Logistic Regression with Many Features

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Hide code cell source 
MAKE_BOOK_FIGURES=Trueimport numpy as npimport scipy.stats as stimport matplotlib as mplimport matplotlib.pyplot as plt%matplotlib inlineimport matplotlib_inlinematplotlib_inline.backend_inline.set_matplotlib_formats('svg')import seaborn as snssns.set_context("paper")sns.set_style("ticks")def set_book_style():    plt.style.use('seaborn-v0_8-white')     sns.set_style("ticks")    sns.set_palette("deep")    mpl.rcParams.update({        # Font settings        'font.family': 'serif',  # For academic publishing        'font.size': 8,  # As requested, 10pt font        'axes.labelsize': 8,        'axes.titlesize': 8,        'xtick.labelsize': 7,  # Slightly smaller for better readability        'ytick.labelsize': 7,        'legend.fontsize': 7,                # Line and marker settings for consistency        'axes.linewidth': 0.5,        'grid.linewidth': 0.5,        'lines.linewidth': 1.0,        'lines.markersize': 4,                # Layout to prevent clipped labels        'figure.constrained_layout.use': True,                # Default DPI (will override when saving)        'figure.dpi': 600,        'savefig.dpi': 600,                # Despine - remove top and right spines        'axes.spines.top': False,        'axes.spines.right': False,                # Remove legend frame        'legend.frameon': False,                # Additional trim settings        'figure.autolayout': True,  # Alternative to constrained_layout        'savefig.bbox': 'tight',    # Trim when saving        'savefig.pad_inches': 0.1   # Small padding to ensure nothing gets cut off    })def set_notebook_style():    plt.style.use('seaborn-v0_8-white')    sns.set_style("ticks")    sns.set_palette("deep")    mpl.rcParams.update({        # Font settings - using default sizes        'font.family': 'serif',        'axes.labelsize': 10,        'axes.titlesize': 10,        'xtick.labelsize': 9,        'ytick.labelsize': 9,        'legend.fontsize': 9,                # Line and marker settings        'axes.linewidth': 0.5,        'grid.linewidth': 0.5,        'lines.linewidth': 1.0,        'lines.markersize': 4,                # Layout settings        'figure.constrained_layout.use': True,                # Remove only top and right spines        'axes.spines.top': False,        'axes.spines.right': False,                # Remove legend frame        'legend.frameon': False,                # Additional settings        'figure.autolayout': True,        'savefig.bbox': 'tight',        'savefig.pad_inches': 0.1    })def save_for_book(fig, filename, is_vector=True, **kwargs):    """    Save a figure with book-optimized settings.        Parameters:    -----------    fig : matplotlib figure        The figure to save    filename : str        Filename without extension    is_vector : bool        If True, saves as vector at 1000 dpi. If False, saves as raster at 600 dpi.    **kwargs : dict        Additional kwargs to pass to savefig    """        # Set appropriate DPI and format based on figure type    if is_vector:        dpi = 1000        ext = '.pdf'    else:        dpi = 600        ext = '.tif'        # Save the figure with book settings    fig.savefig(f"{filename}{ext}", dpi=dpi, **kwargs)def make_full_width_fig():    return plt.subplots(figsize=(4.7, 2.9), constrained_layout=True)def make_half_width_fig():    return plt.subplots(figsize=(2.35, 1.45), constrained_layout=True)if MAKE_BOOK_FIGURES:    set_book_style()else:    set_notebook_style()make_full_width_fig = make_full_width_fig if MAKE_BOOK_FIGURES else lambda: plt.subplots()make_half_width_fig = make_half_width_fig if MAKE_BOOK_FIGURES else lambda: plt.subplots()

Logistic Regression with Many Features#

Let’s repeat what we did for the HMX example. Instead of using a linear model inside the sigmoid, we will use a quadratic model. That is, the probability of an explosion will be:

\[ p(y=1|x,\mathbf{w}) = \operatorname{sigm}\left(w_0 + w_1 x + w_2 x^2\right). \]

Let’s load the data first:

Hide code cell source 
url = "https://github.com/PredictiveScienceLab/data-analytics-se/raw/master/lecturebook/data/hmx_data.csv"
!curl -O $url

import pandas as pd

data = pd.read_csv('hmx_data.csv')
x = data['Height'].values
label_coding = {'E': 1, 'N': 0}
y = np.array([label_coding[r] for r in data['Result']])
data['y'] = y
data.head()
Height Result y
0 40.5 E 1
1 40.5 E 1
2 40.5 E 1
3 40.5 E 1
4 40.5 E 1

Now let’s train a second degree polynomial model:

from sklearn.preprocessing import PolynomialFeatures
from sklearn.linear_model import LogisticRegression

# Design matrix
poly = PolynomialFeatures(2)
Phi = poly.fit_transform(x[:, None])

# Fit
model = LogisticRegression(
    penalty=None,
    fit_intercept=False
).fit(Phi, y)

Here are the model parameters:

model.coef_

array([[-0.02784038, -0.41748133,  0.01335223]])

Let’s plot the predictions:

fig, ax = plt.subplots()
xx = np.linspace(20.0, 45.0, 100)
Phi_xx = poly.fit_transform(xx[:, None])
predictions_xx = model.predict_proba(Phi_xx)
ax.plot(
    xx,
    predictions_xx[:, 0],
    label='Probability of N'
)
ax.plot(
    xx,
    predictions_xx[:, 1],
    label='Probability of E'
)
ax.set_xlabel('$x$ (cm)')
ax.set_ylabel('Probability')
plt.legend(loc='best', frameon=False)
sns.despine(trim=True);
../_images/defb2f6f719a68f1620506360cce046ab692e2c57819be52703cac38514e319b.svg

Questions#

  • Is it worth going to a second-degree model? Can you compare the two models?

  • Rerun the code above with polynomial degrees 3, 4, and 5. What do you observe? Do you trust the results? Why or why not?

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