Difference between revisions of "Python:Logical Masks"

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(Example 1 - Discrete Function)
(Intermediate Results)
 
(16 intermediate revisions by the same user not shown)
Line 16: Line 16:
 
import matplotlib.pyplot as plt
 
import matplotlib.pyplot as plt
 
</source>
 
</source>
 +
 +
== NEW for Fall 2023 ==
 +
There is a new introduction to logical masks on Google Colab at [https://colab.research.google.com/drive/1JEogTGSTIIDrfMQ43fZZYQj5KjHT3ObE?usp=sharing Logical Mask Introduction].  If you have not used Google Colab before, see the Introduction at [[Google_Colab#Introduction]]
 +
  
 
==Piecewise Constant==
 
==Piecewise Constant==
Line 28: Line 32:
 
\mbox{GPA}(\mbox{Grade})=\left\{
 
\mbox{GPA}(\mbox{Grade})=\left\{
 
\begin{array}{rl}
 
\begin{array}{rl}
\mbox{Grade}\geq 90~~~ & 4\\
+
4, & \mbox{Grade}\geq 90~~~\\
80\leq\mbox{Grade} ~\And~ \mbox{Grade}<90~~~ & 3\\
+
3, & 80\leq\mbox{Grade} ~\And~ \mbox{Grade}<90~~~\\
70\leq\mbox{Grade} ~\And~ \mbox{Grade}<80~~~ & 2\\
+
2, & 70\leq\mbox{Grade} ~\And~ \mbox{Grade}<80~~~\\
60\leq\mbox{Grade} ~\And~ \mbox{Grade}<70~~~ & 1\\
+
1, & 60\leq\mbox{Grade} ~\And~ \mbox{Grade}<70~~~\\
\mbox{Grade}<60~~~ & 0
+
0, & \mbox{Grade}<60~~~
 
\end{array}
 
\end{array}
 
\right.
 
\right.
Line 56: Line 60:
 
MaskForA = Grade>=90
 
MaskForA = Grade>=90
 
FunctionForA = 4
 
FunctionForA = 4
GPA = MaskForA*FunctionForA
+
GPA = FunctionForA * MaskForA
 
</source>
 
</source>
 
At the end of this script, the <code>GPA</code> variable will contain a 4
 
At the end of this script, the <code>GPA</code> variable will contain a 4
Line 67: Line 71:
  
 
====Code and Graph====
 
====Code and Graph====
<source lang="python">
+
<html>
import numpy as np
+
<iframe src="https://trinket.io/embed/python3/49f786317f" width="100%" height="600" frameborder="0" marginwidth="0" marginheight="0" allowfullscreen></iframe>
import matplotlib.pyplot as plt
+
</html>
  
 
+
For a further explanation, look at the following code.  It investigates the individual masks for each grade for an array containing eight different grades:
def num_to_gpa(grade):
 
    out = (
 
        (grade >= 90) * (4)
 
        + ((80 <= grade) & (grade < 90)) * (3)
 
        + ((70 <= grade) & (grade < 80)) * (2)
 
        + ((60 <= grade) & (grade < 70)) * (1)
 
        + (grade < 60) * (0)
 
    )
 
    return out
 
 
 
 
 
if __name__ == "__main__":
 
 
 
    grades = np.linspace(0, 100, 1000)
 
    GPA = num_to_gpa(grades)
 
 
 
    fig = plt.figure(num=1, clear=True)
 
    ax = fig.add_subplot(1, 1, 1)
 
    ax.plot(Grade, GPA)
 
    ax.set(
 
        xlabel="Numerical Grade", ylabel="GPA",
 
        title="GPA vs. Grade for EGR 103L (mrg)"
 
    )
 
    fig.tight_layout()
 
    fig.savefig("GPAPlot.png")
 
</source>
 
<center>
 
[[image:GPAPlotPython.png]]
 
</center>
 
 
 
For a further explanation, look at the following script.  It investigates the individual masks for each grade for an array containing eight different grades:
 
 
<source lang="python">
 
<source lang="python">
 
ClassGrades = np.array([95, 70, 68, 38, 94, 84, 92, 87])
 
ClassGrades = np.array([95, 70, 68, 38, 94, 84, 92, 87])
Line 124: Line 97:
  
 
<source lang="python">
 
<source lang="python">
MaskForA*FunctionForA: array([4, 0, 0, 0, 4, 0, 4, 0])
+
FunctionForA * MaskForA: array([4, 0, 0, 0, 4, 0, 4, 0])
MaskForB*FunctionForB: array([0, 0, 0, 0, 0, 3, 0, 3])
+
FunctionForB * MaskForB: array([0, 0, 0, 0, 0, 3, 0, 3])
MaskForC*FunctionForC: array([0, 2, 0, 0, 0, 0, 0, 0])
+
FunctionForC * MaskForC: array([0, 2, 0, 0, 0, 0, 0, 0])
MaskForD*FunctionForD: array([0, 0, 1, 0, 0, 0, 0, 0])
+
FunctionForD * MaskForD: array([0, 0, 1, 0, 0, 0, 0, 0])
MaskForF*FunctionForF: array([0, 0, 0, 0, 0, 0, 0, 0])
+
FunctionForF * MaskForF: array([0, 0, 0, 0, 0, 0, 0, 0])
 
</source>
 
</source>
  
Line 136: Line 109:
 
GPA: array([4, 2, 1, 0, 4, 3, 4, 3])
 
GPA: array([4, 2, 1, 0, 4, 3, 4, 3])
 
</source>
 
</source>
 +
 +
All of that work is being done in the "one line of code" in the function!
  
 
== Non-constant Piecewise Functions ==
 
== Non-constant Piecewise Functions ==
Line 145: Line 120:
 
<center>
 
<center>
 
<math>
 
<math>
f(x)=\left\{
+
f(x)=
\begin{array}{rl}
+
\begin{cases}
x<0 & 0\\
+
0,&x<0\\
0\leq x<5 & x\\
+
x,&0\leq x<5 \\
5\leq x<10 & 5\\
+
5,&5\leq x<10 \\
x\geq 10 & 2x-15
+
2x-15,&x\geq 10
\end{array}
+
\end{cases}
\right.
 
 
</math>
 
</math>
 
</center>
 
</center>
 
which, for integers -2 through 12 can be calculated and graphed:
 
which, for integers -2 through 12 can be calculated and graphed:
 
====Code and Graph====
 
====Code and Graph====
 +
<html>
 +
<iframe src="https://trinket.io/embed/python3/c5dad756b0" width="100%" height="450" frameborder="0" marginwidth="0" marginheight="0" allowfullscreen></iframe>
 +
</html>
  
<source lang="python">
+
==== Intermediate Results ====
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')
 
</source>
 
<center>
 
[[image:PFunctionPython.png]]
 
</center>
 
 
 
=== Intermediate Results ===
 
 
Several of the intermediate results are presented below.  You will ''not'' have any of this in your programs.
 
Several of the intermediate results are presented below.  You will ''not'' have any of this in your programs.
 
Also, note that the spacing
 
Also, note that the spacing
 
in this document is set to make it easier to see which elements in one variable are
 
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,
 
paired up with equivalent elements in another variable (for example,
to determine the element-multiplication of <code>Mask 2</code> and <code>Function 2</code>).
+
to determine the element-multiplication of <code>Function 2</code> and <code>Mask 2</code>).
 +
 
 +
The functions - as calculated over the ''entire''
 +
domain, are:
 +
<source lang="python">
 +
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])
 +
</source>
 +
 
 
Note that the masks are shown below as arrays with 1 representing True and 0  
 
Note that the masks are shown below as arrays with 1 representing True and 0  
 
representing False in order to save space:  
 
representing False in order to save space:  
 
<source lang="python">
 
<source lang="python">
(x>0) * 1:                  array([1, 1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0])
+
(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])
 
((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])
 
((5<=x) & (x<10)) * 1:      array([0, 0, 0, 0, 0, 0, 0, 1, 1, 1, 1, 1, 0, 0, 0])
Line 201: Line 160:
 
</source>
 
</source>
 
Notice how the masks end up spanning the entire domain of <math>x</math> without
 
Notice how the masks end up spanning the entire domain of <math>x</math> without
overlapping. The functions - as calculated over the ''entire''
+
overlapping.  
domain, are:
 
<source lang="python">
 
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])
 
</source>
 
  
 
The individual products of the masks and the functions give:
 
The individual products of the masks and the functions give:
 
<source lang="python">
 
<source lang="python">
Mask 1 *Function 1: array([0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0])
+
Function 1 *Mask 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])
+
Function 2 *Mask 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])
+
Function 3 *Mask 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])
+
Function 4 *Mask 4: array([0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 5, 7, 9])
 
</source>
 
</source>
 
Finally, when these are all added together, none of the individual
 
Finally, when these are all added together, none of the individual
Line 231: Line 183:
 
<math>
 
<math>
 
\mbox{Gravity}(r)=
 
\mbox{Gravity}(r)=
\left\{
+
\begin{cases}
\begin{array}{lr}
+
\frac{4}{3}\pi G\rho r,& r<=R\\
r<=R & \frac{4}{3}\pi G\rho r\\
+
\frac{4}{3}\pi G \rho\frac{R^3}{r^2},&r>R
r>R & \frac{4}{3}\pi G \rho\frac{R^3}{r^2}
+
\end{cases}
\end{array}
 
\right.
 
 
</math>
 
</math>
 
</center>
 
</center>
You can use the same method as the example above, making sure to calculate the values of the function rather than simply using
+
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:
constants.  The following code and graph demonstrate:
+
 
 
====Code and Graph====
 
====Code and Graph====
<source lang="python">
+
<html>
G = 6.67e-11
+
<iframe src="https://trinket.io/embed/python3/2e73c8d116" width="100%" height="450" frameborder="0" marginwidth="0" marginheight="0" allowfullscreen></iframe>
rho = 5515
+
</html>
R = 6371e3
 
Position = np.linspace(0.1, 2*R, 1000)
 
MaskForInside = Position<=R
 
FunctionForInside = 4/3*m.pi*G*rho*Position
 
MaskForOutside = Position>R
 
FunctionForOutside = 4/3*m.pi*G*rho*R**3/Position**2
 
Gravity = \
 
    MaskForInside*FunctionForInside + \
 
    MaskForOutside*FunctionForOutside
 
plt.plot(Position, Gravity, 'k-');
 
plt.xlabel('Position from Center of Earth ($m$)');
 
plt.ylabel('Gravity ($m/s^2$)');
 
plt.title('Gravity vs. Position for Earth (mrg)');
 
plt.savefig('GravityPlot.png')
 
</source>
 
<center>
 
[[image:GravityPlot.png]]
 
</center>
 
 
 
== Using Logical Masks in Functions ==
 
Any of these tasks can be done using functions.  For example, you could use a function in the script to compute the gravity - in this case G could be defined in the function, the density and radius of the planet in question could be passed along (with Earth value as the default, because, astronomical narcissism), and the rest of the work could be in the script itself.
 
 
 
<source lang="python">
 
def grav_fun(r, rho=5515, R=6371e3):
 
    # Note: defaults to Earth
 
    G = 6.67e-11
 
    return (r<=R) * (4/3*m.pi*G*rho*r) + \
 
          (r>R)  * (4/3*m.pi*G*rho*R**3/r**2)
 
 
 
Position = np.linspace(0.1, 2*R, 1000)
 
plt.plot(Position, grav_fun(Position), 'k-')
 
plt.xlabel('Position from Center of Earth (m)')
 
plt.ylabel('Gravity (m/s^2)')
 
plt.title('Gravity vs. Position for Earth (mrg)')
 
plt.savefig('GravityPlot.png') 
 
</source>
 
 
 
  
 
== Questions ==
 
== Questions ==

Latest revision as of 19:46, 17 October 2023

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

NEW for Fall 2023

There is a new introduction to logical masks on Google Colab at Logical Mask Introduction. If you have not used Google Colab before, see the Introduction at Google_Colab#Introduction


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:

\( \mbox{GPA}(\mbox{Grade})=\left\{ \begin{array}{rl} 4, & \mbox{Grade}\geq 90~~~\\ 3, & 80\leq\mbox{Grade} ~\And~ \mbox{Grade}<90~~~\\ 2, & 70\leq\mbox{Grade} ~\And~ \mbox{Grade}<80~~~\\ 1, & 60\leq\mbox{Grade} ~\And~ \mbox{Grade}<70~~~\\ 0, & \mbox{Grade}<60~~~ \end{array} \right. \)

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 = FunctionForA * MaskForA

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:

FunctionForA * MaskForA: array([4, 0, 0, 0, 4, 0, 4, 0])
FunctionForB * MaskForB: array([0, 0, 0, 0, 0, 3, 0, 3])
FunctionForC * MaskForC: array([0, 2, 0, 0, 0, 0, 0, 0])
FunctionForD * MaskForD: array([0, 0, 1, 0, 0, 0, 0, 0])
FunctionForF * MaskForF: 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:

\( f(x)= \begin{cases} 0,&x<0\\ x,&0\leq x<5 \\ 5,&5\leq x<10 \\ 2x-15,&x\geq 10 \end{cases} \)

which, for integers -2 through 12 can be calculated and graphed:

Code and Graph

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 Function 2 and Mask 2).

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])

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 individual products of the masks and the functions give:

Function 1 *Mask 1: array([0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0])
Function 2 *Mask 2: array([0, 0, 0, 1, 2, 3, 4, 0, 0, 0, 0, 0, 0, 0, 0])
Function 3 *Mask 3: array([0, 0, 0, 0, 0, 0, 0, 5, 5, 5, 5, 5, 0, 0, 0])
Function 4 *Mask 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:

\( \mbox{Gravity}(r)= \begin{cases} \frac{4}{3}\pi G\rho r,& r<=R\\ \frac{4}{3}\pi G \rho\frac{R^3}{r^2},&r>R \end{cases} \)

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

Questions

Post your questions by editing the discussion page of this article. Edit the page, then scroll to the bottom and add a question by putting in the characters *{{Q}}, followed by your question and finally your signature (with four tildes, i.e. ~~~~). Using the {{Q}} will automatically put the page in the category of pages with questions - other editors hoping to help out can then go to that category page to see where the questions are. See the page for Template:Q for details and examples.

External Links

References