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Comparison between EM and gradient ascent for tracking a foraging mouse
The code below compares the expectation maximization (EM) and gradient ascent algorithm for tracking a foraging mouse.
import sys
import os.path
import argparse
import configparser
import math
import random
import pickle
import numpy as np
import pandas as pd
import torch
import scipy
import plotly.graph_objects as go
import lds.learning
Define parameters for estimation
skip_estimation_sigma_a = False
skip_estimation_R = False
skip_estimation_m0 = False
skip_estimation_V0 = False
start_position = 0
# number_positions = 10000
number_positions = 7500
# number_positions = 50
lbfgs_max_iter = 2
lbfgs_tolerance_grad = -1
lbfgs_tolerance_change = 1e-3
lbfgs_lr = 1.0
lbfgs_n_epochs = 100
lbfgs_tol = 1e-3
em_max_iter = 200
Qe_reg_param_ga = None
Qe_reg_param_em = 1e-5
Provide initial conditions
pos_x0 = 0.0
pos_y0 = 0.0
vel_x0 = 0.0
vel_y0 = 0.0
ace_x0 = 0.0
ace_y0 = 0.0
sigma_a0 = 1.0
sigma_x0 = 1.0
sigma_y0 = 1.0
sqrt_diag_V0_value = 0.1
if math.isnan(pos_x0):
pos_x0 = y[0, 0]
if math.isnan(pos_y0):
pos_y0 = y[1, 0]
Get mouse positions
data_filename = "http://www.gatsby.ucl.ac.uk/~rapela/svGPFA/data/positions_session003_start0.00_end15548.27.csv"
data = pd.read_csv(data_filename)
data = data.iloc[start_position:start_position+number_positions,:]
y = np.transpose(data[["x", "y"]].to_numpy())
date_times = pd.to_datetime(data["time"])
dt = (date_times.iloc[1]-date_times.iloc[0]).total_seconds()
Build the matrices of the CWPA model
B, _, Z, _, Qe = lds.tracking.utils.getLDSmatricesForTracking(
dt=dt, sigma_a=np.nan, sigma_x=np.nan, sigma_y=np.nan)
m0 = np.array([pos_x0, vel_x0, ace_x0, pos_y0, vel_y0, ace_y0],
dtype=np.double)
vars_to_estimate = {}
if skip_estimation_sigma_a:
vars_to_estimate["sigma_a"] = False
else:
vars_to_estimate["sigma_a"] = True
if skip_estimation_R:
vars_to_estimate["sqrt_diag_R"] = False
vars_to_estimate["R"] = False
else:
vars_to_estimate["sqrt_diag_R"] = True
vars_to_estimate["R"] = True
if skip_estimation_m0:
vars_to_estimate["m0"] = False
else:
vars_to_estimate["m0"] = True
if skip_estimation_V0:
vars_to_estimate["sqrt_diag_V0"] = False
vars_to_estimate["V0"] = False
else:
vars_to_estimate["sqrt_diag_V0"] = True
vars_to_estimate["V0"] = True
Perform gradient ascent optimization
sqrt_diag_R_torch = torch.DoubleTensor([sigma_x0, sigma_y0])
m0_torch = torch.from_numpy(m0.copy())
sqrt_diag_V0_torch = torch.DoubleTensor([sqrt_diag_V0_value
for i in range(len(m0))])
if Qe_reg_param_ga is not None:
Qe_regularized_ga = Qe + Qe_reg_param_ga * np.eye(Qe.shape[0])
else:
Qe_regularized_ga = Qe
y_torch = torch.from_numpy(y.astype(np.double))
B_torch = torch.from_numpy(B.astype(np.double))
Qe_regularized_ga_torch = torch.from_numpy(Qe_regularized_ga.astype(np.double))
Z_torch = torch.from_numpy(Z.astype(np.double))
optim_res_ga = lds.learning.torch_lbfgs_optimize_SS_tracking_diagV0(
y=y_torch, B=B_torch, sigma_a0=sigma_a0,
Qe=Qe_regularized_ga_torch, Z=Z_torch, sqrt_diag_R_0=sqrt_diag_R_torch, m0_0=m0_torch,
sqrt_diag_V0_0=sqrt_diag_V0_torch, max_iter=lbfgs_max_iter, lr=lbfgs_lr,
vars_to_estimate=vars_to_estimate, tolerance_grad=lbfgs_tolerance_grad,
tolerance_change=lbfgs_tolerance_change, n_epochs=lbfgs_n_epochs,
tol=lbfgs_tol)
print("gradient ascent: " + optim_res_ga["termination_info"])
in closure, ll=-4349760.865100649
in closure, ll=-864425.783357166
--------------------------------------------------------------------------------
epoch: 0
likelihood: -4349760.865100649
sigma_a:
tensor([1.0474], dtype=torch.float64, requires_grad=True)
sqrt_diag_R:
tensor([1.0419, 1.1762], dtype=torch.float64, requires_grad=True)
m0:
tensor([2.2156e-04, 5.9511e-05, 7.5925e-06, 4.6697e-04, 1.2752e-04, 1.6344e-05],
dtype=torch.float64, requires_grad=True)
sqrt_diag_V0:
tensor([0.2254, 0.1090, 0.1001, 0.6569, 0.1415, 0.1007], dtype=torch.float64,
requires_grad=True)
in closure, ll=-864425.783357166
in closure, ll=-675903.2550139598
--------------------------------------------------------------------------------
epoch: 1
likelihood: -864425.783357166
sigma_a:
tensor([1.0546], dtype=torch.float64, requires_grad=True)
sqrt_diag_R:
tensor([1.0580, 1.1819], dtype=torch.float64, requires_grad=True)
m0:
tensor([3.6056e-04, 9.6815e-05, 1.2338e-05, 5.2679e-04, 1.4607e-04, 1.8836e-05],
dtype=torch.float64, requires_grad=True)
sqrt_diag_V0:
tensor([0.3178, 0.1123, 0.1002, 0.7057, 0.1428, 0.1007], dtype=torch.float64,
requires_grad=True)
in closure, ll=-675903.2550139598
in closure, ll=-477237.577875291
--------------------------------------------------------------------------------
epoch: 2
likelihood: -675903.2550139598
sigma_a:
tensor([1.0715], dtype=torch.float64, requires_grad=True)
sqrt_diag_R:
tensor([1.0742, 1.1879], dtype=torch.float64, requires_grad=True)
m0:
tensor([6.4423e-04, 1.7359e-04, 2.2145e-05, 7.1291e-04, 2.0678e-04, 2.7092e-05],
dtype=torch.float64, requires_grad=True)
sqrt_diag_V0:
tensor([0.4702, 0.1140, 0.1002, 0.8275, 0.1438, 0.1007], dtype=torch.float64,
requires_grad=True)
in closure, ll=-477237.577875291
in closure, ll=-356966.5981741533
--------------------------------------------------------------------------------
epoch: 3
likelihood: -477237.577875291
sigma_a:
tensor([1.1039], dtype=torch.float64, requires_grad=True)
sqrt_diag_R:
tensor([1.0940, 1.1901], dtype=torch.float64, requires_grad=True)
m0:
tensor([1.0232e-03, 2.7659e-04, 3.5270e-05, 1.0244e-03, 3.1449e-04, 4.1850e-05],
dtype=torch.float64, requires_grad=True)
sqrt_diag_V0:
tensor([0.6188, 0.1134, 0.1002, 0.9843, 0.1422, 0.1007], dtype=torch.float64,
requires_grad=True)
in closure, ll=-356966.5981741533
in closure, ll=-263783.7252192787
--------------------------------------------------------------------------------
epoch: 4
likelihood: -356966.5981741533
sigma_a:
tensor([1.1846], dtype=torch.float64, requires_grad=True)
sqrt_diag_R:
tensor([1.1422, 1.2010], dtype=torch.float64, requires_grad=True)
m0:
tensor([1.6564e-03, 4.4898e-04, 5.7099e-05, 1.6083e-03, 5.3194e-04, 7.1734e-05],
dtype=torch.float64, requires_grad=True)
sqrt_diag_V0:
tensor([0.8021, 0.1110, 0.1002, 1.2044, 0.1371, 0.1006], dtype=torch.float64,
requires_grad=True)
in closure, ll=-263783.7252192787
in closure, ll=-194897.69144144744
--------------------------------------------------------------------------------
epoch: 5
likelihood: -263783.7252192787
sigma_a:
tensor([1.3511], dtype=torch.float64, requires_grad=True)
sqrt_diag_R:
tensor([1.2445, 1.2457], dtype=torch.float64, requires_grad=True)
m0:
tensor([2.6471e-03, 7.1836e-04, 9.0867e-05, 2.5812e-03, 9.2699e-04, 1.2594e-04],
dtype=torch.float64, requires_grad=True)
sqrt_diag_V0:
tensor([1.0194, 0.1069, 0.1001, 1.4763, 0.1278, 0.1004], dtype=torch.float64,
requires_grad=True)
in closure, ll=-194897.69144144744
in closure, ll=-141995.14515118996
--------------------------------------------------------------------------------
epoch: 6
likelihood: -194897.69144144744
sigma_a:
tensor([1.6424], dtype=torch.float64, requires_grad=True)
sqrt_diag_R:
tensor([1.4248, 1.3583], dtype=torch.float64, requires_grad=True)
m0:
tensor([0.0042, 0.0011, 0.0001, 0.0042, 0.0016, 0.0002], dtype=torch.float64,
requires_grad=True)
sqrt_diag_V0:
tensor([1.2894, 0.1007, 0.1000, 1.8129, 0.1123, 0.1001], dtype=torch.float64,
requires_grad=True)
in closure, ll=-141995.14515118996
in closure, ll=-105560.15024555809
--------------------------------------------------------------------------------
epoch: 7
likelihood: -141995.14515118996
sigma_a:
tensor([2.0430], dtype=torch.float64, requires_grad=True)
sqrt_diag_R:
tensor([1.6636, 1.5428], dtype=torch.float64, requires_grad=True)
m0:
tensor([0.0066, 0.0018, 0.0002, 0.0068, 0.0028, 0.0004], dtype=torch.float64,
requires_grad=True)
sqrt_diag_V0:
tensor([1.6078, 0.0926, 0.0999, 2.2095, 0.0883, 0.0997], dtype=torch.float64,
requires_grad=True)
in closure, ll=-105560.15024555809
in closure, ll=-81571.05497658401
--------------------------------------------------------------------------------
epoch: 8
likelihood: -105560.15024555809
sigma_a:
tensor([2.5434], dtype=torch.float64, requires_grad=True)
sqrt_diag_R:
tensor([1.9304, 1.7680], dtype=torch.float64, requires_grad=True)
m0:
tensor([0.0103, 0.0027, 0.0003, 0.0109, 0.0048, 0.0006], dtype=torch.float64,
requires_grad=True)
sqrt_diag_V0:
tensor([1.9838, 0.0824, 0.0997, 2.6900, 0.0530, 0.0992], dtype=torch.float64,
requires_grad=True)
in closure, ll=-81571.05497658401
in closure, ll=-65955.31706052765
--------------------------------------------------------------------------------
epoch: 9
likelihood: -81571.05497658401
sigma_a:
tensor([3.1675], dtype=torch.float64, requires_grad=True)
sqrt_diag_R:
tensor([2.1935, 1.9958], dtype=torch.float64, requires_grad=True)
m0:
tensor([0.0158, 0.0042, 0.0005, 0.0173, 0.0079, 0.0011], dtype=torch.float64,
requires_grad=True)
sqrt_diag_V0:
tensor([2.4329, 0.0699, 0.0995, 3.2824, 0.0036, 0.0985], dtype=torch.float64,
requires_grad=True)
in closure, ll=-65955.31706052765
in closure, ll=-55724.821515104406
--------------------------------------------------------------------------------
epoch: 10
likelihood: -65955.31706052765
sigma_a:
tensor([3.9680], dtype=torch.float64, requires_grad=True)
sqrt_diag_R:
tensor([2.3983, 2.1734], dtype=torch.float64, requires_grad=True)
m0:
tensor([0.0243, 0.0063, 0.0007, 0.0272, 0.0128, 0.0017], dtype=torch.float64,
requires_grad=True)
sqrt_diag_V0:
tensor([ 2.9750, 0.0546, 0.0993, 4.0193, -0.0624, 0.0977],
dtype=torch.float64, requires_grad=True)
in closure, ll=-55724.821515104406
in closure, ll=-48677.46276967638
--------------------------------------------------------------------------------
epoch: 11
likelihood: -55724.821515104406
sigma_a:
tensor([5.0503], dtype=torch.float64, requires_grad=True)
sqrt_diag_R:
tensor([2.4412, 2.2076], dtype=torch.float64, requires_grad=True)
m0:
tensor([0.0373, 0.0094, 0.0011, 0.0424, 0.0207, 0.0027], dtype=torch.float64,
requires_grad=True)
sqrt_diag_V0:
tensor([ 3.6442, 0.0358, 0.0990, 4.9566, -0.1490, 0.0967],
dtype=torch.float64, requires_grad=True)
in closure, ll=-48677.46276967638
in closure, ll=-43126.20433079402
--------------------------------------------------------------------------------
epoch: 12
likelihood: -48677.46276967638
sigma_a:
tensor([6.6675], dtype=torch.float64, requires_grad=True)
sqrt_diag_R:
tensor([2.1361, 1.9359], dtype=torch.float64, requires_grad=True)
m0:
tensor([0.0579, 0.0141, 0.0016, 0.0659, 0.0344, 0.0044], dtype=torch.float64,
requires_grad=True)
sqrt_diag_V0:
tensor([ 4.5150, 0.0122, 0.0986, 6.2094, -0.2669, 0.0955],
dtype=torch.float64, requires_grad=True)
in closure, ll=-43126.20433079402
in closure, ll=-38575.54661835186
--------------------------------------------------------------------------------
epoch: 13
likelihood: -43126.20433079402
sigma_a:
tensor([9.4357], dtype=torch.float64, requires_grad=True)
sqrt_diag_R:
tensor([1.6484, 1.3869], dtype=torch.float64, requires_grad=True)
m0:
tensor([0.0914, 0.0211, 0.0022, 0.1031, 0.0634, 0.0078], dtype=torch.float64,
requires_grad=True)
sqrt_diag_V0:
tensor([ 5.6448, -0.0180, 0.0982, 7.7935, -0.4575, 0.0943],
dtype=torch.float64, requires_grad=True)
in closure, ll=-38575.54661835186
in closure, ll=-35330.423396786195
--------------------------------------------------------------------------------
epoch: 14
likelihood: -38575.54661835186
sigma_a:
tensor([15.3115], dtype=torch.float64, requires_grad=True)
sqrt_diag_R:
tensor([2.3744, 1.9971], dtype=torch.float64, requires_grad=True)
m0:
tensor([0.1637, 0.0355, 0.0034, 0.1824, 0.1377, 0.0159], dtype=torch.float64,
requires_grad=True)
sqrt_diag_V0:
tensor([ 7.8777, -0.0843, 0.0971, 10.4573, -0.9780, 0.0917],
dtype=torch.float64, requires_grad=True)
in closure, ll=-35330.423396786195
in closure, ll=-32321.736077543555
--------------------------------------------------------------------------------
epoch: 15
likelihood: -35330.423396786195
sigma_a:
tensor([18.5797], dtype=torch.float64, requires_grad=True)
sqrt_diag_R:
tensor([1.9159, 1.2629], dtype=torch.float64, requires_grad=True)
m0:
tensor([0.2029, 0.0433, 0.0041, 0.2259, 0.1783, 0.0203], dtype=torch.float64,
requires_grad=True)
sqrt_diag_V0:
tensor([ 9.0688, -0.1161, 0.0967, 12.1607, -1.2467, 0.0905],
dtype=torch.float64, requires_grad=True)
in closure, ll=-32321.736077543555
in closure, ll=-28041.283336008222
--------------------------------------------------------------------------------
epoch: 16
likelihood: -32321.736077543555
sigma_a:
tensor([35.9907], dtype=torch.float64, requires_grad=True)
sqrt_diag_R:
tensor([-1.4661, -1.1189], dtype=torch.float64, requires_grad=True)
m0:
tensor([0.4041, 0.0831, 0.0075, 0.4475, 0.4111, 0.0448], dtype=torch.float64,
requires_grad=True)
sqrt_diag_V0:
tensor([15.4072, -0.2725, 0.0943, 20.5471, -2.9635, 0.0843],
dtype=torch.float64, requires_grad=True)
in closure, ll=-28041.283336008222
in closure, ll=-60845.44854412978
in closure, ll=-28016.72369289751
in closure, ll=-27642.86961457806
--------------------------------------------------------------------------------
epoch: 17
likelihood: -28041.283336008222
sigma_a:
tensor([35.4827], dtype=torch.float64, requires_grad=True)
sqrt_diag_R:
tensor([-1.3597, -0.9811], dtype=torch.float64, requires_grad=True)
m0:
tensor([0.3985, 0.0820, 0.0074, 0.4413, 0.4046, 0.0441], dtype=torch.float64,
requires_grad=True)
sqrt_diag_V0:
tensor([15.2323, -0.2681, 0.0944, 20.2934, -2.9165, 0.0845],
dtype=torch.float64, requires_grad=True)
in closure, ll=-27642.86961457806
in closure, ll=-50348.9578325196
in closure, ll=-28031.70882471392
in closure, ll=-27294.53614617836
--------------------------------------------------------------------------------
epoch: 18
likelihood: -27642.86961457806
sigma_a:
tensor([35.6823], dtype=torch.float64, requires_grad=True)
sqrt_diag_R:
tensor([-1.3666, -0.7355], dtype=torch.float64, requires_grad=True)
m0:
tensor([0.4014, 0.0825, 0.0075, 0.4445, 0.4081, 0.0444], dtype=torch.float64,
requires_grad=True)
sqrt_diag_V0:
tensor([15.3370, -0.2703, 0.0943, 20.3412, -2.9506, 0.0843],
dtype=torch.float64, requires_grad=True)
in closure, ll=-27294.53614617836
in closure, ll=-26992.066136108533
--------------------------------------------------------------------------------
epoch: 19
likelihood: -27294.53614617836
sigma_a:
tensor([37.1363], dtype=torch.float64, requires_grad=True)
sqrt_diag_R:
tensor([-1.1797, -0.7222], dtype=torch.float64, requires_grad=True)
m0:
tensor([0.4197, 0.0861, 0.0078, 0.4646, 0.4266, 0.0465], dtype=torch.float64,
requires_grad=True)
sqrt_diag_V0:
tensor([15.8772, -0.2868, 0.0941, 21.0349, -3.0763, 0.0837],
dtype=torch.float64, requires_grad=True)
in closure, ll=-26992.066136108533
in closure, ll=-26167.47437272276
--------------------------------------------------------------------------------
epoch: 20
likelihood: -26992.066136108533
sigma_a:
tensor([44.6102], dtype=torch.float64, requires_grad=True)
sqrt_diag_R:
tensor([-0.9844, -0.6663], dtype=torch.float64, requires_grad=True)
m0:
tensor([0.5111, 0.1042, 0.0093, 0.5653, 0.5243, 0.0570], dtype=torch.float64,
requires_grad=True)
sqrt_diag_V0:
tensor([18.6658, -0.3660, 0.0929, 24.5325, -3.7736, 0.0806],
dtype=torch.float64, requires_grad=True)
in closure, ll=-26167.47437272276
in closure, ll=-24700.283777922232
--------------------------------------------------------------------------------
epoch: 21
likelihood: -26167.47437272276
sigma_a:
tensor([61.7582], dtype=torch.float64, requires_grad=True)
sqrt_diag_R:
tensor([-1.0137, -0.5894], dtype=torch.float64, requires_grad=True)
m0:
tensor([0.7194, 0.1455, 0.0129, 0.7946, 0.7500, 0.0811], dtype=torch.float64,
requires_grad=True)
sqrt_diag_V0:
tensor([25.0668, -0.5439, 0.0901, 32.5239, -5.4040, 0.0736],
dtype=torch.float64, requires_grad=True)
in closure, ll=-24700.283777922232
in closure, ll=-23851.18246110244
--------------------------------------------------------------------------------
epoch: 22
likelihood: -24700.283777922232
sigma_a:
tensor([81.8113], dtype=torch.float64, requires_grad=True)
sqrt_diag_R:
tensor([-0.9371, -0.6384], dtype=torch.float64, requires_grad=True)
m0:
tensor([0.9630, 0.1937, 0.0170, 1.0629, 1.0134, 0.1093], dtype=torch.float64,
requires_grad=True)
sqrt_diag_V0:
tensor([32.5313, -0.7526, 0.0869, 41.9016, -7.2929, 0.0654],
dtype=torch.float64, requires_grad=True)
in closure, ll=-23851.18246110244
in closure, ll=-23494.35249912862
in closure, ll=-23197.13638543685
--------------------------------------------------------------------------------
epoch: 23
likelihood: -23851.18246110244
sigma_a:
tensor([107.4640], dtype=torch.float64, requires_grad=True)
sqrt_diag_R:
tensor([-0.8749, -0.4626], dtype=torch.float64, requires_grad=True)
m0:
tensor([1.2749, 0.2554, 0.0223, 1.4063, 1.3513, 0.1455], dtype=torch.float64,
requires_grad=True)
sqrt_diag_V0:
tensor([42.1096, -1.0195, 0.0828, 53.8506, -9.7275, 0.0548],
dtype=torch.float64, requires_grad=True)
in closure, ll=-23197.13638543685
in closure, ll=-22650.26086579791
--------------------------------------------------------------------------------
epoch: 24
likelihood: -23197.13638543685
sigma_a:
tensor([134.3731], dtype=torch.float64, requires_grad=True)
sqrt_diag_R:
tensor([-0.8753, -0.4871], dtype=torch.float64, requires_grad=True)
m0:
tensor([1.6015, 0.3200, 0.0278, 1.7659, 1.7053, 0.1834], dtype=torch.float64,
requires_grad=True)
sqrt_diag_V0:
tensor([ 5.2132e+01, -1.2988e+00, 7.8562e-02, 6.6417e+01, -1.2273e+01,
4.3807e-02], dtype=torch.float64, requires_grad=True)
in closure, ll=-22650.26086579791
in closure, ll=-22253.747573382083
--------------------------------------------------------------------------------
epoch: 25
likelihood: -22650.26086579791
sigma_a:
tensor([179.0867], dtype=torch.float64, requires_grad=True)
sqrt_diag_R:
tensor([-0.8171, -0.4377], dtype=torch.float64, requires_grad=True)
m0:
tensor([2.1447, 0.4275, 0.0370, 2.3639, 2.2936, 0.2463], dtype=torch.float64,
requires_grad=True)
sqrt_diag_V0:
tensor([ 6.8797e+01, -1.7635e+00, 7.1441e-02, 8.7286e+01, -1.6503e+01,
2.5486e-02], dtype=torch.float64, requires_grad=True)
in closure, ll=-22253.747573382083
in closure, ll=-22052.708890079248
--------------------------------------------------------------------------------
epoch: 26
likelihood: -22253.747573382083
sigma_a:
tensor([218.3609], dtype=torch.float64, requires_grad=True)
sqrt_diag_R:
tensor([-0.7963, -0.4390], dtype=torch.float64, requires_grad=True)
m0:
tensor([2.6215, 0.5219, 0.0451, 2.8890, 2.8104, 0.3016], dtype=torch.float64,
requires_grad=True)
sqrt_diag_V0:
tensor([ 8.3429e+01, -2.1713e+00, 6.5191e-02, 1.0562e+02, -2.0219e+01,
9.4132e-03], dtype=torch.float64, requires_grad=True)
in closure, ll=-22052.708890079248
in closure, ll=-21962.3996258442
--------------------------------------------------------------------------------
epoch: 27
likelihood: -22052.708890079248
sigma_a:
tensor([258.2953], dtype=torch.float64, requires_grad=True)
sqrt_diag_R:
tensor([-0.7783, -0.4232], dtype=torch.float64, requires_grad=True)
m0:
tensor([3.1064, 0.6178, 0.0533, 3.4229, 3.3360, 0.3578], dtype=torch.float64,
requires_grad=True)
sqrt_diag_V0:
tensor([ 9.8310e+01, -2.5860e+00, 5.8836e-02, 1.2426e+02, -2.3999e+01,
-6.9359e-03], dtype=torch.float64, requires_grad=True)
in closure, ll=-21962.3996258442
in closure, ll=-21934.28696190734
in closure, ll=-22043.622258688818
in closure, ll=-21953.372906392866
in closure, ll=-21932.374703323203
in closure, ll=-21940.424137954924
in closure, ll=-21935.742710205144
in closure, ll=-21933.88841178075
in closure, ll=-21933.08817943135
in closure, ll=-21932.720493706052
in closure, ll=-21932.544848530142
in closure, ll=-21932.459086793122
in closure, ll=-21932.416722561076
in closure, ll=-21932.39566979108
in closure, ll=-21932.38517576563
in closure, ll=-21932.3799368494
in closure, ll=-21932.377319418145
in closure, ll=-21932.376011200173
in closure, ll=-21932.375357218676
in closure, ll=-21932.375030254232
in closure, ll=-21932.374866781967
in closure, ll=-21932.374785051157
in closure, ll=-21932.374744187
in closure, ll=-21932.37472374585
in closure, ll=-21932.37471354182
in closure, ll=-21932.374708425283
in closure, ll=-21932.37470588156
--------------------------------------------------------------------------------
epoch: 28
likelihood: -21962.3996258442
sigma_a:
tensor([320.1212], dtype=torch.float64, requires_grad=True)
sqrt_diag_R:
tensor([-0.7416, -0.4207], dtype=torch.float64, requires_grad=True)
m0:
tensor([3.8572, 0.7664, 0.0660, 4.2495, 4.1506, 0.4449], dtype=torch.float64,
requires_grad=True)
sqrt_diag_V0:
tensor([ 1.2134e+02, -3.2280e+00, 4.8998e-02, 1.5313e+02, -2.9853e+01,
-3.2238e-02], dtype=torch.float64, requires_grad=True)
in closure, ll=-21932.374703323203
gradient ascent: nan generated
Perform EM optimization
sqrt_diag_R = np.array([sigma_x0, sigma_y0])
R_0 = np.diag(sqrt_diag_R)
m0_0 = m0
sqrt_diag_V0 = np.array([sqrt_diag_V0_value for i in range(len(m0))])
V0_0 = np.diag(sqrt_diag_V0)
times = np.arange(0, y.shape[1]*dt, dt)
not_nan_indices_y0 = set(np.where(np.logical_not(np.isnan(y[0, :])))[0])
not_nan_indices_y1 = set(np.where(np.logical_not(np.isnan(y[1, :])))[0])
not_nan_indices = np.array(sorted(not_nan_indices_y0.union(not_nan_indices_y1)))
y_no_nan = y[:, not_nan_indices]
t_no_nan = times[not_nan_indices]
y_interpolated = np.empty_like(y)
tck, u = scipy.interpolate.splprep([y_no_nan[0, :], y_no_nan[1, :]], s=0, u=t_no_nan)
y_interpolated[0, :], y_interpolated[1, :] = scipy.interpolate.splev(times, tck)
if Qe_reg_param_em is not None:
Qe_regularized_em = Qe + Qe_reg_param_em * np.eye(Qe.shape[0])
else:
Qe_regularized_em = Qe
optim_res_em = lds.learning.em_SS_tracking(
y=y_interpolated, B=B, sigma_a0=sigma_a0,
Qe=Qe_regularized_em, Z=Z, R_0=R_0, m0_0=m0_0, V0_0=V0_0,
vars_to_estimate=vars_to_estimate,
max_iter=em_max_iter)
print("EM: " + optim_res_em["termination_info"])
LogLike[0000]=-1532961.666165
sigma_a=1.000000
R:
[[1. 0.]
[0. 1.]]
m0:
[0. 0. 0. 0. 0. 0.]
V0:
[[0.1 0. 0. 0. 0. 0. ]
[0. 0.1 0. 0. 0. 0. ]
[0. 0. 0.1 0. 0. 0. ]
[0. 0. 0. 0.1 0. 0. ]
[0. 0. 0. 0. 0.1 0. ]
[0. 0. 0. 0. 0. 0.1]]
LogLike[0001]=-56083.466132
sigma_a=2.320011
R:
[[25.16113242 31.90946823]
[31.90946823 77.90941043]]
m0:
[188.93507296 47.02895603 5.48547484 397.12406422 103.86097823
12.3310036 ]
V0:
[[ 0.03106149 -0.01705374 -0.00199045 0. 0. 0. ]
[-0.01705374 0.08734047 -0.00277441 0. 0. 0. ]
[-0.00199045 -0.00277441 0.09897246 0. 0. 0. ]
[ 0. 0. 0. 0.03106149 -0.01705374 -0.00199045]
[ 0. 0. 0. -0.01705374 0.08734047 -0.00277441]
[ 0. 0. 0. -0.00199045 -0.00277441 0.09897246]]
LogLike[0002]=-54472.593838
sigma_a=2.564253
R:
[[ 35.37688641 48.73015339]
[ 48.73015339 121.26331529]]
m0:
[189.70957688 47.30563329 5.54917522 398.61986099 104.90457137
12.54064585]
V0:
[[ 2.95260918e-02 -1.74339463e-02 -2.02991792e-03 5.99137707e-04
1.06471579e-04 3.69656031e-06]
[-1.74339463e-02 8.63039357e-02 -3.07862433e-03 1.06471579e-04
2.48348818e-04 4.79567735e-05]
[-2.02991792e-03 -3.07862433e-03 9.88222388e-02 3.69656031e-06
4.79567735e-05 1.14329137e-05]
[ 5.99137707e-04 1.06471579e-04 3.69656031e-06 3.05165026e-02
-1.72579423e-02 -2.02380728e-03]
[ 1.06471579e-04 2.48348818e-04 4.79567735e-05 -1.72579423e-02
8.67144713e-02 -2.99934889e-03]
[ 3.69656031e-06 4.79567735e-05 1.14329137e-05 -2.02380728e-03
-2.99934889e-03 9.88411381e-02]]
LogLike[0003]=-53853.778108
sigma_a=2.770890
R:
[[ 35.76653521 49.25133189]
[ 49.25133189 124.70669138]]
m0:
[190.3200895 47.5038523 5.59217624 399.5385031 105.60763469
12.68412762]
V0:
[[ 2.84831892e-02 -1.76810358e-02 -2.05250346e-03 9.91916125e-04
1.70366217e-04 4.10256195e-06]
[-1.76810358e-02 8.55151446e-02 -3.31461774e-03 1.70345195e-04
4.35952348e-04 8.50568293e-05]
[-2.05250346e-03 -3.31461774e-03 9.87026266e-02 4.09686528e-06
8.50555086e-05 2.05726882e-05]
[ 9.91916125e-04 1.70345195e-04 4.09686528e-06 3.01697537e-02
-1.73909794e-02 -2.04543390e-03]
[ 1.70366217e-04 4.35952348e-04 8.50555086e-05 -1.73909794e-02
8.62571640e-02 -3.16978380e-03]
[ 4.10256195e-06 8.50568293e-05 2.05726882e-05 -2.04543390e-03
-3.16978380e-03 9.87376695e-02]]
LogLike[0004]=-53310.596369
sigma_a=2.973903
R:
[[ 34.40528734 47.75760647]
[ 47.75760647 121.10214082]]
m0:
[190.92662984 47.68113641 5.62804419 400.40124977 106.24206749
12.80940523]
V0:
[[ 2.75804672e-02 -1.78646596e-02 -2.06370956e-03 1.31914998e-03
2.09393020e-04 6.60751004e-07]
[-1.78646596e-02 8.48322773e-02 -3.51616528e-03 2.09380593e-04
5.91914558e-04 1.15038698e-04]
[-2.06370956e-03 -3.51616528e-03 9.86006011e-02 6.57059727e-07
1.15037747e-04 2.78999454e-05]
[ 1.31914998e-03 2.09380593e-04 6.57059727e-07 2.98635810e-02
-1.75021065e-02 -2.06252654e-03]
[ 2.09393020e-04 5.91914558e-04 1.15037747e-04 -1.75021065e-02
8.58571112e-02 -3.31694979e-03]
[ 6.60751004e-07 1.15038698e-04 2.78999454e-05 -2.06252654e-03
-3.31694979e-03 9.86489247e-02]]
LogLike[0005]=-52791.362708
sigma_a=3.182133
R:
[[ 32.83707195 46.32121238]
[ 46.32121238 116.66827267]]
m0:
[191.53414901 47.83481226 5.65743723 401.26345954 106.84225309
12.92287915]
V0:
[[ 2.67344819e-02 -1.80018071e-02 -2.06527408e-03 1.62002464e-03
2.30378860e-04 -6.17290606e-06]
[-1.80018071e-02 8.42152691e-02 -3.69371198e-03 2.30426741e-04
7.30651050e-04 1.40696434e-04]
[-2.06527408e-03 -3.69371198e-03 9.85116762e-02 -6.16169426e-06
1.40698766e-04 3.40763450e-05]
[ 1.62002464e-03 2.30426741e-04 -6.16169426e-06 2.95696166e-02
-1.75991701e-02 -2.07619387e-03]
[ 2.30378860e-04 7.30651050e-04 1.40698766e-04 -1.75991701e-02
8.54930843e-02 -3.44769062e-03]
[-6.17290606e-06 1.40696434e-04 3.40763450e-05 -2.07619387e-03
-3.44769062e-03 9.85712583e-02]]
LogLike[0006]=-52288.500863
sigma_a=3.398887
R:
[[ 31.3297926 45.03311803]
[ 45.03311803 112.29577869]]
m0:
[192.13917459 47.96276156 5.68059404 402.13288377 107.41514769
13.02611575]
V0:
[[ 2.59192207e-02 -1.80988532e-02 -2.05829312e-03 1.90775855e-03
2.36118264e-04 -1.60128416e-05]
[-1.80988532e-02 8.36470512e-02 -3.85236840e-03 2.36266355e-04
8.58750260e-04 1.63469429e-04]
[-2.05829312e-03 -3.85236840e-03 9.84333826e-02 -1.59774015e-05
1.63477211e-04 3.94740993e-05]
[ 1.90775855e-03 2.36266355e-04 -1.59774015e-05 2.92801339e-02
-1.76842103e-02 -2.08679352e-03]
[ 2.36118264e-04 8.58750260e-04 1.63477211e-04 -1.76842103e-02
8.51575895e-02 -3.56495322e-03]
[-1.60128416e-05 1.63469429e-04 3.94740993e-05 -2.08679352e-03
-3.56495322e-03 9.85027694e-02]]
LogLike[0007]=-51798.997842
sigma_a=3.626353
R:
[[ 29.93105007 43.82732987]
[ 43.82732987 108.09165489]]
m0:
[192.73955908 48.06527691 5.69799317 403.01057423 107.96301222
13.1198701 ]
V0:
[[ 2.51261820e-02 -1.81594408e-02 -2.04374856e-03 2.18694758e-03
2.28104430e-04 -2.85065748e-05]
[-1.81594408e-02 8.31179458e-02 -3.99541631e-03 2.28363686e-04
9.79638791e-04 1.84174441e-04]
[-2.04374856e-03 -3.99541631e-03 9.83640070e-02 -2.84449230e-05
1.84188233e-04 4.43118006e-05]
[ 2.18694758e-03 2.28363686e-04 -2.84449230e-05 2.89926875e-02
-1.77580967e-02 -2.09456778e-03]
[ 2.28104430e-04 9.79638791e-04 1.84188233e-04 -1.77580967e-02
8.48464289e-02 -3.67065440e-03]
[-2.85065748e-05 1.84174441e-04 4.43118006e-05 -2.09456778e-03
-3.67065440e-03 9.84421158e-02]]
LogLike[0008]=-51322.712710
sigma_a=3.865215
R:
[[ 28.64157747 42.65967457]
[ 42.65967457 104.048869 ]]
m0:
[193.33419196 48.1438379 5.71022775 403.8966386 108.48688104
13.20466052]
V0:
[[ 2.43531660e-02 -1.81864630e-02 -2.02259315e-03 2.45894926e-03
2.07439553e-04 -4.33071072e-05]
[-1.81864630e-02 8.26214051e-02 -4.12525731e-03 2.07784144e-04
1.09539469e-03 2.03339096e-04]
[-2.02259315e-03 -4.12525731e-03 9.83022460e-02 -4.32257855e-05
2.03357557e-04 4.87312993e-05]
[ 2.45894926e-03 2.07784144e-04 -4.32257855e-05 2.87065542e-02
-1.78214071e-02 -2.09973501e-03]
[ 2.07439553e-04 1.09539469e-03 2.03357557e-04 -1.78214071e-02
8.45566260e-02 -3.76625484e-03]
[-4.33071072e-05 2.03339096e-04 4.87312993e-05 -2.09973501e-03
-3.76625484e-03 9.83882490e-02]]
LogLike[0009]=-50860.320623
sigma_a=4.115591
R:
[[ 27.453166 41.5083522 ]
[ 41.5083522 100.15512917]]
m0:
[193.92246832 48.2004204 5.71792202 404.79118026 108.98740988
13.28091799]
V0:
[[ 2.36003593e-02 -1.81826955e-02 -1.99576339e-03 2.72383439e-03
1.75151626e-04 -6.00675309e-05]
[-1.81826955e-02 8.21523719e-02 -4.24381815e-03 1.75515616e-04
1.20747115e-03 2.21343334e-04]
[-1.99576339e-03 -4.24381815e-03 9.82470326e-02 -5.99816255e-05
2.21363154e-04 5.28329430e-05]
[ 2.72383439e-03 1.75515616e-04 -5.99816255e-05 2.84216434e-02
-1.78746681e-02 -2.10250956e-03]
[ 1.75151626e-04 1.20747115e-03 2.21363154e-04 -1.78746681e-02
8.42857360e-02 -3.85298070e-03]
[-6.00675309e-05 2.21343334e-04 5.28329430e-05 -2.10250956e-03
-3.85298070e-03 9.83402879e-02]]
LogLike[0010]=-50411.286925
sigma_a=4.378348
R:
[[26.35515438 40.36278475]
[40.36278475 96.40150097]]
m0:
[194.5040394 48.23715251 5.72168108 405.69436238 109.46507985
13.34902548]
V0:
[[ 2.28689800e-02 -1.81509837e-02 -1.96417690e-03 2.98107411e-03
1.32301389e-04 -7.84406456e-05]
[-1.81509837e-02 8.17067571e-02 -4.35269067e-03 1.32578804e-04
1.31694008e-03 2.38470481e-04]
[-1.96417690e-03 -4.35269067e-03 9.81974724e-02 -7.83730480e-05
2.38486463e-04 5.66894049e-05]
[ 2.98107411e-03 1.32578804e-04 -7.83730480e-05 2.81381612e-02
-1.79184211e-02 -2.10310532e-03]
[ 1.32301389e-04 1.31694008e-03 2.38486463e-04 -1.79184211e-02
8.40316530e-02 -3.93189234e-03]
[-7.84406456e-05 2.38470481e-04 5.66894049e-05 -2.10310532e-03
-3.93189234e-03 9.82974748e-02]]
LogLike[0011]=-49976.250935
sigma_a=4.653379
R:
[[25.33435536 39.21675759]
[39.21675759 92.77505604]]
m0:
[195.07875561 48.25610179 5.7220514 406.606318 109.92019052
13.40931576]
V0:
[[ 2.21606863e-02 -1.80942464e-02 -1.92871726e-03 3.22983980e-03
8.00039719e-05 -9.80830091e-05]
[-1.80942464e-02 8.12813329e-02 -4.45316519e-03 8.00519711e-05
1.42456654e-03 2.54926069e-04]
[-1.92871726e-03 -4.45316519e-03 9.81528281e-02 -9.80630848e-05
2.54931336e-04 6.03516482e-05]
[ 3.22983980e-03 8.00519711e-05 -9.80630848e-05 2.78564811e-02
-1.79532241e-02 -2.10173176e-03]
[ 8.00039719e-05 1.42456654e-03 2.54931336e-04 -1.79532241e-02
8.37926049e-02 -4.00388883e-03]
[-9.80830091e-05 2.54926069e-04 6.03516482e-05 -2.10173176e-03
-4.00388883e-03 9.82591695e-02]]
LogLike[0012]=-49555.159033
sigma_a=4.940840
R:
[[24.37981546 38.06990831]
[38.06990831 89.26990104]]
m0:
[195.64670731 48.25917907 5.71950856 407.52719934 110.35302292
13.4621071 ]
V0:
[[ 2.14770471e-02 -1.80154584e-02 -1.89021620e-03 3.46926175e-03
1.94225051e-05 -1.18664541e-04]
[-1.80154584e-02 8.08734404e-02 -4.54632245e-03 1.90667752e-05
1.53090937e-03 2.70860693e-04]
[-1.89021620e-03 -4.54632245e-03 9.81124715e-02 -1.18726474e-04
2.70846964e-04 6.38556381e-05]
[ 3.46926175e-03 1.90667752e-05 -1.18726474e-04 2.75770072e-02
-1.79796685e-02 -2.09859358e-03]
[ 1.94225051e-05 1.53090937e-03 2.70846964e-04 -1.79796685e-02
8.35670239e-02 -4.06976083e-03]
[-1.18664541e-04 2.70860693e-04 6.38556381e-05 -2.09859358e-03
-4.06976083e-03 9.82248138e-02]]
LogLike[0013]=-49147.635814
sigma_a=5.240908
R:
[[23.4816108 36.92413184]
[36.92413184 85.88016681]]
m0:
[196.20819013 48.24807083 5.71444779 408.45709109 110.76382463
13.50770193]
V0:
[[ 2.08193543e-02 -1.79175658e-02 -1.84943611e-03 3.69854102e-03
-4.82631316e-05 -1.39876459e-04]
[-1.79175658e-02 8.04809326e-02 -4.63305743e-03 -4.92215893e-05
1.63635419e-03 2.86379984e-04]
[-1.84943611e-03 -4.63305743e-03 9.80758718e-02 -1.40057484e-04
2.86337886e-04 6.72256916e-05]
[ 3.69854102e-03 -4.92215893e-05 -1.40057484e-04 2.73001309e-02
-1.79983592e-02 -2.09388575e-03]
[-4.82631316e-05 1.63635419e-03 2.86337886e-04 -1.79983592e-02
8.33535444e-02 -4.13019612e-03]
[-1.39876459e-04 2.86379984e-04 6.72256916e-05 -2.09388575e-03
-4.13019612e-03 9.81939251e-02]]
LogLike[0014]=-48753.439932
sigma_a=5.553402
R:
[[22.6315643 35.78291829]
[35.78291829 82.60084329]]
m0:
[196.76367367 48.22421279 5.70718599 409.39600935 111.15283137
13.54639034]
V0:
[[ 2.01884563e-02 -1.78033894e-02 -1.80705300e-03 3.91705022e-03
-1.21899686e-04 -1.61440463e-04]
[-1.78033894e-02 8.01020679e-02 -4.71411399e-03 -1.23677767e-04
1.74115855e-03 3.01555907e-04]
[-1.80705300e-03 -4.71411399e-03 9.80425784e-02 -1.61779271e-04
3.01475271e-04 7.04778354e-05]
[ 3.91705022e-03 -1.23677767e-04 -1.61779271e-04 2.70261759e-02
-1.80098952e-02 -2.08778884e-03]
[-1.21899686e-04 1.74115855e-03 3.01475271e-04 -1.80098952e-02
8.31509707e-02 -4.18579405e-03]
[-1.61440463e-04 3.01555907e-04 7.04778354e-05 -2.08778884e-03
-4.18579405e-03 9.81660859e-02]]
LogLike[0015]=-48371.770562
sigma_a=5.878519
R:
[[21.82388511 34.65115173]
[34.65115173 79.42975805]]
m0:
[197.31374436 48.18880282 5.69797288 410.34392512 111.52030033
13.57845591]
V0:
[[ 1.95846995e-02 -1.76755541e-02 -1.76364849e-03 4.12438364e-03
-2.00397408e-04 -1.83114693e-04]
[-1.76755541e-02 7.97353972e-02 -4.79012112e-03 -2.03224100e-04
1.84549470e-03 3.16437009e-04]
[-1.76364849e-03 -4.79012112e-03 9.80122037e-02 -1.83650182e-04
3.16307104e-04 7.36226471e-05]
[ 4.12438364e-03 -2.03224100e-04 -1.83650182e-04 2.67553661e-02
-1.80148542e-02 -2.08046640e-03]
[-2.00397408e-04 1.84549470e-03 3.16307104e-04 -1.80148542e-02
8.29582346e-02 -4.23708354e-03]
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sigma_a=6.567981
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m0:
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LogLike[0018]=-47292.640797
sigma_a=6.933031
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m0:
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LogLike[0019]=-46952.053478
sigma_a=7.312255
R:
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m0:
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LogLike[0020]=-46619.973878
sigma_a=7.705835
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m0:
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LogLike[0021]=-46295.744729
sigma_a=8.114308
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m0:
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LogLike[0022]=-45978.812161
sigma_a=8.538173
R:
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m0:
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LogLike[0023]=-45668.772842
sigma_a=8.977763
R:
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m0:
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LogLike[0024]=-45365.286692
sigma_a=9.433329
R:
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m0:
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LogLike[0025]=-45067.882889
sigma_a=9.905405
R:
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m0:
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V0:
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LogLike[0026]=-44776.082758
sigma_a=10.394594
R:
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m0:
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V0:
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LogLike[0027]=-44489.579638
sigma_a=10.901122
R:
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m0:
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V0:
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LogLike[0028]=-44207.824156
sigma_a=11.425807
R:
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m0:
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V0:
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LogLike[0029]=-43930.345885
sigma_a=11.969322
R:
[[13.60574902 21.34372401]
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m0:
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V0:
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LogLike[0030]=-43656.658116
sigma_a=12.532441
R:
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m0:
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V0:
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LogLike[0031]=-43386.234493
sigma_a=13.116125
R:
[[12.77463475 19.88318091]
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m0:
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V0:
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LogLike[0032]=-43118.576195
sigma_a=13.721353
R:
[[12.3799549 19.18677521]
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m0:
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V0:
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LogLike[0033]=-42853.108744
sigma_a=14.349476
R:
[[11.99767834 18.51105721]
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m0:
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V0:
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LogLike[0034]=-42589.346361
sigma_a=15.001709
R:
[[11.62672985 17.85459047]
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m0:
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V0:
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LogLike[0035]=-42326.772153
sigma_a=15.679544
R:
[[11.26620362 17.2161289 ]
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m0:
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V0:
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LogLike[0036]=-42064.905449
sigma_a=16.384541
R:
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m0:
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V0:
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sigma_a=17.118285
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m0:
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LogLike[0038]=-41541.578533
sigma_a=17.882516
R:
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m0:
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LogLike[0039]=-41279.289629
sigma_a=18.679211
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m0:
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LogLike[0040]=-41016.084028
sigma_a=19.510403
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m0:
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LogLike[0041]=-40751.669528
sigma_a=20.378070
R:
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m0:
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V0:
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LogLike[0042]=-40485.798004
sigma_a=21.284247
R:
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m0:
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V0:
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LogLike[0043]=-40218.256874
sigma_a=22.231053
R:
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m0:
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V0:
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LogLike[0044]=-39948.866309
sigma_a=23.220691
R:
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m0:
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V0:
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LogLike[0045]=-39677.495752
sigma_a=24.255328
R:
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m0:
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V0:
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LogLike[0046]=-39404.042189
sigma_a=25.337171
R:
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m0:
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LogLike[0047]=-39128.437210
sigma_a=26.468374
R:
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m0:
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LogLike[0048]=-38850.588060
sigma_a=27.651363
R:
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m0:
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V0:
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LogLike[0049]=-38570.428827
sigma_a=28.888455
R:
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m0:
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V0:
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LogLike[0050]=-38287.879057
sigma_a=30.182082
R:
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m0:
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V0:
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LogLike[0051]=-38002.815762
sigma_a=31.534959
R:
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m0:
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LogLike[0052]=-37715.098574
sigma_a=32.949873
R:
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m0:
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LogLike[0053]=-37424.546970
sigma_a=34.429850
R:
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m0:
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V0:
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LogLike[0054]=-37130.931391
sigma_a=35.978224
R:
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m0:
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LogLike[0055]=-36833.978861
sigma_a=37.598627
R:
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m0:
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V0:
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LogLike[0056]=-36533.363878
sigma_a=39.295110
R:
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m0:
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V0:
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LogLike[0057]=-36228.723407
sigma_a=41.072062
R:
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m0:
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V0:
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LogLike[0058]=-35919.653939
sigma_a=42.934308
R:
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m0:
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V0:
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LogLike[0059]=-35605.712575
sigma_a=44.887182
R:
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m0:
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V0:
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LogLike[0060]=-35286.435152
sigma_a=46.936413
R:
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m0:
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V0:
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7.88750679]
V0:
[[ 0.00594938 -0.00920894 -0.00035916 0.00401579 -0.00207572 -0.00037048]
[-0.00920894 0.06619451 -0.00640152 -0.00278354 0.00640434 0.00067469]
[-0.00035916 -0.00640152 0.09759614 -0.00038994 0.00061387 0.00011615]
[ 0.00401579 -0.00278354 -0.00038994 0.0098564 -0.01033883 -0.00070984]
[-0.00207572 0.00640434 0.00061387 -0.01033883 0.07344064 -0.00557872]
[-0.00037048 0.00067469 0.00011615 -0.00070984 -0.00557872 0.09776433]]
LogLike[0086]=-24649.616234
sigma_a=155.537059
R:
[[0.62468581 0.27445944]
[0.27445944 0.32754846]]
m0:
[238.39179069 30.32574361 2.58223168 493.89848695 88.30026576
7.78646833]
V0:
[[ 0.0058722 -0.00913021 -0.00035179 0.00391754 -0.00200885 -0.00036164]
[-0.00913021 0.06609159 -0.00640855 -0.00271817 0.00637115 0.00066852]
[-0.00035179 -0.00640855 0.09759536 -0.00038018 0.00060692 0.00011529]
[ 0.00391754 -0.00271817 -0.00038018 0.00964071 -0.01015843 -0.00069089]
[-0.00200885 0.00637115 0.00060692 -0.01015843 0.07326795 -0.00559413]
[-0.00036164 0.00066852 0.00011529 -0.00069089 -0.00559413 0.09776258]]
LogLike[0087]=-191385.912354
sigma_a=161.163137
R:
[[0.57496949 0.23616374]
[0.23616374 0.28184554]]
m0:
[239.01130467 29.80539128 2.52253721 494.98672426 87.46706754
7.68980274]
V0:
[[ 0.00579694 -0.00905256 -0.00034465 0.00382179 -0.00194264 -0.00035306]
[-0.00905256 0.06599005 -0.00641544 -0.0026523 0.00633606 0.00066234]
[-0.00034465 -0.00641544 0.0975946 -0.00037073 0.0006001 0.00011445]
[ 0.00382179 -0.0026523 -0.00037073 0.00943343 -0.0099836 -0.00067277]
[-0.00194264 0.00633606 0.0006001 -0.0099836 0.07309973 -0.00560897]
[-0.00035306 0.00066234 0.00011445 -0.00067277 -0.00560897 0.09776092]]
EM: log likelihood decreased
Plot convergence
fig = go.Figure()
trace = go.Scatter(x=optim_res_ga["elapsed_time"], y=optim_res_ga["log_like"],
name="Gradient ascent", mode="lines+markers")
fig.add_trace(trace)
trace = go.Scatter(x=optim_res_em["elapsed_time"], y=optim_res_em["log_like"],
name="EM", mode="lines+markers")
fig.add_trace(trace)
fig.update_layout(xaxis_title="Elapsed Time (sec)",
yaxis_title="Log Likelihood")
fig
Perform smoothing with optimized parameters
Gradient ascent
Perform batch filtering
View source code of lds.inference.filterLDS_SS_withMissingValues_np
Q_ga = optim_res_ga["estimates"]["sigma_a"].item()**2*Qe
m0_ga = optim_res_ga["estimates"]["m0"].numpy()
V0_ga = np.diag(optim_res_ga["estimates"]["sqrt_diag_V0"].numpy()**2)
R_ga = np.diag(optim_res_ga["estimates"]["sqrt_diag_R"].numpy()**2)
filterRes_ga = lds.inference.filterLDS_SS_withMissingValues_np(
y=y, B=B, Q=Q_ga, m0=m0_ga, V0=V0_ga, Z=Z, R=R_ga)
Perform batch smoothing
View source code of lds.inference.smoothLDS_SS
smoothRes_ga = lds.inference.smoothLDS_SS(
B=B, xnn=filterRes_ga["xnn"], Vnn=filterRes_ga["Vnn"],
xnn1=filterRes_ga["xnn1"], Vnn1=filterRes_ga["Vnn1"], m0=m0_ga, V0=V0_ga)
EM
Perform batch filtering
View source code of lds.inference.filterLDS_SS_withMissingValues_np
Q_em = optim_res_em["estimates"]["sigma_a"].item()**2*Qe
m0_em = optim_res_em["estimates"]["m0"]
V0_em = optim_res_em["estimates"]["V0"]
R_em = optim_res_em["estimates"]["R"]
filterRes_em = lds.inference.filterLDS_SS_withMissingValues_np(
y=y, B=B, Q=Q_em, m0=m0_em, V0=V0_em, Z=Z, R=R_em)
Perform batch smoothing
View source code of lds.inference.smoothLDS_SS
smoothRes_em = lds.inference.smoothLDS_SS(
B=B, xnn=filterRes_em["xnn"], Vnn=filterRes_em["Vnn"],
xnn1=filterRes_em["xnn1"], Vnn1=filterRes_em["Vnn1"], m0=m0_em, V0=V0_em)
Plot smoothing results
def get_fig_kinematics_vs_time(
time,
measured_x, measured_y,
finite_diff_x, finite_diff_y,
ga_mean_x, ga_mean_y,
ga_ci_x_upper, ga_ci_y_upper,
ga_ci_x_lower, ga_ci_y_lower,
em_mean_x, em_mean_y,
em_ci_x_upper, em_ci_y_upper,
em_ci_x_lower, em_ci_y_lower,
cb_alpha,
color_true,
color_measured,
color_finite_diff,
color_ga_pattern,
color_em_pattern,
xlabel, ylabel):
fig = go.Figure()
if measured_x is not None:
trace_mes_x = go.Scatter(
x=time, y=measured_x,
mode="markers",
marker={"color": color_measured},
name="measured x",
showlegend=True,
)
fig.add_trace(trace_mes_x)
if measured_y is not None:
trace_mes_y = go.Scatter(
x=time, y=measured_y,
mode="markers",
marker={"color": color_measured},
name="measured y",
showlegend=True,
)
fig.add_trace(trace_mes_y)
if finite_diff_x is not None:
trace_fd_x = go.Scatter(
x=time, y=finite_diff_x,
mode="markers",
marker={"color": color_finite_diff},
name="finite difference x",
showlegend=True,
)
fig.add_trace(trace_fd_x)
if finite_diff_y is not None:
trace_fd_y = go.Scatter(
x=time, y=finite_diff_y,
mode="markers",
marker={"color": color_finite_diff},
name="finite difference y",
showlegend=True,
)
fig.add_trace(trace_fd_y)
trace_ga_x = go.Scatter(
x=time, y=ga_mean_x,
mode="markers",
marker={"color": color_ga_pattern.format(1.0)},
name="grad. ascent x",
showlegend=True,
legendgroup="ga_x",
)
fig.add_trace(trace_ga_x)
trace_ga_x_cb = go.Scatter(
x=np.concatenate([time, time[::-1]]),
y=np.concatenate([ga_ci_x_upper, ga_ci_x_lower[::-1]]),
fill="toself",
fillcolor=color_ga_pattern.format(cb_alpha),
line=dict(color=color_ga_pattern.format(0.0)),
showlegend=False,
legendgroup="ga_x",
)
fig.add_trace(trace_ga_x_cb)
trace_ga_y = go.Scatter(
x=time, y=ga_mean_y,
mode="markers",
marker={"color": color_ga_pattern.format(1.0)},
name="grad. ascent y",
showlegend=True,
legendgroup="ga_y",
)
fig.add_trace(trace_ga_y)
trace_ga_y_cb = go.Scatter(
x=np.concatenate([time, time[::-1]]),
y=np.concatenate([ga_ci_y_upper, ga_ci_y_lower[::-1]]),
fill="toself",
fillcolor=color_ga_pattern.format(cb_alpha),
line=dict(color=color_ga_pattern.format(0.0)),
showlegend=False,
legendgroup="ga_y",
)
fig.add_trace(trace_ga_y_cb)
trace_em_x = go.Scatter(
x=time, y=em_mean_x,
mode="markers",
marker={"color": color_em_pattern.format(1.0)},
name="EM x",
showlegend=True,
legendgroup="em_x",
)
fig.add_trace(trace_em_x)
trace_em_x_cb = go.Scatter(
x=np.concatenate([time, time[::-1]]),
y=np.concatenate([em_ci_x_upper, em_ci_x_lower[::-1]]),
fill="toself",
fillcolor=color_em_pattern.format(cb_alpha),
line=dict(color=color_em_pattern.format(0.0)),
showlegend=False,
legendgroup="em_x",
)
fig.add_trace(trace_em_x_cb)
trace_em_y = go.Scatter(
x=time, y=em_mean_y,
mode="markers",
marker={"color": color_em_pattern.format(1.0)},
name="EM y",
showlegend=True,
legendgroup="em_y",
)
fig.add_trace(trace_em_y)
trace_em_y_cb = go.Scatter(
x=np.concatenate([time, time[::-1]]),
y=np.concatenate([em_ci_y_upper, em_ci_y_lower[::-1]]),
fill="toself",
fillcolor=color_em_pattern.format(cb_alpha),
line=dict(color=color_em_pattern.format(0.0)),
showlegend=False,
legendgroup="em_y",
)
fig.add_trace(trace_em_y_cb)
fig.update_layout(xaxis_title=xlabel,
yaxis_title=ylabel,
paper_bgcolor='rgba(0,0,0,0)',
plot_bgcolor='rgba(0,0,0,0)',
)
return fig
N = y.shape[1]
time = np.arange(0, N*dt, dt)
smoothed_means_ga = smoothRes_ga["xnN"]
smoothed_covs_ga = smoothRes_ga["VnN"]
smoothed_std_x_y_ga = np.sqrt(np.diagonal(a=smoothed_covs_ga, axis1=0, axis2=1))
smoothed_means_em = smoothRes_em["xnN"]
smoothed_covs_em = smoothRes_em["VnN"]
smoothed_std_x_y_em = np.sqrt(np.diagonal(a=smoothed_covs_em, axis1=0, axis2=1))
color_true = "blue"
color_measured = "black"
color_finite_diff = "blue"
color_ga_pattern = "rgba(255,0,0,{:f})"
color_em_pattern = "rgba(255,165,0,{:f})"
cb_alpha = 0.3
Gradient ascent
Plot true, measured and smoothed positions (with 95% confidence band)
measured_x = y[0, :]
measured_y = y[1, :]
finite_diff_x = None
finite_diff_y = None
smoothed_mean_x_ga = smoothed_means_ga[0, 0, :]
smoothed_mean_y_ga = smoothed_means_ga[3, 0, :]
smoothed_mean_x_em = smoothed_means_em[0, 0, :]
smoothed_mean_y_em = smoothed_means_em[3, 0, :]
smoothed_ci_x_upper_ga = smoothed_mean_x_ga + 1.96*smoothed_std_x_y_ga[:, 0]
smoothed_ci_x_lower_ga = smoothed_mean_x_ga - 1.96*smoothed_std_x_y_ga[:, 0]
smoothed_ci_y_upper_ga = smoothed_mean_y_ga + 1.96*smoothed_std_x_y_ga[:, 3]
smoothed_ci_y_lower_ga = smoothed_mean_y_ga - 1.96*smoothed_std_x_y_ga[:, 3]
smoothed_ci_x_upper_em = smoothed_mean_x_em + 1.96*smoothed_std_x_y_em[:, 0]
smoothed_ci_x_lower_em = smoothed_mean_x_em - 1.96*smoothed_std_x_y_em[:, 0]
smoothed_ci_y_upper_em = smoothed_mean_y_em + 1.96*smoothed_std_x_y_em[:, 3]
smoothed_ci_y_lower_em = smoothed_mean_y_em - 1.96*smoothed_std_x_y_em[:, 3]
fig = get_fig_kinematics_vs_time(
time=time,
measured_x=measured_x, measured_y=measured_y,
finite_diff_x=finite_diff_x, finite_diff_y=finite_diff_y,
ga_mean_x=smoothed_mean_x_ga, ga_mean_y=smoothed_mean_y_ga,
ga_ci_x_upper=smoothed_ci_x_upper_ga,
ga_ci_y_upper=smoothed_ci_y_upper_ga,
ga_ci_x_lower=smoothed_ci_x_lower_ga,
ga_ci_y_lower=smoothed_ci_y_lower_ga,
em_mean_x=smoothed_mean_x_em, em_mean_y=smoothed_mean_y_em,
em_ci_x_upper=smoothed_ci_x_upper_em,
em_ci_y_upper=smoothed_ci_y_upper_em,
em_ci_x_lower=smoothed_ci_x_lower_em,
em_ci_y_lower=smoothed_ci_y_lower_em,
cb_alpha=cb_alpha,
color_true=color_true, color_measured=color_measured,
color_finite_diff=color_finite_diff,
color_ga_pattern=color_ga_pattern,
color_em_pattern=color_em_pattern,
xlabel="Time (sec)", ylabel="Position (pixels)")
# fig_filename_pattern = "../../figures/smoothed_pos.{:s}"
# fig.write_image(fig_filename_pattern.format("png"))
# fig.write_html(fig_filename_pattern.format("html"))
fig
Plot true and smoothed velocities (with 95% confidence band)
measured_x = None
measured_y = None
finite_diff_x = np.diff(y[0, :])/dt
finite_diff_y = np.diff(y[1, :])/dt
smoothed_mean_x_ga = smoothed_means_ga[1, 0, :]
smoothed_mean_y_ga = smoothed_means_ga[4, 0, :]
smoothed_mean_x_em = smoothed_means_em[1, 0, :]
smoothed_mean_y_em = smoothed_means_em[4, 0, :]
smoothed_ci_x_upper_ga = smoothed_mean_x_ga + 1.96*smoothed_std_x_y_ga[:, 1]
smoothed_ci_x_lower_ga = smoothed_mean_x_ga - 1.96*smoothed_std_x_y_ga[:, 1]
smoothed_ci_y_upper_ga= smoothed_mean_y_ga + 1.96*smoothed_std_x_y_ga[:, 4]
smoothed_ci_y_lower_ga = smoothed_mean_y_ga - 1.96*smoothed_std_x_y_ga[:, 4]
smoothed_ci_x_upper_em = smoothed_mean_x_em + 1.96*smoothed_std_x_y_em[:, 1]
smoothed_ci_x_lower_em = smoothed_mean_x_em - 1.96*smoothed_std_x_y_em[:, 1]
smoothed_ci_y_upper_em= smoothed_mean_y_em + 1.96*smoothed_std_x_y_em[:, 4]
smoothed_ci_y_lower_em = smoothed_mean_y_em - 1.96*smoothed_std_x_y_em[:, 4]
fig = get_fig_kinematics_vs_time(
time=time,
measured_x=measured_x, measured_y=measured_y,
finite_diff_x=finite_diff_x, finite_diff_y=finite_diff_y,
ga_mean_x=smoothed_mean_x_ga, ga_mean_y=smoothed_mean_y_ga,
ga_ci_x_upper=smoothed_ci_x_upper_ga,
ga_ci_y_upper=smoothed_ci_y_upper_ga,
ga_ci_x_lower=smoothed_ci_x_lower_ga,
ga_ci_y_lower=smoothed_ci_y_lower_ga,
em_mean_x=smoothed_mean_x_em, em_mean_y=smoothed_mean_y_em,
em_ci_x_upper=smoothed_ci_x_upper_em,
em_ci_y_upper=smoothed_ci_y_upper_em,
em_ci_x_lower=smoothed_ci_x_lower_em,
em_ci_y_lower=smoothed_ci_y_lower_em,
cb_alpha=cb_alpha,
color_true=color_true, color_measured=color_measured,
color_finite_diff=color_finite_diff,
color_ga_pattern=color_ga_pattern,
color_em_pattern=color_em_pattern,
xlabel="Time (sec)", ylabel="Velocity (pixels/sec)")
# fig_filename_pattern = "../../figures/smoothed_vel.{:s}"
# fig.write_image(fig_filename_pattern.format("png"))
# fig.write_html(fig_filename_pattern.format("html"))
fig
Plot true and smoothed accelerations (with 95% confidence band)
measured_x = None
measured_y = None
finite_diff_x = np.diff(np.diff(y[0, :]))/dt**2
finite_diff_y = np.diff(np.diff(y[1, :]))/dt**2
smoothed_mean_x_ga = smoothed_means_ga[2, 0, :]
smoothed_mean_y_ga = smoothed_means_ga[5, 0, :]
smoothed_mean_x_em = smoothed_means_em[2, 0, :]
smoothed_mean_y_em = smoothed_means_em[5, 0, :]
smoothed_ci_x_upper_ga = smoothed_mean_x_ga + 1.96*smoothed_std_x_y_ga[:, 2]
smoothed_ci_x_lower_ga = smoothed_mean_x_ga - 1.96*smoothed_std_x_y_ga[:, 2]
smoothed_ci_y_upper_ga = smoothed_mean_y_ga + 1.96*smoothed_std_x_y_ga[:, 5]
smoothed_ci_y_lower_ga = smoothed_mean_y_ga - 1.96*smoothed_std_x_y_ga[:, 5]
smoothed_ci_x_upper_em = smoothed_mean_x_em + 1.96*smoothed_std_x_y_em[:, 2]
smoothed_ci_x_lower_em = smoothed_mean_x_em - 1.96*smoothed_std_x_y_em[:, 2]
smoothed_ci_y_upper_em = smoothed_mean_y_em + 1.96*smoothed_std_x_y_em[:, 5]
smoothed_ci_y_lower_em = smoothed_mean_y_em - 1.96*smoothed_std_x_y_em[:, 5]
fig = get_fig_kinematics_vs_time(
time=time,
measured_x=measured_x, measured_y=measured_y,
finite_diff_x=finite_diff_x, finite_diff_y=finite_diff_y,
ga_mean_x=smoothed_mean_x_ga, ga_mean_y=smoothed_mean_y_ga,
ga_ci_x_upper=smoothed_ci_x_upper_ga,
ga_ci_y_upper=smoothed_ci_y_upper_ga,
ga_ci_x_lower=smoothed_ci_x_lower_ga,
ga_ci_y_lower=smoothed_ci_y_lower_ga,
em_mean_x=smoothed_mean_x_em, em_mean_y=smoothed_mean_y_em,
em_ci_x_upper=smoothed_ci_x_upper_em,
em_ci_y_upper=smoothed_ci_y_upper_em,
em_ci_x_lower=smoothed_ci_x_lower_em,
em_ci_y_lower=smoothed_ci_y_lower_em,
cb_alpha=cb_alpha,
color_true=color_true, color_measured=color_measured,
color_finite_diff=color_finite_diff,
color_ga_pattern=color_ga_pattern,
color_em_pattern=color_em_pattern,
xlabel="Time (sec)", ylabel="Acceleration (pixels/sec^2)")
# fig_filename_pattern = "../../figures/smoothed_acc.{:s}"
# fig.write_image(fig_filename_pattern.format("png"))
# fig.write_html(fig_filename_pattern.format("html"))
fig
Total running time of the script: (8 minutes 2.825 seconds)
