Develop Reference

'FuncAnimation' object has no attribute '_resize_id'

I am trying to plot a single pendulum using Eulers method and with given theta values and formula in python but I am getting an Attribute error on FuncAnimation saying 'FuncAnimation' object has no attribute '_resize_id'. Does anyone know what I'm doing wrong here?
# Liður 2
import numpy as np
import matplotlib.pyplot as plt
from matplotlib.animation import FuncAnimation
def ydot(t, y):
g = 9.81
l = 1
z1 = y[1]
z2 = -g/l*np.sin(y[0])
return np.array([z1, z2])
def eulerstep(t, x, h):
return ([x[j]+h*ydot(t,x)[j] for j in range(len(x))])
def eulersmethod(Theta0, T, n):
z = Theta0
h = T/n
t = [i*h for i in range(n)]
theta = [[],[]]
for i in range(n):
z = eulerstep(t[i], z, h)
theta[0].append(z[0])
theta[1].append(z[1])
return(t, theta[0], theta[1])
def animate_pendulum(x, y, h):
fig = plt.figure(figsize=(8,8))
ax = fig.add_subplot(autoscale_on = False, xlim=(-2.2, 2.2), ylim = (-2.2, 2.2))
ax.grid()
line = ax.plot([],[], 'o', c='blue', lw=1)
time_text = ax.text(0.05, 0.9, '', transform = ax.transAxes)
def animate(i):
xline = [0, x[1]]
yline = [0, y[1]]
line.set_data(xline, yline)
time_text.set_text(f"time = {i*h:1f}s")
return line, time_text
ani = FuncAnimation(
fig, animate, len(x), interval = h*1000, blit = True, repeat = False
)
plt.show()
def min():
L=2
T=20
n=500
h=T/n
y_0 = [np.pi/12, 0]
t, angle, velocity = eulersmethod(y_0, T, n)
x, y = L*np.sin(angle[:]), -L*np.cos(angle[:])
animate_pendulum(x, y, h)
min()

Axes.plot() returns a list of lines, see the docs. You Can unpack the list by adding a comma at the variable assignment. This code runs on my machine:
# Liður 2
import numpy as np
import matplotlib.pyplot as plt
from matplotlib.animation import FuncAnimation
def ydot(t, y):
g = 9.81
l = 1
z1 = y[1]
z2 = -g/l*np.sin(y[0])
return np.array([z1, z2])
def eulerstep(t, x, h):
return ([x[j]+h*ydot(t,x)[j] for j in range(len(x))])
def eulersmethod(Theta0, T, n):
z = Theta0
h = T/n
t = [i*h for i in range(n)]
theta = [[],[]]
for i in range(n):
z = eulerstep(t[i], z, h)
theta[0].append(z[0])
theta[1].append(z[1])
return(t, theta[0], theta[1])
def animate_pendulum(x, y, h):
fig = plt.figure(figsize=(8,8))
ax = fig.add_subplot(autoscale_on = False, xlim=(-2.2, 2.2), ylim = (-2.2, 2.2))
ax.grid()
line, = ax.plot([],[], 'o', c='blue', lw=1)
time_text = ax.text(0.05, 0.9, '', transform = ax.transAxes)
def animate(i):
xline = [0, x[1]]
yline = [0, y[1]]
line.set_data(xline, yline)
time_text.set_text(f"time = {i*h:1f}s")
return line, time_text
ani = FuncAnimation(
fig, animate, len(x), interval = h*1000, blit = True, repeat = False
)
plt.show()
def min():
L=2
T=20
n=500
h=T/n
y_0 = [np.pi/12, 0]
t, angle, velocity = eulersmethod(y_0, T, n)
x, y = L*np.sin(angle[:]), -L*np.cos(angle[:])
animate_pendulum(x, y, h)
min()

3D Quadratic Plane of best fit

Can someone show me the code on how to make this work for 4th degree?
import numpy as np
import scipy.linalg
import matplotlib.pyplot as plt
# some 3-dim points
x = []
y=[]
z=[]
data = np.c_[x,y,z]
# regular grid covering the domain of the data
mn = np.min(data, axis=0)
mx = np.max(data, axis=0)
X,Y = np.meshgrid(np.linspace(mn[0], mx[0], 20), np.linspace(mn[1], mx[1], 20))
XX = X.flatten()
YY = Y.flatten()
order = 2 # 1: linear, 2: quadratic
if order == 1:
# best-fit linear plane
A = np.c_[data[:,0], data[:,1], np.ones(data.shape[0])]
C,_,_,_ = scipy.linalg.lstsq(A, data[:,2]) # coefficients
# evaluate it on grid
# Z = C[0]*X + C[1]*Y + C[2]
# or expressed using matrix/vector product
Z = np.dot(np.c_[XX, YY, np.ones(XX.shape)], C).reshape(X.shape)
elif order == 2:
# best-fit quadratic curve
# M = [ones(size(x)), x, y, x.*y, x.^2 y.^2]
A = np.c_[np.ones(data.shape[0]), data[:,:2], np.prod(data[:,:2], axis=1), data[:,:2]**2]
C,_,_,_ = scipy.linalg.lstsq(A, data[:,2])
# evaluate it on a grid
Z = np.dot(np.c_[np.ones(XX.shape), XX, YY, XX*YY, XX**2, YY**2], C).reshape(X.shape)
elif order == 3:
# M = [ones(size(x)), x, y, x.^2, x.*y, y.^2, x.^3, x.^2.*y, x.*y.^2, y.^3]
A = np.c_[np.ones(data.shape[0]), data[:,:2], data[:,0]**2, np.prod(data[:,:2], axis=1), \
data[:,1]**2, data[:,0]**3, np.prod(np.c_[data[:,0]**2,data[:,1]],axis=1), \
np.prod(np.c_[data[:,0],data[:,1]**2],axis=1), data[:,1]**3]
C,_,_,_ = scipy.linalg.lstsq(A, data[:,2])
Z = np.dot(np.c_[np.ones(XX.shape), XX, YY, XX**2, XX*YY, YY**2, XX**3, XX**2*YY, XX*YY**2, YY**3], C).reshape(X.shape)
# best-fit quadratic curve
# plot points and fitted surface
fig = plt.figure()
ax = fig.gca(projection='3d')
ax.plot_surface(X, Y, Z, rstride=1, cstride=1, alpha=0.2)
ax.scatter(data[:,0], data[:,1], data[:,2], c='r', s=50)
plt.xlabel('X')
plt.ylabel('Y')
ax.set_zlabel('Z')
ax.axis('auto')
ax.axis('tight')
print(C)
plt.show()
I would also like an explanation after the fact if someone knows how accurate this method is. Or if there is a better way to find a plane of best fit. I am trying to find a plane that would best fit three wheel paths. The first wheel path is at y=0, the second at y=2, and the third at y=4. They all go from 0-94 in the x direction, but they all have different z values.

Inverse the spline interpolation SciPy

I have an array of X and Y data, and I also have a spline that outputs Y values based on X values.
I need to get a spline that will perform the reverse operation, i.e. output the X value over the Y value.
(interpolate.splrep(y, x, s=0) do not suggest)
def f(x):
return math.sin(x)
def compare_func(func, a, b, eps):
x_for_plot = list(np.arange(a, b, eps / 10))
x = list(np.arange(a, b, eps))
y_for_plot = [func(i) for i in x_for_plot]
y = [func(i) for i in x]
plt.plot(x_for_plot, y_for_plot, label='Orig')
plt.scatter(x, y, label='Points', color='blue')
spline = interpolate.splrep(x, y, s=0)
y_interpolated = interpolate.splev(x_for_plot, spline, der=0)
spline_inv = ???
x_calc = interpolate.splev(y_interpolated, spline_inv, der=0)
print(x_calc)
#x_calc and x_for_plot must be equal
plt.plot(x_for_plot, y_interpolated, label='Interpolated')
plt.xlabel('X')
plt.ylabel('Y')
plt.legend()
plt.show()
compare_func(f, -3, 3, 0.5)

How do I modify my contourplot to display a region with an huge gradient better?

I'm working on a contourplot with matplotlib and for my data I have a region where I have a strong gradient - Now I have the problem that matplotlib will display the regions with different colors, according to the selected colormap, and distribute the colors linear over the whole spectrum.
Since 90% of my datapoints are within one end of the spectra, and only this small region acts completely differently, my contourplot looks kind of
monochrome, as you can see in the attached picture
Also, I've added some contours to make the differences in the values more visible. Since we have a huge gradient at a specific spot, there area lot of contours and it is super hard to see the underlying colors or the values.
Is there a good way how to handle such "problematic" regions with matplotlib? Maybe to define another colormap there? I've tried to set some manual levels and to "cut out" the specific region, but it would be nice to find a way to display the value of this region
Just to get a feeling: My minimal value to display is around 7, the maximum value is 145 and the average
Here is the important part of my code:
z = [] # z is a list of values that i've read in before from a file
X = np.arange(0, 61, 1)
Y = np.arange(0, 151, 1)
z = z.reshape((len(Y), len(X)))
blurred = gaussian_filter(z, sigma=2) # applies a gaussian filter to smooth the plot
xx, yy = np.meshgrid(X, Y) # gets the grid for the plot
procent = np.arange(np.min(z), np.max(z), 5) # levels for the contourlines
newlevels = [5,10,15,20,30,40,50, 80, 100, 120, 140] # sets manual levels for the plot, where I've tried to set a stronger focus on the first part of the spectra
plusmin = plt.contourf(xx,yy,z, origin='lower', extend='both', levels=procent,)
levels = np.arange(np.min(z), np.max(z), 3)
CS = plt.contourf(xx, yy, z, levels=newlevels, extend="both", cmap=cm.viridis)
s = plt.contour(xx, yy, blurred, plusmin.levels, colors='white', linewidths=2)
cbar = plt.colorbar(CS, fraction=0.042, pad=0.04)
ax.clabel(s, fontsize=12, inline=1, colors ='white')

A solution might be to scale the colormap so that each colors is equally displayed.
Here is a piece of code I use to handle this kind of problem. There is certainly more proper way to do it with matplotlib, but I do not know it.
import matplotlib.pyplot as plt
from matplotlib import colors
import numpy as np
import copy
#-----------------------
def repartition(z):
"""compute the repartition function of an array"""
hist, bin_edges = np.histogram(z.flat[:], bins = 1000)
x = 0.5 * (bin_edges[1:] + bin_edges[:-1])
y = np.cumsum(np.array(hist, float))
y = (y - y[0]) / (y[-1] - y[0])
return x, y
#-----------------------
def adjustcmap2data(Z, cmap, N = 16384):
"""scale the colormap so that all colors are equally displayed"""
def cmap2xs(cmap):
"convert cmap to matrix"
sd = cmap._segmentdata
xr = [tup[0] for tup in sd['red']]
xg = [tup[0] for tup in sd['green']]
xb = [tup[0] for tup in sd['blue']]
return tuple([np.array(x) for x in xr, xg, xb])
def xs2cmap(cmap, xr, xg, xb):
"convert matrix to cmap"
sd = cmap._segmentdata
for k in 'red', 'green', 'blue':
sd[k] = list(sd[k])
for i in xrange(len(sd[k])): sd[k][i] = list(sd[k][i])
for i in xrange(len(sd['red'])): sd['red'][i] = (xr[i], sd['red'][i][1], sd['red'][i][2])
for i in xrange(len(sd['green'])): sd['green'][i] = (xg[i], sd['green'][i][1], sd['green'][i][2])
for i in xrange(len(sd['blue'])): sd['blue'][i] = (xb[i], sd['blue'][i][1], sd['blue'][i][2])
for k in 'red', 'green', 'blue':
sd[k] = tuple(sd[k])
return colors.LinearSegmentedColormap('mycmap_%010.0f' % (np.random.randn() * 1.e10), sd, N)
x, y = repartition(Z)
x = (x - x[0]) / (x[-1] - x[0])
xr, xg, xb = cmap2xs(cmap)
xrr = np.interp(xr, xp = y, fp = x)
xgg = np.interp(xg, xp = y, fp = x)
xbb = np.interp(xb, xp = y, fp = x)
for x in xrr, xgg, xbb:
x[x < 0.] = 0.
x[x > 1.] = 1.
x[0], x[-1] = 0., 1.
x.sort()
mycmap = xs2cmap(copy.deepcopy(cmap), xrr, xgg, xbb)
return mycmap
#---------------------
def fake_data():
"""generate a fake dataset"""
x = np.linspace(-1., 1., 256)
y = np.linspace(-1., 1., 256)
X, Y = np.meshgrid(x, y)
Z = np.zeros_like(X)
#create background noise
for _ in xrange(100):
x0 = np.random.randn()
y0 = np.random.randn()
Z += 0.05 * np.exp(-0.5 * (((X - x0) / 0.1) ** 2. + ((Y - y0) / 0.1) ** 2.))
#add strong peak
Z += np.exp(-0.5 * (((X - 0.5) / 0.3) ** 2. + ((Y - 0.5) / 0.02) ** 2.))
return X, Y, Z
#---------------------
if __name__ == "__main__":
X, Y, Z = fake_data()
plt.figure()
cmap = plt.cm.spectral
plt.pcolormesh(X, Y, Z, cmap = cmap)
plt.colorbar()
plt.contour(X, Y, Z, colors = "w")
plt.gcf().show()
plt.figure()
scaledcmap = adjustcmap2data(Z, cmap = cmap)
plt.pcolormesh(X, Y, Z, cmap = scaledcmap)
plt.colorbar()
plt.gcf().show()
raw_input('pause')
which should give you the following results
1) linear colorbar
2) scaled colorbar

Plotting a decision boundary separating 2 classes using Matplotlib's pyplot

I could really use a tip to help me plotting a decision boundary to separate to classes of data. I created some sample data (from a Gaussian distribution) via Python NumPy. In this case, every data point is a 2D coordinate, i.e., a 1 column vector consisting of 2 rows. E.g.,
[ 1
2 ]
Let's assume I have 2 classes, class1 and class2, and I created 100 data points for class1 and 100 data points for class2 via the code below (assigned to the variables x1_samples and x2_samples).
mu_vec1 = np.array([0,0])
cov_mat1 = np.array([[2,0],[0,2]])
x1_samples = np.random.multivariate_normal(mu_vec1, cov_mat1, 100)
mu_vec1 = mu_vec1.reshape(1,2).T # to 1-col vector
mu_vec2 = np.array([1,2])
cov_mat2 = np.array([[1,0],[0,1]])
x2_samples = np.random.multivariate_normal(mu_vec2, cov_mat2, 100)
mu_vec2 = mu_vec2.reshape(1,2).T
When I plot the data points for each class, it would look like this:
Now, I came up with an equation for an decision boundary to separate both classes and would like to add it to the plot. However, I am not really sure how I can plot this function:
def decision_boundary(x_vec, mu_vec1, mu_vec2):
g1 = (x_vec-mu_vec1).T.dot((x_vec-mu_vec1))
g2 = 2*( (x_vec-mu_vec2).T.dot((x_vec-mu_vec2)) )
return g1 - g2
I would really appreciate any help!
EDIT:
Intuitively (If I did my math right) I would expect the decision boundary to look somewhat like this red line when I plot the function...

Your question is more complicated than a simple plot : you need to draw the contour which will maximize the inter-class distance. Fortunately it's a well-studied field, particularly for SVM machine learning.
The easiest method is to download the scikit-learn module, which provides a lot of cool methods to draw boundaries: scikit-learn: Support Vector Machines
Code :
# -*- coding: utf-8 -*-
import numpy as np
import matplotlib
from matplotlib import pyplot as plt
import scipy
from sklearn import svm
mu_vec1 = np.array([0,0])
cov_mat1 = np.array([[2,0],[0,2]])
x1_samples = np.random.multivariate_normal(mu_vec1, cov_mat1, 100)
mu_vec1 = mu_vec1.reshape(1,2).T # to 1-col vector
mu_vec2 = np.array([1,2])
cov_mat2 = np.array([[1,0],[0,1]])
x2_samples = np.random.multivariate_normal(mu_vec2, cov_mat2, 100)
mu_vec2 = mu_vec2.reshape(1,2).T
fig = plt.figure()
plt.scatter(x1_samples[:,0],x1_samples[:,1], marker='+')
plt.scatter(x2_samples[:,0],x2_samples[:,1], c= 'green', marker='o')
X = np.concatenate((x1_samples,x2_samples), axis = 0)
Y = np.array([0]*100 + [1]*100)
C = 1.0 # SVM regularization parameter
clf = svm.SVC(kernel = 'linear', gamma=0.7, C=C )
clf.fit(X, Y)
Linear Plot
w = clf.coef_[0]
a = -w[0] / w[1]
xx = np.linspace(-5, 5)
yy = a * xx - (clf.intercept_[0]) / w[1]
plt.plot(xx, yy, 'k-')
MultiLinear Plot
C = 1.0 # SVM regularization parameter
clf = svm.SVC(kernel = 'rbf', gamma=0.7, C=C )
clf.fit(X, Y)
h = .02 # step size in the mesh
# create a mesh to plot in
x_min, x_max = X[:, 0].min() - 1, X[:, 0].max() + 1
y_min, y_max = X[:, 1].min() - 1, X[:, 1].max() + 1
xx, yy = np.meshgrid(np.arange(x_min, x_max, h),
np.arange(y_min, y_max, h))
# Plot the decision boundary. For that, we will assign a color to each
# point in the mesh [x_min, m_max]x[y_min, y_max].
Z = clf.predict(np.c_[xx.ravel(), yy.ravel()])
# Put the result into a color plot
Z = Z.reshape(xx.shape)
plt.contour(xx, yy, Z, cmap=plt.cm.Paired)
Implementation
If you want to implement it yourself, you need to solve the following quadratic equation:
The Wikipedia article
Unfortunately, for non-linear boundaries like the one you draw, it's a difficult problem relying on a kernel trick but there isn't a clear cut solution.

Based on the way you've written decision_boundary you'll want to use the contour function, as Joe noted above. If you just want the boundary line, you can draw a single contour at the 0 level:
f, ax = plt.subplots(figsize=(7, 7))
c1, c2 = "#3366AA", "#AA3333"
ax.scatter(*x1_samples.T, c=c1, s=40)
ax.scatter(*x2_samples.T, c=c2, marker="D", s=40)
x_vec = np.linspace(*ax.get_xlim())
ax.contour(x_vec, x_vec,
decision_boundary(x_vec, mu_vec1, mu_vec2),
levels=[0], cmap="Greys_r")
Which makes:

Those were some great suggestions, thanks a lot for your help! I ended up solving the equation analytically and this is the solution I ended up with (I just want to post it for future reference:
# 2-category classification with random 2D-sample data
# from a multivariate normal distribution
import numpy as np
from matplotlib import pyplot as plt
def decision_boundary(x_1):
""" Calculates the x_2 value for plotting the decision boundary."""
return 4 - np.sqrt(-x_1**2 + 4*x_1 + 6 + np.log(16))
# Generating a Gaussion dataset:
# creating random vectors from the multivariate normal distribution
# given mean and covariance
mu_vec1 = np.array([0,0])
cov_mat1 = np.array([[2,0],[0,2]])
x1_samples = np.random.multivariate_normal(mu_vec1, cov_mat1, 100)
mu_vec1 = mu_vec1.reshape(1,2).T # to 1-col vector
mu_vec2 = np.array([1,2])
cov_mat2 = np.array([[1,0],[0,1]])
x2_samples = np.random.multivariate_normal(mu_vec2, cov_mat2, 100)
mu_vec2 = mu_vec2.reshape(1,2).T # to 1-col vector
# Main scatter plot and plot annotation
f, ax = plt.subplots(figsize=(7, 7))
ax.scatter(x1_samples[:,0], x1_samples[:,1], marker='o', color='green', s=40, alpha=0.5)
ax.scatter(x2_samples[:,0], x2_samples[:,1], marker='^', color='blue', s=40, alpha=0.5)
plt.legend(['Class1 (w1)', 'Class2 (w2)'], loc='upper right')
plt.title('Densities of 2 classes with 25 bivariate random patterns each')
plt.ylabel('x2')
plt.xlabel('x1')
ftext = 'p(x|w1) ~ N(mu1=(0,0)^t, cov1=I)\np(x|w2) ~ N(mu2=(1,1)^t, cov2=I)'
plt.figtext(.15,.8, ftext, fontsize=11, ha='left')
# Adding decision boundary to plot
x_1 = np.arange(-5, 5, 0.1)
bound = decision_boundary(x_1)
plt.plot(x_1, bound, 'r--', lw=3)
x_vec = np.linspace(*ax.get_xlim())
x_1 = np.arange(0, 100, 0.05)
plt.show()
And the code can be found here
EDIT:
I also have a convenience function for plotting decision regions for classifiers that implement a fit and predict method, e.g., the classifiers in scikit-learn, which is useful if the solution cannot be found analytically. A more detailed description how it works can be found here.

You can create your own equation for the boundary:
where you have to find the positions x0 and y0, as well as the constants ai and bi for the radius equation. So, you have 2*(n+1)+2 variables. Using scipy.optimize.leastsq is straightforward for this type of problem.
The code attached below builds the residual for the leastsq penalizing the points outsize the boundary. The result for your problem, obtained with:
x, y = find_boundary(x2_samples[:,0], x2_samples[:,1], n)
ax.plot(x, y, '-k', lw=2.)
x, y = find_boundary(x1_samples[:,0], x1_samples[:,1], n)
ax.plot(x, y, '--k', lw=2.)
using n=1:
using n=2:
usng n=5:
using n=7:
import numpy as np
from numpy import sin, cos, pi
from scipy.optimize import leastsq
def find_boundary(x, y, n, plot_pts=1000):
def sines(theta):
ans = np.array([sin(i*theta) for i in range(n+1)])
return ans
def cosines(theta):
ans = np.array([cos(i*theta) for i in range(n+1)])
return ans
def residual(params, x, y):
x0 = params[0]
y0 = params[1]
c = params[2:]
r_pts = ((x-x0)**2 + (y-y0)**2)**0.5
thetas = np.arctan2((y-y0), (x-x0))
m = np.vstack((sines(thetas), cosines(thetas))).T
r_bound = m.dot(c)
delta = r_pts - r_bound
delta[delta>0] *= 10
return delta
# initial guess for x0 and y0
x0 = x.mean()
y0 = y.mean()
params = np.zeros(2 + 2*(n+1))
params[0] = x0
params[1] = y0
params[2:] += 1000
popt, pcov = leastsq(residual, x0=params, args=(x, y),
ftol=1.e-12, xtol=1.e-12)
thetas = np.linspace(0, 2*pi, plot_pts)
m = np.vstack((sines(thetas), cosines(thetas))).T
c = np.array(popt[2:])
r_bound = m.dot(c)
x_bound = popt[0] + r_bound*cos(thetas)
y_bound = popt[1] + r_bound*sin(thetas)
return x_bound, y_bound

I like the mglearn library to draw decision boundaries. Here is one example from the book "Introduction to Machine Learning with Python" by A. Mueller:
fig, axes = plt.subplots(1, 3, figsize=(10, 3))
for n_neighbors, ax in zip([1, 3, 9], axes):
clf = KNeighborsClassifier(n_neighbors=n_neighbors).fit(X, y)
mglearn.plots.plot_2d_separator(clf, X, fill=True, eps=0.5, ax=ax, alpha=.4)
mglearn.discrete_scatter(X[:, 0], X[:, 1], y, ax=ax)
ax.set_title("{} neighbor(s)".format(n_neighbors))
ax.set_xlabel("feature 0")
ax.set_ylabel("feature 1")
axes[0].legend(loc=3)

If you want to use scikit learn, you can write your code like this:
import numpy as np
import pandas as pd
import matplotlib.pyplot as plt
from sklearn.linear_model import LogisticRegression
# read data
data = pd.read_csv('ex2data1.txt', header=None)
X = data[[0,1]].values
y = data[2]
# use LogisticRegression
log_reg = LogisticRegression()
log_reg.fit(X, y)
# Coefficient of the features in the decision function. (from theta 1 to theta n)
parameters = log_reg.coef_[0]
# Intercept (a.k.a. bias) added to the decision function. (theta 0)
parameter0 = log_reg.intercept_
# Plotting the decision boundary
fig = plt.figure(figsize=(10,7))
x_values = [np.min(X[:, 1] -5 ), np.max(X[:, 1] +5 )]
# calcul y values
y_values = np.dot((-1./parameters[1]), (np.dot(parameters[0],x_values) + parameter0))
colors=['red' if l==0 else 'blue' for l in y]
plt.scatter(X[:, 0], X[:, 1], label='Logistics regression', color=colors)
plt.plot(x_values, y_values, label='Decision Boundary')
plt.show()
see: Building-a-Logistic-Regression-with-Scikit-learn

Just solved a very similar problem with a different approach (root finding) and wanted to post this alternative as answer here for future reference:
def discr_func(x, y, cov_mat, mu_vec):
"""
Calculates the value of the discriminant function for a dx1 dimensional
sample given covariance matrix and mean vector.
Keyword arguments:
x_vec: A dx1 dimensional numpy array representing the sample.
cov_mat: numpy array of the covariance matrix.
mu_vec: dx1 dimensional numpy array of the sample mean.
Returns a float value as result of the discriminant function.
"""
x_vec = np.array([[x],[y]])
W_i = (-1/2) * np.linalg.inv(cov_mat)
assert(W_i.shape[0] > 1 and W_i.shape[1] > 1), 'W_i must be a matrix'
w_i = np.linalg.inv(cov_mat).dot(mu_vec)
assert(w_i.shape[0] > 1 and w_i.shape[1] == 1), 'w_i must be a column vector'
omega_i_p1 = (((-1/2) * (mu_vec).T).dot(np.linalg.inv(cov_mat))).dot(mu_vec)
omega_i_p2 = (-1/2) * np.log(np.linalg.det(cov_mat))
omega_i = omega_i_p1 - omega_i_p2
assert(omega_i.shape == (1, 1)), 'omega_i must be a scalar'
g = ((x_vec.T).dot(W_i)).dot(x_vec) + (w_i.T).dot(x_vec) + omega_i
return float(g)
#g1 = discr_func(x, y, cov_mat=cov_mat1, mu_vec=mu_vec_1)
#g2 = discr_func(x, y, cov_mat=cov_mat2, mu_vec=mu_vec_2)
x_est50 = list(np.arange(-6, 6, 0.1))
y_est50 = []
for i in x_est50:
y_est50.append(scipy.optimize.bisect(lambda y: discr_func(i, y, cov_mat=cov_est_1, mu_vec=mu_est_1) - \
discr_func(i, y, cov_mat=cov_est_2, mu_vec=mu_est_2), -10,10))
y_est50 = [float(i) for i in y_est50]
Here is the result:
(blue the quadratic case, red the linear case (equal variances)

I know this question has been answered in a very thorough way analytically. I just wanted to share a possible 'hack' to the problem. It is unwieldy but gets the job done.
Start by building a mesh grid of the 2d area and then based on the classifier just build a class map of the entire space. Subsequently detect changes in the decision made row-wise and store the edges points in a list and scatter plot the points.
def disc(x): # returns the class of the point based on location x = [x,y]
temp = 0.5 + 0.5*np.sign(disc0(x)-disc1(x))
# disc0() and disc1() are the discriminant functions of the respective classes
return 0*temp + 1*(1-temp)
num = 200
a = np.linspace(-4,4,num)
b = np.linspace(-6,6,num)
X,Y = np.meshgrid(a,b)
def decColor(x,y):
temp = np.zeros((num,num))
print x.shape, np.size(x,axis=0)
for l in range(num):
for m in range(num):
p = np.array([x[l,m],y[l,m]])
#print p
temp[l,m] = disc(p)
return temp
boundColorMap = decColor(X,Y)
group = 0
boundary = []
for x in range(num):
group = boundColorMap[x,0]
for y in range(num):
if boundColorMap[x,y]!=group:
boundary.append([X[x,y],Y[x,y]])
group = boundColorMap[x,y]
boundary = np.array(boundary)
Sample Decision Boundary for a simple bivariate gaussian classifier

Given two bi-variate normal distributions, you can use Gaussian Discriminant Analysis (GDA) to come up with a decision boundary as the difference between the log of the 2 pdf's.
Here's a way to do it using scipy multivariate_normal (the code is not optimized):
import numpy as np
import matplotlib.pyplot as plt
from scipy.stats import multivariate_normal
from numpy.linalg import norm
from numpy.linalg import inv
from scipy.spatial.distance import mahalanobis
def normal_scatter(mean, cov, p):
size = 100
sigma_x = cov[0,0]
sigma_y = cov[1,1]
mu_x = mean[0]
mu_y = mean[1]
x_ps, y_ps = np.random.multivariate_normal(mean, cov, size).T
x,y = np.mgrid[mu_x-3*sigma_x:mu_x+3*sigma_x:1/size, mu_y-3*sigma_y:mu_y+3*sigma_y:1/size]
grid = np.empty(x.shape + (2,))
grid[:, :, 0] = x; grid[:, :, 1] = y
z = p*multivariate_normal.pdf(grid, mean, cov)
return x_ps, y_ps, x,y,z
# Dist 1
mu_1 = np.array([1, 1])
cov_1 = .5*np.array([[1, 0], [0, 1]])
p_1 = .5
x_ps, y_ps, x,y,z = normal_scatter(mu_1, cov_1, p_1)
plt.plot(x_ps,y_ps,'x')
plt.contour(x, y, z, cmap='Blues', levels=3)
# Dist 2
mu_2 = np.array([2, 1])
#cov_2 = np.array([[2, -1], [-1, 1]])
cov_2 = cov_1
p_2 = .5
x_ps, y_ps, x,y,z = normal_scatter(mu_2, cov_2, p_2)
plt.plot(x_ps,y_ps,'.')
plt.contour(x, y, z, cmap='Oranges', levels=3)
# Decision Boundary
X = np.empty(x.shape + (2,))
X[:, :, 0] = x; X[:, :, 1] = y
g = np.log(p_1*multivariate_normal.pdf(X, mu_1, cov_1)) - np.log(p_2*multivariate_normal.pdf(X, mu_2, cov_2))
plt.contour(x, y, g, [0])
plt.grid()
plt.axhline(y=0, color='k')
plt.axvline(x=0, color='k')
plt.plot([mu_1[0], mu_2[0]], [mu_1[1], mu_2[1]], 'k')
plt.show()
If p_1 != p_2, then you get non-linear boundary. The decision boundary is given by g above.
Then to plot the decision hyper-plane (line in 2D), you need to evaluate g for a 2D mesh, then get the contour which will give a separating line.
You can also assume to have equal co-variance matrices for both distributions, which will give a linear decision boundary. In this case, you can replace the calculation of g in the above code with the following:
W = inv(cov_1).dot(mu_1-mu_2)
x_0 = 1/2*(mu_1+mu_2) - cov_1.dot(np.log(p_1/p_2)).dot((mu_1-mu_2)/mahalanobis(mu_1, mu_2, cov_1))
X = np.empty(x.shape + (2,))
X[:, :, 0] = x; X[:, :, 1] = y
g = (X-x_0).dot(W)

i use this method from this book python-machine-learning-2nd.pdf URL
from matplotlib.colors import ListedColormap
import matplotlib.pyplot as plt
def plot_decision_regions(X, y, classifier, test_idx=None, resolution=0.02):
# setup marker generator and color map
markers = ('s', 'x', 'o', '^', 'v')
colors = ('red', 'blue', 'lightgreen', 'gray', 'cyan')
cmap = ListedColormap(colors[:len(np.unique(y))])
# plot the decision surface
x1_min, x1_max = X[:, 0].min() - 1, X[:, 0].max() + 1
x2_min, x2_max = X[:, 1].min() - 1, X[:, 1].max() + 1
xx1, xx2 = np.meshgrid(np.arange(x1_min, x1_max, resolution),
np.arange(x2_min, x2_max, resolution))
Z = classifier.predict(np.array([xx1.ravel(), xx2.ravel()]).T)
Z = Z.reshape(xx1.shape)
plt.contourf(xx1, xx2, Z, alpha=0.3, cmap=cmap)
plt.xlim(xx1.min(), xx1.max())
plt.ylim(xx2.min(), xx2.max())
for idx, cl in enumerate(np.unique(y)):
plt.scatter(x=X[y == cl, 0],
y=X[y == cl, 1],
alpha=0.8,
c=colors[idx],
marker=markers[idx],
label=cl,
edgecolor='black')
# highlight test samples
if test_idx:
# plot all samples
X_test, y_test = X[test_idx, :], y[test_idx]
plt.scatter(X_test[:, 0],
X_test[:, 1],
c='',
edgecolor='black',
alpha=1.0,
linewidth=1,
marker='o',
s=100,
label='test set')

Since version 1.1, sklearn has a function for this:
https://scikit-learn.org/stable/modules/generated/sklearn.inspection.DecisionBoundaryDisplay.html#sklearn.inspection.DecisionBoundaryDisplay