Classwork 8

Linear Regression I

Author

Byeong-Hak Choe

Published

April 14, 2025

Modified

April 14, 2025

Setup for PySpark, UDFs, and Plots

Required Libraries and SparkSession Entry Point

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

from pyspark.sql import SparkSession
from pyspark.sql.functions import rand, col, pow, mean, when, log
from pyspark.ml.feature import VectorAssembler
from pyspark.ml.regression import LinearRegression

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

UDF for Regression Tables

def regression_table(model, assembler):
    """
    Creates a formatted regression table from a fitted LinearRegression model and its VectorAssembler.

    If the model’s labelCol (retrieved using getLabelCol()) starts with "log", an extra column showing np.exp(coeff)
    is added immediately after the beta estimate column for predictor rows. Additionally, np.exp() of the 95% CI
    Lower and Upper bounds is also added unless the predictor's name includes "log_". The Intercept row does not
    include exponentiated values.

    When labelCol starts with "log", the columns are ordered as:
        y: [label] | Beta | Exp(Beta) | Sig. | Std. Error | p-value | 95% CI Lower | Exp(95% CI Lower) | 95% CI Upper | Exp(95% CI Upper)

    Otherwise, the columns are:
        y: [label] | Beta | Sig. | Std. Error | p-value | 95% CI Lower | 95% CI Upper

    Parameters:
        model: A fitted LinearRegression model (with a .summary attribute and a labelCol).
        assembler: The VectorAssembler used to assemble the features for the model.

    Returns:
        A formatted string containing the regression table.
    """
    # Determine if we should display exponential values for coefficients.
    is_log = model.getLabelCol().lower().startswith("log")

    # Extract coefficients and standard errors as NumPy arrays.
    coeffs = model.coefficients.toArray()
    std_errors_all = np.array(model.summary.coefficientStandardErrors)

    # Check if the intercept's standard error is included (one extra element).
    if len(std_errors_all) == len(coeffs) + 1:
        intercept_se = std_errors_all[0]
        std_errors = std_errors_all[1:]
    else:
        intercept_se = None
        std_errors = std_errors_all

    # Use provided tValues and pValues.
    df = model.summary.numInstances - len(coeffs) - 1
    t_critical = stats.t.ppf(0.975, df)
    p_values = model.summary.pValues

    # Helper: significance stars.
    def significance_stars(p):
        if p < 0.01:
            return "***"
        elif p < 0.05:
            return "**"
        elif p < 0.1:
            return "*"
        else:
            return ""

    # Build table rows for each feature.
    table = []
    for feature, beta, se, p in zip(assembler.getInputCols(), coeffs, std_errors, p_values):
        ci_lower = beta - t_critical * se
        ci_upper = beta + t_critical * se

        # Check if predictor contains "log_" to determine if exponentiation should be applied
        apply_exp = is_log and "log_" not in feature.lower()

        exp_beta = np.exp(beta) if apply_exp else ""
        exp_ci_lower = np.exp(ci_lower) if apply_exp else ""
        exp_ci_upper = np.exp(ci_upper) if apply_exp else ""

        if is_log:
            table.append([
                feature,            # Predictor name
                beta,               # Beta estimate
                exp_beta,           # Exponential of beta (or blank)
                significance_stars(p),
                se,
                p,
                ci_lower,
                exp_ci_lower,       # Exponential of 95% CI lower bound
                ci_upper,
                exp_ci_upper        # Exponential of 95% CI upper bound
            ])
        else:
            table.append([
                feature,
                beta,
                significance_stars(p),
                se,
                p,
                ci_lower,
                ci_upper
            ])

    # Process intercept.
    if intercept_se is not None:
        intercept_p = model.summary.pValues[0] if model.summary.pValues is not None else None
        intercept_sig = significance_stars(intercept_p)
        ci_intercept_lower = model.intercept - t_critical * intercept_se
        ci_intercept_upper = model.intercept + t_critical * intercept_se
    else:
        intercept_sig = ""
        ci_intercept_lower = ""
        ci_intercept_upper = ""
        intercept_se = ""

    if is_log:
        table.append([
            "Intercept",
            model.intercept,
            "",                    # Removed np.exp(model.intercept)
            intercept_sig,
            intercept_se,
            "",
            ci_intercept_lower,
            "",
            ci_intercept_upper,
            ""
        ])
    else:
        table.append([
            "Intercept",
            model.intercept,
            intercept_sig,
            intercept_se,
            "",
            ci_intercept_lower,
            ci_intercept_upper
        ])

    # Append overall model metrics.
    if is_log:
        table.append(["Observations", model.summary.numInstances, "", "", "", "", "", "", "", ""])
        table.append(["R²", model.summary.r2, "", "", "", "", "", "", "", ""])
        table.append(["RMSE", model.summary.rootMeanSquaredError, "", "", "", "", "", "", "", ""])
    else:
        table.append(["Observations", model.summary.numInstances, "", "", "", "", ""])
        table.append(["R²", model.summary.r2, "", "", "", "", ""])
        table.append(["RMSE", model.summary.rootMeanSquaredError, "", "", "", "", ""])

    # Format the table rows.
    formatted_table = []
    for row in table:
        formatted_row = []
        for i, item in enumerate(row):
            # Format Observations as integer with commas.
            if row[0] == "Observations" and i == 1 and isinstance(item, (int, float, np.floating)) and item != "":
                formatted_row.append(f"{int(item):,}")
            elif isinstance(item, (int, float, np.floating)) and item != "":
                if is_log:
                    # When is_log, the columns are:
                    # 0: Metric, 1: Beta, 2: Exp(Beta), 3: Sig, 4: Std. Error, 5: p-value,
                    # 6: 95% CI Lower, 7: Exp(95% CI Lower), 8: 95% CI Upper, 9: Exp(95% CI Upper).
                    if i in [1, 2, 4, 6, 7, 8, 9]:
                        formatted_row.append(f"{item:,.3f}")
                    elif i == 5:
                        formatted_row.append(f"{item:.3f}")
                    else:
                        formatted_row.append(f"{item:.3f}")
                else:
                    # When not is_log, the columns are:
                    # 0: Metric, 1: Beta, 2: Sig, 3: Std. Error, 4: p-value, 5: 95% CI Lower, 6: 95% CI Upper.
                    if i in [1, 3, 5, 6]:
                        formatted_row.append(f"{item:,.3f}")
                    elif i == 4:
                        formatted_row.append(f"{item:.3f}")
                    else:
                        formatted_row.append(f"{item:.3f}")
            else:
                formatted_row.append(item)
        formatted_table.append(formatted_row)

    # Set header and column alignment based on whether label starts with "log"
    if is_log:
        headers = [
            f"y: {model.getLabelCol()}",
            "Beta", "Exp(Beta)", "Sig.", "Std. Error", "p-value",
            "95% CI Lower", "Exp(95% CI Lower)", "95% CI Upper", "Exp(95% CI Upper)"
        ]
        colalign = ("left", "right", "right", "center", "right", "right", "right", "right", "right", "right")
    else:
        headers = [f"y: {model.getLabelCol()}", "Beta", "Sig.", "Std. Error", "p-value", "95% CI Lower", "95% CI Upper"]
        colalign = ("left", "right", "center", "right", "right", "right", "right")

    table_str = tabulate(
        formatted_table,
        headers=headers,
        tablefmt="pretty",
        colalign=colalign
    )

    # Insert a dashed line after the Intercept row.
    lines = table_str.split("\n")
    dash_line = '-' * len(lines[0])
    for i, line in enumerate(lines):
        if "Intercept" in line and not line.strip().startswith('+'):
            lines.insert(i+1, dash_line)
            break

    return "\n".join(lines)

# Example usage:
# print(regression_table(model_1, assembler_1))

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)


Coefficient Plots

terms = assembler3.getInputCols()
coefs = model3.coefficients.toArray()[:len(terms)]
stdErrs = model3.summary.coefficientStandardErrors[:len(terms)]

df_summary = pd.DataFrame({
    "term": terms,
    "estimate": coefs,
    "std_error": stdErrs
})

# Filter df_summary if needed

# Plot using the DataFrame columns
plt.errorbar(df_summary["term"], df_summary["estimate"],
             yerr = 1.96 * df_summary["std_error"], fmt='o', capsize=5)
plt.xlabel("Terms")
plt.ylabel("Coefficient Estimate")
plt.title("Coefficient Estimates (Model 2)")
plt.axhline(0, color="red", linestyle="--")  # Add horizontal line at 0
plt.xticks(rotation=45)
plt.show()


Residual Plots

# Convert test predictions to Pandas
dfpd = TEST_DATAFRAME.select(["prediction", "Y_VARIABLE"]).toPandas()
dfpd["residual"] = dfpd["Y_VARIABLE"] - dfpd["prediction"]
plt.scatter(dfpd["prediction"], dfpd["residual"], alpha=0.2, color="darkgray")

# Use lowess smoothing for the trend line
smoothed = sm.nonparametric.lowess(dfpd["residual"], dfpd["prediction"])
plt.plot(smoothed[:, 0], smoothed[:, 1], color="darkblue")
plt.axhline(y=0, color="red", linestyle="--")
plt.xlabel("Predicted y (Model)")
plt.ylabel("Residuals")
plt.title("Residual Plot for Model")
plt.show()


Histogram

# Create a histogram
dfpd = DATAFRAME.select(["Y_VARIABLE"]).toPandas()
sns.histplot(dfpd["Y_VARIABLE"], bins=10, kde=True)



Question 1

  • Read the data file, https://bcdanl.github.io/data/bikeshare_cleaned.csv, as the PySpark DataFrame object with the name, bikeshare.
    • What are continuous variables? What are categorical variables?
    • For each continuous variable, provide descriptive statistics.
    • For each categorical variable, provide the frequency of each distinct value’s occurrence.


Variable description

Variable Description
cnt Count of total rental bikes
year Year
month Month
date Date
hr Hour
wkday Weekday
holiday Holiday indicator (1 if holiday, 0 otherwise)
seasons Season
weather_cond Weather condition
temp Temperature (measured in standard deviations from average)
hum Humidity (measured in standard deviations from average)
windspeed Wind speed (measured in standard deviations from average)



Question 2

  • Divide the bikeshare DataFrame into training and test DataFrames.
    • Use dtrain and dtest for training and test DataFrames, respectively.
    • 60% of observations in the bikeshare are assigned to dtrain; the rest is assigned to dtest.



Question 3

Train the following linear regression model. Provide the summary of the regression result.

\[ \begin{align} \text{cnt}_{i} =\ &\beta_{\text{intercept}}\\ &+ \beta_{\text{temp}} \, \text{temp}_{i} + \beta_{\text{hum}} \, \text{hum}_{i} + \beta_{\text{windspeed}} \, \text{windspeed}_{i} \nonumber \\ &+ \beta_{\text{year\_2012}} \, \text{year\_2012}_{i}\\ &+ \beta_{\text{month\_2}} \, \text{month\_2}_{i} + \beta_{\text{month\_3}} \, \text{month\_3}_{i} + \beta_{\text{month\_4}} \, \text{month\_4}_{i} \nonumber \\ &+ \beta_{\text{month\_5}} \, \text{month\_5}_{i} + \beta_{\text{month\_6}} \, \text{month\_6}_{i} + \beta_{\text{month\_7}} \, \text{month\_7}_{i} + \beta_{\text{month\_8}} \, \text{month\_8}_{i} \nonumber \\ &+ \beta_{\text{month\_9}} \, \text{month\_9}_{i} + \beta_{\text{month\_10}} \, \text{month\_10}_{i} + \beta_{\text{month\_11}} \, \text{month\_11}_{i} + \beta_{\text{month\_12}} \, \text{month\_12}_{i} \nonumber \\ &+ \beta_{\text{hr\_1}} \, \text{hr\_1}_{i} + \beta_{\text{hr\_2}} \, \text{hr\_2}_{i} + \beta_{\text{hr\_3}} \, \text{hr\_3}_{i} + \beta_{\text{hr\_4}} \, \text{hr\_4}_{i} \nonumber \\ &+ \beta_{\text{hr\_5}} \, \text{hr\_5}_{i} + \beta_{\text{hr\_6}} \, \text{hr\_6}_{i} + \beta_{\text{hr\_7}} \, \text{hr\_7}_{i} + \beta_{\text{hr\_8}} \, \text{hr\_8}_{i} \nonumber \\ &+ \beta_{\text{hr\_9}} \, \text{hr\_9}_{i} + \beta_{\text{hr\_10}} \, \text{hr\_10}_{i} + \beta_{\text{hr\_11}} \, \text{hr\_11}_{i} + \beta_{\text{hr\_12}} \, \text{hr\_12}_{i} \nonumber \\ &+ \beta_{\text{hr\_13}} \, \text{hr\_13}_{i} + \beta_{\text{hr\_14}} \, \text{hr\_14}_{i} + \beta_{\text{hr\_15}} \, \text{hr\_15}_{i} + \beta_{\text{hr\_16}} \, \text{hr\_16}_{i} \nonumber \\ &+ \beta_{\text{hr\_17}} \, \text{hr\_17}_{i} + \beta_{\text{hr\_18}} \, \text{hr\_18}_{i} + \beta_{\text{hr\_19}} \, \text{hr\_19}_{i} + \beta_{\text{hr\_20}} \, \text{hr\_20}_{i} \nonumber \\ &+ \beta_{\text{hr\_21}} \, \text{hr\_21}_{i} + \beta_{\text{hr\_22}} \, \text{hr\_22}_{i} + \beta_{\text{hr\_23}} \, \text{hr\_23}_{i} \nonumber \\ &+ \beta_{\text{wkday\_monday}} \, \text{wkday\_monday}_{i} + \beta_{\text{wkday\_tuesday}} \, \text{wkday\_tuesday}_{i} + \beta_{\text{wkday\_wednesday}} \, \text{wkday\_wednesday}_{i} \nonumber \\ &+ \beta_{\text{wkday\_thursday}} \, \text{wkday\_thursday}_{i} + \beta_{\text{wkday\_friday}} \, \text{wkday\_friday}_{i} + \beta_{\text{wkday\_saturday}} \, \text{wkday\_saturday}_{i} \nonumber \\ &+ \beta_{\text{holiday\_1}} \, \text{holiday\_1}_{i} \nonumber \\ &+ \beta_{\text{seasons\_summer}} \, \text{seasons\_summer}_{i} + \beta_{\text{seasons\_fall}} \, \text{seasons\_fall}_{i} + \beta_{\text{seasons\_winter}} \, \text{seasons\_winter}_{i} \nonumber \\ &+ \beta_{\text{weather\_cond\_Light\_Snow\_or\_Light\_Rain}} \, \text{weather\_cond\_Light\_Snow\_or\_Light\_Rain}_{i}\nonumber \\ &+ \beta_{\text{weather\_cond\_Mist\_or\_Cloudy}} \, \text{weather\_cond\_Mist\_or\_Cloudy}_{i}\\ &+ \epsilon_{i} \end{align} \]

Note that all predictors are dummy variables, except for temp, hum, and windspeed.



Question 4

Make a prediction on the outcome variable using the test DataFrame and the regression result from Question 3.



Question 5

Interpret the beta estimate of windspeed.



Question 6

  • Which hr is most strongly associated with changes in cnt?
  • Interpret the beta estimate of that hr.



Question 7

  • Draw a coefficient plot for temp, hum and windspeed variables.
  • Draw a coefficient plot for month variables.
  • Draw a coefficient plot for hr variables.
  • Draw a coefficient plot for wkday variables.
  • Draw a coefficient plot for seasons variables.
  • Draw a coefficient plot for weather_cond variables.



Question 8

  • Draw a residual plot.
    • On average, are the predictions correct in the model in Question 3? Are there systematic errors?



Question 9

Draw a histogram of \(\text{cnt}_{i}\) and a histogram of \(\log(\text{cnt}_{i})\).



Question 10

Train the following linear regression model. Provide the summary of the regression result. Repeat Questions 4-8.

\[ \begin{align} \log(\text{cnt}_{i}) =\ &\beta_{\text{intercept}}\\ &+ \beta_{\text{temp}} \, \text{temp}_{i} + \beta_{\text{hum}} \, \text{hum}_{i} + \beta_{\text{windspeed}} \, \text{windspeed}_{i} \nonumber \\ &+ \beta_{\text{year\_2012}} \, \text{year\_2012}_{i}\\ &+ \beta_{\text{month\_2}} \, \text{month\_2}_{i} + \beta_{\text{month\_3}} \, \text{month\_3}_{i} + \beta_{\text{month\_4}} \, \text{month\_4}_{i} \nonumber \\ &+ \beta_{\text{month\_5}} \, \text{month\_5}_{i} + \beta_{\text{month\_6}} \, \text{month\_6}_{i} + \beta_{\text{month\_7}} \, \text{month\_7}_{i} + \beta_{\text{month\_8}} \, \text{month\_8}_{i} \nonumber \\ &+ \beta_{\text{month\_9}} \, \text{month\_9}_{i} + \beta_{\text{month\_10}} \, \text{month\_10}_{i} + \beta_{\text{month\_11}} \, \text{month\_11}_{i} + \beta_{\text{month\_12}} \, \text{month\_12}_{i} \nonumber \\ &+ \beta_{\text{hr\_1}} \, \text{hr\_1}_{i} + \beta_{\text{hr\_2}} \, \text{hr\_2}_{i} + \beta_{\text{hr\_3}} \, \text{hr\_3}_{i} + \beta_{\text{hr\_4}} \, \text{hr\_4}_{i} \nonumber \\ &+ \beta_{\text{hr\_5}} \, \text{hr\_5}_{i} + \beta_{\text{hr\_6}} \, \text{hr\_6}_{i} + \beta_{\text{hr\_7}} \, \text{hr\_7}_{i} + \beta_{\text{hr\_8}} \, \text{hr\_8}_{i} \nonumber \\ &+ \beta_{\text{hr\_9}} \, \text{hr\_9}_{i} + \beta_{\text{hr\_10}} \, \text{hr\_10}_{i} + \beta_{\text{hr\_11}} \, \text{hr\_11}_{i} + \beta_{\text{hr\_12}} \, \text{hr\_12}_{i} \nonumber \\ &+ \beta_{\text{hr\_13}} \, \text{hr\_13}_{i} + \beta_{\text{hr\_14}} \, \text{hr\_14}_{i} + \beta_{\text{hr\_15}} \, \text{hr\_15}_{i} + \beta_{\text{hr\_16}} \, \text{hr\_16}_{i} \nonumber \\ &+ \beta_{\text{hr\_17}} \, \text{hr\_17}_{i} + \beta_{\text{hr\_18}} \, \text{hr\_18}_{i} + \beta_{\text{hr\_19}} \, \text{hr\_19}_{i} + \beta_{\text{hr\_20}} \, \text{hr\_20}_{i} \nonumber \\ &+ \beta_{\text{hr\_21}} \, \text{hr\_21}_{i} + \beta_{\text{hr\_22}} \, \text{hr\_22}_{i} + \beta_{\text{hr\_23}} \, \text{hr\_23}_{i} \nonumber \\ &+ \beta_{\text{wkday\_monday}} \, \text{wkday\_monday}_{i} + \beta_{\text{wkday\_tuesday}} \, \text{wkday\_tuesday}_{i} + \beta_{\text{wkday\_wednesday}} \, \text{wkday\_wednesday}_{i} \nonumber \\ &+ \beta_{\text{wkday\_thursday}} \, \text{wkday\_thursday}_{i} + \beta_{\text{wkday\_friday}} \, \text{wkday\_friday}_{i} + \beta_{\text{wkday\_saturday}} \, \text{wkday\_saturday}_{i} \nonumber \\ &+ \beta_{\text{holiday\_1}} \, \text{holiday\_1}_{i} \nonumber \\ &+ \beta_{\text{seasons\_summer}} \, \text{seasons\_summer}_{i} + \beta_{\text{seasons\_fall}} \, \text{seasons\_fall}_{i} + \beta_{\text{seasons\_winter}} \, \text{seasons\_winter}_{i} \nonumber \\ &+ \beta_{\text{weather\_cond\_Light\_Snow\_or\_Light\_Rain}} \, \text{weather\_cond\_Light\_Snow\_or\_Light\_Rain}_{i}\nonumber \\ &+ \beta_{\text{weather\_cond\_Mist\_or\_Cloudy}} \, \text{weather\_cond\_Mist\_or\_Cloudy}_{i}\\ &+ \epsilon_{i} \end{align} \]



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