Classwork 11

Addressing Quasi-Separation in Logistic Regression with Regularization

Setup for PySpark, UDFs, and Plots

Required Libraries and SparkSession Entry Point

# Below is for an interactive display of Pandas DataFrame in Colab
from google.colab import data_table
data_table.enable_dataframe_formatter()

import pandas as pd
import numpy as np
from tabulate import tabulate  # for table summary
import scipy.stats as stats
from scipy.stats import norm
import matplotlib.pyplot as plt
import seaborn as sns
import statsmodels.api as sm  # for lowess smoothing
from sklearn.metrics import precision_recall_curve
from sklearn.metrics import roc_curve

from pyspark.sql import SparkSession
from pyspark.sql.functions import rand, col, pow, mean, avg, when, log, sqrt, exp
from pyspark.ml.feature import VectorAssembler
from pyspark.ml.regression import LinearRegression, GeneralizedLinearRegression
from pyspark.ml.evaluation import BinaryClassificationEvaluator

spark = SparkSession.builder.master("local[*]").getOrCreate()

UDF for Adding Dummy Variables

def add_dummy_variables(var_name, reference_level, category_order=None):
    """
    Creates dummy variables for the specified column in the global DataFrames dtrain and dtest.
    Allows manual setting of category order.

    Parameters:
        var_name (str): The name of the categorical column (e.g., "borough_name").
        reference_level (int): Index of the category to be used as the reference (dummy omitted).
        category_order (list, optional): List of categories in the desired order. If None, categories are sorted.

    Returns:
        dummy_cols (list): List of dummy column names excluding the reference category.
        ref_category (str): The category chosen as the reference.
    """
    global dtrain, dtest

    # Get distinct categories from the training set.
    categories = dtrain.select(var_name).distinct().rdd.flatMap(lambda x: x).collect()

    # Convert booleans to strings if present.
    categories = [str(c) if isinstance(c, bool) else c for c in categories]

    # Use manual category order if provided; otherwise, sort categories.
    if category_order:
        # Ensure all categories are present in the user-defined order
        missing = set(categories) - set(category_order)
        if missing:
            raise ValueError(f"These categories are missing from your custom order: {missing}")
        categories = category_order
    else:
        categories = sorted(categories)

    # Validate reference_level
    if reference_level < 0 or reference_level >= len(categories):
        raise ValueError(f"reference_level must be between 0 and {len(categories) - 1}")

    # Define the reference category
    ref_category = categories[reference_level]
    print("Reference category (dummy omitted):", ref_category)

    # Create dummy variables for all categories
    for cat in categories:
        dummy_col_name = var_name + "_" + str(cat).replace(" ", "_")
        dtrain = dtrain.withColumn(dummy_col_name, when(col(var_name) == cat, 1).otherwise(0))
        dtest = dtest.withColumn(dummy_col_name, when(col(var_name) == cat, 1).otherwise(0))

    # List of dummy columns, excluding the reference category
    dummy_cols = [var_name + "_" + str(cat).replace(" ", "_") for cat in categories if cat != ref_category]

    return dummy_cols, ref_category


# Example usage without category_order:
# dummy_cols_year, ref_category_year = add_dummy_variables('year', 0)

# Example usage with category_order:
# custom_order_wkday = ['sunday', 'monday', 'tuesday', 'wednesday', 'thursday', 'friday', 'saturday']
# dummy_cols_wkday, ref_category_wkday = add_dummy_variables('wkday', reference_level=0, category_order = custom_order_wkday)

Setup for scikit-learn and Plots

from sklearn.model_selection import train_test_split
from sklearn.linear_model import LogisticRegression, LogisticRegressionCV
from sklearn.metrics import (confusion_matrix, accuracy_score, precision_score, 
                             recall_score, roc_curve, roc_auc_score)

Question 1

dfpd = pd.read_csv('https://bcdanl.github.io/data/car-data.csv')

• Convert the dfpd Pandas DataFrame into the PySpark DataFrame object with the name, df.

Variable description

Variable	Description
buying	Buying price of the car (vhigh, high, med, low)
maint	Maintenance cost (vhigh, high, med, low)
doors	Number of doors (2, 3, 4, 5more)
persons	Capacity in terms of persons to carry (2, 4, more)
lug_boot	Size of luggage boot (small, med, big)
safety	Estimated safety of the car (low, med, high)
rating	Car acceptability (unacc, acc, good, vgood)
fail	TRUE if the car is unacceptable (unacc), otherwise FALSE

Question 2

• Divide the df DataFrame into training and test DataFrames.
  • Use dtrain and dtest for training and test DataFrames, respectively.
  • 70% of observations in the df are assigned to dtrain; the rest is assigned to dtest.

Question 3

Fit the following regression model:

\[ \begin{align} &\quad\;\; \text{Prob}(\text{fail}_{i} = 1) \\ &= G\Big(\beta_{0} \\ &\qquad\quad\;\;\; \,+\, \beta_{4} \text{buying\_med}_{i} \,+\, \beta_{4} \text{buying\_high}_{i} \,+\, \beta_{4} \text{buying\_vhigh}_{i} \\ &\qquad\quad\;\;\; \,+\, \beta_{4} \text{maint\_med}_{i} \,+\, \beta_{4} \text{maint\_high}_{i} \,+\, \beta_{4} \text{maint\_vhigh}_{i} \\ &\qquad\quad\;\;\; \,+\, \beta_{7} \text{persons\_4}_{i} \,+\, \beta_{8} \text{persons\_more}_{i} \\ &\qquad\quad\;\;\; \,+\, \beta_{10} \text{lug\_boot\_med}_{i}\,+\, \beta_{10} \text{lug\_boot\_big}_{i} \\ &\qquad\quad\;\;\; \,+\, \beta_{11} \text{safety\_med}_{i}\,+\, \beta_{11} \text{safety\_high}_{i} \Big), \end{align} \]

where \(G(\,\cdot\,)\) is

\[ G(\,\cdot\,) = \frac{\exp(\,\cdot\,)}{1 + \exp(\,\cdot\,)}. \]

Provide the summary of the regression result.

• Set the reference levels accordingly.

Question 4

• How are coefficient estimates?

Question 5

• Calculate the followings:
  • Confusion matrix with the appropriate threshold level.
  • Accuracy
  • Precision
  • Recall
  • Specificity
  • Average rate of at-risk babies
  • Enrichment

Question 6

Visualize the variation in recall and enrichment across different threshold levels.

Question 7

• Draw the receiver operating characteristic (ROC) curve.
• Calculate the area under the curve (AUC).

Question 8

• Use sklearn to fit a Lasso logistic regression.
  • Repeat Questions 3-7.

Question 9

• Use sklearn to fit a Ridge logistic regression.
  • Repeat Questions 3-7.

Question 10

• Use sklearn to fit a Elastic Net logistic regression.
  • Repeat Questions 3-7.

Discussion

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