List Comprehensions and Simulations

Transcription

1 List Comprehensions and Simulations 1 List Comprehensions examples in the Python shell zipping, filtering, and reducing 2 Monte Carlo Simulations testing the normal distribution the Mean Time Between Failures (MTBF) problem MCS 507 Lecture 5 Mathematical, Statistical and Scientific Software Jan Verschelde, 6 September 2013 Scientific Software (MCS 507 L-5) Lists Comprehensions and Simulations 6 September / 24

2 List Comprehensions and Simulations 1 List Comprehensions examples in the Python shell zipping, filtering, and reducing 2 Monte Carlo Simulations testing the normal distribution the Mean Time Between Failures (MTBF) problem Scientific Software (MCS 507 L-5) Lists Comprehensions and Simulations 6 September / 24

3 using for in lists Computing the square of some numbers: >>> L = range(5) >>> L [0, 1, 2, 3, 4] >>> [x**2 for x in L] [0, 1, 4, 9, 16] The general syntax of a list comprehension is [ expression(x) for x in somelist ] Scientific Software (MCS 507 L-5) Lists Comprehensions and Simulations 6 September / 24

4 range versus xrange With range we create a list. So: >>> L = [x**2 for x in range(0,5)] 1 creates the list [0, 1, 2, 3, 4] 2 lets x run through that list. If we are interested only in the end result, then generating [0, 1, 2, 3, 4] is wasteful. With xrange, the element in the range is generated only when needed and the entire range is not stored. >>> L = [x**2 for x in xrange(0,5)] >>> L [0, 1, 4, 9, 16] Scientific Software (MCS 507 L-5) Lists Comprehensions and Simulations 6 September / 24

5 listing all characters Every character is encoded by one byte: >>> ord( a ) 97 >>> chr(97) a To list all characters: >>> a = ord( a ) >> z = ord( z ) >>> [chr(i) for i in xrange(a,z+1)] [ a, b, c,.., y, z ] Scientific Software (MCS 507 L-5) Lists Comprehensions and Simulations 6 September / 24

6 string representations The str() function is defined on a list: >>> L = range(5) >>> s = str(l) >>> L [0, 1, 2, 3, 4] >>> s [0, 1, 2, 3, 4] >>> eval(s) [0, 1, 2, 3, 4] We may prompt for a list: >>> K = input( enter a list : ) enter a list : [1,2,3] >>> K [1, 2, 3] Scientific Software (MCS 507 L-5) Lists Comprehensions and Simulations 6 September / 24

7 List Comprehensions and Simulations 1 List Comprehensions examples in the Python shell zipping, filtering, and reducing 2 Monte Carlo Simulations testing the normal distribution the Mean Time Between Failures (MTBF) problem Scientific Software (MCS 507 L-5) Lists Comprehensions and Simulations 6 September / 24

8 zipping lists >>> L = range(5) >>> a = ord( a ); a 97 >>> chr(a) a The name a refers to the string a, which is encoded by the byte with value 97. >>> K = [chr(i) for i in xrange(a,a+5)] >>> K [ a, b, c, d, e ] >>> L [0, 1, 2, 3, 4] >>> Z = [[a,b] for a, b in zip(l,k)] >>> Z [[0, a ], [1, b ], [2, c ], [3, d ], [4, e ]] Scientific Software (MCS 507 L-5) Lists Comprehensions and Simulations 6 September / 24

9 lambda expressions We can make short functions with lambda: >>> f = lambda x: x**2 >>> type(f) <type function > >>> f(3) 9 Lambda expressions are used in other functions, for example: numerical integration, and are in list processing functions. Scientific Software (MCS 507 L-5) Lists Comprehensions and Simulations 6 September / 24

10 filtering lists Find all numbers between 0 and 99 that are divisible by 3; and that have 2 as a last digit. First we define a filter function: >>> f = lambda x: x % 3 == 0 and x % 10 == 2 >>> f(12) True >>> f(11) False Then we apply the filter f to a list: >>> L = range(100) >>> F = filter(f,l) >>> F [12, 42, 72] Scientific Software (MCS 507 L-5) Lists Comprehensions and Simulations 6 September / 24

11 reducing lists To compute the sum of the elements in a list L: sum(l). The factorial of 10 is : >>> L = range(1,11) >>> reduce(lambda x,y: x*y, L) The reduce() calculated (..((1 2) 3)..) 10 The function given as argument to reduce() must take two elements on input, return one single element. reduce() repeatedly replaces the first two elements of the list by the result of the function, applied to those first two elements, until only one element in the list is left. Scientific Software (MCS 507 L-5) Lists Comprehensions and Simulations 6 September / 24

12 the Horner form The Horner form of a polynomial is 2x 2 8x + 3 = ((2x 8)x + 3. To evaluate a dense polynomial of degree d we need d additions and d multiplications. We apply reduce() to a polynomial with integer coefficients stored in a list: >>> c = [2, -8, 3] >>> h = lambda x,y: ( + str(x) + *x \... + ( %+d % y) + ) >>> h(2,-8) (2*x-8) >>> h(2,8) (2*x+8) >>> reduce(h,c) ((2*x-8)*x+3) Scientific Software (MCS 507 L-5) Lists Comprehensions and Simulations 6 September / 24

13 List Comprehensions and Simulations 1 List Comprehensions examples in the Python shell zipping, filtering, and reducing 2 Monte Carlo Simulations testing the normal distribution the Mean Time Between Failures (MTBF) problem Scientific Software (MCS 507 L-5) Lists Comprehensions and Simulations 6 September / 24

14 testing random.gauss The normal distribution is provided by the function gauss in the builtin module random of Python. To test gauss, given mean µ and standard deviation σ, we generate a number n of samples, and for the n samples, we compute the mean µ and standard deviation σ. For large enough n, the computed µ µ and σ σ. We count the number of samples within the standard deviation of the mean, that is: inside [σ µ,σ + µ]. Scientific Software (MCS 507 L-5) Lists Comprehensions and Simulations 6 September / 24

15 running testgauss.py $ python testgauss.py testing the normal distribution give the mean : 5 give the standard deviation : 2 give the number of samples : average of samples : standard deviation : #samples in [3.00,7.00] : 6856 smallest sample : largest sample : Scientific Software (MCS 507 L-5) Lists Comprehensions and Simulations 6 September / 24

16 the script testgauss.py from random import gauss from math import sqrt print testing the normal distribution MU = input( give the mean : ) SIGMA = input( give the standard deviation : ) DIM = input( give the number of samples : ) L = [gauss(mu, SIGMA) for i in xrange(dim)] Scientific Software (MCS 507 L-5) Lists Comprehensions and Simulations 6 September / 24

17 testgauss.py continued AVG = sum(l)/dim SQR = [(e - AVG)**2 for e in L] STD = sqrt(sum(sqr)/dim) A = MU - SIGMA B = MU + SIGMA F = filter(lambda x: A < x < B, L) print average of samples :, AVG print standard deviation :, STD print #samples in [%.2f, %.2f] : %d % (A, B, len(f)) print smallest sample :, min(l) print largest sample :, max(l) Scientific Software (MCS 507 L-5) Lists Comprehensions and Simulations 6 September / 24

18 deprecated-lambda in map/filter pylint warns about F = filter(lambda x: A < x < B, L) In particular: W: 26, 4: Used builtin function filter (bad-builtin) W: 26, 4: map/filter on lambda could be replaced by comprehension (deprecated-lambda) A better construction (if only interested in len(f)) is TEST = lambda x: A < x < B F = [TEST(e) for e in L] print sum(f) Scientific Software (MCS 507 L-5) Lists Comprehensions and Simulations 6 September / 24

19 List Comprehensions and Simulations 1 List Comprehensions examples in the Python shell zipping, filtering, and reducing 2 Monte Carlo Simulations testing the normal distribution the Mean Time Between Failures (MTBF) problem Scientific Software (MCS 507 L-5) Lists Comprehensions and Simulations 6 September / 24

20 mean time between failures The Mean Time Between Failures (MTBF) problem asks for the expected life span of a product made of components. Every component is critical. The multi-component product fails as soon as one of its components fails. For every component we assume that the life span follows a known normal distribution, given by µ and σ. For example, consider 3 components with respective means 11, 12, 13 and corresponding standard deviations 1, 2, 3. Running 10,000 simulations, we compute the average life span of the composite product. Scientific Software (MCS 507 L-5) Lists Comprehensions and Simulations 6 September / 24

21 the script mtbf.py from random import gauss from math import sqrt (MEAN1, SIGMA1) = (11, 1) (MEAN2, SIGMA2) = (12, 2) (MEAN3, SIGMA3) = (13, 3) DIM = L1 = [gauss(mean1, SIGMA1) for i in xrange(dim)] L2 = [gauss(mean2, SIGMA2) for i in xrange(dim)] L3 = [gauss(mean3, SIGMA3) for i in xrange(dim)] L = [min(x, y, z) for x, y, z in zip(l1, L2, L3)] AVG = sum(l)/dim SQR = [(e - AVG)**2 for e in L] STD = sqrt(sum(sqr)/dim) print average of samples :, AVG print standard deviation :, STD Scientific Software (MCS 507 L-5) Lists Comprehensions and Simulations 6 September / 24

22 running mtbf.py $ python mtbf.py average of samples : standard deviation : $ Some extensions to the example: product fails if two of its components fail; or product fails if component one and two both fail or if component three fails. Scientific Software (MCS 507 L-5) Lists Comprehensions and Simulations 6 September / 24

23 Summary + Exercises We covered more of chapter 2 of the text book. Exercises: 1 Make a list of all the digits of math.pi. 2 A sieve method to find all primes in a range 2 to n removes all multiples of 2, 3, 5, etc. Apply the filter function to sieve for all primes between 2 and Use list comprehensions to generate points (x, y) uniformly distributed on the circle: x 2 + y 2 = 1. Use random.uniform(0,2*math.pi) for random angles t: x = cos(t), y = sin(t), cos 2 (t) + sin 2 (t) = 1. Scientific Software (MCS 507 L-5) Lists Comprehensions and Simulations 6 September / 24

24 more exercises 4 Generate a sequence of cubes x 3, for x ranging over all natural numbers between 1 and 70. To determine which cubes are squares of natural numbers, take the square root of those cubes and select those numbers which have a natural number as their square root. 5 Extend mtbf.py so that it works for any number of components, prompting the user for a list of means and a list of corresponding standard deviations. The first homework collection is on Monday 9 September, at 9AM. Bring to class your answers to exercise 3 of Lecture 1; exercises 1, 2 of Lecture 2; and exercises 1, 3 of Lecture 3. Along with a paper version, also me your scripts. Scientific Software (MCS 507 L-5) Lists Comprehensions and Simulations 6 September / 24