1. EE 194/BIO 196: Modeling,simulating and optimizing biological systems
Spring 2018
Tufts University
Instructor: Joel Grodstein
Arrays (part 1)

2. What is a an array
An array is an ordered, indexed group of variables that share the same name.
It's like a variable, but with a bunch of buckets in a row that are all numbered.
Demo with using a 1D egg carton and eggs to represent an array.
EE 194/Bio 196 Joel Grodstein

3. What is a an array Example:
The name of the variable
Arrays are in the numpy package. We’ll talk about packages more in the functions lecture
Example:
import numpy as np
arr1 = np.array ([1, 3, 7])
Our variable arr1 has three little buckets in it
We can access each bucket individually:
arr1[0] is 1; arr1[1] is 3; arr1[2] is 7. 1 3 7
arr1[0]
arr1[1]
arr1[2]
EE 194/Bio 196 Joel Grodstein

4. What is an array Terminology: Later on we’ll do 2-dimensional arrays 1
the individual elements of the array are called, well, elements
Our arr1 has 3 elements.
Later on we’ll do 2-dimensional arrays 1 3 7
arr1[0]
arr1[1]
arr1[2]
EE 194/Bio 196 Joel Grodstein

5. So what?
An array lets you collect a bunch of similar things, refer to them by one name, and operate on all of them at once.
Consider the grades of a 4-student class for HW1:
Which format would you rather work with?
What if there were 100 students rather than 4?
HW1_a=90
HW1_b=99
HW1_c=60
HW1_d=90
HW1=np.array([90,99,60,90])
EE 194/Bio 196 Joel Grodstein

6. But wait, there’s more… Consider the same deal for HW2: HW2_a=91
HW2_b=80
HW2_c=55
HW2_d=93
HW2 = np.array([91,80,55,93])
What if you want each student’s average?
avg_a=(HW1_a+HW2_a)/2
avg_b=(HW1_b+HW2_b)/2
avg_c=(HW1_c+HW2_c)/2
avg_d=(HW1_d+HW2_d)/2
avg=(HW1+HW2)/2
Works no matter how many students there are! Yes it does!
EE 194/Bio 196 Joel Grodstein

7. Reading and writing array elements
HW1=np.array([90,99,60,90])
print (HW1)
HW1[2]
HW1[2]=99
[90,99,60,90] 60
[90,99,99,90]
EE 194/Bio 196 Joel Grodstein

8. array-scalar operations
arr1=np.array([1.0, 3.0, 7.0, 5.0])
arr1 = arr1 * 1.1
Perhaps we’re storing how much money everyone has, and they all got 10% interest.
What do you think this would do?
arr2 = arr
Now arr1=[1.1, 3.3, 7.7, 5.5]
Why might we do this?
arr2=[2.2, 4.4, 8.8, 6.6]
EE 194/Bio 196 Joel Grodstein

9. array-array operations
arr1=np.array([1, 3, 7, 5])
arr2=np.array([2, 2, 1, 1])
What do you think these would do?
arr3 = arr1 + arr2
arr3 = arr1 * arr2
arr3=[3, 5, 8, 6]
[2, 6, 7, 5]
EE 194/Bio 196 Joel Grodstein

10. Dot product Arrays support a dot product operation
arr1=np.array ([1, 3, 7, 5])
arr2=np.array ([2, 2, 1, 3])
x = arr1.dot(arr2)
How about x = arr2.dot(arr1)?
Will arr1.dot(arr2) and arr2.dot(arr1) always give the same answer?
Yes, because multiplication is commutative
Why do I care about dot products?
They’re extremely useful in linear algebra
They also help in population modeling… as we will soon see
= (1*2) + (3*2) + (7*1) + (5*3)
= = 30
= (2*1) + (2*3) + (1*7) + (3*5)
= = 30
EE 194/Bio 196 Joel Grodstein

11. Where else might arrays be useful?
Any other ideas where arrays might be useful?
Height of 100 patients (or weight)
Temperature of each day of the year
Really, most any time you want to keep track of a big bunch of items
EE 194/Bio 196 Joel Grodstein

12. Other ways to make an array
np.array ([1,3,7])
np.zeros (5)
np.ones(3)
np.array(range(6))
[1,3,7] as we’ve already seen
[0,0,0,0,0] i.e., 5 zeros
[1,1,1] i.e., 3 ones
[0,1,2,3,4,5] just like “for”
EE 194/Bio 196 Joel Grodstein

13. Getting information about an array
a1=np.array ([1,3,7])
a2=np.zeros (5)
a1.size
a2.size
3 (because a1 has 3 elements)
5 (because a2 has 5 elements)
EE 194/Bio 196 Joel Grodstein

14. sum() and its friends a1=np.array ([1,3,7]) a2=np.zeros (5) a1.sum()
a1.prod()
a2.prod()
a1.min()
a1.max()
11 (i.e., 1+3+7)
21 (i.e., 1*3*7)
0 (i.e., 0*0*0*0*0)
1 (i.e., the smallest number of 1, 3 and7)
7 (i.e., the largest number of 1, 3 and7)
EE 194/Bio 196 Joel Grodstein

15. Slices
Array slices let us refer to a piece of an array, just as if it were an entire array.
a1=np.array ([1,3,5,7,9])
a1[1:3]
a1[2:5]
a1[2:5].sum()
a2 = np.array ([2,3,4])
a1 + a2
a1[1:4]+a2
a1[1:4].dot(a2)
[3,5]
[5,7,9]
5+7+9=21 (remember this for later…)
Illegal; a1 has 5 elements & a2 has 3
[5,8,11]
3*2 + 5*3 + 7*4 =49
EE 194/Bio 196 Joel Grodstein

16. Inconsistent syntax Note the similar (but not quite identical) syntax
for i in range (1,9,2):
a1[1:9:2]
Why does Python use “,” for the range and “:” for arrays?
Well, it just does. Who ever really knows why?
range() is a function, so it uses “,” to separate its arguments
Using “,” for arrays wouldn’t work for multi-dimensional arrays
i becomes 1, 3, 5 and 7
Accesses the array slice with elements a[1], a[3], a[5] and a[7]
EE 194/Bio 196 Joel Grodstein

17. Writing slices You can write into slices, too a=np.array ([1,3,5,7,9])
Now a is [1,10,11,12,9]
error; cannot assign a[1:4] with just 2 elements
this actually works. Python figures out what you really mean; it assigns 5 to all 3 elements of a[1:4]. And it could be useful on HW2…
EE 194/Bio 196 Joel Grodstein

18. More slices a=np.array ([1,3,5,7,9]) a[-1] a[-3] a[4:0:-1] a[-1:0:-1]
But how do you get [9,7,5,3,1]?
Leave it as a challenge
9 (counts from the end) 5
trick: a has 5 elements, and 5-1=4, so use a[4]
[9,7,5,3]
[9,7,5,3]
EE 194/Bio 196 Joel Grodstein

19. for loops and arrays a1=np.array ([1,3,7]) for data in a1:
print (data) 1 3 7
And of course, slices work as usual
for data in a1[1:3]:
print (data) 3 7
In fact, this works too
for data in [1,3,7]:
print (data) 1 3 7
EE 194/Bio 196 Joel Grodstein

20. Pitfall #1: arrays have a type
a=np.array ([1,3,5])
a[1]=4.1 a
What happened?
‘a’ is an array of integers, not an array of real numbers
It cannot hold 4.1, so 4.1 gets truncated to 4.
Python is not like Matlab in this regard
a = np.array ([1.0, 3.0, 5.0])
print(a)
In fact, all numbers are either integers or floating point; we just never had to deal with it until now
[1,3,4]
[1,3,4.1]
Now it’s an array of real numbers (i.e., “float”)!
EE 194/Bio 196 Joel Grodstein

21. Pitfall #1: arrays have a type
a=np.array ([1,3,5])
a=np.array ([1.0,3.0,5.0])
a=np.array ([1,3,5.0])
all elements of an array will have the same type
5.0 is a float, so everything else becomes one too
a=np.array ([1,3,5],dtype=float)
a.dtype
array of integers
array of float
array of float
array of float
float64
int64
EE 194/Bio 196 Joel Grodstein

22. Pitfall #2: Arrays are weird
a1 = np.array ([2,4,6])
a2 = a1
print (a1)
print (a2)
a2[1]=1
Now a2=[2, 4, 6]
[2, 4, 6]
[2, 4, 6]
[2, 1, 6]
[2, 1, 6]. What happened???!!
EE 194/Bio 196 Joel Grodstein

23. Explain this
Do a demo with cups and egg cartons. A carton cannot fit in a cup, so we use a pointer instead
And a2=a1 does not make a new egg carton; just adds a string to the same one.
EE 194/Bio 196 Joel Grodstein

24. Beating pitfall #2
a1 = np.array ([2,4,6]) a2 = a1 a2[1]=1 print (a2) print (a1) a2 = np.array ([2,4,6]) We’ll learn more strategies to beat this problem later
[2, 4, 6]
Now a2=[2, 4, 6]
[2, 1, 6]
[2, 1, 6]
Now we have two different egg cartons
[2, 1, 6]
[2, 4, 6]
EE 194/Bio 196 Joel Grodstein

25. Follow the bouncing ball
a1 = np.array ([2,4,6])
a2 = a1
a2[1]=1
a2 = np.array ([2,4,6]) U U a1 a2 2 4 1 6 2 4 1 6 2 4 6
EE 194/Bio 196 Joel Grodstein

26. Do slices have the same issue?
a1 = np.array ([2,4,6,8,9]) a2 = a1[1:3] a1[1]=1 print (a2) What happens?
EE 194/Bio 196 Joel Grodstein

27. Group exercise Remember lx-mx and new births?
n1,n+1 = n0,n * p0
n2,n+1 = n1,n * p1
n3,n+1 = n2,n * p2
n0,n+1= (n1,n+1* m1)+(n2,n+1* m2)+ (n3,n+1 * m3)
Can we use arrays for this?
n = np.array ([n0, n1, n2, n3])
p = np.array ([p0, p1, p2, p3])
How do we make an array n_next?
Hint: use slices and dot products
EE 194/Bio 196 Joel Grodstein

28. Advanced indexing
See for documentation
Arrays and Booleans
In addition to integers and floats, Python also has Boolean: either True or False
ar=np.array ([1,3,5,9,4])
ar>4
ar[ar>4]
ar[ar>4] = 10
np.all (ar>4)
np.any (ar>4)
Note that this is quite similar to Matlab
EE 194/Bio 196 Joel Grodstein

29. Advanced indexing
You can index an array with another array. What do these all do?
ar=np.array ([1,3,5,9,4])
ar [ np.array([2,2,1]) ]
ar [ np.array([2,2,1]) ] += 1
np.flatnonzero (ar>4)
ar2 = np.array ([10,20,30,40,50,60])
ar2 [ np.flatnonzero (ar>4) ] = 100
ar3 = np.array ([10,20,30,40,50,60])
ar3 [ 1+np.flatnonzero (ar>4) ] = 100
EE 194/Bio 196 Joel Grodstein

30. Advanced indexing Is advanced indexing weird? What does this code do?
I.e., does advanced indexing copy by reference?
What does this code do?
ar=np.array ([1,3,5,9,4])
ar2 = ar [ np.array([2,1]) ]
ar[1] = 10
print (ar2)
EE 194/Bio 196 Joel Grodstein

31. A few more group exercises
Print an array backwards (so if ar=[1,3,7] then print [7,3,1])
You can do it with a for loop
You can also do it in one statement with slices (but there’s a trick that we haven’t learned yet)
This will be useful for the Manduca HW
Take an array p. Multiply every element by a random number r, and then bound them all to be ≥0 and ≤1.
You may want to Google “numpy clip”
This will be useful for HW3
EE 194/Bio 196 Joel Grodstein

32. Follow-up activities Try the examples from this lecture yourself
Vary them, or even mis-type some to see what happens
More exercises. Write a program that…
determines the largest and smallest numbers from an array
creates a 5-element histogram from a given array of integers. So if ar=[1,4,1] then it creates an array hist=[0,2,0,0,1].
creates an array of 10 zeros, and update sixth value to 11
Use advanced indexing (if you choose) to
print all elements in an array that are even
multiply all odd numbers in an array by 10.
reverse the order of all even numbers in an array, so that [0,1,3,4,5] becomes [4,1,3,0,5]
You can do all of these in one line of code
EE 194/Bio 196 Joel Grodstein

33. Random numbers
We’ve talked about stochastic models. How do you get random numbers in Python?
random.randrange(0,10)
Returns a random integer ≥ 0 and <10
exact same syntax as range()
import random
for i in range(3):
print (random.randrange(0,10)) 5 2 8 1 3 9 5
Observations?
Yes, the numbers seem pretty random
No reason you can’t get the same number twice in a row
Every time you run the program, you potentially get a different sequence
EE 194/Bio 196 Joel Grodstein

34. Random repeatability Random behavior can be really hard to debug
Consider the following program to print small fractions in decimal form
import random
num = (random.randrange(0,10))
den = (random.randrange(0,10))
print (num / den)
Any obvious bugs? Let’s run it a few times .2 3
What combination of numbers could have given .2?
1 / 5 (not 2 / 10)
Traceback (most recent call last):
File "Q:/test.py", line 4, in <module>
print (num/den)
ZeroDivisionError: division by zero
It’s hard to debug this issue, if we can never predict when it will occur
EE 194/Bio 196 Joel Grodstein

35. Seeding Random-number generators can be seeded.
If you plant tomato seeds → tomatoes will grow
import random
random.seed(5)
for i in range(3):
print (random.randrange(0,10)) 5 8 9 3 6
A different seed gives us a different sequence (which, again, is the same every time we run the program)
Now we get the same sequence every time
(1)
EE 194/Bio 196 Joel Grodstein

36. Seeding and debugging Still doesn’t quite solve our problem Ideas?
import random
random.seed(5)
num = (random.randrange(0,10))
den = (random.randrange(0,10))
print (num / den)
Run over and over… get 0.5 every time. It never breaks!
0.5
Ideas?
Keep changing the “5” to other numbers and re-running; eventually it will fail repeatably
2 1.0 …
Traceback… print (i, num/den)
ZeroDivisionError: division by zero
import random
for i in range(1,20):
random.seed(i)
num = (random.randrange(0,10))
den = (random.randrange(0,10))
print (i, num / den)
EE 194/Bio 196 Joel Grodstein

37. rnd = random.gauss (mean, sigma)
Normal distribution
Normal distribution (a.k.a., Gaussian or bell-shaped curve)
used quite commonly in biology, of course
Python can draw numbers from it:
rnd = random.gauss (mean, sigma)
Everything we already said about seeds still applies
EE 194/Bio 196 Joel Grodstein

38. Quantization How do you round off numbers in Python? round(4.3) % 4.0
ar = numpy.array ([4.3, 4.7])
round (ar)
numpy.round (ar)
You can also round to nearest 0.1, .01, etc.
% 4.0
% 5.0
% error
% [4.0, 5.0]
EE 194/Bio 196 Joel Grodstein