Contour plots

A contour plot is a graphical method to visualize the 3-D surface by plotting constant Z slices called contours in a 2-D format. The contour plot is an alternative to a 3-D surface plot

The contour plot is formed by:

• Vertical axis: Independent variable 2
• Horizontal axis: Independent variable 1
• Lines: iso-response values, can be calculated with the help (x,y).

The independent variable usually restricted to a regular grid. The actual techniques for determining the correct iso-response values are rather complex and almost always computer-generated.

The contour plot is used to depict the change in Z values as compared to X and Y values. If the data (or function) do not form a regular grid, you typically need to perform a 2-D interpolation to form a regular grid.

For one variable data, a run sequence/ histogram is considered necessary. For two-variable data, a scatter plot is considered necessary. The contour plots can also polar co-ordinates (r,theta) instead of traditional rectangular (x, y, z) coordinates. 

Types of Contour Plot:

• Rectangular Contour plot: A projection of 2D-plot in 2D-rectangular canvas. It is the most common form of the contour plot.
• Polar contour plot: Polar contour plot is plotted by using the polar coordinates r and theta. The response variable here is the collection of values generated while passing r and theta into the given function, where r is the distance from origin and theta is the angle from the positive x axis.
• Ternary contour plot: Ternary contour plot is used to represent the relationship between 3 explanatory variables and the response variable in the form of a filled triangle.

Contour plot can be plotted in different programming languages:

• Python/ Matplotlib: Contour plot can be plotted using plt.contour or plt.contourf functions, where plt is matplotlib.pyplot. The difference between these two that plot.contour generates hollow contour plot, the plt.contourf generated filled.
• Matlab: functions such as contourf (2d-plot) and contour3 (3D-contour) can be used for contour plotting
• R: can create a contour plot with filled.contour functions in R.

Implementations:

• Rectangular Contour Plot: Below is the sample code for plotting rectangular contour plots in Python and matplotlib.

#  imports
import numpy as np
import matplotlib.pyplot as plt
#
# define a function
def func(x, y):
    return np.sin(x) ** 2 +  np.cos(y) **2
# generate 50 values b/w 0 a5
x = np.linspace(0, 5, 50)
y = np.linspace(0, 5, 50)

# Generate combination of grids
X, Y = np.meshgrid(x, y)
Z = func(X, Y)

# Draw rectangular contour plot
plt.contour(X, Y, Z, cmap='gist_rainbow_r');

• Polar Contour plot: For plotting polar contour plot we need to define first r and theta. Below is the sample code for plotting polar contour plots using matplotlib subplots.

# generate r and theta arrays
rad_arr = np.radians(np.linspace(0, 360, 20))
r_arr = np.arange(0, 1, .1)
# define function
def func(r, theta):
  return r * np.sin(theta) 

r, theta = np.meshgrid(r_arr, rad_arr)
# get the values of response variables
values = func(r,theta)

# plot the polar coordinates
fig, ax = plt.subplots(subplot_kw=dict(projection='polar'))
ax.contourf(theta, r, values, cmap='Spectral_r')

plt.show()

• Ternary Contour Plot: Matplotlib does not provide a definitive API for plotting Ternary Contour plot, however, there are many other package which does that. IN this example, we will be using Plotly library.

# install & import plotly
! pip install plotly
import plotly.figure_factory as ff

# Define variables
a = np.array([0. , 0. , 0., 0., 1./3, 1./3, 1./3, 2./3, 2./3, 1.])
b = np.array([0., 1./3, 2./3, 1., 0., 1./3, 2./3, 0., 1./3, 0.])
c = 1 - a - b
# Define function that generates response variable
func = (a - 0.02) * b * (a - 0.5) * (b - 0.4) * (c - 1)**2

# plot ternary contour
fig = ff.create_ternary_contour(np.array([a, b, c]), func,
                                pole_labels=['a', 'b', 'c'],
                                interp_mode='cartesian',
                                colorscale='Viridis',)
fig.show()

References:

• NIST Handbook
• Plotly Library