Markov model, numpy introduction

Matt Valeriote

4 March 2019

Markov models (dictionary example)

I A Markov model is one where the future state of a systemdepends only on its current state (not on any previous states).

I They’re widely used in lots of fields: physics, chemistry, queuingtheory, economics, genetics, mathematical biology, sports, . . . .

I Here is an example from this wikipedia page:https://en.wikipedia.org/wiki/Markov_chain.

I Suppose that you start with $10, and you wager $1 on anunending, fair, coin toss indefinitely, or until you lose all of yourmoney. If Xn represents the number of dollars you have after ntosses, with X0 = 10, then the sequence {Xn : n ∈ N} is aMarkov process. If I know that you have $12 now, then itwould be expected that with even odds, you will either have$11 or $13 after the next toss. This guess is not improved bythe added knowledge that you started with $10, then went upto $11, down to $10, up to $11, and then to $12.

I Markov models can be used for text analysis.

I A Markov model of text would say that the next word in apiece of text (or letter, depending on what scale we’re workingat) depends only on the current word.

I We will create some code that will analyse some text and basedon the frequency of word pairs, will produce a short “sentence”from the words in the text, using the Markov model.

I The key element of this exercise will be a dictionary withkey:value pairs of the form word:list, where word is aword from the text and list is the list of words in the textthat immediately follow word.

Some issues

I The text that we use, for example Kafka’s Metamorphosis(http://www.gutenberg.org/files/5200/5200.txt) or Melville’sMoby Dick (http://www.gutenberg.org/files/2701/2701-0.txt),will contain lots of symbols, such as punctuation, that weshould remove first.

I We should also convert all words to lower case.I The text that we use will either be in a file stored locally, or

maybe accessed using its url.I There is a random element to Markov processes and so we will

need to be able to generate numbers randomly (orpseudo-randomly).

Cleaning strings

I Strings have a lower() method that can be used to convertall upper case letters to lower case.

I text/data cleaning is an inevitable part of dealing with text filesor data sets.

I python has a function called translate() that can be used toscrub certain characters from a string, but it is a littlecomplicated.

I For a discussion of this, check out:https://machinelearningmastery.com/clean-text-machine-learning-python/

I here is a simple function that will delete from a given string scharacters that appear in the string delete_chars.

I The python string string.punctuation is equal to!"#$%&\'()*+,-./:;<=>?@[\\]ˆ_{|}~‘.

I To use it import string must be executed.

def clean_string(s,delete_chars):for i in delete_chars:

s = s.replace(i,"")return(s)

x = "ab,Cde!?Q@#$I"import stringprint(clean_string(x,string.punctuation))

## abCdeQI

Description of the Markov text model algorithm

1. Open and read the text file.2. Clean the file.3. Create the text dictionary with items word: list of words

that follow.4. Randomly select a word from the text and then create a

“sentence” using randomly selected words from the dictionary.

markov_create function

def markov_create(file_name, sentence_length = 20):# open the file and store its contents in a string.text_file = open(file_name, 'r')text = text_file.read()# clean the text and then split it into wordsclean_text = clean_string(text)word_list = clean_text.split()# create the markov dictionarytext_dict = markov_dict(word_list)# Produce a sentence (a list of strings) of length# sentence_length using the dictionarysentence = markov_sentence(text_dict, sentence_length)# print out the sentence as a string using# the .join() method.return " ".join(sentence)

To complete this exercise, we need to produce the followingfunctions:

def clean_string(s,delete_chars = string.punctuation):

that strips the text of punctuation and converts upper case wordsinto lower case.

def markov_dict(word_list):

a function that creates a dictionary from a list of words, and

def markov_sentence(text_dict, sentence_length):

a function that randomly produces a sentence using the dictionary.

the random moduleI The module random can be used to generate pseudo-random

numbers or to pseudo-randomly select items.I Doc: https://docs.python.org/3/library/random.htmlI A random integer from a prescribed range can be generated

using the randrange method.I The method choice(seq) randomly chooses an element from

a sequence, such as a list or tuple.I shuffle and sample can be used to shuffle the items in a list

and to sample elements from a list, tuple, or set.

import randomrandom.randrange(2, 102, 2) # random even numbersrandom.choice([1, 2, 3, 4, 5]) # random choice from listrandom.sample([1, 2, 3, 4, 5], 3) # rand. sample of 3 itemsrandom.random() # random float between 0 and 1random.uniform(3, 7) # random num between 3 and 7

NumPy Installation

I NumPy is the fundamental package for scientific computingwith Python. It contains among other things:

I a powerful N-dimensional array objectI sophisticated (broadcasting) functionsI useful linear algebra and random number capabilities

I NumPy should already be installed on the spyder IDE. If notyou will need to install it.

I Good NumPy documentation can be found here (as pdf) andhere.

Arrays

I The array() is numpy’s main data structure.I It is similar to a python list, butI while a list can contain items of various types,I arrays contain a homogeneous set of values (e.g. floating point

(float64) or integer (int64) or str).

import numpy as npx = [1, 2, 3]a = np.array([1, 4, 5, 8], float)print(type(a)), print(a), print(a.shape)

## <class 'numpy.ndarray'>## [1. 4. 5. 8.]## (4,)

I The shape of an array is a tuple that lists its dimensions.I np.array([1,2]) produces a 1-dimensional (1-D) array of

length 2 whose entries have type intI np.array([1,2], float) produces a 1-dimensional (1-D)

array of length 2 whose entries have type float.

a1 = np.array([1,2])print(a1.dtype), print(a1.shape), print(len(a1))

## int32## (2,)## 2

a2 = np.array([1,2],float)print(a2.dtype)

## float64

I arrays can be created from lists or tuples.I arrays can also be created using the range function.I NumPy has a similar function, called arange that can be used

to create arrays.I an array of all zeroes or ones can be created using the zeros()

and ones() functions.

x = [1, 'a', 3]a = np.array(x)b = np.array(range(10), float)c = np.arange(5, dtype=float)d = np.arange(2,4, 0.5, dtype=float)np.ones(10)np.zeros(4)

I array slicing and indexing of 1-D arrays works the same way aslists/tuples/strings

I arrays are mutable like lists/dictionaries, so we can setelements (e.g. a[1]=0)

I or use the .copy() method (works for lists etc. too!)

a1 = np.array([1.0, 2, 3, 4, 5, 6])a1[1]a1[:-3]b1 = a1c1 = a1.copy()b1[1] = 23a1[1]c1[1]

Multi-dimensional arrays

I we have seen nested lists of lists and have used them torepresent matrices.

I in numpy, 2-dimensional arrays serve the same purpose but areeasier to work with.

I they can be created by passing a list of lists/tuple of tuples tothe array() function.

I Elements of an array are indexed via a[i,j] rather thana[i][j]

nested = [[1, 2, 3], [4, 5, 6]]a = np.array(nested, float)nested[0][2]a[0,2]aa.shape

slicing, reshapingI slicing of multiple dimensional arrays works in a similar way as

for lists and strings.I for each dimension, we can specify a particular sliceI : indicates that everything along a dimension will be used.

a = np.array([[1, 2, 3], [4, 5, 6]], float)a[1, :]a[:, 2]a[-1:, -2:]

I An array can be reshaped using the reshape(tuple) function,where tuple gives the new dimensions of the array.

a = np.array(range(10), float)a = a.reshape((5,2))print(a)

I An array with a given shape can be flattened into a 1-D array:

a = np.array([[1, 2, 3], [4, 5, 6], [7, 8, 9]])print(a), print(a.flatten())

## [[1 2 3]## [4 5 6]## [7 8 9]]## [1 2 3 4 5 6 7 8 9]

I arrays consisting of just 0’s or 1’s that have the shape of agiven array can also be easily created using the fill(),zeros_like() or ones_like() functions:

b = np.ones_like(a)b.fill(33)

I The identity matrix (with 1’s down the diagonal) can becreated using the identity() or the eye() function.

I With the eye() function, ones can also be placed onoff-diagonals;

print(np.identity(4, dtype=float)),

## [[1. 0. 0. 0.]## [0. 1. 0. 0.]## [0. 0. 1. 0.]## [0. 0. 0. 1.]]

print(np.eye(4, k = -1, dtype=int))

## [[0 0 0 0]## [1 0 0 0]## [0 1 0 0]## [0 0 1 0]]

array mathematicsI for lists (or tuples or strings), the + operation is used to

concatenate two lists to create a longer one.I this operation for arrays plays a different role.I to concatenate two arrays of suitable shapes, the

concatenate() function can be used:

a = np.array([[1, 2, 3], [4, 5, 6], [7, 8, 9]])b = np.array([[10, 11,12], [13, 14, 15], [16, 17, 18]])print(np.concatenate((a,b)))

## [[ 1 2 3]## [ 4 5 6]## [ 7 8 9]## [10 11 12]## [13 14 15]## [16 17 18]]

I When the + operation is used on arrays, it is applied on anelement-by-element basis.

I This also applies to most other standard mathematicaloperations.

print(a+b), print(a*b), print(a/b)

## [[11 13 15]## [17 19 21]## [23 25 27]]## [[ 10 22 36]## [ 52 70 90]## [112 136 162]]## [[0.1 0.18181818 0.25 ]## [0.30769231 0.35714286 0.4 ]## [0.4375 0.47058824 0.5 ]]

I To add a number, say 1, to every element of an array a, type a+ 1

I similarly for other operations, like -, *, **, /, . . .

print(a + 1), print(a/2), print(a ** 3)

## [[ 2 3 4]## [ 5 6 7]## [ 8 9 10]]## [[0.5 1. 1.5]## [2. 2.5 3. ]## [3.5 4. 4.5]]## [[ 1 8 27]## [ 64 125 216]## [343 512 729]]

I NumPy comes with a large library of common functions, suchas sin, cos, log, exp, . . .

I There are functions that can be applied to arrays to extractvarious properties,

I for example a.sum() and a.prod() will produce the sum andthe product of the items in a:

print(np.sin(a)), print(a.sum()), print(a.prod()), \print(a.mean())

## [[ 0.84147098 0.90929743 0.14112001]## [-0.7568025 -0.95892427 -0.2794155 ]## [ 0.6569866 0.98935825 0.41211849]]## 45## 362880## 5.0