from dsc80_utils import *

Lecture 13 – Linear Regression

DSC 80, Fall 2024

Announcements 📣

• Midterm Survey due tonight: https://forms.gle/8pMeYeHk6ktLa4867
  • If ≥80% of the class fills it out, everyone will get +1% EC on the midterm.
• Lab 6 is due tomorrow.
• The Final Project will be released next week.

Agenda 📆

• Last bits of TF-IDF.
• Modeling.
• Case study: Restaurant tips 🧑‍🍳.
• Regression in sklearn.
• Announcement about HDSI career services.

Conceptually, today will mostly be review from DSC 40A, but we'll introduce a few new practical tools that we'll build upon next week.

State of the Union addresses

from pathlib import Path
sotu_txt = Path('data') / 'stateoftheunion1790-2023.txt'
sotu = sotu_txt.read_text()
speeches = sotu.split('\n***\n')[1:]

import re
def extract_struct(speech):
    L = speech.strip().split('\n', maxsplit=3)
    L[3] = re.sub(r"[^A-Za-z' ]", ' ', L[3]).lower()
    return dict(zip(['speech', 'president', 'date', 'contents'], L))

speeches_df = pd.DataFrame(list(map(extract_struct, speeches)))
speeches_df

	speech	president	date	contents
0	State of the Union Address	George Washington	January 8, 1790	fellow citizens of the senate and house of re...
1	State of the Union Address	George Washington	December 8, 1790	fellow citizens of the senate and house of re...
2	State of the Union Address	George Washington	October 25, 1791	fellow citizens of the senate and house of re...
...	...	...	...	...
230	State of the Union Address	Joseph R. Biden Jr.	April 28, 2021	thank you thank you thank you good to be b...
231	State of the Union Address	Joseph R. Biden Jr.	March 1, 2022	madam speaker madam vice president and our ...
232	State of the Union Address	Joseph R. Biden Jr.	February 7, 2023	mr speaker madam vice president our firs...

233 rows × 4 columns

Finding the most important words in each speech

Here, a "document" is a speech. We have 233 documents.

speeches_df

	speech	president	date	contents
0	State of the Union Address	George Washington	January 8, 1790	fellow citizens of the senate and house of re...
1	State of the Union Address	George Washington	December 8, 1790	fellow citizens of the senate and house of re...
2	State of the Union Address	George Washington	October 25, 1791	fellow citizens of the senate and house of re...
...	...	...	...	...
230	State of the Union Address	Joseph R. Biden Jr.	April 28, 2021	thank you thank you thank you good to be b...
231	State of the Union Address	Joseph R. Biden Jr.	March 1, 2022	madam speaker madam vice president and our ...
232	State of the Union Address	Joseph R. Biden Jr.	February 7, 2023	mr speaker madam vice president our firs...

233 rows × 4 columns

A rough sketch of what we'll compute:

for each word t:
    for each speech d:
        compute tfidf(t, d)

unique_words = speeches_df['contents'].str.split().explode().value_counts()
# Take the top 500 most common words for speed
unique_words = unique_words.iloc[:500].index
unique_words

Index(['the', 'of', 'to', 'and', 'in', 'a', 'that', 'for', 'be', 'our',
       ...
       'desire', 'call', 'submitted', 'increasing', 'months', 'point', 'trust',
       'throughout', 'set', 'object'],
      dtype='object', name='contents', length=500)

💡 Pro-Tip: Using tqdm

This code takes a while to run, so we'll use the tdqm package to track its progress. (Install with mamba install tqdm if needed).

from tqdm.notebook import tqdm

tfidf_dict = {}
tf_denom = speeches_df['contents'].str.split().str.len()

# Wrap the sequence with `tqdm()` to display a progress bar
for word in tqdm(unique_words):
    re_pat = fr' {word} ' # Imperfect pattern for speed.
    tf = speeches_df['contents'].str.count(re_pat) / tf_denom
    idf = np.log(len(speeches_df) / speeches_df['contents'].str.contains(re_pat).sum())
    tfidf_dict[word] =  tf * idf

  0%|          | 0/500 [00:00<?, ?it/s]

tfidf = pd.DataFrame(tfidf_dict)
tfidf.head()

	the	of	to	and	...	trust	throughout	set	object
0	0.0	0.0	0.0	0.0	...	4.29e-04	0.00e+00	0.00e+00	2.04e-03
1	0.0	0.0	0.0	0.0	...	0.00e+00	0.00e+00	0.00e+00	1.06e-03
2	0.0	0.0	0.0	0.0	...	4.06e-04	0.00e+00	3.48e-04	6.44e-04
3	0.0	0.0	0.0	0.0	...	6.70e-04	2.17e-04	0.00e+00	7.09e-04
4	0.0	0.0	0.0	0.0	...	2.38e-04	4.62e-04	0.00e+00	3.77e-04

5 rows × 500 columns

Note that the TF-IDFs of many common words are all 0!

Summarizing speeches

By using idxmax, we can find the word with the highest TF-IDF in each speech.

summaries = tfidf.idxmax(axis=1)
summaries

0          object
1      convention
2       provision
          ...    
230          it's
231       tonight
232          it's
Length: 233, dtype: object

What if we want to see the 5 words with the highest TF-IDFs, for each speech?

def five_largest(row):
    return ', '.join(row.index[row.argsort()][-5:])

keywords = tfidf.apply(five_largest, axis=1)
keywords_df = pd.concat([
    speeches_df['president'],
    speeches_df['date'],
    keywords
], axis=1)

keywords_df

	president	date	0
0	George Washington	January 8, 1790	your, proper, regard, ought, object
1	George Washington	December 8, 1790	case, established, object, commerce, convention
2	George Washington	October 25, 1791	community, upon, lands, proper, provision
...	...	...	...
230	Joseph R. Biden Jr.	April 28, 2021	get, americans, percent, jobs, it's
231	Joseph R. Biden Jr.	March 1, 2022	let, jobs, americans, get, tonight
232	Joseph R. Biden Jr.	February 7, 2023	down, percent, jobs, tonight, it's

233 rows × 3 columns

Uncomment the cell below to see every single row of keywords_df.

# display_df(keywords_df, rows=233)

Aside: What if we remove the $\log$ from $\text{idf}(t)$?

Let's try it and see what happens.

tfidf_nl_dict = {}
tf_denom = speeches_df['contents'].str.split().str.len()

for word in tqdm(unique_words):
    re_pat = fr' {word} ' # Imperfect pattern for speed.
    tf = speeches_df['contents'].str.count(re_pat) / tf_denom
    idf_nl = len(speeches_df) / speeches_df['contents'].str.contains(re_pat).sum()
    tfidf_nl_dict[word] =  tf * idf_nl

  0%|          | 0/500 [00:00<?, ?it/s]

tfidf_nl = pd.DataFrame(tfidf_nl_dict)
tfidf_nl.head()

	the	of	to	and	...	trust	throughout	set	object
0	0.09	0.06	0.05	0.04	...	1.47e-03	0.00e+00	0.00e+00	5.78e-03
1	0.09	0.06	0.03	0.03	...	0.00e+00	0.00e+00	0.00e+00	2.99e-03
2	0.11	0.07	0.04	0.03	...	1.39e-03	0.00e+00	1.30e-03	1.82e-03
3	0.09	0.07	0.04	0.03	...	2.29e-03	7.53e-04	0.00e+00	2.01e-03
4	0.09	0.07	0.04	0.02	...	8.12e-04	1.60e-03	0.00e+00	1.07e-03

5 rows × 500 columns

keywords_nl = tfidf_nl.apply(five_largest, axis=1)
keywords_nl_df = pd.concat([
    speeches_df['president'],
    speeches_df['date'],
    keywords_nl
], axis=1)
keywords_nl_df

	president	date	0
0	George Washington	January 8, 1790	a, and, to, of, the
1	George Washington	December 8, 1790	in, and, to, of, the
2	George Washington	October 25, 1791	a, and, to, of, the
...	...	...	...
230	Joseph R. Biden Jr.	April 28, 2021	of, it's, and, to, the
231	Joseph R. Biden Jr.	March 1, 2022	we, of, to, and, the
232	Joseph R. Biden Jr.	February 7, 2023	a, of, and, to, the

233 rows × 3 columns

The role of $\log$ in $\text{idf}(t)$

$$ \begin{align*} \text{tfidf}(t, d) &= \text{tf}(t, d) \cdot \text{idf}(t) \\\ &= \frac{\text{\# of occurrences of $t$ in $d$}}{\text{total \# of words in $d$}} \cdot \log \left(\frac{\text{total \# of documents}}{\text{\# of documents in which $t$ appears}} \right) \end{align*} $$

• Remember, for any positive input $x$, $\log(x)$ is (much) smaller than $x$.
• In $\text{idf}(t)$, the $\log$ "dampens" the impact of the ratio $\frac{\text{\# documents}}{\text{\# documents with $t$}}$.

• If a word is very common, the ratio will be close to 1. The log of the ratio will be close to 0.

(1000 / 999)

1.001001001001001

np.log(1000 / 999)

np.float64(0.001000500333583622)

• If a word is very common (e.g. 'the'), removing the log multiplies the statistic by a large factor.
• If a word is very rare, the ratio will be very large. However, for instance, a word being seen in 2 out of 50 documents is not very different than being seen in 2 out of 500 documents (it is very rare in both cases), and so $\text{idf}(t)$ should be similar in both cases.

(50 / 2)

25.0

(500 / 2)

250.0

np.log(50 / 2)

np.float64(3.2188758248682006)

np.log(500 / 2)

np.float64(5.521460917862246)

Question 🤔 (Answer at dsc80.com/q)

Code: tfidf

From the Fa23 final: Consider the following corpus:

Document number Content
1               yesterday rainy today sunny
2               yesterday sunny today sunny
3               today rainy yesterday today
4               yesterday yesterday today today

Which words have a TF-IDF score of 0 for all four documents?

Modeling

Reflection

So far this quarter, we've learned how to:

• Extract information from tabular data using pandas and regular expressions.

• Clean data so that it best represents an underlying data generating process.

  • Missingness analyses and imputation.

• Collect data from the internet through scraping and APIs, and parse it using BeautifulSoup.

• Perform exploratory data analysis through aggregation, visualization, and the computation of summary statistics like TF-IDF.

• Infer about the relationships between samples and populations through hypothesis and permutation testing.

• Now, let's make predictions.

Modeling

• A model is a set of assumptions about how data were generated.

• George Box, a famous statistician, once said "All models are wrong, but some are useful." What did he mean?

Philosophy

"It has been said that "all models are wrong but some models are useful." In other words, any model is at best a useful fiction—there never was, or ever will be, an exactly normal distribution or an exact linear relationship. Nevertheless, enormous progress has been made by entertaining such fictions and using them as approximations."

"Since all models are wrong the scientist cannot obtain a "correct" one by excessive elaboration. On the contrary following William of Occam he should seek an economical description of natural phenomena. Just as the ability to devise simple but evocative models is the signature of the great scientist so overelaboration and overparameterization is often the mark of mediocrity."

Goals of modeling

1. To make accurate predictions regarding unseen data.

   • Given this dataset of past UCSD data science students' salaries, can we predict your future salary? (regression)
   • Given this dataset of images, can we predict if this new image is of a dog, cat, or zebra? (classification)

2. To make inferences about complex phenomena in nature.

   • Is there a linear relationship between the heights of children and the heights of their biological mothers?
   • The weights of smoking and non-smoking mothers' babies babies in my sample are different – how confident am I that this difference exists in the population?

• Of the two focuses of models, we will focus on prediction.

• In the above taxonomy, we will focus on supervised learning.

• We'll start with regression before moving to classification.

Features

• A feature is a measurable property of a phenomenon being observed.

  • Other terms for "feature" include "(explanatory) variable" and "attribute".
  • Typically, features are the inputs to models.

• In DataFrames, features typically correspond to columns, while rows typically correspond to different individuals.

• Some features come as part of a dataset, e.g. weight and height, but others we need to create given existing features, for example: $$\text{BMI} = \frac{\text{weight (kg)}}{\text{[height (m)]}^2}$$

• Example: TF-IDF creates features that summarize documents!

Example: Restaurant tips 🧑‍🍳

About the data

What features does the dataset contain? Is this likely a recent dataset, or an older one?

# The dataset is built into plotly!
tips = px.data.tips()
tips

	total_bill	tip	sex	smoker	day	time	size
0	16.99	1.01	Female	No	Sun	Dinner	2
1	10.34	1.66	Male	No	Sun	Dinner	3
2	21.01	3.50	Male	No	Sun	Dinner	3
...	...	...	...	...	...	...	...
241	22.67	2.00	Male	Yes	Sat	Dinner	2
242	17.82	1.75	Male	No	Sat	Dinner	2
243	18.78	3.00	Female	No	Thur	Dinner	2

244 rows × 7 columns

Predicting tips

• Goal: Given various information about a table at a restaurant, we want to predict the tip that a server will earn.

• Why might a server be interested in doing this?
  • To determine which tables are likely to tip the most (inference).
  • To predict earnings over the next month (prediction).

Exploratory data analysis

• The most natural feature to look at first is total bills.

• As such, we should explore the relationship between total bills and tips. Moving forward:
  • $x$: Total bills.
  • $y$: Tips.

fig = tips.plot(kind='scatter', x='total_bill', y='tip', title='Tip vs. Total Bill')
fig.update_layout(xaxis_title='Total Bill', yaxis_title='Tip')