Python Programming for Business Analytics Lecture 6 PDF

BMAN737 01 Programming in for Business Python Analytics Week 3: Lecture 2 Numerical Analysis Prof. Manuel López-Ibáñez manuel.lopez-ibanez@manchester. ac.uk Office hours: Mon 4pm-5pm, Fri 9am-10am  Contact Manuel Ló...

BMAN737 01 Programming in for Business Python Analytics Week 3: Lecture 2 Numerical Analysis Prof. Manuel López-Ibáñez manuel.lopez-ibanez@manchester. ac.uk Office hours: Mon 4pm-5pm, Fri 9am-10am  Contact Manuel López-Ibáñez (or Lopez-Ibanez) Discussion Board in Blackboard Office hours (AMBS 3.050): Mon 4pm-5pm, Fri 9am- 10am https://calendly.com/manuel-lopez-ibanez Email: [email protected] – 22 711 emails received last year ! – Allow me 2-3 days to reply – BMAN7370 3  What is new? From basics to advanced analytics Complete Python code available in BB Labs: Less exercise-like, more case- study-like BMAN7370 9  Plan Week 3: Numerical Analysis (NumPy) Week 4, Lecture 1: Data Exploration and Visualization (Pandas + Matplotlib) Week 4, Lecture 2: Data preprocessing and preparation (Pandas + Scikit-learn) Week 5: Machine learning (Scikit-learn) BMAN7370 10 BMAN737 01 Programming in for Business Python Analytics Week 3: Lecture 2 Numerical Analysis Part 1: Vectors and matrices Part 2: Broadcasting Part 3: Aggregations and other useful operations  Summing up 10M numbers a = list(range(10**7)) For- sum() over a list loop np.sum() s = 0 s = sum(a) x = np.array(a ) for i in a: s = np.sum(x) s += i Time: 0.1 Time: 0.05 Time: 0.0009 seconds seconds seconds BMAN7370 1  Good integration with Pandas and Matplotlib Scikit-learn (ML) is built on top of NumPy Fast computations Large multi-dimensional arrays (vectors and matrices) Reference docs: https://docs.scipy.org/doc/numpy/reference/index.html import numpy as # Just a np shortcut. BMAN7370 14  NumPy arrays import numpy as np Vector Matrice 𝑎Ԧ 0 1 0 1 s s 𝐴 = 2 2 = 3 4 In: a = In: A = 5 np.array([[0, 1, 2], np.array([0,1,2]) [3, 4, 5]] Out: array([0, 1, 2]) ) Out: 1, 2] In: a.shape array([[0, In: A.shape Out: (3,) # 1D (3 [3, 4, 5] ]) Out: (2,3) # 2D (rows, columns) columns) BMAN7370 15  NumPy arrays ≠ Lists In: a = In: a = np.arange(4) list(range(4)) Out: Out: array([0, 1, 2, [0, 1, 2, 3] 3]) In: a * In: a * Out 2 In:[0, a1, [0, 1, 2, 3, 2,0, 3, 1, 2,0, 3] 1, 2, 2 In: a array([0, 2, 4, 6]) + a 3] : + a In: Out: a.append(10 array([0 )[0, 1, a 2, 3, 10] Out , 2, 4, array([0, 1, 2, 3, 10]) Out: [0, 1, 2, : 6]) 3, 10] In: a = np.append(a, BMAN7370 10) a 16  Indexing: List vs NumPy 0 1 𝐴 2 = # A list of lists 3 4 A = [[0,1,2], 5 [3,4,5]] Return first row, first column  A A Return first row  A ？？ Return first column  ?? # A NumPy matrix A = np.array([[0,1,2], [3,4,5]]) Return first row, first column  A[0, 0] Return first row  A[0, :] BMAN7370 1  Slice Notation x[:] is the same as x[0:len(x):1] x[START:END:STEP] Start counting at START (default 0) Stop counting before END (default len or num. rows/columns), Increment by STEP (default 1) x[0:2] same as x[[0,1]] or x[range(2)] x[1:] same as x[range(1,len(x))] x[:5] same as x[range(5)] x[:] same as x[range(len(x))] and default END is BMAN73701  Slice Notation 0 1 2 Expression Shape 0 A[0:2, 1:3] (2,2) 1 A[:2, 1:] (2,2) 2 A (3,) A[2, :] (3,) A[2:, :] (1,3) A[:, :2] (3,2) A[:, [0,1]] (3,2) A[:2, 1] (2,) A[:2, 1:2] (2,1) BMAN7370 22  Quiz z x = np.array([ [3,2,1], [4,5,6] ]) 3 2 𝑋 1 = In: x[:, 1] 4 5 Out: array([2, 6 5]) In: x[-1, :] Out: array([4, 5, 6]) In: x[:2, 1:] Out: array([[2, 1], [5, 6]]) In: x[::-1, [2,1,0]] Out: array([[6, 5, 4], [1, 2, 3]]) BMAN7370 23  Boolean Indexing # A NumPy vector x = np.array([0,3,1,4,2,5]) x > 2  [False,True,False,True,False,True] x[x > 2]  [3,4,5] # A list x = [0,3,1,4,2,5] error x[x > 2]  Error !!! # A NumPy Matrix A = np.array([[0,3,1], [4,2,5]]) [3,4,5] A[A > 2]  [3,4,5] BMAN7370 24  𝐴 0 1 2 Copies vs Views = A  Creates a view 3 4 B 5 B = = A.copy()  Creates a copy B[0, 0] = 10 B[0, 0] = 20 A[0, 0]  value ? A[0, 0]  value ? Slices are views not copies! Same as B = A A[0, row0 0] = 1 :] = A[0,  returns first row row0 = 5  change first element of first row A[0, 0] slice_copy = A[0, value ? :].copy() # Creates a copy! BMAN7370 26  Example: Total gains and losses Given a matrix 𝑋 where 𝑋𝑡𝑗 is the net profit of department 𝑗 in time period 𝑡, calculate: 𝑔𝑎𝑖𝑛𝑠 = 𝑡 σ 𝑇 if 𝑋𝑡𝑗 > 𝑙𝑜𝑠𝑠𝑒𝑠 =𝑗 σ 𝑇 0 σ𝐷 𝑋 σ 𝑋𝑡𝑗 𝐷 𝑡𝑗 𝑡 if 𝑋𝑡𝑗 < 0 𝑗 for-loop No for t in loops! range(X.shape): for j in range(X.shape): gains = np.sum(X[X > if X[t,j] > 0: 0]) losses= gains += np.sum(X[X < 0]) X[t,j] Time: 0.004 else: seconds losses += X[t,j] BMAN7370 27  Element-wise operators (Most) operations are element-wise (+ - / * **) A * 2 # multiply each element of A by 2 A ** 2 # square each element of A by 2 A * B # Element-wise (not matrix product) np.dot(A, B) # Matrix Broadcastin A × A + x # Matrix (2,3) + Vector g! 3 2 product: (3,) B + 0 1 2 + 1 3 2 1 0 1 2 4 5 =? 4 5 6 0 1 2 BMAN7370 28  Recap NumPy arrays represent vectors and matrices ≠ Lists ! Indexing Numpy arrays: – Slicing similar to lists – More powerful than lists – Boolean indexing Element-wise mathematical operations Mathematical operations that apply to many elements of a Numpy array are much faster than indexing with for- BMAN7370 loops 29 BMAN737 01 Programming in for Business Python Analytics Week 3: Lecture 2 Numerical Analysis Part 1: Vectors and matrices Part 2: Broadcasting Part 3: Aggregations and other useful operations  Broadcasting http://www.scipy-lectures.org/intro/numpy/operations.htm l#broadcasting Creative Commons Attribution 4.0 International License (CC-by) http://creativecommons.org/licenses/b BMAN7370 3  Broadcasting x = np.array([0,10,20,30]) y = np.array([0,1,2]) In: x + y Out: ValueError: operands could not be broadcast together with shapes (4,) (3,) BMAN7370 32  Broadcasting 0 10 20 30 ? x = np.array([0,10,20,30]) y = np.array([0,1,2]) In: x + y Out: ValueError: operands could not be broadcast together with shapes (4,) (3,) BMAN7370 33  Broadcasting x = np.array([0,10,20,30]) y = np.array([0,1,2]) In: x + y Out: ValueError: operands could not be broadcast together with shapes (4,) (3,) In: x.reshape((4,1)) + y Out: array([[ 0, 1, 2], [10, 11, 12], [20, 21, 22], [30, 31, 32]]) BMAN7370 34  Reshaping x = np.array([0,10,20,30]) In: x Out: array([ 0, 10, 20, 30]) 0 10 20 30 In: x.reshape((4,1)) Out: array([[ 0], , , ]) In: x.reshape((2,2)) Out: array([[ 0, 10], 0 10 [20, 30]]) 20 30 BMAN7370 3  Example: Total of outer product Given two vectors 𝑥, 𝑦 calculate the 𝑥 = 1 10 20 30 𝑦 sum of every pairwise product of 2 3 4 5 𝑡𝑜𝑡𝑎𝑙 = 𝑖 =1σ 𝑛 𝑥𝑖 ⋅ 𝑦𝑗 = σ 𝑥 = their elements: 𝑖 σ 𝑛 𝑗 =1 × 𝑦𝑇 𝑗 for-loops No loops! n = total = np.sum( x.reshape((n,1)) * x.shape y) total = 0 1 1 11 2 3 4 5 1 5 for i in 10 10 10 10 * 2 3 4 5 range(n): 20 20 20 20 2 3 4 5 for j in 30 30 30 30 2 3 4 5 range(n): total BMAN7370 3  Broadcasting and reshape In: X = np.array([[1,2,3],[4,5,6]]) a = np.array([0,1,0]) print(X * a) Out: array([[0 2 0], [0 5 0]]) In: b = np.array([0,1]) print(X * b) Out: Error: operands could not be broadcast together with shapes (2,3) print(X (2,) * b) Out: array([[ 0 0], 0 In: b = [4 5 6]] BMAN7370 40  Recap Broadcasting allows mathematical operations between NumPy arrays of different shapes If shapes cannot be broadcast  Error! Mathematical operations using broadcasting are: than for- Faster – Shorter to to execute loops write BMAN7370 41 BMAN737 01 Programming in for Business Python Analytics Week 3: Lecture 2 Numerical Analysis Part 1: Vectors and matrices Part 2: Broadcasting Part 3: Aggregations and  Aggregations (Reductions) Some operations aggregate: sum, min, max, mean, all, any,... 0 1 2 axis = 1 3 4 5 axis = 0 np.min(A)  returns 1 number np.min(A, axis = 0)  min along rows, returns 1 number per column np.min(A, axis = 1)  min along columns, returns 1 number per row BMAN7370 43  Maximum Mean Squared Error 𝑃11 ⋯ 𝑃1𝑛 P ⋮ ⋱ ⋮ each row gives 𝑛 predictions by k ML = 𝑃𝑘1 ⋯ 𝑃𝑘𝑛 methods obs = 𝑜𝑏𝑠1 ⋯ 𝑜𝑏𝑠𝑛 the actual observed values Calculate the maximum Mean Squared Error (MSE) 1 𝑛 given as: 𝑚𝑎𝑥𝑀𝑆𝐸 = ෍ 𝑃𝑘𝑖 − 2 max 𝑜𝑏𝑠𝑖 𝑘=1,…𝑛 𝑖=1 maxMSE = np.max(np.mean((P-obs)**2, axis = 1)) BMAN7370 44  Functions vs. Methods Most functions in NumPy have an equivalent method np.min(A)  A.min() np.min(A, axis = 0)  A.min(axis = 0) Methods can only be applied to NumPy arrays! np.min([1,2,3])  OK [1,2,3].min()  Error Some methods modify the array in- place! BMAN7370 46  Boolean arrays: Any vs. All b = np.array([1, 1, 0, 0]) # 1 is True, 0 is False np.logical_not(b) np.logical_and(b, b) np.logical_or(b, b) np.all(b # all ) True? # np.any(b axis = any True? ) 1 1 0 B = np.array([[1,0],[0,1]]) axis = 0 1 0 np.all(B, axis = 0)  all True along rows ? returns 1 value per column np.any(B, axis = 1)  any True along columns ? returns BMAN7370 1 value per row 47  −1 2 𝐴 3 = −4 −5 6  np.any(A Are there negative values? < 0) Are all negative values?  np.all(A < 0) Which columns have only negative values?  np.all(A < 0, axis = 0) Which rows have at least BMAN7370 one negative 48  Sorting Direct np.sort(array, axis=) sorting  sort ascending np.sort(a) -np.sort(-a)  sort descending np.sort(A, axis=0)  sort each column (along rows) x = np.array([11,12,10,9]) In: np.sort(x) Out: [9,10,11,12] In: -np.sort(-x) Out: [12,11,10,9] BMAN7370 50  Indirect Sorting Direct sorting (ascending) np.sort(array, axis=) np.argsort(array, Indirect sorting axis=) np.argmin(array, axis=) x = np.argmax(array, np.array([12,11,10,9]) axis=) In: In: np.sort(x) np.max(x) Out: [9,10,11,12] In: Out: 12 np.argsort(x) In: Out: [3,2,1,0] In: np.argmax(x) x[np.argsort(x)] Out: 0 BMAN7370 51  Minimisation by indirect sorting Find the x value that produces the minimum of cos(x) between [0, 6] with a precision of 0.0000001 x = np.arange(0 6, 0.0000001 , ) y = np.cos(x) i = np.argmin(y ) print(x[i]) Out: 3.1415926999999 BMAN7370 52  numpy.random Sub-module of NumPy with lots of functions for random number generation, random distributions, etc. Different from built-in module (import random).  Better not mix the two to avoid confusion! Basic functions: np.random.permutation(array) np.random.rand(N) np.random.randn(N) np.random.seed(N) BMAN7370 5  Library for numerical analysis and scientific computations scipy.optimize (BMAN60101) Multi-variate numerical optimization, linear programming, non-linear optimization, differential evolution scipy.linalg Faster (than np.linalg) linear algebra, matrix inversion determinant, norms, decompositions, Eigen-vectors, etc. scipy.stats (BMAN71791) Statistical distributions, tests, trimmed mean, geometric mean, interquartile BMAN7370 range, etc. 56  Recap Aggregations (reductions): sum, min, max, mean, all, any,... Along axis=0 or axis=1 np.sort() np.argsort()  indirect sorting np.argmin() np.argmax() Scipy: lots of useful mathematical and statistical functions using Numpy arrays BMAN7370 57  Going further NumPy User Guide: https://docs.scipy.org/doc/numpy/user/inde x.html Advanced numerical tutorial: http://www.scipy-lectures.org BMAN7370 58  Next week Python library for data manipulation and analysis Advanced customisation requires using Matplotlib functions Complex plots require Matplotlib concepts BMAN7370 59