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EE 194/ BIO 196: Modeling,simulating and optimizing biological systems

Learn about arrays in modeling, simulating, and optimizing biological systems. Understand the basics of arrays, their operations, and their applications in biology. Explore examples and gain practical knowledge in using arrays effectively.

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EE 194/ BIO 196: Modeling,simulating and optimizing biological systems

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  1. EE 194/BIO 196: Modeling,simulating and optimizing biological systems Spring 2018 Tufts University Instructor: Joel Grodstein joel.grodstein@tufts.edu 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 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: • 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 • print (HW1) [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 = arr1 + 1.1 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) = 2 + 6 + 7 + 15 = 30 = (2*1) + (2*3) + (1*7) + (3*5) = 2 + 6 + 7 + 15 = 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]) • a[1:4]=[10,11,12] • a[1:4]=[10,11] • a[1:4] = 5 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]) a[1]=4.1 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 • a=np.array ([1,3,5]) • 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 • print (a2) • print (a1) 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) a1 = np.array ([2,4,6]) a2 = np.array ([2,4,6]) a2[1]=1 print (a2) print (a1) 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 a1 = np.array ([2,4,6]) a2 = np.array ([2,4,6]) a2[1]=1 U U a2 a1 1 2 4 6 2 4 6 1 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 https://docs.scipy.org/doc/numpy-1.13.0/user/basics.indexing.html 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? • 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() 5 2 8 1 3 1 0 9 5 import random for i in range(3): print (random.randrange(0,10)) • 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 3 .2 • 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)) 3 6 6 Now we get the same sequence every time 5 8 9 A different seed gives us a different sequence (which, again, is the same every time we run the program) (1) EE 194/Bio 196 Joel Grodstein

  36. Seeding and debugging • Still doesn’t quite solve our problem 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 1 0.625 2 1.0 3 0.8888888888888888 … 9 0.8333333333333334 10 2.25 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. 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) • round(4.7) • 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

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