Python:Logical Masks
Contents
Introduction
Many times, you will see functions that are written in some kind of piecewise fashion. That is, depending upon that value of the input argument, the output argument may be calculated in one of several different ways. In these cases, you can use Python and NumPy's logical and relational arguments to create logical masks - arrays containing either 0's or 1's - to help determine the appropriate calculation to perform to determine the values in an array.
Important Notes
- All of the example codes assume you have already run (or have included in your script)
import math as m
import numpy as np
import matplotlib.pyplot as plt
Piecewise Constant
Sometimes, piecewise functions are broken up into several regions, with each region being defined by a single constant. These are called piecewise constant functions and can be easily generated in Python with NumPy.
Example - GPA Calculation
Take for example GPA as a function of numerical grade (assume \(\pm\)do not exist for the moment...). The mathematical statement of GPA might be:
\( \mbox{GPA}(\mbox{Grade})=\left\{ \begin{array}{rl} \mbox{Grade}\geq 90~~~ & 4\\ 80\leq\mbox{Grade} ~\And~ \mbox{Grade}<90~~~ & 3\\ 70\leq\mbox{Grade} ~\And~ \mbox{Grade}<80~~~ & 2\\ 60\leq\mbox{Grade} ~\And~ \mbox{Grade}<70~~~ & 1\\ \mbox{Grade}<60~~~ & 0 \end{array} \right. \)
If you want to write this table as a function that can accept multiple
inputs at once, you really should not use an if
-tree
(because if
-trees can only run one program segment for the entire matrix) or a
for
loop (because for
loops are slower and more complex
for this particular situation).
Instead, use NumPy's ability to create logical masks. Look
at the following code:
Grade=np.linspace(0, 100, 1000)
MaskForA = Grade>=90
The variable MaskForA
will be the same size of the Grade
variable, and will contain only True and False values. Given that, you can use
this in conjunction with the function for an "A" grade to start
building the GPA
variable if you recall that True evaluates to 1 and False to 0:
Grade=np.linspace(0, 100, 1000)
MaskForA = Grade>=90
FunctionForA = 4
GPA = MaskForA*FunctionForA
At the end of this script, the GPA
variable will contain a 4
wherever the logical mask MaskForA
was True and it will contain a
0 wherever the logical mask was False. All that is required is to
extend this to the rest of the possible GPA's - the full code for a function is in the section below,
along with an image of the graph it will create. Note the use of an outer set of parentheses
to write the commands in the same visual way the function definition is written -
this is not required, but does make the code easier to interpret.
Code and Graph
For a further explanation, look at the following code. It investigates the individual masks for each grade for an array containing eight different grades:
ClassGrades = np.array([95, 70, 68, 38, 94, 84, 92, 87])
MaskForA = ClassGrades>=90
MaskForB = (80<=ClassGrades) & (ClassGrades<90)
MaskForC = (70<=ClassGrades) & (ClassGrades<80)
MaskForD = (60<=ClassGrades) & (ClassGrades<70)
MaskForF = ClassGrades<60
The masks now look like:
MaskForA: array([ True, False, False, False, True, False, True, False])
MaskForB: array([False, False, False, False, False, True, False, True])
MaskForC: array([False, True, False, False, False, False, False, False])
MaskForD: array([False, False, True, False, False, False, False, False])
MaskForF: array([False, False, False, True, False, False, False, False])
and the products of each of the masks with each of the relevant GPA values would produce:
MaskForA*FunctionForA: array([4, 0, 0, 0, 4, 0, 4, 0])
MaskForB*FunctionForB: array([0, 0, 0, 0, 0, 3, 0, 3])
MaskForC*FunctionForC: array([0, 2, 0, 0, 0, 0, 0, 0])
MaskForD*FunctionForD: array([0, 0, 1, 0, 0, 0, 0, 0])
MaskForF*FunctionForF: array([0, 0, 0, 0, 0, 0, 0, 0])
These five matrices are added together to give:
GPA: array([4, 2, 1, 0, 4, 3, 4, 3])
All of that work is being done in the "one line of code" in the function!
Non-constant Piecewise Functions
Sometimes, piecewise functions are defined as following different formulas involving the independent variable over a range of inputs rather than being piece-wise constant.
Example 1 - Discrete Function
For example, the expression:
\( f(x)=\left\{ \begin{array}{rl} x<0 & 0\\ 0\leq x<5 & x\\ 5\leq x<10 & 5\\ x\geq 10 & 2x-15 \end{array} \right. \)
which, for integers -2 through 12 can be calculated and graphed:
Code and Graph
import numpy as np
import matplotlib.pyplot as plt
def fun(x):
out = (
(x < 0) * (0)
+ ((0 <= x) & (x < 5)) * (x)
+ ((5 <= x) & (x < 10)) * (5)
+ (x >= 10) * (2 * x - 15)
)
return out
if __name__ == "__main__":
xvals = np.arange(-2, 13)
yvals = fun(xvals)
fig = plt.figure(num=1, clear=True)
ax = fig.add_subplot(1, 1, 1)
ax.stem(xvals, yvals)
ax.set(xlabel="x", ylabel="f(x)", title="Piecewise Function (NetID)")
fig.tight_layout()
fig.savefig("PFunction.png")
Intermediate Results
Several of the intermediate results are presented below. You will not have any of this in your programs.
Also, note that the spacing
in this document is set to make it easier to see which elements in one variable are
paired up with equivalent elements in another variable (for example,
to determine the element-multiplication of Mask 2
and Function 2
).
Note that the masks are shown below as arrays with 1 representing True and 0
representing False in order to save space:
(x>0) * 1: array([1, 1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0])
((0<=x) & (x<5)) * (x) * 1: array([0, 0, 1, 1, 1, 1, 1, 0, 0, 0, 0, 0, 0, 0, 0])
((5<=x) & (x<10)) * 1: array([0, 0, 0, 0, 0, 0, 0, 1, 1, 1, 1, 1, 0, 0, 0])
(x>=10) * 1: array([0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 1, 1])
Notice how the masks end up spanning the entire domain of \(x\) without overlapping. The functions - as calculated over the entire domain, are:
Function 1: 0
Function 2: array([-2, -1, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12])
Function 3: 5
Function 4: array([-19, -17, -15, -13, -11, -9, -7, -5, -3, -1, 1, 3, 5, 7, 9])
The individual products of the masks and the functions give:
Mask 1 *Function 1: array([0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0])
Mask 2 *Function 2: array([0, 0, 0, 1, 2, 3, 4, 0, 0, 0, 0, 0, 0, 0, 0])
Mask 3 *Function 3: array([0, 0, 0, 0, 0, 0, 0, 5, 5, 5, 5, 5, 0, 0, 0])
Mask 4 *Function 4: array([0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 5, 7, 9])
Finally, when these are all added together, none of the individual products interfere with each other, so:
f: array([0, 0, 0, 1, 2, 3, 4, 5, 5, 5, 5, 5, 5, 7, 9])
Example 2 - Acceleration Due To Gravity
The following example determines the acceleration due to gravity for a sphere a particular distance away from the center of a sphere of constant density \(\rho\) and radius R. The formula changes depending on whether you are inside the sphere's surface or not. That is:
\( \mbox{Gravity}(r)= \left\{ \begin{array}{lr} r<=R & \frac{4}{3}\pi G\rho r\\ r>R & \frac{4}{3}\pi G \rho\frac{R^3}{r^2} \end{array} \right. \)
You can use the same method as the example above, making sure to calculate the values of the function rather than simply using constants. The following code and graph demonstrate this:
Code and Graph
import numpy as np
import matplotlib.pyplot as plt
def grav_fun(r, rho=5515, R=6371e3):
# Note: defaults to Earth
G = 6.67e-11
return (
(r <= R) * (4 / 3 * np.pi * G * rho * r) +
(r > R) * (4 / 3 * np.pi * G * rho * R ** 3 / r ** 2)
)
if __name__ == "__main__":
pos = np.linspace(0.1, 2 * 6371e3, 1000)
fig = plt.figure(num=1, clear=True)
ax = fig.add_subplot(1, 1, 1)
ax.plot(pos, grav_fun(pos), "k-")
ax.set(
xlabel="Position from Center of Earth (m)",
ylabel="Gravity (m/s$^2$)",
title="Gravity vs. Position for Earth (mrg)",
)
fig.tight_layout()
fig.savefig("GravityPlot.png")
Questions
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