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functions.py
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Mon, May 20, 13:25

functions.py
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## NOTE: we have to impport everything needed in this file as well, otherwsie pd, np, plt
## are not defined, even if they are in the notebook
import os
import random
import pandas as pd
import numpy as np
import math
from scipy.integrate import solve_ivp
import matplotlib.pyplot as plt
def _fix_seed(seed):
random.seed(seed)
np.random.seed(seed)
def unicycle(t, x):
# paramater values
v = 0.5
omega = -math.pi / 3
dp0_dt = v * np.cos(x[2])
dp1_dt = v * np.sin(x[2])
dphi_dt = omega
return np.array([dp0_dt, dp1_dt, dphi_dt])
def generate_trajectories(number_trajectories, noise):
dfs = []
for i in range(number_trajectories):
solution = solve_ivp(unicycle, y0=[np.random.random_sample() * 2 - 1, np.random.random_sample() * 2 - 1, np.random.random_sample() * 2 * math.pi - math.pi], t_span=[0,50], t_eval=np.arange(0,50,0.02), method="RK45")
steps = solution.t.size
t = solution.t
p_0 = solution.y[0,:] + np.random.normal(loc=0, scale=noise * 2, size=steps)
p_1 = solution.y[1,:] + np.random.normal(loc=0, scale=noise * 2, size=steps)
phi = solution.y[2,:] + np.random.normal(loc=0, scale=noise * 2 * math.pi, size=steps)
df = pd.DataFrame(data={'t': t, 'p_0': p_0, 'p_1': p_1, 'phi': phi})
dfs.append(df)
# Return the last trajectory as example
return dfs
def plot_trajectories(df_, labels=None):
if not isinstance(df_, list):
df_ = [df_]
plt.figure()
for j, X in enumerate(df_):
plt.scatter(X.iloc[:,1], X.iloc[:,2], s=1, marker=".")
plt.xlabel("$p_0$", size=15)
plt.xticks(size=15)
plt.ylabel("$p_1$", size=15)
plt.yticks(size=15)
plt.axis('square')
plt.grid()
if labels is not None:
plt.legend(labels, prop={'size':15}, markerscale=10)
plt.show()
plt.figure()
for j, df in enumerate(df_):
plt.scatter(df.iloc[:,0], df.iloc[:,3], s=1, marker=".")
plt.ylabel("$\\varphi$", size=15)
plt.xticks(size=15)
plt.xlabel("$t$", size=15)
plt.yticks(size=15)
plt.grid()
if labels is not None:
plt.legend(labels, prop={'size':15}, markerscale=10)
plt.show()
def generate_X_Y(dfs_train_, dfs_test_, generate_target):
dfs = []
for j in range(2):
Xs = []
Ys = []
ts = []
for i in range(len(dfs_train_) if j==0 else len(dfs_test_)):
df = dfs_train_[i] if j == 0 else dfs_test_[i]
X, Y, t = generate_target(df)
Xs.append(X)
Ys.append(Y)
ts.append(t)
X = np.concatenate(Xs)
Y = np.concatenate(Ys)
t = np.concatenate(ts)
df = pd.DataFrame(data={'t': t, 'p0': X[:,0], 'p1': X[:,1],'phi': X[:,2], 'dp0dt': Y[:,0], 'dp1dt': Y[:,1],'dphidt': Y[:,2],})
dfs.append(df)
# return the train and test data
return dfs
def separate_trajectories(df_test, number_trajectories):
a = len(df_test) // number_trajectories
t_test = [df_test[['t']].to_numpy()[i*a:(i+1)*a,:].flatten() for i in range(number_trajectories)]
X_test = [df_test[['p0', 'p1', 'phi']].to_numpy()[i*a:(i+1)*a,:] for i in range(number_trajectories)]
Y_test = [df_test[['dp0dt', 'dp1dt', 'dphidt']].to_numpy()[i*a:(i+1)*a,:] for i in range(number_trajectories)]
return t_test, X_test, Y_test
def plot_data(t, X, Y, labels=None):
fig, ax = plt.subplots(2, 2, figsize=(16,9))
ax[0,0].scatter(X[:,0], X[:,1])
ax[0,0].set_xlabel("$p_0$", size=15)
ax[0,0].set_ylabel("$p_1$", size=15)
ax[0,0].axis('square')
ax[0,0].set_title('$\mathbf{x}_{1:2}$', size=18)
ax[0,0].grid()
ax[1,0].scatter(Y[:,0], Y[:,1])
ax[1,0].set_xlabel("$\dot{p}_0$", size=15)
ax[1,0].set_ylabel("$\dot{p}_1$", size=15)
ax[1,0].axis('square')
ax[1,0].set_title('$\mathbf{y}_{1:2}$', size=18)
ax[1,0].grid()
ax[0,1].scatter(t, X[:,2])
ax[0,1].set_xlabel("t", size=15)
ax[0,1].set_ylabel("$\\varphi$", size=15)
ax[0,1].set_title('$\mathbf{x}_{3}$', size=18)
ax[0,1].grid()
ax[1,1].scatter(t, Y[:,2])
ax[1,1].set_xlabel("t", size=15)
ax[1,1].set_ylabel("$\dot{\\varphi}$", size=15)
ax[1,1].set_title('$\mathbf{y}_{3}$', size=18)
ax[1,1].grid()
fig.tight_layout()
plt.show()
def predict_and_plot(X_, Y_, t_, w_1, w_2, w_3, feature_map):
if not isinstance(t_, list):
t_ = [t_]
if not isinstance(Y_, list):
Y_ = [Y_]
if not isinstance(X_, list):
X_ = [X_]
# define vector field
def f_vector(x, w_1, w_2, w_3):
return np.array([w_1.T @ feature_map(np.expand_dims(x, axis=0))[0,:], w_2.T @ feature_map(np.expand_dims(x, axis=0))[0,:], w_3.T @ feature_map(np.expand_dims(x, axis=0))[0,:]])
fig, ((ax1, ax2), (ax3, ax4)) = plt.subplots(2, 2, figsize=(7+len(t_),7+len(t_)))
for t,X,Y in zip(t_,X_,Y_):
# integrate
sol = solve_ivp(lambda t,x: f_vector(x, w_1, w_2, w_3), y0=X[0,:], t_span=[t[0],t[-1]], t_eval=t, method="RK45")
Y_pred = feature_map(sol.y.T) @ np.stack([w_1, w_2, w_3], axis=1)
# plotting
ax1.plot(sol.y[0,:], sol.y[1,:], label="$(p_0, p_1)$ predicted")
ax1.scatter(X[:,0], X[:,1],s=1, marker=".", c="k", label="$(p_0, p_1)$ data")
ax2.plot(sol.t,sol.y[2,:], label="$\\varphi$ prediction")
ax2.scatter(t, X[:,2], s=1, marker=".", c="k", label="$\\varphi$ data")
ax3.plot(Y_pred[:, 0], Y_pred[:, 1], label="$(\dot{p}_0, \dot{p}_1)$ predicted")
ax3.scatter(Y[:,0], Y[:,1],s=1, marker=".", c="k", label="$(\dot{p}_0, \dot{p}_1)$ data")
ax4.plot(sol.t, Y_pred[:,2], label="$\dot{\\varphi}$ prediction")
ax4.scatter(t, Y[:,2], s=1, marker=".", c="k", label="$\dot{\\varphi}$ data")
# Cosmetics
ax1.set_xlabel("$p_0$")
ax1.set_ylabel("$p_1$")
ax2.set_ylabel("$\\varphi$")
ax2.set_xlabel("$t$")
ax3.set_xlabel("$\dot{p}_0$")
ax3.set_ylabel("$\dot{p}_1$")
ax4.set_ylabel("$\dot{\\varphi}$")
ax4.set_xlabel("$t$")
ax1.axis('equal')
ax2.axis('equal')
ax3.axis('equal')
ax4.axis('equal')
if len(t_) == 1:
ax1.legend(bbox_to_anchor=(0.5,-0.25), loc='upper center', markerscale=5)
ax2.legend(bbox_to_anchor=(0.5,-0.25), loc='upper center', markerscale=5)
ax3.legend(bbox_to_anchor=(0.5,-0.25), loc='upper center', markerscale=5)
ax4.legend(bbox_to_anchor=(0.5,-0.25), loc='upper center', markerscale=5)
fig.tight_layout()
plt.show()

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