from dsc80_utils import *
import lec15_util as util

Lecture 15 – Pipelines, Multicollinearity, and Generalization

DSC 80, Spring 2025

Agenda 📆

• Pipelines.
• Multicollinearity.
• Generalization.
  • Bias and variance.
  • Train-test splits.

Pipelines

So far, we've used transformers for feature engineering and models for prediction. We can combine these steps into a single Pipeline.

Pipelines in sklearn

• Pipeline allows you to sequentially apply a list of transformers to preprocess the data and, if desired, conclude the sequence with a final predictor for predictive modeling.

• General template: pl = Pipeline([trans_1, trans_2, ..., model])
  • Note that the model is optional.

• Once a Pipeline is instantiated, you can fit all steps (transformers and model) using pl.fit(X, y).

• To make predictions using raw, untransformed data, use pl.predict(X).

• The actual list we provide Pipeline with must be a list of tuples, where
  • The first element is a "name" (that we choose) for the step.
  • The second element is a transformer or estimator instance.

Our first Pipeline

Let's build a Pipeline that:

• One hot encodes the categorical features in tips.
• Fits a regression model on the one hot encoded data.

tips = px.data.tips()

tips_cat = tips[['sex', 'smoker', 'day', 'time']]
tips_cat.head()

	sex	smoker	day	time
0	Female	No	Sun	Dinner
1	Male	No	Sun	Dinner
2	Male	No	Sun	Dinner
3	Male	No	Sun	Dinner
4	Female	No	Sun	Dinner

from sklearn.pipeline import Pipeline

from sklearn.preprocessing import OneHotEncoder
from sklearn.linear_model import LinearRegression

pl = Pipeline([
    ('one-hot', OneHotEncoder()),
    ('lin-reg', LinearRegression())
])

Now that pl is instantiated, we fit it the same way we would fit the individual steps.

pl.fit(tips_cat, tips['tip'])

Pipeline(steps=[('one-hot', OneHotEncoder()), ('lin-reg', LinearRegression())])

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Now, to make predictions using raw data, all we need to do is use pl.predict:

pl.predict(tips_cat.iloc[:5])

array([3.1 , 3.27, 3.27, 3.27, 3.1 ])

pl performs both feature transformation and prediction with just a single call to predict!

We can access individual "steps" of a Pipeline through the named_steps attribute:

pl.named_steps

{'one-hot': OneHotEncoder(), 'lin-reg': LinearRegression()}

pl.named_steps['one-hot'].transform(tips_cat).toarray()

array([[1., 0., 1., ..., 0., 1., 0.],
       [0., 1., 1., ..., 0., 1., 0.],
       [0., 1., 1., ..., 0., 1., 0.],
       ...,
       [0., 1., 0., ..., 0., 1., 0.],
       [0., 1., 1., ..., 0., 1., 0.],
       [1., 0., 1., ..., 1., 1., 0.]])

pl.named_steps['one-hot'].get_feature_names_out()

array(['sex_Female', 'sex_Male', 'smoker_No', 'smoker_Yes', 'day_Fri',
       'day_Sat', 'day_Sun', 'day_Thur', 'time_Dinner', 'time_Lunch'],
      dtype=object)

pl.named_steps['lin-reg'].coef_

array([-0.09,  0.09, -0.04,  0.04, -0.2 , -0.13,  0.14,  0.19,  0.25,
       -0.25])

pl also has a score method, the same way a fit LinearRegression instance does:

# Why is this so low?
pl.score(tips_cat, tips['tip'])

0.02749679020147555

More sophisticated Pipelines

• In the previous example, we one hot encoded every input column. What if we want to perform different transformations on different columns?

• Solution: Use a ColumnTransformer.
  • Instantiate a ColumnTransformer using a list of tuples, where:
    • The first element is a "name" we choose for the transformer.
    • The second element is a transformer instance (e.g. OneHotEncoder()).
    • The third element is a list of relevant column names.

Planning our first ColumnTransformer

from sklearn.compose import ColumnTransformer

Let's perform different transformations on the quantitative and categorical features of tips (note that we are not transforming 'tip').

tips_features = tips.drop('tip', axis=1)
tips_features.head()

	total_bill	sex	smoker	day	time	size
0	16.99	Female	No	Sun	Dinner	2
1	10.34	Male	No	Sun	Dinner	3
2	21.01	Male	No	Sun	Dinner	3
3	23.68	Male	No	Sun	Dinner	2
4	24.59	Female	No	Sun	Dinner	4

• We will leave the 'total_bill' column untouched.

• To the 'size' column, we will apply the Binarizer transformer with a threshold of 2 (big tables vs. small tables).

• To the categorical columns, we will apply the OneHotEncoder transformer.

• In essence, we will create a transformer that reproduces the following DataFrame:

	size	x0_Female	x0_Male	x1_No	x1_Yes	x2_Fri	x2_Sat	x2_Sun	x2_Thur	x3_Dinner	x3_Lunch	total_bill
0	0	1.0	0.0	1.0	0.0	0.0	0.0	1.0	0.0	1.0	0.0	16.99
1	1	0.0	1.0	1.0	0.0	0.0	0.0	1.0	0.0	1.0	0.0	10.34
2	1	0.0	1.0	1.0	0.0	0.0	0.0	1.0	0.0	1.0	0.0	21.01
3	0	0.0	1.0	1.0	0.0	0.0	0.0	1.0	0.0	1.0	0.0	23.68
4	1	1.0	0.0	1.0	0.0	0.0	0.0	1.0	0.0	1.0	0.0	24.59

Building a Pipeline using a ColumnTransformer

Let's start by creating our ColumnTransformer.

from sklearn.preprocessing import Binarizer

preproc = ColumnTransformer(
    transformers=[
        ('size', Binarizer(threshold=2), ['size']),
        ('categorical_cols', OneHotEncoder(), ['sex', 'smoker', 'day', 'time'])
    ],
    # Specify what to do with all other columns ('total_bill' here) – drop or passthrough.
    remainder='passthrough',
    # Keep original dtypes for remaining columns
    force_int_remainder_cols=False,
)

Now, let's create a Pipeline using preproc as a transformer, and fit it:

pl = Pipeline([
    ('preprocessor', preproc), 
    ('lin-reg', LinearRegression())
])

pl.fit(tips_features, tips['tip'])

Pipeline(steps=[('preprocessor',
                 ColumnTransformer(force_int_remainder_cols=False,
                                   remainder='passthrough',
                                   transformers=[('size',
                                                  Binarizer(threshold=2),
                                                  ['size']),
                                                 ('categorical_cols',
                                                  OneHotEncoder(),
                                                  ['sex', 'smoker', 'day',
                                                   'time'])])),
                ('lin-reg', LinearRegression())])

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Prediction is as easy as calling predict:

tips_features.head()

	total_bill	sex	smoker	day	time	size
0	16.99	Female	No	Sun	Dinner	2
1	10.34	Male	No	Sun	Dinner	3
2	21.01	Male	No	Sun	Dinner	3
3	23.68	Male	No	Sun	Dinner	2
4	24.59	Female	No	Sun	Dinner	4

# Note that we fit the Pipeline using tips_features, not tips_features.head()!
pl.predict(tips_features.head())

array([2.74, 2.32, 3.37, 3.37, 3.75])

Aside: FunctionTransformer

A transformer you'll often use as part of a ColumnTransformer is the FunctionTransformer, which enables you to use your own functions on entire columns. Think of it as the sklearn equivalent of apply.

from sklearn.preprocessing import FunctionTransformer

f = FunctionTransformer(np.sqrt)
f.transform([1, 2, 3])

array([1.  , 1.41, 1.73])

💡 Pro-Tip: Using make_pipeline and make_column_transformer

Instead of using Pipeline and ColumnTransformer classes directly, scikit-learn provides nifty shortcut methods called make_pipeline and make_column_transformer:

# Old code

preproc = ColumnTransformer(
    transformers=[
        ('size', Binarizer(threshold=2), ['size']),
        ('categorical_cols', OneHotEncoder(), ['sex', 'smoker', 'day', 'time'])
    ],
    remainder='passthrough' 
)

pl = Pipeline([
    ('preprocessor', preproc), 
    ('lin-reg', LinearRegression())
])
pl

Pipeline(steps=[('preprocessor',
                 ColumnTransformer(remainder='passthrough',
                                   transformers=[('size',
                                                  Binarizer(threshold=2),
                                                  ['size']),
                                                 ('categorical_cols',
                                                  OneHotEncoder(),
                                                  ['sex', 'smoker', 'day',
                                                   'time'])])),
                ('lin-reg', LinearRegression())])

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from sklearn.pipeline import make_pipeline
from sklearn.compose import make_column_transformer

preproc = make_column_transformer(
    (Binarizer(threshold=2), ['size']),
    (OneHotEncoder(), ['sex', 'smoker', 'day', 'time']),
    remainder='passthrough',
)

pl = make_pipeline(preproc, LinearRegression())
# Notice that the steps in the pipeline and column transformer are
# automatically named
pl

Pipeline(steps=[('columntransformer',
                 ColumnTransformer(remainder='passthrough',
                                   transformers=[('binarizer',
                                                  Binarizer(threshold=2),
                                                  ['size']),
                                                 ('onehotencoder',
                                                  OneHotEncoder(),
                                                  ['sex', 'smoker', 'day',
                                                   'time'])])),
                ('linearregression', LinearRegression())])

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An example Pipeline

One of the transformers we used was the StandardScaler transformer, which standardizes columns.

$$z(x_i) = \frac{x_i - \text{mean of } x}{\text{SD of } x}$$

Let's build a Pipeline that:

• Takes in the 'total_bill' and 'size' features of tips.
• Standardizes those features.
• Uses the resulting standardized features to fit a linear model that predicts 'tip'.

# Let's define these once, since we'll use them repeatedly.
X = tips[['total_bill', 'size']]
y = tips['tip']

from sklearn.preprocessing import StandardScaler

model_with_std = make_pipeline(
    StandardScaler(),
    LinearRegression(),
)

model_with_std.fit(X, y)

Pipeline(steps=[('standardscaler', StandardScaler()),
                ('linearregression', LinearRegression())])

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How well does our model do? We can compute its $R^2$ and RMSE.

model_with_std.score(X, y)

0.46786930879612587

from sklearn.metrics import root_mean_squared_error

root_mean_squared_error(y, model_with_std.predict(X))

np.float64(1.007256127114662)

Does this model perform any better than one that doesn't standardize its features? Let's find out.

model_without_std = LinearRegression()
model_without_std.fit(X, y)

LinearRegression()

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model_without_std.score(X, y)

0.46786930879612587

root_mean_squared_error(y, model_without_std.predict(X))

np.float64(1.007256127114662)

No!

The purpose of standardizing features

If you're performing "vanilla" linear regression – that is, using the LinearRegression object – then standardizing your features will not change your model's error.

• There are other models where standardizing your features will improve performance, because the methods assume features are standardized.
  • Regularized linear regression (see DSC 140A).
  • PCA (assumes centered data, not necessarily standardized: see DSC 140B).
  • Clustering algorithms, e.g. $k$-means clustering (saw in DSC 40A!).

• There is a benefit to standardizing features when performing vanilla linear regression, as we saw in DSC 40A: the features are brought to the same scale, so the coefficients can be compared directly.

# Total bill, table size.
model_without_std.coef_

array([0.09, 0.19])

# Total bill, table size.
model_with_std.named_steps['linearregression'].coef_

array([0.82, 0.18])

Aside: Pipelines of just transformers

If you want to apply multiple transformations to the same column in a dataset, you can create a Pipeline just for that column.

For example, suppose we want to:

• One hot encode the 'sex', 'smoker', and 'time' columns.
• One hot encode the 'day' column, but as either 'Weekday', 'Sat', or 'Sun'.
• Binarize the 'size' column.

Here's how we might do that:

def is_weekend(s):
    # The input to is_weekend is a Series!
    return s.replace({'Thur': 'Weekday', 'Fri': 'Weekday'})

pl_day = make_pipeline(
    FunctionTransformer(is_weekend),
    OneHotEncoder(),
)

col_trans = make_column_transformer(
    (pl_day, ['day']),
    (OneHotEncoder(drop='first'), ['sex', 'smoker', 'time']),
    (Binarizer(threshold=2), ['size']),
    remainder='passthrough',
    force_int_remainder_cols=False,
)

pl = make_pipeline(
    col_trans,
    LinearRegression(),
)

pl.fit(tips.drop('tip', axis=1), tips['tip'])

Pipeline(steps=[('columntransformer',
                 ColumnTransformer(force_int_remainder_cols=False,
                                   remainder='passthrough',
                                   transformers=[('pipeline',
                                                  Pipeline(steps=[('functiontransformer',
                                                                   FunctionTransformer(func=<function is_weekend at 0x283024180>)),
                                                                  ('onehotencoder',
                                                                   OneHotEncoder())]),
                                                  ['day']),
                                                 ('onehotencoder',
                                                  OneHotEncoder(drop='first'),
                                                  ['sex', 'smoker', 'time']),
                                                 ('binarizer',
                                                  Binarizer(threshold=2),
                                                  ['size'])])),
                ('linearregression', LinearRegression())])

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Question 🤔 (Answer at dsc80.com/q)

Code: weights

How many weights does this linear model have?

pl.named_steps

{'columntransformer': ColumnTransformer(force_int_remainder_cols=False, remainder='passthrough',
                   transformers=[('pipeline',
                                  Pipeline(steps=[('functiontransformer',
                                                   FunctionTransformer(func=<function is_weekend at 0x283024180>)),
                                                  ('onehotencoder',
                                                   OneHotEncoder())]),
                                  ['day']),
                                 ('onehotencoder', OneHotEncoder(drop='first'),
                                  ['sex', 'smoker', 'time']),
                                 ('binarizer', Binarizer(threshold=2),
                                  ['size'])]),
 'linearregression': LinearRegression()}

Multicollinearity

Heights and weights

We have a dataset containing the weights and heights of 25,000 18 year olds, taken from here.

people_path = Path('data') / 'SOCR-HeightWeight.csv'
people = pd.read_csv(people_path).drop(columns=['Index'])
people.head()

	Height (Inches)	Weight (Pounds)
0	65.78	112.99
1	71.52	136.49
2	69.40	153.03
3	68.22	142.34
4	67.79	144.30

people.plot(kind='scatter', x='Height (Inches)', y='Weight (Pounds)', 
            title='Weight vs. Height for 25,000 18 Year Olds')