Delay Embedding and the MFPT

Here, we give an example script, showing the effect of Delay Embedding on a Brownian motion on the Muller-Brown potential, projeted onto its y-axis. This script may take a long time to run, as considerable data is required to accurately reconstruct the hidden degrees of freedom.



In [1]:

    
import matplotlib.pyplot as plt
import numpy as np
import pyedgar
from pyedgar.data_manipulation import tlist_to_flat, flat_to_tlist, delay_embed, lift_function

%matplotlib inline

Load Data and set Hyperparameters

We first load in the pre-sampled data. The data consists of 400 short trajectories, each with 30 datapoints. The precise sampling procedure is described in "Galerkin Approximation of Dynamical Quantities using Trajectory Data" by Thiede et al. Note that this is a smaller dataset than in the paper. We use a smallar dataset to ensure the diffusion map basis construction runs in a reasonably short time.

Set Hyperparameters

Here we specify a few hyperparameters. Thes can be varied to study the behavior of the scheme in various limits by the user.



In [2]:

    
ntraj = 700
trajectory_length = 40
lag_values = np.arange(1, 37, 2)
embedding_values = lag_values[1:] - 1

Load and format the data



In [3]:

    
trajs_2d = np.load('data/muller_brown_trajs.npy')[:ntraj, :trajectory_length] # Raw trajectory
trajs = trajs_2d[:, :, 1] # Only keep y coordinate
stateA = (trajs > 1.15).astype('float')
stateB = (trajs < 0.15).astype('float')

# Convert to list of trajectories format
trajs = [traj_i.reshape(-1, 1) for traj_i in trajs]
stateA = [A_i for A_i in stateA]
stateB = [B_i for B_i in stateB]

# Load the true results
true_mfpt = np.load('data/htAB_1_0_0_1.npy')

We also convert the data into the flattened format. This converts the data into a 2D array, which allows the data to be passed into many ML packages that require a two-dimensional dataset. In particular, this is the format accepted by the Diffusion Atlas object. Trajectory start/stop points are then stored in the traj_edges array.



In [4]:

    
flattened_trajs, traj_edges = tlist_to_flat(trajs)
flattened_stateA = np.hstack(stateA)
flattened_stateB = np.hstack(stateB)
print("Flattened Shapes are: ", flattened_trajs.shape, flattened_stateA.shape, flattened_stateB.shape,)









    



Flattened Shapes are:  (28000, 1) (28000,) (28000,)

Construct DGA MFPT by increasing lag times

We first construct the MFPT with increasing lag times.



In [5]:

    
# Build the basis set
diff_atlas = pyedgar.basis.DiffusionAtlas.from_sklearn(alpha=0, k=500, bandwidth_type='-1/d', epsilon='bgh_generous')
diff_atlas.fit(flattened_trajs)
flat_basis = diff_atlas.make_dirichlet_basis(200, in_domain=(1. - flattened_stateA))
basis = flat_to_tlist(flat_basis, traj_edges)
flat_basis_no_boundaries = diff_atlas.make_dirichlet_basis(200)
basis_no_boundaries = flat_to_tlist(flat_basis_no_boundaries, traj_edges)



In [6]:

    
# Perform DGA calculation
mfpt_BA_lags = []
for lag in lag_values:
    mfpt = pyedgar.galerkin.compute_mfpt(basis, stateA, lag=lag)
    pi = pyedgar.galerkin.compute_change_of_measure(basis_no_boundaries, lag=lag)
    flat_pi = np.array(pi).ravel()
    flat_mfpt = np.array(mfpt).ravel()
    mfpt_BA = np.mean(flat_mfpt * flat_pi * np.array(stateB).ravel()) / np.mean(flat_pi * np.array(stateB).ravel())
    mfpt_BA_lags.append(mfpt_BA)

Construct DGA MFPT with increasing Delay Embedding

We now construct the MFPT using delay embedding.



In [7]:

    
mfpt_BA_embeddings = []
for lag in embedding_values:
    # Perform delay embedding
    debbed_traj = delay_embed(trajs, n_embed=lag)
    lifted_A = lift_function(stateA, n_embed=lag)
    lifted_B = lift_function(stateB, n_embed=lag)
    
    flat_debbed_traj, embed_edges = tlist_to_flat(debbed_traj)
    flat_lifted_A = np.hstack(lifted_A)
        
    # Build the basis 
    diff_atlas = pyedgar.basis.DiffusionAtlas.from_sklearn(alpha=0, k=500, bandwidth_type='-1/d',
                                                           epsilon='bgh_generous', neighbor_params={'algorithm':'brute'})
    diff_atlas.fit(flat_debbed_traj)
    flat_deb_basis = diff_atlas.make_dirichlet_basis(200, in_domain=(1. - flat_lifted_A))
    deb_basis = flat_to_tlist(flat_deb_basis, embed_edges)
    
    flat_pi_basis = diff_atlas.make_dirichlet_basis(200)
    pi_basis = flat_to_tlist(flat_deb_basis, embed_edges)
    
    
    # Construct the Estimate
    deb_mfpt = pyedgar.galerkin.compute_mfpt(deb_basis, lifted_A, lag=1)
    pi = pyedgar.galerkin.compute_change_of_measure(pi_basis)
    flat_pi = np.array(pi).ravel()
    flat_mfpt = np.array(deb_mfpt).ravel()
    deb_mfpt_BA = np.mean(flat_mfpt * flat_pi * np.array(lifted_B).ravel()) / np.mean(flat_pi * np.array(lifted_B).ravel())
    mfpt_BA_embeddings.append(deb_mfpt_BA)

Plot the Results

We plot the results of our calculation, against the true value (black line, with the standard deviation in stateB given by the dotted lines). We see that increasing the lag time causes the mean-first-passage time to grow unboundedly. In contrast, with delay embedding the mean-first-passage time converges. We do, however, see one bad fluction at a delay length of 16, and that as the the delay length gets sufficiently long, the calculation blows up.



In [12]:

    
plt.plot(embedding_values, mfpt_BA_embeddings, label="Delay Embedding")
plt.plot(lag_values, mfpt_BA_lags, label="Lags")
plt.axhline(true_mfpt[0] * 10, color='k', label='True')
plt.axhline((true_mfpt[0] + true_mfpt[1]) * 10., color='k', linestyle=':')
plt.axhline((true_mfpt[0] - true_mfpt[1]) * 10., color='k', linestyle=':')

plt.legend()
plt.ylim(0, 100)

plt.xlabel("Lag / Delay Length")
plt.ylabel("Estimated MFPT")









    Out[12]:





Text(0,0.5,'Estimated MFPT')



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