From 030b29a444fb8c8b6fae8d040936adcd220529ae Mon Sep 17 00:00:00 2001
From: Dave <davepreiss@gmail.com>
Date: Thu, 13 Apr 2023 11:22:33 -0400
Subject: [PATCH] week 9

---
 week7/img/animation.mp4 | Bin 0 -> 117170 bytes
 week7/week7_9_2.py      | 163 ----------------------------------------
 week9/README.MD         |  23 ++++++
 week9/img/11_1.png      | Bin 0 -> 16879 bytes
 week9/img/11_2.png      | Bin 0 -> 43630 bytes
 week9/img/animation.gif | Bin 0 -> 138962 bytes
 week9/week9_11_1.py     |  80 ++++++++++++++++++++
 week9/week9_11_2.py     |  91 ++++++++++++++++++++++
 8 files changed, 194 insertions(+), 163 deletions(-)
 create mode 100644 week7/img/animation.mp4
 create mode 100644 week9/README.MD
 create mode 100644 week9/img/11_1.png
 create mode 100644 week9/img/11_2.png
 create mode 100644 week9/img/animation.gif
 create mode 100644 week9/week9_11_1.py
 create mode 100644 week9/week9_11_2.py

diff --git a/week7/img/animation.mp4 b/week7/img/animation.mp4
new file mode 100644
index 0000000000000000000000000000000000000000..ec6bca7e480c6a61797cffc1cc089e9214caadbf
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diff --git a/week7/week7_9_2.py b/week7/week7_9_2.py
index 8ba9556..e69de29 100644
--- a/week7/week7_9_2.py
+++ b/week7/week7_9_2.py
@@ -1,163 +0,0 @@
-import scipy.spatial as spatial
-from scipy import integrate as integrate
-import numpy as np
-import matplotlib.pyplot as plt
-import matplotlib.animation as animation
-import time
-
-# import jit from numba
-# from numba import jit
-
-# @jit(nopython=True)
-def calc_force(P1, P2):
-    # calculate the force between points P1 and P2 (both 2D numpy arrays)
-    # the neutral distance is the variable neutral_distance
-    dist = np.linalg.norm(P1 - P2) # we should have this from scipy already but recalculating for now...
-    force = spring_constant * (dist - neutral_distance) * (P1 - P2) / dist
-    # repulsive forces only
-    # force = np.where(dist < neutral_distance, force, force*0.1)
-    return force
-
-# @jit(nopython=True)
-def solve_forces(particles, velocities):
-    particles_tree = spatial.cKDTree(particles)
-    # For each point, calculate the force due to the nearest neighbors
-    F = np.zeros((x_num*y_num, 2))
-    for particle in range(x_num*y_num):
-        # return the index of all particles within radius of radius_check
-        neighbors = particles_tree.query_ball_point(particles[particle, :], radius_check)
-        # Moving local damping here to use average velocity to avoid overapplying it
-        avg_vel = np.mean(velocities[neighbors, :], axis=0) # axis = 0 should average the columns
-        F_damping = damping_constant * (velocities[particle, :] - avg_vel)
-        # calculate the force on the particle due its neighbors
-        for neighbor in neighbors:
-            if neighbor != particle:
-                F[particle, :] -= calc_force(particles[particle, :], particles[neighbor, :])
-        F[particle, :] -= F_damping # leaving this out here for now
-    # Add some global damping
-    F -= damping_constant_global * velocities
-    F[:, 1] -= gravity  # gravity
-    np.clip(F, -force_threshold, force_threshold, out=F) # clip the force to be within limits of the threshold
-    return F
-
-# @jit(nopython=True)
-def solve_velocities(F, particles, dt):
-    # update the velocity of each particle from V_prev according to the force acting on it
-    V = dt * F / particle_mass * 0.95  # some damping here as well
-    # if we contact the ground, reverse the y velocity and dampen it
-    V[:, 1] = np.where((particles[:, 1] < 0), -V[:, 1]*coef_of_restitution, V[:, 1]) # if we cross into the floor, reverse the y velocity and apply a damping factor
-    # apply friction to the ground
-    V[:, 0] = np.where((particles[:, 1] < 0), V[:, 0]*0.9, V[:, 0])
-    return V
-
-# @jit(nopython=True)
-def solve_positions(particles, V):
-    # update the position of each particle from particles according to the velocity acting on it
-    particles_new = particles + dt * V
-    particles_new[:, 1] = np.where((particles_new[:, 1] < 0), 0, particles_new[:, 1]) # if we cross into the floor, reverse the y velocity and apply a damping factor
-    # CANTILEVERED BEAM - set all particles in the first column to have a position of zero
-    '''
-    for particle_num in range(len(particles)):
-        # if the particle is in the first row:
-        if particle_num <= y_num:
-            # keep that particle in place
-            particles_new[particle_num, 0] = particles[particle_num, 0]
-            particles_new[particle_num, 1] = particles[particle_num, 1]
-    '''
-    return particles_new
-
-# Constants:
-x_num = 40 # 10 or 40
-y_num = 40 # 10 or 40
-t_final = 1.0
-dt = 0.005
-sub_increments = 20 # how many timesteps to calculate between each frame
-mesh_spacing = 1
-neutral_distance = mesh_spacing
-height_offset = 8 # 3 for 10x10, 8 for 40x40
-
-# Physical Constants (these alter the particle physics)
-radius_check = neutral_distance*1.35
-spring_constant = 5000 # 1000
-damping_constant = 25 # 10
-damping_constant_global = 0.1 # air resistance
-coef_of_restitution = .75 # 0.5
-gravity = 1
-particle_mass = 0.5
-force_threshold_distance = 0.75 # springs shouldn't get any stronger past this distance
-force_threshold =  force_threshold_distance*neutral_distance * spring_constant
-
-
-# Create our triangular mesh of points
-particles = np.zeros((x_num*y_num, 2))
-velocities = np.zeros((x_num*y_num, 2))
-forces = np.zeros((x_num*y_num, 2))
-
-# create a mesh of particles, indexes start at point 0,0 and grow up the y-axis, then move to the next column and grow up that one
-for i in range(x_num):
-    for j in range(y_num):
-        particles[i*y_num+j, 0] = i
-        particles[i*y_num+j, 1] = j * np.sqrt(3) / 2 # height of equilateral triangle
-        if j % 2 == 1:  # add an offset to every other point
-            particles[i*y_num+j, 0] += 0.5
-
-# particles[-1, 0] -= mesh_spacing*.5 # offset a particle for chaos
-particles *= mesh_spacing # scale the mesh to the desired spacing
-particles[:,1] += mesh_spacing*height_offset
-centroid = np.mean(particles, axis=0) # find the centroid of the mesh
-
-# rotate the mesh of particles about the origin
-theta = np.radians(30)
-rot_matrix = np.array([[np.cos(theta), -np.sin(theta)], [np.sin(theta), np.cos(theta)]])
-particles = np.dot(particles-centroid, rot_matrix) + centroid
-
-# Create a scatterplot of X
-my_dpi = 96
-fig = plt.figure(figsize=(800/my_dpi, 400/my_dpi), dpi=my_dpi)
-ax = fig.add_axes([0, 0, 1, 1], frameon=False)
-# numpy array of colors for each particle in shades from red to blue
-colors = np.zeros((x_num*y_num, 4))
-colors[:, 0] = np.linspace(1, 0, x_num*y_num)
-colors[:, 2] = np.linspace(0, 1, x_num*y_num)
-colors[:, 3] = 1
-plt.axis('equal')   # ensure axes are equally scaled
-plt.plot([-mesh_spacing*10, x_num*mesh_spacing+mesh_spacing*10], [0, 0], color = 'k') # Draw a line representing the ground at y = 0;
-plot = ax.scatter(particles[:, 0], particles[:, 1],
-                  marker = 'o',
-                  s = 20,
-                  color = colors,
-)
-
-time1 = time.time()
-
-
-# @jit(nopython=True)
-def animate(w):
-    global particles, velocities, forces
-    for i in range(sub_increments):
-        # Semi-implicit Euler integration
-#        velocities = velocities + solve_velocities(forces, particles)
-#        particles = solve_positions(particles, velocities)
-#        forces = solve_forces(particles, velocities)
-        # Velocity Verlet integration
-        velocities_half = velocities + solve_velocities(forces, particles, dt/2)
-        particles = solve_positions(particles, velocities_half)
-        forces = solve_forces(particles, velocities)
-        velocities = velocities_half + solve_velocities(forces, particles, dt/2)
-    plot.set_offsets(np.c_[particles[:, 0],particles[:, 1]])
-
-anim = animation.FuncAnimation(fig, animate, frames=int(2*t_final/dt/2))
-# plt.show()
-solve_time = time.time() - time1
-
-
-time2 = time.time()
-import matplotlib as mpl 
-mpl.rcParams['animation.ffmpeg_path'] = r'C:\ffmpeg_2023_03_30\bin\ffmpeg.exe'
-path = r"c://Users/david/Desktop/NMM/week7/img/animation.mp4" 
-writervideo = animation.FFMpegWriter(fps=30, extra_args=['-vcodec', 'libx264']) 
-anim.save(path, writer=writervideo, dpi=my_dpi)
-mp4_time = time.time() - time2
-
-print("Solve time: ", solve_time)
-print("MP4 time: ", mp4_time)
\ No newline at end of file
diff --git a/week9/README.MD b/week9/README.MD
new file mode 100644
index 0000000..c0966a2
--- /dev/null
+++ b/week9/README.MD
@@ -0,0 +1,23 @@
+# Week 9 - Function Fitting
+
+## 11.1
+
+<img src="img/11_1.png" height="600">
+
+Fit line: y = 1.648 + 3.772x
+Known error: slope error: 0.3401
+Known error: intercept error: 0.1034
+Bootstrap: slope error: 0.0278
+Bootstrap: intercept error: 0.0090
+Ensemble: slope error: 0.0543
+Ensemble: intercept error: 0.0172
+
+## 11.2
+
+The top plot is our noisy data + fit and the bottom are the a and b coefficients converging over 100 steps of Levenberg-Marquardt iteration.
+
+<img src="img/11_2.png" height="600">
+
+Here's an animation of the coefficients actually updating and converging to fit.
+
+<img src="img/animation.gif" height="600">
\ No newline at end of file
diff --git a/week9/img/11_1.png b/week9/img/11_1.png
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diff --git a/week9/week9_11_1.py b/week9/week9_11_1.py
new file mode 100644
index 0000000..c842817
--- /dev/null
+++ b/week9/week9_11_1.py
@@ -0,0 +1,80 @@
+import numpy as np
+import matplotlib.pyplot as plt
+
+def generate_data(size, sigma):
+    # Generate 100 points uniformly distributed between 0 and 1
+    x = np.random.rand(100)
+    # Ley y - 2+3x + C, where C is a gaussian random variable with standard deviation of 0.5
+    C = np.random.normal(0, sigma, size)
+    y = 2 + 3*x + C
+    return x, y
+
+# Use an SVD to fit y = a + bx to this data set, finding a and b. 
+# https://ltetrel.github.io/data-science/2018/06/08/line_svd.html
+
+
+def fit_line(x, y):
+    # Make a matrix A with the x values in the first column and 1's in the second column
+    A = np.vstack([x, y]).T
+    avg = np.mean(A, axis = 0)
+    u, s, V = np.linalg.svd(A - avg) # V is a 2x2 matrix with the eigenvectors as columns, with leftmost columns providing the best fit
+    # get a rise over run slope value from the direction vector V
+    slope = V[0,1]/V[0,0]
+    # find the y intercept by plugging in the x value of the average
+    intercept = avg[1] - slope*avg[0]
+    return slope, intercept, V, s
+
+# Generate the data
+size = 100
+sigma = 0.5
+x, y = generate_data(size, sigma)
+slope, intercept, V, s = fit_line(x, y)
+
+slope = round(slope, 3)
+intercept = round(intercept, 3)
+print("Fit line: y = {0} + {1}x".format(intercept, slope))
+
+# Plot the data and the fit
+plt.title("Fit line: y = {0} + {1}x".format(intercept, slope))
+plt.plot(x, y, 'o', label='Original data', markersize=3)
+plt.plot(x, slope*x + intercept, 'r', label='Fitted line')
+# set the limits of the plot
+plt.show()
+
+'''
+Evaluate the errors in a and b
+'''
+
+# Part A) According to Equation 12.34
+errors = []
+for i in range(2):
+    error = 0
+    for j in range(2):
+        error += (V[j, i] / s[j]) ** 2
+    errors.append(error)
+print("Known error: slope error: %.4f" % np.sqrt(errors[0] * (sigma ** 2)))
+print("Known error: intercept error: %.4f" % np.sqrt(errors[1] * (sigma ** 2)))
+
+# Part B) Bootstrap Sampling
+space = np.linspace(0, 99, 100) # set up a linespace for each x value
+A, B = [], []
+for i in range(100):
+    sample = np.random.choice(space, 100)
+    x_b, y_b = x[sample.astype(int)], y[sample.astype(int)]
+    a, b = np.polyfit(x_b, y_b, 1)
+    A.append(a)
+    B.append(b)
+print("Bootstrap: slope error: %.4f" % np.var(A))
+print("Bootstrap: intercept error: %.4f" % np.var(B))
+
+# Part C) From fitting an ensemble of 100 independent data sets
+# Here we are literaly just going to run the same code as part B 100 times
+# and then take the variance of the results
+A, B = [], []
+for i in range(100):
+    x, y = generate_data(size, sigma)
+    a,b,V,s = fit_line(x, y)
+    A.append(a)
+    B.append(b)
+print("Ensemble: slope error: %.4f" % np.var(A))
+print("Ensemble: intercept error: %.4f" % np.var(B))
\ No newline at end of file
diff --git a/week9/week9_11_2.py b/week9/week9_11_2.py
new file mode 100644
index 0000000..e4ce973
--- /dev/null
+++ b/week9/week9_11_2.py
@@ -0,0 +1,91 @@
+import numpy as np
+import matplotlib.pyplot as plt
+import matplotlib.animation as animation
+
+'''
+Generate 100 points x uniformly distributed between 0 and 1, and let y = sin(2+ 3x) + ζ, where ζ is a Gaussian random variable
+with a standard deviation of 0.1. Write a Levenberg-Marquardt routine to fit y = sin(a+bx) to this data set starting from a = b = 1 
+(remembering that the second-derivative term can be dropped in the Hessian), and investigate the convergence for both fixed and adaptively 
+adjusted λ values.
+'''
+
+def generate_data(size, sigma):
+    # Generate 100 points uniformly distributed between 0 and 1
+    x = np.random.rand(100)
+    # Ley y - 2+3x + C, where C is a gaussian random variable with standard deviation of 0.5
+    C = np.random.normal(0, sigma, size)
+    y = np.sin(2 + 3*x) + C
+    return x, y
+
+# Levenberg-Marquardt routine to fit y = sin(a+bx) to this data set starting from a = b = 1
+def levenberg_step(x, y, a, b, l):
+    # Calculate the Jacobian
+    J = np.array([np.ones(len(x)), x]).T
+    # Calculate the Hessian
+    H = np.dot(J.T, J)
+    # Calculate the gradient
+    g = np.dot(J.T, y - np.sin(a + b*x))
+    # update = (H + l*I)^-1 * g (where I is the identity matrix np.eye(2)
+    coefficients = np.dot(np.linalg.inv(H + l*np.eye(2)), g)
+    return coefficients
+
+a = 1
+b = 1
+l = 0.1
+steps = 100
+
+size = 100
+sigma = 0.1
+x, y = generate_data(size, sigma)
+
+A = []
+B = []
+
+for i in range(steps):
+    coefficients = levenberg_step(x, y, a, b, l)
+    A.append(a)
+    B.append(b)
+    a += coefficients[0]
+    b += coefficients[1]
+    # print("a: %.4f, b: %.4f" % (a, b))
+
+'''
+# Here we plot the data and the fit line
+fig, (ax1, ax2) = plt.subplots(2, 1)
+# slim plot
+fig.subplots_adjust(hspace=0.5)
+ax1.plot(x, y, 'o')
+x_sorted = np.sort(x)
+ax1.plot(x_sorted, np.sin(a + b*x_sorted))
+# print inital parameters
+ax1.text(0,-1, 'Fit line: y = sin(a+bx)\ny = sin(%.2f + %.2fx)' % (a, b), fontsize=10)
+ax1.set_title('Levenberg-Marquardt Fit\nsteps = %.2f, sigma = %.2f' % (steps, sigma))
+# Plot A and B as an XY scatter plot with color associated with the value of the iteration number
+ax2.scatter(A, B, c=range(steps))
+ax2.set_title('Converging a and b values')
+plt.show()
+'''
+
+# Here we animate the fit converging over the scatter plot
+my_dpi = 96
+fig = plt.figure(figsize=(800/my_dpi, 400/my_dpi), dpi=my_dpi)
+ax = fig.add_subplot(111)
+# Scatter plot
+scatter = ax.plot(x, y, 'o')
+# Fit line
+x_sorted = np.sort(x)
+fit = ax.plot(x_sorted, np.sin(a + b*x_sorted))
+
+# Animate the fit converging over the scatter plot
+def animate(w):
+    global A, B
+    x = A[w]
+    y = B[w]
+    fit[0].set_data(x_sorted, np.sin(x + y*x_sorted))
+
+anim = animation.FuncAnimation(fig, animate, frames=int(20))
+# plt.show()
+
+path = r"c://Users/david/Desktop/NMM/week9/img/animation.gif" 
+# save as a gif
+anim.save(path, writer='imagemagick', fps=8, dpi=my_dpi)
-- 
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