be 0 unless the parameter is the inital condition
+    # Entry i,j is partial of y[i] wrt to p[j]
+    dydp_vec = tt.vector('dydp', dtype=theano.config.floatX)
+
+    dydp = dydp_vec.reshape((n, m))
+
+    # Stack the results of the ode_func
+    # TODO: Does this behave the same of ODE is scalar?
+    f_tensor = tt.stack(ode_func(t_y, t_t, t_p))
+
+    # Now compute gradients
+    J = tt.jacobian(f_tensor, t_y)
+
+    Jdfdy = tt.dot(J, dydp)
+
+    grad_f = tt.jacobian(f_tensor, t_p)
+
+    # This is the time derivative of dydp
+    ddt_dydp = (Jdfdy + grad_f).flatten()
+
+    system = theano.function(
+        inputs=[t_y, t_t, t_p, dydp_vec],
+        outputs=[f_tensor, ddt_dydp],
+        on_unused_input='ignore')
+
+    return system
+
+
+
+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