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utilities.py
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import numpy as np
import scipy
from numpy import random
import matplotlib.pyplot as plt
import matplotlib
from typing import Dict
from pykoop.lmi_regressors import LmiRegressor
from pykoop import lmi_regressors
from typing import Any, Dict, Optional
from pykoop import tsvd
import logging
import tempfile
import joblib
import picos
import scipy.signal
import sklearn.base
import os
from random import randint
import pykoop
import shutil
polite_stop = False
# Create logger
log = logging.getLogger(__name__)
log.addHandler(logging.NullHandler())
# Create temporary cache directory for memoized computations
_cachedir = tempfile.TemporaryDirectory(prefix='pykoop_')
log.info(f'Temporary directory created at `{_cachedir.name}`')
memory = joblib.Memory(_cachedir.name, verbose=0)
color_dict = {
'EDMD': (0.90, 0.60, 0.00),
'EDMD-AS': (0.80, 0.40, 0.00),
'FBEDMD': (0.35, 0.70, 0.90),
'FBEDMD-AS': (0.00, 0.45, 0.70)
}
color_list = [(0.00, 0.45, 0.70), (0.90, 0.60, 0.00), (0.00, 0.60, 0.50),
(0.35, 0.70, 0.90), (0.60, 0.60, 0.60)]
linestyle_dict = {
'EDMD': '-',
'EDMD-AS': '--',
'FBEDMD': '-',
'FBEDMD-AS': '--'
}
def Q_gen(n_states: int = 2,
add_noise: list = [0, 1],
noise: int = 0) -> np.ndarray:
Q = np.zeros((n_states, n_states))
for i in add_noise:
Q[i, i] = noise
return Q
def add_noise(data, Q, mu, n, seed: int = 3) -> np.ndarray:
v = np.zeros((Q.shape[0], data.shape[1]))
e, u = np.linalg.eig(Q)
rng = random.default_rng(seed=seed)
r_noise = rng.normal(0, 1, (data.shape[0], data.shape[1])) + mu
v = np.diag(e)**0.5 @ r_noise
noise = u @ v
Data = noise + data
n = data.shape[1]
s_mean = np.mean(data)
n_mean = np.mean(noise)
s_pow = np.sum((data - s_mean)**2) / n
n_pow = np.sum((noise - n_mean)**2) / n
snr = 10 * np.log10(s_pow / n_pow)
print('Avg signal power: {}, avg noise power: {}, SNR: {}'.format(
s_pow, n_pow, snr))
return Data.T, snr
def plot_rms_and_avg_error_paper(
val_data: Dict[str, np.ndarray],
true_val_data: np.ndarray,
path: str,
norm_params: np.ndarray,
robot: str,
val: int = 0,
n: int = None,
n_bins: int = 20,
**kwargs,
) -> None:
plt.rcParams.update(**kwargs)
usetex = True if shutil.which('latex') else False
if usetex:
plt.rc('text', usetex=True)
fig, ax = plt.subplots(
1,
2,
constrained_layout=True,
figsize=(5.3348, 2.5),
sharey=True,
)
i = 0
conv_param = 2.54 if robot == 'soft_robot' else 1
if robot == 'soft_robot':
if n is None:
n = val_data[list(val_data.keys())[0]].shape[0]
n_states = val_data[list(val_data.keys())[0]].shape[1] - 1
n_val_eps = int(np.max(true_val_data[:, 0]) + 1)
error_metrics = [r'Root-mean-square error', r'Mean error']
for metric in error_metrics:
avg_error = {}
error = {}
ep_val = {}
for tag in val_data.keys():
if tag == 'EDMD' or tag == 'FBEDMD':
continue
avg_error[tag] = 0
for val in range(n_val_eps):
ep_val['{}'.format(val)] = {}
max_error = 0
regressors = [r'EDMD-AS', r'FBEDMD-AS']
for ep in range(n_val_eps):
for tag, data in val_data.items():
if tag == r'EDMD' or tag == r'FBEDMD':
continue
if metric == r'Root-mean-square error':
temp = np.sqrt(
np.sum(
(data[data[:, 0] == ep, 1:][:n] *
(conv_param * norm_params[0])
- true_val_data[true_val_data[:, 0] == ep, 1:
(n_states + 1)][:n] *
(conv_param * norm_params[0]))**2) / n)
elif metric == r'Mean error':
temp = np.abs(
np.sum(data[data[:, 0] == ep, 1:][:n] *
(conv_param * norm_params[0])
- true_val_data[true_val_data[:, 0] == ep,
1:(n_states + 1)][:n] *
(conv_param * norm_params[0])) / n)
max_error = temp if temp > max_error else max_error
error[tag] = temp
avg_error[tag] += temp / n_val_eps
ep_val['{}'.format(ep)] = list(error.values())
for tag in val_data.keys():
if tag == r'EDMD' or tag == r'FBEDMD':
continue
print('Average {} for {}: {}'.format(metric, tag,
avg_error[tag]))
x_ticks = np.arange(0, len(error.keys()), 1)
y_ticks = np.arange(0, max_error, max_error / 10)
width = 0.1
multiplier = 0
for ep_tag, data in ep_val.items():
offset = width * multiplier
rects = ax[i].bar(x_ticks + offset,
np.abs(data),
width,
edgecolor='k',
linewidth=0.5,
color=color_list[int(ep_tag)],
label=r'test ep. {}'.format(int(ep_tag) + 1),
zorder=3)
multiplier += 1
offset = width * multiplier
rects = ax[i].bar(x_ticks + offset,
np.abs(list(avg_error.values())),
width,
edgecolor='k',
linewidth=0.5,
hatch='//',
color=color_list[int(ep_tag) + 1],
label='average',
zorder=3)
if robot == 'soft_robot':
ax[i].set_ylabel('{}'.format(metric) + r' (cm)')
else:
ax[i].set_ylabel('{}'.format(metric) + r' (mm)')
if metric == r'Root-mean-square error':
ax[i].set_ylim(0, 1.2)
ax[i].set_yticks(np.linspace(0, 1.2, 7))
ax[i].grid(axis='y', linestyle='--', zorder=0)
i = +1
ax[1].set_xticks(x_ticks + width * 2.0, regressors)
ax[0].set_xticks(x_ticks + width * 2.0, regressors)
handles, labels = ax[0].get_legend_handles_labels()
fig.legend(
handles,
labels,
bbox_to_anchor=(0.5, 0.03),
loc='upper center',
ncol=3,
)
fig.text(0.034, 1.015, r'$(a)$')
fig.text(0.54, 1.015, r'$(b)$')
fig.savefig('build/figures/paper/{}_error_bars.pdf'.format(robot),
bbox_inches='tight')
fig.savefig('build/figures/paper/{}_error_bars.png'.format(robot),
bbox_inches='tight')
fig.tight_layout()
def plot_trajectory_error_paper(
val_data: Dict[str, np.ndarray],
true_val_data: np.ndarray,
path: str,
norm_params: np.ndarray,
robot: str,
val: int = 2,
n: int = None,
**kwargs,
) -> None:
if n is None:
n = val_data[list(val_data.keys())[0]].shape[0]
n_val_eps = int(np.max(true_val_data[:, 0]) + 1)
plt.rcParams.update(**kwargs)
usetex = True if shutil.which('latex') else False
if usetex:
plt.rc('text', usetex=True)
fig_dict = {}
conv_param = 2.54 if robot == 'soft_robot' else 1
for i in range(n_val_eps):
fig, axs = plt.subplot_mosaic(
[['x1_err', 'traj'], ['x2_err', 'traj']],
figsize=(5.3348, 2.2),
layout='constrained',
)
max_y = 0
min_y = 0
max_x = 0
min_x = 0
ax = axs['traj']
# Plot trajectory
for tag, data in val_data.items():
if np.isnan(data[data[:, 0] == i, 1][:n]).any() or np.isnan(
data[data[:, 0] == i, 2][:n]).any():
print('NaN detected in {}.'.format(tag))
continue
else:
ax.plot(data[data[:, 0] == i, 1][:n] *
(conv_param * norm_params[0]),
data[data[:, 0] == i, 2][:n] *
(conv_param * norm_params[1]),
label=tag,
color=color_dict[tag],
linestyle=linestyle_dict[tag])
temp_x = data[data[:, 0] == i,
1][:n] * (conv_param * norm_params[0])
temp_y = data[data[:, 0] == i,
2][:n] * (conv_param * norm_params[1])
max_x = np.max(temp_x) if (np.max(temp_x) > max_x) else max_x
min_x = np.min(temp_x) if (np.min(temp_x) < min_x
and np.min(temp_x) < 0) else min_x
max_y = np.max(temp_y) if (np.max(temp_y) > max_y) else max_y
min_y = np.min(temp_y) if (np.min(temp_y) < min_y
and np.min(temp_y) < 0) else min_y
ax.plot(true_val_data[true_val_data[:, 0] == i, 1][:n] *
(conv_param * norm_params[0]),
true_val_data[true_val_data[:, 0] == i, 2][:n] *
(conv_param * norm_params[1]),
label='Ground truth',
color='k',
linestyle=':',
zorder=5)
if i == val:
ax.scatter(true_val_data[true_val_data[:, 0] == i, 1][0] *
(conv_param * norm_params[0]),
true_val_data[true_val_data[:, 0] == i, 2][0] *
(conv_param * norm_params[1]),
marker='x',
s=25,
color='k',
zorder=4)
ax.scatter(true_val_data[true_val_data[:, 0] == i, 1][n - 1] *
(conv_param * norm_params[0]),
true_val_data[true_val_data[:, 0] == i, 2][n - 1] *
(conv_param * norm_params[1]),
marker='x',
s=25,
color='k',
zorder=4)
if robot == 'soft_robot':
ax.set(ylabel=r'$x_{}$ (cm)'.format(2))
ax.set(xlabel=r'$x_{}$ (cm)'.format(1))
else:
ax.set(ylabel=r'$x_{}$ (mm)'.format(2))
ax.set(xlabel=r'$x_{}$ (mm)'.format(1))
# ax.set_ylim(min_y - np.abs(min_y / 10), max_y + np.abs(max_y / 10))
# ax.set_xlim(min_x - np.abs(min_x / 10), max_x + np.abs(max_x / 10))
if robot == 'soft_robot':
ax.set_ylim(0, 20)
ax.set_xlim(0, 20)
ax.set_xticks(np.linspace(0, 20, 11))
ax.set_yticks(np.linspace(0, 20, 11))
elif robot == 'nl_msd':
ax.set_ylim(-4, 6)
ax.set_xlim(-4, 6)
ax.set_xticks(np.linspace(-4, 6, 6))
ax.set_yticks(np.linspace(-4, 6, 6))
ax.set_aspect('equal')
ax.grid(linestyle='--')
# Plot prediction errors
ax = [axs['x1_err'], axs['x2_err']]
t = np.linspace(0, n * 0.01, n) if robot == 'nl_msd' else np.linspace(
0, n * 0.0829, n)
conv_param = 2.54 if robot == 'soft_robot' else 1
for tag, data in val_data.items():
for k in range(2):
true_temp = true_val_data[true_val_data[:, 0] == i,
k + 1][:n] * (conv_param
* norm_params[k])
pred_temp = data[data[:, 0] == i,
k + 1][:n] * (conv_param * norm_params[k])
if np.isnan(pred_temp).any():
print('NaN detected in {} state {}'.format(tag, k))
continue
else:
if i == val:
ax[k].plot(
t,
pred_temp - true_temp,
color=color_dict[tag],
label=tag,
linestyle=linestyle_dict[tag],
)
else:
ax[k].plot(
pred_temp - true_temp,
color=color_dict[tag],
label=tag,
)
if robot == 'soft_robot':
ax[k].set(ylabel='$\Delta x_{}$ (cm)'.format(k + 1))
else:
ax[k].set(ylabel='$\Delta x_{}$ (mm)'.format(k + 1))
if robot == 'nl_msd':
ax[0].set_ylim(-3, 3)
ax[0].set_yticks(np.linspace(-3, 3, 7))
ax[1].set_ylim(-3, 3)
ax[1].set_yticks(np.linspace(-3, 3, 7))
ax[1].set_xlim(0, n * 0.01)
ax[1].set_xticks(np.linspace(0, 20, 11))
fig.text(0.73, 0.88, r'$t = 0$' + ' s')
fig.text(0.745, 0.305, r'$t = 21$' + ' s')
else:
ax[0].set_ylim(-3, 3)
ax[0].set_yticks(np.linspace(-3, 3, 7))
ax[1].set_ylim(-3, 3)
ax[1].set_yticks(np.linspace(-3, 3, 7))
ax[1].set_xlim(0, n * 0.0829)
ax[1].set_xticks(np.arange(0, 26, 5))
fig.text(0.64, 0.55, r'$t = 0$' + ' s')
fig.text(0.74, 0.51, r'$t = 24.5$' + ' s')
ax[1].set(xlabel=r'$t$' + ' (s)')
ax[0].grid(linestyle='--')
ax[1].grid(linestyle='--')
ax[0].tick_params(axis='y', which='major', pad=2)
ax[0].sharex(ax[1])
ax[0].tick_params(labelbottom=False)
handles, labels = axs['traj'].get_legend_handles_labels()
fig.legend(
handles,
labels,
bbox_to_anchor=(0.5, 0.03),
loc='upper center',
ncol=3,
)
fig.text(0.034, 1.015, r'$(a)$')
fig.text(0.56, 1.015, r'$(b)$')
fig_dict["F_{}".format(i)] = fig
fig_dict['F_{}'.format(val)].savefig(
'build/figures/paper/{}_trajectory_err.png'.format(robot),
bbox_inches='tight')
fig_dict['F_{}'.format(val)].savefig(
'build/figures/paper/{}_trajectory_err.pdf'.format(robot),
bbox_inches='tight')
def plot_frob_err(frob_error: Dict[str, np.ndarray], variances: np.ndarray,
snr: np.ndarray, path: str, **kwargs) -> None:
plt.rcParams.update(**kwargs)
usetex = True if shutil.which('latex') else False
if usetex:
plt.rc('text', usetex=True)
i = 0
fig, axs = plt.subplots(1, 3, layout='constrained', figsize=(5.3348, 2))
# for (k,v), (k2,v2) in zip(d.items(), d2.items()):
for (matrix, err), ax in zip(frob_error.items(), range(len(axs))):
for tag, data in err.items():
axs[ax].plot(snr,
data,
color=color_dict[tag],
label=tag,
linestyle=linestyle_dict[tag])
if usetex == True:
axs[ax].set(
ylabel=
r'$\frac{{\left\|\mathbf{{{}}}_\mathrm{{true}} - \bf{{{}}}_\mathrm{{approx}}\right\|_\ensuremath{{\mathsf{{F}}}}}}{{\left\|\mathbf{{{}}}_\mathrm{{true}}\right\|_\ensuremath{{\mathsf{{F}}}}}}$'
.format(matrix, matrix, matrix))
else:
axs[ax].set(ylabel='rel. frob. norm of {}'.format(matrix))
axs[ax].grid(linestyle='--')
axs[ax].set_xlim(np.min(snr), np.max(snr))
axs[ax].set_xticks(np.arange(10, 60, 10))
axs[ax].set(xlabel=r'SNR')
axs[0].set_yticks(np.linspace(0, 2.5, 6))
axs[1].sharey(axs[0])
axs[1].tick_params(labelleft=False)
axs[2].set_yticks(np.linspace(0, 0.20, 6))
handles, labels = axs[-1].get_legend_handles_labels()
fig.legend(
handles,
labels,
bbox_to_anchor=(0.5, 0.04),
loc='upper center',
ncol=4,
)
fig.savefig('{}/frob_norm_sqrd.png'.format(path), bbox_inches='tight')
fig.savefig('{}/frob_norm_sqrd.pdf'.format(path), bbox_inches='tight')
def plot_polar(koop_matrices: Dict[str, np.ndarray], path: str, robot: str,
**kwargs):
plt.rcParams.update(**kwargs)
usetex = True if shutil.which('latex') else False
if usetex:
plt.rc('text', usetex=True)
p_theta, p = koop_matrices[list(koop_matrices.keys())[0]].shape
n_inputs = p - p_theta
fig = plt.figure(figsize=(5.3348, 3.5), constrained_layout=True)
ax = plt.subplot(projection='polar')
ax.set_xlabel(r'$\mathrm{Re}(\lambda)$')
ax.set_ylabel(r'$\mathrm{Im}(\lambda)$', labelpad=30)
theta_min = -12
theta_max = 12
axin = ax.inset_axes([1.05, -0.3, 0.7, 0.7], projection='polar')
axes = [ax, axin]
sub_eig = 0
sup_eig = 2
max_eig = 0
i = 0
koop_matrices_new = {}
koop_matrices_new['EDMD'] = koop_matrices['EDMD']
koop_matrices_new['EDMD-AS'] = koop_matrices['EDMD-AS']
koop_matrices_new['FBEDMD'] = koop_matrices['FBEDMD']
koop_matrices_new['FBEDMD-AS'] = koop_matrices['FBEDMD-AS']
zorder = {}
zorder['EDMD'] = 7
zorder['EDMD-AS'] = 8
zorder['FBEDMD'] = 5
zorder['FBEDMD-AS'] = 6
markers = ['s', 'D', 'v', 'o'] + ['o'] * (len(koop_matrices.keys()) - 4)
# plot eigenvalues
for tag, U in koop_matrices_new.items():
eigv = scipy.linalg.eig(U[:, :-n_inputs])[0]
marker = markers.pop()
for axx in axes:
axx.scatter(np.angle(eigv),
np.absolute(eigv),
zorder=zorder[tag],
marker=marker,
s=60,
color=color_dict[tag],
linewidths=0.5,
edgecolors='w',
label=tag)
i += 1
unit_angles = np.linspace(0, 2 * np.pi, 100)
unit_radius = np.ones_like(unit_angles)
axin.plot(unit_angles, unit_radius, color='k', linestyle='--', linewidth=1)
axin.set_thetamin(theta_min)
axin.set_thetamax(theta_max)
sub = 0.999
sup = 1.0001
sub = float("{:.5f}".format(sub))
sup = float("{:.5f}".format(sup))
axin.set_rticks([sub - (1 - sub), sub, 1.0, sup])
axin.set_rmin(sub)
axin.set_rmax(sup)
axin.set_xticks([
theta_min * np.pi / 180, theta_min / 2 * np.pi / 180, 0,
theta_max / 2 * np.pi / 180, theta_max * np.pi / 180
])
ax.plot(unit_angles, unit_radius, color='k', linestyle='--', linewidth=1)
ax.set_rlim(0, 1.5)
ax.set_rticks([0, 0.5, 1, 1.5])
ax.grid(linestyle='--')
axin.set_rticks([sub, sup])
axin.set_rmin(sub)
axin.set_rmax(sup)
axin.grid(linestyle='--')
# Create lines linking border to zoomed plot
axin.annotate(
'',
xy=(0, sup),
xycoords=ax.transData,
xytext=(theta_max * np.pi / 180, sup),
textcoords=axin.transData,
arrowprops={
'arrowstyle': '-',
'color': 'k',
'shrinkA': 0,
'shrinkB': 0,
},
)
axin.annotate(
'',
xy=(0, sub),
xycoords=ax.transData,
xytext=(0, sub),
textcoords=axin.transData,
arrowprops={
'arrowstyle': '-',
'color': 'k',
'shrinkA': 0,
'shrinkB': 0,
},
)
axin.set_zorder(15)
ax.legend(
bbox_to_anchor=(0.74, -0.2),
loc='upper center',
ncol=4,
)
fig.savefig('build/figures/paper/{}_polar.pdf'.format(robot))
fig.savefig('build/figures/paper/{}_polar.png'.format(robot))
def summary_fig(
val_data: Dict[str, np.ndarray],
true_val_data: np.ndarray,
path: str,
norm_params: np.ndarray,
robot: str,
val: int = 2,
n: int = None,
**kwargs,
) -> None:
if n is None:
n = val_data[list(val_data.keys())[0]].shape[0]
n_val_eps = int(np.max(true_val_data[:, 0]) + 1)
plt.rcParams.update(**kwargs)
plt.rc('font', size=12)
usetex = True if shutil.which('latex') else False
if usetex:
plt.rc('text', usetex=True)
fig_dict = {}
conv_param = 2.54 if robot == 'soft_robot' else 1
for i in range(n_val_eps):
fig, ax = plt.subplots(
figsize=(5, 2.2),
layout='constrained',
)
max_y = 0
min_y = 0
max_x = 0
min_x = 0
ax.plot(true_val_data[true_val_data[:, 0] == i, 1][:(n - 1000)] *
(conv_param * norm_params[0]),
true_val_data[true_val_data[:, 0] == i, 2][:(n - 1000)] *
(conv_param * norm_params[1]),
label='Ground truth',
color='k',
linestyle=':',
zorder=5,
linewidth=2.5)
for tag, data in val_data.items():
if np.isnan(data[data[:, 0] == i,
1][:(n - 1000)]).any() or np.isnan(
data[data[:, 0] == i, 2][:(n - 1000)]).any():
print('NaN detected in {}.'.format(tag))
continue
else:
if tag == 'EDMD' or tag == 'FBEDMD':
continue
else:
if tag == "EDMD-AS":
ax.plot(data[data[:, 0] == i, 1][:(n - 1000)] *
(conv_param * norm_params[0]),
data[data[:, 0] == i, 2][:(n - 1000)] *
(conv_param * norm_params[1]),
label=tag + " (biased)",
color=color_list[2],
linestyle=linestyle_dict['EDMD'],
zorder=4,
linewidth=2.5)
else:
ax.plot(data[data[:, 0] == i, 1][:(n - 1000)] *
(conv_param * norm_params[0]),
data[data[:, 0] == i, 2][:(n - 1000)] *
(conv_param * norm_params[1]),
label=tag + " (unbiased)",
color=color_list[1],
linestyle=linestyle_dict['EDMD'],
zorder=3,
linewidth=2.5)
temp_x = data[data[:, 0] == i,
1][:n] * (conv_param * norm_params[0])
temp_y = data[data[:, 0] == i,
2][:n] * (conv_param * norm_params[1])
max_x = np.max(temp_x) if (np.max(temp_x) > max_x) else max_x
min_x = np.min(temp_x) if (np.min(temp_x) < min_x
and np.min(temp_x) < 0) else min_x
max_y = np.max(temp_y) if (np.max(temp_y) > max_y) else max_y
min_y = np.min(temp_y) if (np.min(temp_y) < min_y
and np.min(temp_y) < 0) else min_y
if robot == 'soft_robot':
ax.set(ylabel=r'$x_{}$ (cm)'.format(2))
ax.set(xlabel=r'$x_{}$ (cm)'.format(1))
else:
ax.set(ylabel=r'$x_{}$'.format(2))
ax.set(xlabel=r'$x_{}$'.format(1))
if robot == 'soft_robot':
ax.set_ylim(0, 20)
ax.set_xlim(0, 20)
ax.set_xticks(np.linspace(0, 20, 11))
ax.set_yticks(np.linspace(0, 20, 11))
ax.set_aspect('equal')
ax.tick_params(labelleft=False, labelbottom=False)
ax.tick_params(axis='both',
which='both',
bottom=False,
top=False,
labelbottom=False,
left=False,
labelleft=False)
fig.text(0.49, 0.8, r'$\mathbf{U}_\mathrm{f}$')
fig.text(0.595, 0.8, r'$\tilde{\mathbf{U}}$')
handles, labels = ax.get_legend_handles_labels()
fig_dict["F_{}".format(i)] = fig
fig_dict['F_{}'.format(val)].savefig(
'build/figures/paper/{}_summary_trajectory.png'.format(robot),
bbox_inches='tight')
fig_dict['F_{}'.format(val)].savefig(
'build/figures/paper/{}_summary_trajectory.pdf'.format(robot),
bbox_inches='tight')
def print_koop_matrices(koop_matrices: Dict[str, np.ndarray], **kwargs):
n_inputs = koop_matrices[list(
koop_matrices.keys())[0]].shape[1] - koop_matrices[list(
koop_matrices.keys())[0]].shape[0]
for tag, U in koop_matrices.items():
print('Eigenvalues of A_{}: '.format(tag),
scipy.linalg.eig(U[:, :-n_inputs])[0])
class LmiEdmd(LmiRegressor):
"""LMI-based EDMD with regularization. Inspired from the pykoop package.
"""
def __init__(
self,
alpha: float = 0,
ratio: float = 1,
reg_method: str = 'tikhonov',
inv_method: str = 'svd',
tsvd: Optional[tsvd.Tsvd] = None,
square_norm: bool = False,
picos_eps: float = 0,
solver_params: Optional[Dict[str, Any]] = None,
) -> None:
"""Instantiate :class:`LmiEdmd`. Inspired from the pykoop package.
"""
self.alpha = alpha
self.ratio = ratio
self.reg_method = reg_method
self.inv_method = inv_method
self.tsvd = tsvd
self.square_norm = square_norm
self.picos_eps = picos_eps
self.solver_params = solver_params
def _fit_regressor(self, X_unshifted: np.ndarray,
X_shifted: np.ndarray) -> np.ndarray:
# Set solver parameters
self.solver_params_ = self._default_solver_params.copy()
if self.solver_params is not None:
self.solver_params_.update(self.solver_params)
# Compute regularization coefficients
if self.reg_method == 'tikhonov':
self.alpha_tikhonov_ = self.alpha
self.alpha_other_ = 0.0
else:
self.alpha_tikhonov_ = self.alpha * (1.0 - self.ratio)
self.alpha_other_ = self.alpha * self.ratio
# Clone TSVD
self.tsvd_ = (sklearn.base.clone(self.tsvd)
if self.tsvd is not None else tsvd.Tsvd())
# Form optimization problem. Regularization coefficients must be scaled
# because of how G and H are defined.
q = X_unshifted.shape[0]
problem = self._create_base_problem(X_unshifted, X_shifted,
self.alpha_tikhonov_ / q,
self.inv_method, self.tsvd_,
self.picos_eps)
if self.reg_method == 'twonorm':
problem = lmi_regressors._add_twonorm(problem,
problem.variables['U'],
self.alpha_other_ / q,
self.square_norm,
self.picos_eps)
elif self.reg_method == 'nuclear':
problem = lmi_regressors._add_nuclear(problem,
problem.variables['U'],
self.alpha_other_ / q,
self.square_norm,
self.picos_eps)
# Solve optimization problem
problem.solve(**self.solver_params_)
# Save solution status
self.solution_status_ = problem.last_solution.claimedStatus
# Extract solution from ``Problem`` object
coef = self._extract_solution(problem)
return coef
def _validate_parameters(self) -> None:
# Check regularization methods
valid_reg_methods = ['tikhonov', 'twonorm', 'nuclear']
if self.reg_method not in valid_reg_methods:
raise ValueError('`reg_method` must be one of '
f'{valid_reg_methods}.')
# Check ratio
if (self.ratio <= 0) or (self.ratio > 1):
raise ValueError('`ratio` must be positive and less than one.')
@staticmethod
def _create_base_problem(
X_unshifted: np.ndarray,
X_shifted: np.ndarray,
alpha_tikhonov: float,
inv_method: str,
tsvd: tsvd.Tsvd,
picos_eps: float,
) -> picos.Problem:
"""Create optimization problem."""
# Validate ``alpha``
if alpha_tikhonov < 0:
raise ValueError('Parameter `alpha` must be positive or zero.')
# Validate ``inv_method``
valid_inv_methods = [
'inv', 'pinv', 'eig', 'ldl', 'chol', 'sqrt', 'svd'
]
if inv_method not in valid_inv_methods:
raise ValueError('`inv_method` must be one of '
f'{valid_inv_methods}.')
# Validate ``picos_eps``
if picos_eps < 0:
raise ValueError('Parameter `picos_eps` must be positive or zero.')
var = np.var(X_shifted, axis=0)
W = np.diag(1 / var)
W = np.eye(W.shape[0])
# Compute ``G`` and ``H``. ``alpha_tikhonov`` must already be scaled
# by ``q`` if applicable.
c, G, H, _ = lmi_regressors._calc_c_G_H(X_unshifted, X_shifted @ W.T,
alpha_tikhonov)
# Optimization problem
problem = picos.Problem()
# Constants
G_T = picos.Constant('G^T', G.T)
W = picos.Constant('W', W)
q = X_shifted.shape[0]
gamma = 10
# Variables
U = picos.RealVariable('U', (G.shape[0], H.shape[0]))
Z = picos.SymmetricVariable('Z', (G.shape[0], G.shape[0]))
# W = picos.RealVariable('W', (G.shape[0], G.shape[0]))
v = picos.RealVariable('v', 1)
Q = picos.SymmetricVariable('Q', (G.shape[1], G.shape[1]))
# Constraints
problem.add_constraint(Z >> picos_eps)
# problem.add_constraint(Q >> picos_eps)
# Choose method to handle inverse of H
if inv_method == 'inv':
H_inv = picos.Constant('H^-1', lmi_regressors._calc_Hinv(H))
problem.add_constraint(
picos.block([
[Z, U],
[U.T, H_inv],
]) >> picos_eps)
elif inv_method == 'pinv':
H_inv = picos.Constant('H^+', lmi_regressors._calc_Hpinv(H))
problem.add_constraint(
picos.block([
[Z, U],
[U.T, H_inv],
]) >> picos_eps)
elif inv_method == 'eig':
VsqrtLmb = picos.Constant('(V Lambda^(1/2))',
lmi_regressors._calc_VsqrtLmb(H))
problem.add_constraint(
picos.block([
[Z, U * VsqrtLmb],
[VsqrtLmb.T * U.T, 'I'],
]) >> picos_eps)
elif inv_method == 'ldl':
LsqrtD = picos.Constant('(L D^(1/2))',
lmi_regressors._calc_LsqrtD(H))
problem.add_constraint(
picos.block([
[Z, U * LsqrtD],
[LsqrtD.T * U.T, 'I'],
]) >> picos_eps)
elif inv_method == 'chol':
L = picos.Constant('L', lmi_regressors._calc_L(H))
problem.add_constraint(
picos.block([
[Z, U * L],
[L.T * U.T, 'I'],
]) >> picos_eps)
elif inv_method == 'sqrt':
sqrtH = picos.Constant('sqrt(H)', lmi_regressors._calc_sqrtH(H))
problem.add_constraint(
picos.block([
[Z, U * sqrtH],
[sqrtH.T * U.T, 'I'],
]) >> picos_eps)
elif inv_method == 'svd':
QSig = picos.Constant(
'Q Sigma',
lmi_regressors._calc_QSig(X_unshifted, alpha_tikhonov, tsvd))
problem.add_constraint(
picos.block([
[Z, W * U * QSig],
[QSig.T * U.T * W.T, 'I'],
]) >> picos_eps)
else:
# Should never, ever get here.
assert False
# Set objective
obj = c - 2 * picos.trace(W * U * G_T * W.T) + picos.trace(Z)
problem.set_objective('min', obj)
return problem
@staticmethod
def _create_new_forw_base_problem(
# Variablesm(
X_unshifted: np.ndarray,
X_shifted: np.ndarray,
alpha_tikhonov: float,
spectral_radius: float,
picos_eps: float,
new_eps: np.ndarray,
) -> picos.Problem:
"""Create optimization problem."""
# Validate ``alpha``
if alpha_tikhonov < 0:
raise ValueError('Parameter `alpha` must be positive or zero.')
# Validate ``inv_method``
# Validate ``picos_eps``
if picos_eps < 0:
raise ValueError('Parameter `picos_eps` must be positive or zero.')
# Compute ``G`` and ``H``. ``alpha_tikhonov`` must already be scaled
# by ``q`` if applicable.
# c, G, H, _ = lmi_regressors._calc_c_G_H(X_unshifted, X_shifted, alpha_tikhonov)
# G = picos.Constant('G', G)
# H = picos.Constant('H', H)
# Optimization problem
problem = picos.Problem()
# Constants
q = X_shifted.shape[0]
p_theta = X_shifted.shape[1]
Psi = X_unshifted.T
Theta_plus = X_shifted.T
n_inputs = Psi.shape[0] - Theta_plus.shape[0]
_G = 1 / q * Theta_plus @ Psi.T
# _G = _G + np.eye(_G.shape[0])*1e-5
_H = 1 / q * Psi @ Psi.T
# _H = _H + np.eye(_H.shape[0])*1e-5
H_inv = scipy.linalg.pinv(_H)
U_edmd = scipy.linalg.lstsq(_H.T, _G.T)[0].T
# _R = scipy.linalg.cholesky(_Hb, lower=True)
W = picos.Constant(
'W',
picos.block([[np.eye(p_theta),
np.zeros((p_theta, n_inputs))],
[np.zeros((n_inputs, p_theta)),
np.eye(n_inputs)]]))
rho = picos.Constant('rho', spectral_radius)
alpha = picos.Constant('alpha', 1)
# Variables
P = picos.SymmetricVariable('P', p_theta)
B = picos.RealVariable('B', (_G.shape[0], _H.shape[0] - _G.shape[0]))
K = picos.RealVariable('K', (p_theta, p_theta))
P_tilde = picos.block(
[[P, np.zeros((P.shape[0], H_inv.shape[0] - P.shape[0]))],
[
np.zeros((H_inv.shape[0] - P.shape[0], P.shape[0])),
alpha * np.eye(H_inv.shape[0] - P.shape[0])
]])