supporting variable dt with optimize

Author: pavelkomarov

pynumdiff/finite_difference/_finite_difference.py

@@ -36,7 +36,7 @@ def finitediff(x, dt, num_iterations, order):
             dxdt_hat[0] = x_hat[1] - x_hat[0]
             dxdt_hat[-1] = x_hat[-1] - x_hat[-2] # use first-order endpoint formulas so as not to amplify noise. See #104
 
-        x_hat = utility.integrate_dxdt_hat(dxdt_hat, dt=1) # estimate new x_hat by integrating derivative
+        x_hat = utility.integrate_dxdt_hat(dxdt_hat, 1) # estimate new x_hat by integrating derivative
         # We can skip dividing by dt here and pass dt=1, because the integration multiplies dt back in.
         # No need to find integration constant until the very end, because we just differentiate again.
         # Note that I also tried integrating with Simpson's rule here, and it seems to do worse. See #104

pynumdiff/linear_model/_linear_model.py

@@ -252,7 +252,7 @@ def rbfdiff(x, _t, sigma=1, lmbd=0.01):
         if len(x) != len(_t): raise ValueError("If `_t` is given as array-like, must have same length as `x`.")
         t = _t
 
-    # The below does the approximate equivalent of this code, but sparsely in O(N sigma), since the rbf falls off rapidly
+    # The below does the approximate equivalent of this code, but sparsely in O(N sigma^2), since the rbf falls off rapidly
     # t_i, t_j = np.meshgrid(t,t)
     # r = t_j - t_i # radius
     # rbf = np.exp(-(r**2) / (2 * sigma**2)) # radial basis function kernel

pynumdiff/tests/test_optimize.py

@@ -6,6 +6,7 @@
 from ..linear_model import spectraldiff
 from ..polynomial_fit import polydiff, savgoldiff, splinediff
 from ..total_variation_regularization import velocity, acceleration, iterative_velocity
+from ..kalman_smooth import rtsdiff
 from ..optimize import optimize
 from ..utils.simulate import pi_cruise_control
 
@@ -92,3 +93,12 @@ def test_polydiff():
     params2, val2 = optimize(polydiff, x, dt, tvgamma=tvgamma, search_space_updates={'step_size':1}, padding='auto')
     assert (params1['degree'], params1['window_size'], params1['kernel']) == (6, 50, 'friedrichs')
     assert (params2['degree'], params2['window_size'], params2['kernel']) == (3, 10, 'gaussian')
+
+def test_rtsdiff_with_irregular_step():
+    t = np.arange(len(x))*dt
+    t_irreg = t + np.random.uniform(-dt/10, dt/10, *t.shape) # add jostle
+    params1, val1 = optimize(rtsdiff, x, t, dxdt_truth=dxdt_truth)
+    params2, val2 = optimize(rtsdiff, x, t_irreg, dxdt_truth=dxdt_truth)
+    assert val2 < 1.1*val1 # optimization works and comes out similar, since jostle is small
+    assert params1['qr_ratio']*0.9 < params2['qr_ratio'] < params1['qr_ratio']*1.1
+

pynumdiff/utils/utility.py

@@ -1,5 +1,7 @@
-import os, sys, copy, scipy
+import os, sys, copy
 import numpy as np
+from scipy.integrate import cumulative_trapezoid
+from scipy.optimize import minimize
 
 
 def hankel_matrix(x, num_delays, pad=False): # fixed delay step of 1
@@ -95,16 +97,16 @@ def peakdet(x, delta, t=None):
 
 
 # Trapazoidal integration, with 0 first value so that the lengths match. See #88.
-def integrate_dxdt_hat(dxdt_hat, dt):
-    """Wrapper for scipy.integrate.cumulative_trapezoid to integrate dxdt_hat that ensures
-    the integral has the same length
+def integrate_dxdt_hat(dxdt_hat, _t):
+    """Wrapper for scipy.integrate.cumulative_trapezoid
 
     :param np.array[float] dxdt_hat: estimate derivative of timeseries
-    :param float dt: time step in seconds
+    :param float _t: stepsize if given as a scalar or a vector of sample locations
 
     :return: **x_hat** (np.array[float]) -- integral of dxdt_hat
     """
-    return np.hstack((0, scipy.integrate.cumulative_trapezoid(dxdt_hat)))*dt
+    return cumulative_trapezoid(dxdt_hat, initial=0)*_t if np.isscalar(_t) \
+            else cumulative_trapezoid(dxdt_hat, x=_t, initial=0)
 
 
 # Optimization routine to estimate the integration constant.
@@ -118,7 +120,7 @@ def estimate_integration_constant(x, x_hat):
 
     :return: **integration constant** (float) -- initial condition that best aligns x_hat with x
     """
-    return scipy.optimize.minimize(lambda x0, x, xhat: np.linalg.norm(x - (x_hat+x0)), # fn to minimize in 1st argument
+    return minimize(lambda x0, x, xhat: np.linalg.norm(x - (x_hat+x0)), # fn to minimize in 1st argument
         0, args=(x, x_hat), method='SLSQP').x[0] # result is a vector, even if initial guess is just a scalar