import pandas as pd
import numpy as np
import re

pd.options.plotting.backend = 'plotly'

import util

Lecture 18 – Regular Expressions, Bag of Words

DSC 80, Winter 2023

📣 Announcements

• Lab 7 (regular expressions and text features) is due on Monday, February 27th at 11:59PM.
  • We won't cover the necessary ideas for Question 3 until Monday; see this old lecture if you'd like to finish it over the weekend.
• Nice work on Project 3! Project 4 (Language Models 🗣) will be released over the weekend.
  • The checkpoint will be due on Thursday, March 2nd at 11:59PM.
  • The full project will be due on Thursday, March 9th at 11:59PM.

Agenda

• More regular expressions, including how to use them in Python.
  • Example: Log parsing.
  • Limitations.
  • Remember to look here](dsc80.com/resources/#regular-expressions for resources.
• Text features.
• Bag of words 💰.

More regular expressions

Even more regex syntax

operation	example	matches ✅	does not match ❌
escape character	ucsd\.edu	'ucsd.edu'	'ucsd!edu'
beginning of line	^ark	'ark two' 'ark o ark'	'dark'
end of line	ark$	'dark' 'ark o ark'	'ark two'
zero or one	cat?	'ca' 'cat'	'cart' (matches 'ca' only)
built-in character classes*	\w+ \d+	'billy' '231231'	'this person' '858 people'
character class negation	[^a-z]+	'KINGTRITON551' '1721$$'	'porch' 'billy.edu'

Example (built-in character classes)

*Note: in Python's implementation of regex,

• \d refers to digits.
• \w refers to alphanumeric characters ([A-Z][a-z][0-9]_).
• \s refers to whitespace.
• \b is a word boundary.

• What does \d{3} \d{3}-\d{4} match?
• What does \bcat\b match? Does it find a match in 'my cat is hungry'? What about 'concatenate'?

Exercise

Write a regular expression that matches any string that:

• is between 5 and 10 characters long, and
• is made up of only vowels (either uppercase or lowercase, including 'Y' and 'y'), periods, and spaces.

Examples include 'yoo.ee.IOU' and 'AI.I oey'.

✅ Click here to see the answer after you've tried it yourself at regex101.com.

One answer: ^[aeiouyAEIOUY. ]{5,10}$
Key idea: Within a character class (i.e. [...]), special characters do not generally need to be escaped.

Regex in Python

re in Python

The re package is built into Python. It allows us to use regular expressions to find, extract, and replace strings.

import re

re.search takes in a string regex and a string text and returns the location and substring corresponding to the first match of regex in text.

re.search('AB*A', 
          'here is a string for you: ABBBA. here is another: ABBBBBBBA')

<re.Match object; span=(26, 31), match='ABBBA'>

re.findall takes in a string regex and a string text and returns a list of all matches of regex in text. You'll use this most often.

re.findall('AB*A', 
           'here is a string for you: ABBBA. here is another: ABBBBBBBA')

['ABBBA', 'ABBBBBBBA']

re.sub takes in a string regex, a string repl, and a string text, and replaces all matches of regex in text with repl.

re.sub('AB*A', 
       'billy', 
       'here is a string for you: ABBBA. here is another: ABBBBBBBA')

'here is a string for you: billy. here is another: billy'

Raw strings

When using regular expressions in Python, it's a good idea to use raw strings, denoted by an r before the quotes, e.g. r'exp'.

re.findall('\bcat\b', 'my cat is hungry')

[]

re.findall(r'\bcat\b', 'my cat is hungry')

['cat']

# Huh?
print('\bcat\b')

cat

Capture groups

• Surround a regex with ( and ) to define a capture group within a pattern.
• Capture groups are useful for extracting relevant parts of a string.

re.findall(r'\w+@(\w+)\.edu', 
           'my old email was billy@notucsd.edu, my new email is notbilly@ucsd.edu')

['notucsd', 'ucsd']

• Notice what happens if we remove the ( and )!

re.findall(r'\w+@\w+\.edu', 
           'my old email was billy@notucsd.edu, my new email is notbilly@ucsd.edu')

['billy@notucsd.edu', 'notbilly@ucsd.edu']

• Earlier, we also saw that parentheses can be used to group parts of a regex together. When using re.findall, all groups are treated as capturing groups.

# A regex that matches strings with two of the same vowel followed by 3 digits
# We only want to capture the digits, but...
re.findall(r'(aa|ee|ii|oo|uu)(\d{3})', 'eeoo124')

[('oo', '124')]

Example: Log parsing

Web servers typically record every request made of them in the "logs".

s = '''132.249.20.188 - - [24/Feb/2023:12:26:15 -0800] "GET /my/home/ HTTP/1.1" 200 2585'''

Let's use our new regex syntax (including capturing groups) to extract the day, month, year, and time from the log string s.

exp = '\[(.+)\/(.+)\/(.+):(.+):(.+):(.+) .+\]'
re.findall(exp, s)

[('24', 'Feb', '2023', '12', '26', '15')]

While above regex works, it is not very specific. It works on incorrectly formatted log strings.

other_s = '[adr/jduy/wffsdffs:r4s4:4wsgdfd:asdf 7]'
re.findall(exp, other_s)

[('adr', 'jduy', 'wffsdffs', 'r4s4', '4wsgdfd', 'asdf')]

The more specific, the better!

• Be as specific in your pattern matching as possible – you don't want to match and extract strings that don't fit the pattern you care about.
  • .* matches every possible string, but we don't use it very often.

• A better date extraction regex:

  \[(\d{2})\/([A-Z]{1}[a-z]{2})\/(\d{4}):(\d{2}):(\d{2}):(\d{2}) -\d{4}\]

  • \d{2} matches any 2-digit number.
  • [A-Z]{1} matches any single occurrence of any uppercase letter.
  • [a-z]{2} matches any 2 consecutive occurrences of lowercase letters.
  • Remember, special characters ([, ], /) need to be escaped with \.

s

'132.249.20.188 - - [24/Feb/2023:12:26:15 -0800] "GET /my/home/ HTTP/1.1" 200 2585'

new_exp = '\[(\d{2})\/([A-Z]{1}[a-z]{2})\/(\d{4}):(\d{2}):(\d{2}):(\d{2}) -\d{4}\]'
re.findall(new_exp, s)

[('24', 'Feb', '2023', '12', '26', '15')]

A benefit of new_exp over exp is that it doesn't capture anything when the string doesn't follow the format we specified.

other_s

'[adr/jduy/wffsdffs:r4s4:4wsgdfd:asdf 7]'

re.findall(new_exp, other_s)

[]

Limitations

Limitations of regexes

Writing a regular expression is like writing a program.

• You need to know the syntax well.
• They can be easier to write than to read.
• They can be difficult to debug.

Regular expressions are terrible at certain types of problems. Examples:

• Anything involving counting (same number of instances of a and b).
• Anything involving complex structure (palindromes).
• Parsing highly complex text structure (HTML, for instance).

Below is a regular expression that validates email addresses in Perl. See this article for more details.

StackOverflow crashed due to regex! See this article for the details.

Text features

Review: Regression and features

• In DSC 40A, our running example was to use regression to predict a data scientist's salary, given their GPA, years of experience, and years of education.

• After minimizing empirical risk to determine optimal parameters, $w_0^*, \dots, w_3^*$, we made predictions using:

$$\text{predicted salary} = w_0^* + w_1^* \cdot \text{GPA} + w_2^* \cdot \text{experience} + w_3^* \cdot \text{education}$$

• GPA, years of experience, and years of education are features – they represent a data scientist as a vector of numbers.
  • e.g. Your feature vector may be [3.5, 1, 7].

• This approach requires features to be numerical.

Moving forward

Suppose we'd like to predict the sentiment of a piece of text from 1 to 10.

• 10: Very positive (happy).
• 1: Very negative (sad, angry).

Example:

• Input: "DSC 80 is a pretty good class."
• Output: 7.

• We can frame this as a regression problem, but we can't directly use what we learned in 40A, because here our inputs are text, not numbers.

Text features

• Big question: How do we represent a text document as a feature vector of numbers?

• If we can do this, we can:
  • use a text document as input in a regression or classification model (in a few lectures).
  • quantify the similarity of two text documents (today).

Example: San Diego employee salaries

• Transparent California publishes the salaries of all City of San Diego employees.
• The latest available data is from 2021.

salaries = pd.read_csv('https://transcal.s3.amazonaws.com/public/export/san-diego-2021.csv')
util.anonymize_names(salaries)

salaries.head()

	Employee Name	Job Title	Base Pay	Overtime Pay	Other Pay	Benefits	Total Pay	Pension Debt	Total Pay & Benefits	Year	Notes	Agency	Status
0	Mara Xxxx	City Attorney	218759.0	0.0	-2560.00	108652.0	216199.0	427749.18	752600.18	2021	NaN	San Diego	FT
1	Todd Xxxx	Mayor	218759.0	0.0	-81.00	95549.0	218678.0	427749.18	741976.18	2021	NaN	San Diego	FT
2	Elizabeth Xxxx	Investment Officer	259732.0	0.0	-870.00	71438.0	258862.0	221041.09	551341.09	2021	NaN	San Diego	FT
3	Terence Xxxx	Police Officer	212837.0	0.0	39683.00	56569.0	252520.0	222375.06	531464.06	2021	NaN	San Diego	FT
4	Andrea Xxxx	Independent Budget Analyst	224312.0	0.0	59819.00	54213.0	284131.0	192126.79	530470.79	2021	NaN	San Diego	FT

Aside on privacy and ethics

• Even though the data we downloaded is publicly available, employee names still correspond to real people.

• Be careful when dealing with PII (personably identifiable information).
  • Only work with the data that is needed for your analysis.
  • Even when data is public, people have a reasonable right to privacy.

• Remember to think about the impacts of your work outside of your Jupyter Notebook.

Goal: Quantifying similarity

• Our goal is to describe, numerically, how similar two job titles are.

• For instance, our similarity metric should tell us that 'Deputy Fire Chief' and 'Fire Battalion Chief' are more similar than 'Deputy Fire Chief' and 'City Attorney'.

• Idea: Two job titles are similar if they contain shared words, regardless of order. So, to measure the similarity between two job titles, let's count the number of words they share in common.

• Before we do this, we need to be confident that the job titles are clean and consistent – let's explore.

Exploring job titles

jobtitles = salaries['Job Title']
jobtitles.head()

0                 City Attorney
1                         Mayor
2            Investment Officer
3                Police Officer
4    Independent Budget Analyst
Name: Job Title, dtype: object

How many employees are in the dataset? How many unique job titles are there?

jobtitles.shape[0], jobtitles.nunique()

(12305, 588)

What are the most common job titles?

jobtitles.value_counts().iloc[:100]

Police Officer                   2123
Fire Fighter Ii                   331
Assistant Engineer - Civil        284
Grounds Maintenance Worker Ii     250
Fire Captain                      248
                                 ... 
Grounds Maintenance Manager        27
Electrician                        27
Executive Assistant                26
Paralegal                          26
Librarian Iv                       25
Name: Job Title, Length: 100, dtype: int64

jobtitles.value_counts().iloc[:25].sort_values().plot(kind='barh')

Are there any missing job titles?

jobtitles.isna().sum()

2

There aren't many. To avoid having to deal with missing values later on, let's just drop the two missing job titles now.

jobtitles = jobtitles[jobtitles.notna()]

Canonicalization

Remember, our goal is ultimately to count the number of shared words between job titles. But before we start counting the number of shared words, we need to consider the following:

• Some job titles may have punctuation, like '-' and '&', which may count as words when they shouldn't.
  • 'Assistant - Manager' and 'Assistant Manager' should count as the same job title.

• Some job titles may have "glue" words, like 'to' and 'the', which (we can argue) also shouldn't count as words.
  • 'Assistant To The Manager' and 'Assistant Manager' should count as the same job title.

Let's address the above issues. The process of converting job titles so that they are always represented the same way is called canonicalization.

Punctuation

Are there job titles with unnecessary punctuation that we can remove?

• To find out, we can write a regular expression that looks for characters other than letters, numbers, and spaces.

• We can use regular expressions with the .str methods we learned earlier in the quarter just by using regex=True.

# Uses character class negation
jobtitles.str.contains(r'[^A-Za-z0-9 ]', regex=True).sum()

845

jobtitles[jobtitles.str.contains(r'[^A-Za-z0-9 ]', regex=True)].head()

281          Park & Recreation Director
539     Associate Engineer - Mechanical
1023         Associate Engineer - Civil
1376       Associate Engineer - Traffic
1460       Budget/Legislative Analyst I
Name: Job Title, dtype: object

It seems like we should replace these pieces of punctuation with a single space.

"Glue" words

Are there job titles with "glue" words in the middle, such as 'Assistant to the Manager'?

To figure out if any titles contain the word 'to', we can't just do the following, because it will evaluate to True for job titles that have 'to' anywhere in them, even if not as a standalone word.

# Why are we converting to lowercase?
jobtitles.str.lower().str.contains('to').sum()

1541

jobtitles[jobtitles.str.lower().str.contains('to')]

0                             City Attorney
10       Assistant Retirement Administrator
25                      Department Director
26                  Assistant City Attorney
27             Fire Prevention Inspector Ii
                        ...                
12162                       Test Monitor Ii
12185              Word Processing Operator
12190                       Deputy Director
12210            City Attorney Investigator
12267                       Test Monitor Ii
Name: Job Title, Length: 1541, dtype: object

Instead, we need to look for 'to' separated by word boundaries.

jobtitles.str.lower().str.contains(r'\bto\b', regex=True).sum()

11

jobtitles[jobtitles.str.lower().str.contains(r'\bto\b', regex=True)]

664                            Assistant To The Fire Chief
1403                  Principal Assistant To City Attorney
2358                             Assistant To The Director
4336                Confidential Secretary To Police Chief
4459                             Assistant To The Director
5196     Confidential Secretary To Chief Operating Officer
5563               Confidential Secretary To City Attorney
5685                             Assistant To The Director
7544                       Confidential Secretary To Mayor
9627                  Principal Assistant To City Attorney
12061                            Assistant To The Director
Name: Job Title, dtype: object

We can look for other filler words too, like 'the' and 'for'.

jobtitles[jobtitles.str.lower().str.contains(r'\bthe\b', regex=True)]

664      Assistant To The Fire Chief
2358       Assistant To The Director
4459       Assistant To The Director
5685       Assistant To The Director
12061      Assistant To The Director
Name: Job Title, dtype: object

jobtitles[jobtitles.str.lower().str.contains(r'\bfor\b', regex=True)]

3676     Assistant For Community Outreach
4451     Assistant For Community Outreach
11010    Assistant For Community Outreach
Name: Job Title, dtype: object

We should probably remove these "glue" words.

Fixing punctuation and removing "glue" words

Let's put the following two steps together, and canonicalize job titles by:

• converting to lowercase,
• removing each occurrence of 'to', 'the', and 'for',
• replacing each non-letter/digit/space character with a space, and
• replacing each sequence of multiple spaces with a single space.

jobtitles = (
    jobtitles
    .str.lower()
    .str.replace(r'\bto\b|\bthe\b|\bfor\b', '', regex=True)
    .str.replace('[^A-Za-z0-9 ]', ' ', regex=True)
    .str.replace(' +', ' ', regex=True)               # ' +' matches 1 or more occurrences of a space.
    .str.strip()                                      # Removes leading/trailing spaces if present.
)

jobtitles.sample(10)

7755                     paralegal
3775                police officer
11323        clerical assistant ii
9372                  greenskeeper
7221          pesticide applicator
10655    assistant center director
7010      recycling specialist iii
11046                  lifeguard i
8452         library assistant iii
10363         library assistant ii
Name: Job Title, dtype: object

Possible issue: inconsistent representations

Another possible issue is that some job titles may have inconsistent representations of the same word (e.g. 'Asst.' vs 'Assistant').

jobtitles[jobtitles.str.contains('asst')].value_counts()

Series([], Name: Job Title, dtype: int64)

jobtitles[jobtitles.str.contains('assistant')].value_counts().head()

assistant engineer civil    284
library assistant i         127
library assistant ii        116
library assistant iii       107
clerical assistant ii       100
Name: Job Title, dtype: int64

The 2020 salaries dataset had several of these issues, but fortunately they appear to be fixed for us in the 2021 dataset (thanks, Transparent California).

Bag of words 💰

Text similarity

Recall, our idea is to measure the similarity of two job titles by counting the number of shared words between the job titles. How do we actually do that, for all of the job titles we have?

A counts matrix

Let's create a "counts" matrix, such that:

• there is 1 row per job title,
• there is 1 column per unique word that is used in job titles, and
• the value in row title and column word is the number of occurrences of word in title.

Such a matrix might look like:

	senior	lecturer	teaching	professor	assistant	associate
senior lecturer	1	1	0	0	0	0
assistant teaching professor	0	0	1	1	1	0
associate professor	0	0	0	1	0	1
senior assistant to the assistant professor	1	0	0	1	2	0

Creating a counts matrix

First, we need to determine all words that are used across all job titles.

jobtitles.str.split()

0                      [city, attorney]
1                               [mayor]
2                 [investment, officer]
3                     [police, officer]
4        [independent, budget, analyst]
                      ...              
12300           [recreation, leader, i]
12301               [fire, fighter, ii]
12302                   [fire, captain]
12303       [fleet, repair, supervisor]
12304                  [fire, engineer]
Name: Job Title, Length: 12303, dtype: object

all_words = jobtitles.str.split().sum()
all_words[:10]

['city',
 'attorney',
 'mayor',
 'investment',
 'officer',
 'police',
 'officer',
 'independent',
 'budget',
 'analyst']

Next, to determine the columns of our matrix, we need to find a list of all unique words used in titles. We can do this with np.unique, but value_counts shows us the distribution, which is interesting.

unique_words = pd.Series(all_words).value_counts()
unique_words.head(10)

officer       2343
ii            2305
police        2294
i             1449
assistant     1193
fire          1158
engineer      1032
civil          667
iii            625
technician     616
dtype: int64

len(unique_words)

327

Note that in unique_words.index, job titles are sorted by number of occurrences!

For each of the 327 unique words that are used in job titles, we can count the number of occurrences of the word in each job title.

• 'deputy fire chief' contains the word 'deputy' once, the word 'fire' once, and the word 'chief' once.
• 'assistant managers assistant' contains the word 'assistant' twice and the word 'managers' once.

# Created using a dictionary to avoid a "DataFrame is highly fragmented" warning.
counts_dict = {}
for word in unique_words.index:
    re_pat = fr'\b{word}\b'
    counts_dict[word] = jobtitles.str.count(re_pat).astype(int).tolist()
    
counts_df = pd.DataFrame(counts_dict)

counts_df.head()

	officer	ii	police	i	assistant	fire	engineer	civil	iii	technician	...	estate	stores	assets	treasurer	risk	security	geologist	utilities	gardener	principle
0	0	0	0	0	0	0	0	0	0	0	...	0	0	0	0	0	0	0	0	0	0
1	0	0	0	0	0	0	0	0	0	0	...	0	0	0	0	0	0	0	0	0	0
2	1	0	0	0	0	0	0	0	0	0	...	0	0	0	0	0	0	0	0	0	0
3	1	0	1	0	0	0	0	0	0	0	...	0	0	0	0	0	0	0	0	0	0
4	0	0	0	0	0	0	0	0	0	0	...	0	0	0	0	0	0	0	0	0	0

5 rows × 327 columns

counts_df has one row for all 12303 employees, and one column for each unique word that is used in a job title.

counts_df.shape

(12303, 327)

To put into context what the numbers in counts_df mean, we can show the actual job title for each row.

counts_df = counts_df.set_index(jobtitles)
counts_df

	officer	ii	police	i	assistant	fire	engineer	civil	iii	technician	...	estate	stores	assets	treasurer	risk	security	geologist	utilities	gardener	principle
Job Title
city attorney	0	0	0	0	0	0	0	0	0	0	...	0	0	0	0	0	0	0	0	0	0
mayor	0	0	0	0	0	0	0	0	0	0	...	0	0	0	0	0	0	0	0	0	0
investment officer	1	0	0	0	0	0	0	0	0	0	...	0	0	0	0	0	0	0	0	0	0
police officer	1	0	1	0	0	0	0	0	0	0	...	0	0	0	0	0	0	0	0	0	0
independent budget analyst	0	0	0	0	0	0	0	0	0	0	...	0	0	0	0	0	0	0	0	0	0
...	...	...	...	...	...	...	...	...	...	...	...	...	...	...	...	...	...	...	...	...	...
recreation leader i	0	0	0	1	0	0	0	0	0	0	...	0	0	0	0	0	0	0	0	0	0
fire fighter ii	0	1	0	0	0	1	0	0	0	0	...	0	0	0	0	0	0	0	0	0	0
fire captain	0	0	0	0	0	1	0	0	0	0	...	0	0	0	0	0	0	0	0	0	0
fleet repair supervisor	0	0	0	0	0	0	0	0	0	0	...	0	0	0	0	0	0	0	0	0	0
fire engineer	0	0	0	0	0	1	1	0	0	0	...	0	0	0	0	0	0	0	0	0	0

12303 rows × 327 columns

The fourth row tells us that the fourth job title contains 'police' once and 'officer' once.

Interpreting the counts matrix

counts_df.head()

	officer	ii	police	i	assistant	fire	engineer	civil	iii	technician	...	estate	stores	assets	treasurer	risk	security	geologist	utilities	gardener	principle
Job Title
city attorney	0	0	0	0	0	0	0	0	0	0	...	0	0	0	0	0	0	0	0	0	0
mayor	0	0	0	0	0	0	0	0	0	0	...	0	0	0	0	0	0	0	0	0	0
investment officer	1	0	0	0	0	0	0	0	0	0	...	0	0	0	0	0	0	0	0	0	0
police officer	1	0	1	0	0	0	0	0	0	0	...	0	0	0	0	0	0	0	0	0	0
independent budget analyst	0	0	0	0	0	0	0	0	0	0	...	0	0	0	0	0	0	0	0	0	0

5 rows × 327 columns

The Series below describes the 20 most common words used in job titles, along with the number of times they appeared in all job titles (including repeats). We will call these words "top 20" words.

# Remember, the columns of counts_df are ordered by number of occurrences.
counts_df.iloc[:, :20].sum()

officer       2343
ii            2305
police        2294
i             1449
assistant     1193
fire          1158
engineer      1032
civil          667
iii            625
technician     616
senior         567
associate      558
analyst        527
worker         496
fighter        470
management     409
manager        393
operator       380
recreation     371
supervisor     362
dtype: int64

The Series below describes the number of top 20 words used in each job title.

counts_df.iloc[:, :20].sum(axis=1)

Job Title
city attorney                 0
mayor                         0
investment officer            1
police officer                2
independent budget analyst    1
                             ..
recreation leader i           2
fire fighter ii               3
fire captain                  1
fleet repair supervisor       1
fire engineer                 2
Length: 12303, dtype: int64

Question: What job titles are most similar to 'deputy fire chief'?

• Remember, our idea was to count the number of shared words between two job titles.

• We now have access to counts_df, which contains a row vector for each job title.

• How can we use it to count the number of shared words between two job titles, i.e. the similarity of two job titles?

To start, let's compare the row vectors for 'deputy fire chief' and 'fire battalion chief'.

dfc = counts_df.loc['deputy fire chief'].iloc[0]
dfc

officer      0
ii           0
police       0
i            0
assistant    0
            ..
security     0
geologist    0
utilities    0
gardener     0
principle    0
Name: deputy fire chief, Length: 327, dtype: int64

fbc = counts_df.loc['fire battalion chief'].iloc[0]
fbc

officer      0
ii           0
police       0
i            0
assistant    0
            ..
security     0
geologist    0
utilities    0
gardener     0
principle    0
Name: fire battalion chief, Length: 327, dtype: int64

We can stack these two vectors horizontally.

pair_counts = (
    pd.concat([dfc, fbc], axis=1)
    .sort_values(by=['deputy fire chief', 'fire battalion chief'], ascending=False)
    .head(10)
    .T
)

pair_counts

	fire	chief	deputy	battalion	officer	ii	police	i	assistant	engineer
deputy fire chief	1	1	1	0	0	0	0	0	0	0
fire battalion chief	1	1	0	1	0	0	0	0	0	0

One way to measure how similar the above two vectors are is through their dot product.

np.sum(pair_counts.iloc[0] * pair_counts.iloc[1])

2

Here, since both vectors consist only of 1s and 0s, the dot product is equal to the number of shared words between the two job titles.

Aside: Dot product

• Recall, if $\vec{a} = \begin{bmatrix} a_1 & a_2 & ... & a_n \end{bmatrix}^T$ and $\vec{b} = \begin{bmatrix} b_1 & b_2 & ... & b_n \end{bmatrix}^T$ are two vectors, then their dot product $\vec{a} \cdot \vec{b}$ is defined as:

$$\vec{a} \cdot \vec{b} = a_1b_1 + a_2b_2 + ... + a_nb_n$$

• The dot product also has a geometric interpretation. If $|\vec{a}|$ and $|\vec{b}|$ are the $L_2$ norms (lengths) of $\vec{a}$ and $\vec{b}$, and $\theta$ is the angle between $\vec{a}$ and $\vec{b}$, then:

$$\vec{a} \cdot \vec{b} = |\vec{a}| |\vec{b}| \cos \theta$$(source)

• $\cos \theta$ is equal to its maximum value (1) when $\theta = 0$, i.e. when $\vec{a}$ and $\vec{b}$ point in the same direction.

• 🚨 Key idea: The more similar two unit vectors are, the larger their dot product is!

Computing similarities

To find the job title that is most similar to 'deputy fire chief', we can compute the dot product of the 'deputy fire chief' word vector with all other titles' word vectors, and find the title with the highest dot product.

counts_df.head()

	officer	ii	police	i	assistant	fire	engineer	civil	iii	technician	...	estate	stores	assets	treasurer	risk	security	geologist	utilities	gardener	principle
Job Title
city attorney	0	0	0	0	0	0	0	0	0	0	...	0	0	0	0	0	0	0	0	0	0
mayor	0	0	0	0	0	0	0	0	0	0	...	0	0	0	0	0	0	0	0	0	0
investment officer	1	0	0	0	0	0	0	0	0	0	...	0	0	0	0	0	0	0	0	0	0
police officer	1	0	1	0	0	0	0	0	0	0	...	0	0	0	0	0	0	0	0	0	0
independent budget analyst	0	0	0	0	0	0	0	0	0	0	...	0	0	0	0	0	0	0	0	0	0

5 rows × 327 columns

dfc

officer      0
ii           0
police       0
i            0
assistant    0
            ..
security     0
geologist    0
utilities    0
gardener     0
principle    0
Name: deputy fire chief, Length: 327, dtype: int64

To do so, we can apply np.dot to each row that doesn't correspond to 'deputy fire chief'.

dots = (
    counts_df[counts_df.index != 'deputy fire chief']
    .apply(lambda s: np.dot(s, dfc), axis=1)
    .sort_values(ascending=False)
)

dots

Job Title
fire battalion chief                           2
fire battalion chief                           2
assistant fire chief                           2
fire battalion chief                           2
fire battalion chief                           2
                                              ..
finance analyst iii                            0
associate engineer traffic                     0
supervising procurement contracting officer    0
sanitation driver ii                           0
city attorney                                  0
Length: 12292, dtype: int64

The unique job titles that are most similar to 'deputy fire chief' are given below.

np.unique(dots.index[dots == dots.max()])

array(['assistant deputy chief operating officer', 'assistant fire chief',
       'deputy chief operating officer', 'fire battalion chief',
       'fire chief'], dtype=object)

Note that they all share two words in common with 'deputy fire chief'.

Note: To truly use the dot product as a measure of similarity, we should normalize by the lengths of the word vectors. More on this next time.

Bag of words

• The bag of words model represents texts (e.g. job titles, sentences, documents) as vectors of word counts.
  • The "counts" matrices we have worked with so far were created using the bag of words model.
  • The bag of words model defines a vector space in $\mathbb{R}^{\text{number of unique words}}$.
• It is called "bag of words" because it doesn't consider order.

(source)

Aside: Interactive bag of words demo

Check this site out – it automatically generates a bag of words matrix for you!

(source)

Summary, next time

Summary

• pandas .str methods can use regular expressions; just set regex=True.
• One way to turn texts, like 'deputy fire chief', into feature vectors, is to count the number of occurrences of each word in the text, ignoring order. This is done using the bag of words model.
  • It allows you to measure the "similarity" of two words by taking the dot product of their word vectors.

Next time

• More on the bag of words model and its pitfalls.
• An improvement to the bag of words model.