Reflection on Regression and Classification Models with Group Project Data

Analytics Objective: Predict Retail Sales Revenue from Discounting, Marketing, and Seasonal Factors

Contents

• Overview
• 1 Analytics Objective (AO)
  • 1.1 Revised Analytics Objective
  • 1.2 Supervised or Unsupervised?
  • 1.3 Variables and Their Roles
  • 1.4 Broader Category of Machine Learning Method
• 2 Setup
• 3 The Group Project Dataset
  • 3.1 Data Import and Cleaning
  • 3.2 Feature Engineering
  • 3.3 Exploratory Data Analysis
• 4 Train/Test Split and Shared Recipe
  • 4.1 Why This Method Is a Good Starting Point
  • 4.2 Split
  • 4.3 Shared Recipe
• 5 The Full Machine Learning Process
  • 5.1 Baseline — OLS Linear Regression
  • 5.2 Ridge Regression
  • 5.3 Lasso Regression
  • 5.4 Elastic Net
  • 5.5 Polynomial Regression
  • 5.6 Tuning the Penalty
• 6 Comparing All Models
  • 6.1 Cross-Validated Comparison
  • 6.2 Test Set Performance Summary
  • 6.3 Interpretation
• 7 Variable Importance
• 8 Reflection Questions (Module 4 Exercises, Applied to Our Data)
  • 8.1 Exercise 4.1 — Ridge vs. Lasso Coefficient Behavior
  • 8.2 Exercise 4.2 — Elastic Net Mixing Parameter
  • 8.3 Exercise 4.3 — Polynomial Degree Selection
  • 8.4 Exercise 4.4 — Tuning Elastic Net
  • 8.5 Exercise 4.5 — Business Interpretation
• 9 Summary
• Appendix
  • 9.1 Data Source
  • 9.2 Codebook Reference
  • 9.3 Group Project Repository (Data Source)
  • 9.4 This Assignment (Individual Repo)

Overview

Prompt: Apply what you learned from Step 1–Step 3 (regression-focused methods and classification models) to your group project data. State your Analytics Objective (AO), work through the full machine learning process for your chosen method, compare alternative methods, produce a variable importance chart, and address the reflection questions at the end of the relevant codebook.

This report applies the regression methods from Module 4—including Ridge, Lasso, Elastic Net, Polynomial Regression, and hyperparameter tuning—to the IBM 6540 group project dataset. Seis Leches (Group 1) has been using the Kaggle Retail Sales Data with Seasonal Trends and Marketing dataset¹ for our project, “Analyzing the Effect of Discounts on Retail Sales Volume and Revenue Using Seasonal Retail Data.” Since this is an individual reflection assignment, the data comes from our shared group repository, but this QMD file and its rendered HTML are stored in my personal GitHub repository (jakevns/RStudio). I also followed the same data/Retail_sales.csv file structure that I used for my M10 dashboard assignment.

A note on my approach: Rather than repeating the CLV example from the codebook, I built every model, recipe, and interpretation below using our group project dataset. This keeps the analysis focused on our actual research objective and makes the results more relevant to the work Group 1 is already completing.

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1 Analytics Objective (AO)

Prompt: State your AO for your project. If your AO didn’t involve machine learning, revise it so it is appropriately stated for a machine learning task. Describe: (1) supervised vs. unsupervised learning, (2) the variables to be included and their roles, and (3) the broader category of machine learning method (regression vs. classification).

1.1 Revised Analytics Objective

Our group’s original project framing—“analyzing the effect of discounts on retail sales volume and revenue”—was more focused on hypothesis testing than prediction. For this assignment, I reframed it as a clear machine learning objective:

Analytics Objective: Predict a retail transaction’s Sales Revenue (USD) using the discount percentage, marketing spend, product category, day of the week, season, and holiday status. This would help store planners estimate how a proposed discount or marketing budget may translate into revenue before a promotion launches, rather than only measuring its impact afterward.

This keeps the group’s original discount/marketing focus but reframes it as a prediction problem with a clear, continuous target, which is what makes it usable as a supervised machine learning task.

1.2 Supervised or Unsupervised?

This is a supervised learning problem because every row already includes a known value for Sales Revenue (USD). The model can learn from these labeled examples, where the predictors are paired with an observed outcome, and its predictions can then be compared with the actual revenue values in a held-out test set.

1.3 Variables and Their Roles

Variable (raw)	Role	Type	Included?	Rationale
Sales Revenue (USD)	Outcome (target)	Continuous	✅	What we want to predict
Discount Percentage	Feature	Numeric	✅	Core lever in our group’s discount question
Marketing Spend (USD)	Feature	Numeric	✅	Core lever in our group’s marketing question
Product Category	Feature	Categorical (4 levels)	✅	Category-level demand differences
Day of the Week	Feature	Categorical (7 levels)	✅	Weekly seasonality
Holiday Effect	Feature	Categorical (binary)	✅	Holiday demand lift
Date	Engineered → Feature	Derived to Month	✅ (derived)	Captures yearly seasonality without treating each of the 731 unique dates as its own category
Units Sold	Excluded	Numeric	❌	Revenue is mechanically driven by units sold (Revenue ≈ Units × Price); including it would leak the answer and defeat the purpose of predicting revenue before a promotion runs
Store ID	Excluded	ID	❌	Constant across all 30,000 rows in this dataset (only one store), so it carries zero variance
Product ID	Excluded	ID	❌	42 near-unique codes that are already summarized by Product Category; including both would be redundant
Store Location	Excluded	Categorical (243 levels)	❌	Far too high-cardinality relative to the sample to encode meaningfully (would explode into 242 dummy columns), and it does not correspond to a stable notion of “location” for a single-store dataset

1.4 Broader Category of Machine Learning Method

Because the outcome, Sales Revenue (USD), is continuous, this is a regression-focused problem, not classification. I am using the Module 4 — Regression-Focused Methods codebook² as the reference for the full workflow below.

TipWhy regression and not classification here

Our group project dataset does not include a natural or meaningful class label like the purchase categories used in the classification codebook’s e-commerce example. Since sales revenue is already measured as a dollar amount, treating it as a continuous outcome preserves more information than forcing it into artificial categories such as “high,” “medium,” or “low.” This makes regression the more accurate and honest way to frame the business question.

---

2 Setup

# install.packages(c("tidymodels", "tidyverse", "glmnet", "patchwork",
#                     "janitor", "lubridate", "gt", "ggrepel"))

# vip is no longer on CRAN directly -- install from the archived source once:
# install.packages(
#   "https://cran.r-project.org/src/contrib/Archive/vip/vip_0.4.6.tar.gz",
#   repos = NULL,
#   type = "source"
# )

library(tidymodels)
library(tidyverse)
library(glmnet)     # Ridge, Lasso, Elastic Net engine
library(patchwork)  # combining plots
library(vip)        # variable importance plots
library(janitor)    # clean_names()
library(lubridate)  # date handling
library(gt)         # publication-quality tables

tidymodels_prefer()
set.seed(2025)

---

3 The Group Project Dataset

3.1 Data Import and Cleaning

The QMD searches for data/Retail_sales.csv relative to the project folder, matching the group repo’s file structure.

sales_raw <- read_csv("data/Retail_sales.csv", show_col_types = FALSE) |>
  clean_names()

glimpse(sales_raw)

Rows: 30,000
Columns: 11
$ store_id            <chr> "Spearsland", "Spearsland", "Spearsland", "Spearsl…
$ product_id          <chr> "52372247", "52372247", "52372247", "52372247", "5…
$ date                <date> 2022-01-01, 2022-01-02, 2022-01-03, 2022-01-04, 2…
$ units_sold          <dbl> 9, 7, 1, 4, 2, 8, 6, 9, 7, 1, 4, 6, 3, 6, 3, 2, 8,…
$ sales_revenue_usd   <dbl> 2741.69, 2665.53, 380.79, 1523.16, 761.58, 3046.32…
$ discount_percentage <dbl> 20, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 15, 0, 15, 0,…
$ marketing_spend_usd <dbl> 81, 0, 0, 0, 0, 41, 0, 83, 0, 164, 61, 0, 197, 0, …
$ store_location      <chr> "Tanzania", "Mauritania", "Saint Pierre and Miquel…
$ product_category    <chr> "Furniture", "Furniture", "Furniture", "Furniture"…
$ day_of_the_week     <chr> "Saturday", "Sunday", "Monday", "Tuesday", "Wednes…
$ holiday_effect      <lgl> FALSE, FALSE, FALSE, FALSE, FALSE, FALSE, FALSE, F…

3.2 Feature Engineering

I created a month variable from date to capture seasonal patterns without treating all 731 individual calendar dates as separate categories. I also converted the remaining categorical variables to factors so the models would handle them correctly.

sales_data <- sales_raw |>
  mutate(
    date = ymd(date),
    month = factor(month(date, label = TRUE), ordered = FALSE),
    product_category = factor(product_category),
    day_of_the_week = factor(
      day_of_the_week,
      levels = c(
        "Monday", "Tuesday", "Wednesday", "Thursday",
        "Friday", "Saturday", "Sunday"
      )
    ),
    holiday_effect = factor(holiday_effect, levels = c("False", "True"))
  )

sales_model_data <- sales_data |>
  select(
    sales_revenue_usd,
    discount_percentage,
    marketing_spend_usd,
    product_category,
    day_of_the_week,
    holiday_effect,
    month
  )

glimpse(sales_model_data)

Rows: 30,000
Columns: 7
$ sales_revenue_usd   <dbl> 2741.69, 2665.53, 380.79, 1523.16, 761.58, 3046.32…
$ discount_percentage <dbl> 20, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 15, 0, 15, 0,…
$ marketing_spend_usd <dbl> 81, 0, 0, 0, 0, 41, 0, 83, 0, 164, 61, 0, 197, 0, …
$ product_category    <fct> Furniture, Furniture, Furniture, Furniture, Furnit…
$ day_of_the_week     <fct> Saturday, Sunday, Monday, Tuesday, Wednesday, Thur…
$ holiday_effect      <fct> NA, NA, NA, NA, NA, NA, NA, NA, NA, NA, NA, NA, NA…
$ month               <fct> Jan, Jan, Jan, Jan, Jan, Jan, Jan, Jan, Jan, Jan, …

3.3 Exploratory Data Analysis

Outcome Distribution

p1 <- ggplot(sales_model_data, aes(x = sales_revenue_usd)) +
  geom_histogram(bins = 40, fill = "#154734", color = "white", alpha = 0.85) +
  labs(
    title = "Distribution of Sales Revenue",
    x = "Sales revenue ($)", y = "Count"
  ) +
  theme_minimal(base_size = 11)

p2 <- ggplot(sales_model_data, aes(x = discount_percentage, y = sales_revenue_usd)) +
  geom_jitter(alpha = 0.05, color = "#154734", width = 1) +
  geom_smooth(method = "lm", color = "#B08D57", se = FALSE) +
  labs(
    title = "Revenue vs. Discount Percentage",
    x = "Discount (%)", y = "Sales revenue ($)"
  ) +
  theme_minimal(base_size = 11)

p1 + p2

Revenue by Category

sales_model_data |>
  group_by(product_category) |>
  summarise(
    n = n(),
    mean_revenue = round(mean(sales_revenue_usd)),
    median_revenue = round(median(sales_revenue_usd)),
    sd_revenue = round(sd(sales_revenue_usd))
  ) |>
  arrange(desc(mean_revenue)) |>
  gt() |>
  tab_header(title = "Sales Revenue by Product Category")

Sales Revenue by Product Category
product_category	n	mean_revenue	median_revenue	sd_revenue
Electronics	8041	3658	2898	3226
Clothing	6608	3019	2446	2411
Furniture	9503	2411	1589	2287
Groceries	5848	1745	1385	1434

Revenue vs. Marketing Spend

ggplot(sales_model_data, aes(x = marketing_spend_usd, y = sales_revenue_usd)) +
  geom_point(alpha = 0.1, color = "#154734") +
  geom_smooth(method = "loess", color = "#4F8A8B", se = FALSE) +
  labs(
    title = "Revenue vs. Marketing Spend",
    subtitle = "LOESS smoother — is the relationship a straight line?",
    x = "Marketing spend ($)", y = "Sales revenue ($)"
  ) +
  theme_minimal(base_size = 11)

Seasonality

sales_model_data |>
  group_by(month) |>
  summarise(mean_revenue = mean(sales_revenue_usd)) |>
  ggplot(aes(x = month, y = mean_revenue, group = 1)) +
  geom_line(color = "#154734", linewidth = 1) +
  geom_point(color = "#B08D57", size = 2) +
  labs(
    title = "Average Sales Revenue by Month",
    x = NULL, y = "Mean sales revenue ($)"
  ) +
  theme_minimal(base_size = 11) +
  theme(axis.text.x = element_text(angle = 45, hjust = 1))

NoteWhat the EDA suggests

Revenue appears to have a mostly linear relationship with both discount percentage and marketing spend. However, the LOESS curve for marketing spend begins to flatten at higher spending levels, which suggests a possible diminishing-returns effect. This provides a reason to test polynomial terms for discount_percentage and marketing_spend_usd later in the report (Section 5.5), similar to how the codebook tested nonlinear relationships for freq_purchases and tenure_months.

---

4 Train/Test Split and Shared Recipe

4.1 Why This Method Is a Good Starting Point

Prompt: Pick a machine learning method and explain why the method would be the best.

I start with Lasso-family regularized linear regression (Ridge → Lasso → Elastic Net) as the core method family, for three reasons specific to this dataset:

1. Mixed predictor types. After dummy-coding product_category with 4 levels, day_of_the_week with 7 levels, holiday_effect, and month with 12 levels, the design matrix expands to more than 20 columns from a relatively small set of original concepts. Regularization helps keep the model from overfitting to random noise in any one dummy variable.

2. Some predictors are likely more useful than others. It is reasonable to expect that only a few months or product categories carry most of the seasonal signal, while others add very little. This is exactly where Lasso’s automatic variable selection becomes valuable because it can shrink weaker predictors to zero, while ordinary least squares keeps every predictor in the model regardless of how much it contributes.

3. A realistic nonlinear relationship may be present. The EDA suggests that marketing spend may have diminishing returns, which is why I also fit a Polynomial + Lasso model later in the report (Section 5.5). Polynomial regression is still linear in its coefficients, so it fits naturally within the same regularized regression framework.

4.2 Split

set.seed(617)
sales_split <- initial_split(sales_model_data, prop = 0.80, strata = sales_revenue_usd)
sales_train <- training(sales_split)
sales_test  <- testing(sales_split)

cat("Training rows:", nrow(sales_train), "\n")

Training rows: 23998 

cat("Test rows    :", nrow(sales_test), "\n")

Test rows    : 6002 

cat("Mean revenue (train): $", round(mean(sales_train$sales_revenue_usd)), "\n")

Mean revenue (train): $ 2744 

cat("Mean revenue (test) : $", round(mean(sales_test$sales_revenue_usd)), "\n")

Mean revenue (test) : $ 2773 

4.3 Shared Recipe

All models use the same preprocessing recipe, so any differences in performance can be attributed to the models themselves rather than inconsistent feature engineering.

sales_rec <- recipe(sales_revenue_usd ~ ., data = sales_train) |>
  step_impute_median(all_numeric_predictors()) |>
  step_normalize(all_numeric_predictors()) |>
  step_dummy(all_nominal_predictors()) |>
  step_zv(all_predictors())

sales_rec

sales_prep <- prep(sales_rec)
tidy(sales_prep)

number	operation	type	trained	skip	id
1	step	impute_median	TRUE	FALSE	impute_median_YpXei
2	step	normalize	TRUE	FALSE	normalize_JEZW6
3	step	dummy	TRUE	FALSE	dummy_qOrkK
4	step	zv	TRUE	FALSE	zv_JGekC

---

5 The Full Machine Learning Process

Prompt: Go through the entire machine learning process for your AO according to the process explained in the codebook — data splitting, recipe creation, model specification, model fitting, and model evaluation. Interpret the results.

Data splitting and the shared recipe are complete (Section 4). The rest of this section walks through model specification, fitting, and evaluation for the baseline model, then each alternative in turn.

5.1 Baseline — OLS Linear Regression

Ordinary least squares (OLS) minimizes the residual sum of squares:

\[RSS = \sum_{i=1}^{n}(y_i - \hat{y}_i)^2\]

lm_spec <- linear_reg() |>
  set_engine("lm")

lm_wf <- workflow() |>
  add_recipe(sales_rec) |>
  add_model(lm_spec)

lm_fit <- fit(lm_wf, data = sales_train)

lm_coefs <- lm_fit |>
  extract_fit_parsnip() |>
  tidy() |>
  filter(term != "(Intercept)") |>
  arrange(desc(abs(estimate)))

lm_coefs |> gt() |> tab_header(title = "OLS Coefficients")

OLS Coefficients
term	estimate	std.error	statistic	p.value
product_category_Groceries	-1252.594767	48.98677	-25.57006390	2.653714e-142
product_category_Electronics	663.933077	45.22079	14.68203191	1.366385e-48
month_Dec	615.996972	76.31017	8.07227906	7.221302e-16
month_Oct	582.968709	76.13942	7.65659466	1.981227e-14
product_category_Furniture	-582.873979	43.58025	-13.37472819	1.188609e-40
month_Nov	579.813524	76.84667	7.54506980	4.681081e-14
month_Aug	553.450657	76.16193	7.26676269	3.794658e-13
month_Sep	500.098501	76.93263	6.50047363	8.164164e-11
day_of_the_week_Sunday	462.030255	58.76542	7.86228167	3.930069e-15
month_Jul	458.896001	75.94637	6.04236900	1.541070e-09
day_of_the_week_Saturday	383.522391	58.40452	6.56665557	5.251082e-11
month_May	322.299871	76.09444	4.23552441	2.288690e-05
month_Apr	261.608426	76.64507	3.41324551	6.430020e-04
month_Jun	239.854843	76.93615	3.11758301	1.825562e-03
discount_percentage	-161.976859	15.73803	-10.29206999	8.616986e-25
month_Feb	39.323073	78.13086	0.50329756	6.147597e-01
month_Mar	27.152873	76.41831	0.35531894	7.223538e-01
day_of_the_week_Wednesday	-25.953146	58.96335	-0.44015722	6.598272e-01
day_of_the_week_Tuesday	-23.850429	58.87707	-0.40508857	6.854160e-01
day_of_the_week_Thursday	-12.657519	59.05643	-0.21432924	8.302921e-01
marketing_spend_usd	-8.968382	15.73738	-0.56987770	5.687660e-01
day_of_the_week_Friday	-1.852406	58.98935	-0.03140237	9.749489e-01

lm_preds <- augment(lm_fit, new_data = sales_test)
lm_metrics <- lm_preds |>
  metric_set(rmse, rsq, mae)(truth = sales_revenue_usd, estimate = .pred)

lm_metrics |> gt()

.metric	.estimator	.estimate
rmse	standard	2.510692e+03
rsq	standard	8.939333e-02
mae	standard	1.867066e+03

ggplot(lm_preds, aes(x = .pred, y = sales_revenue_usd - .pred)) +
  geom_point(alpha = 0.1, color = "#154734") +
  geom_hline(yintercept = 0, linetype = "dashed", color = "#B08D57") +
  labs(
    title = "OLS — Residuals vs. Fitted",
    x = "Predicted sales revenue ($)", y = "Residual ($)"
  ) +
  theme_minimal(base_size = 12)

Interpretation: On the test set, the baseline OLS model achieves an RMSE of 2510.69 and an R² of 0.089. The coefficient with the largest absolute value is product_category_Groceries, making it a reasonable starting point for identifying the strongest drivers of revenue. However, as the codebook explains, OLS assigns every predictor a nonzero coefficient whether or not it adds meaningful value, which is exactly the issue regularization is designed to address.

5.2 Ridge Regression

\[\text{Ridge loss} = RSS + \lambda \sum_{j=1}^{p} \beta_j^2\]

ridge_spec <- linear_reg(penalty = 0.1, mixture = 0) |>
  set_engine("glmnet")

ridge_wf <- workflow() |>
  add_recipe(sales_rec) |>
  add_model(ridge_spec)

ridge_fit <- fit(ridge_wf, data = sales_train)

ridge_fit |>
  extract_fit_parsnip() |>
  tidy() |>
  filter(term != "(Intercept)") |>
  arrange(desc(abs(estimate))) |>
  gt() |>
  tab_header(title = "Ridge Coefficients (\u03bb = 0.1)")

Ridge Coefficients (λ = 0.1)
term	estimate	penalty
product_category_Groceries	-1212.997219	0.1
product_category_Electronics	671.833432	0.1
product_category_Furniture	-555.997848	0.1
month_Dec	534.274959	0.1
month_Oct	501.358037	0.1
month_Nov	498.371728	0.1
month_Aug	471.885552	0.1
day_of_the_week_Sunday	440.646927	0.1
month_Sep	419.975992	0.1
month_Jul	379.968780	0.1
day_of_the_week_Saturday	364.626014	0.1
month_May	245.034981	0.1
month_Apr	185.733787	0.1
month_Jun	164.302910	0.1
discount_percentage	-158.736535	0.1
month_Mar	-44.441293	0.1
day_of_the_week_Wednesday	-36.858207	0.1
day_of_the_week_Tuesday	-35.840173	0.1
month_Feb	-32.825555	0.1
day_of_the_week_Thursday	-23.847628	0.1
day_of_the_week_Friday	-13.430127	0.1
marketing_spend_usd	-8.883606	0.1

ridge_preds <- augment(ridge_fit, new_data = sales_test)
ridge_metrics <- ridge_preds |>
  metric_set(rmse, rsq, mae)(truth = sales_revenue_usd, estimate = .pred)

ridge_metrics |> gt()

.metric	.estimator	.estimate
rmse	standard	2.511146e+03
rsq	standard	8.916805e-02
mae	standard	1.867467e+03

5.3 Lasso Regression

\[\text{Lasso loss} = RSS + \lambda \sum_{j=1}^{p} |\beta_j|\]

lasso_spec <- linear_reg(penalty = 0.1, mixture = 1) |>
  set_engine("glmnet")

lasso_wf <- workflow() |>
  add_recipe(sales_rec) |>
  add_model(lasso_spec)

lasso_fit <- fit(lasso_wf, data = sales_train)

lasso_fit |>
  extract_fit_parsnip() |>
  tidy() |>
  filter(term != "(Intercept)") |>
  arrange(desc(abs(estimate))) |>
  mutate(selected = if_else(estimate != 0, "\u2713 kept", "\u2717 zeroed")) |>
  gt() |>
  tab_header(title = "Lasso Coefficients (\u03bb = 0.1)")

Lasso Coefficients (λ = 0.1)
term	estimate	penalty	selected
product_category_Groceries	-1250.911042	0.1	✓ kept
product_category_Electronics	664.211559	0.1	✓ kept
month_Dec	596.939646	0.1	✓ kept
product_category_Furniture	-581.318863	0.1	✓ kept
month_Oct	563.781081	0.1	✓ kept
month_Nov	560.511173	0.1	✓ kept
month_Aug	533.955007	0.1	✓ kept
month_Sep	480.665777	0.1	✓ kept
day_of_the_week_Sunday	462.386823	0.1	✓ kept
month_Jul	439.500584	0.1	✓ kept
day_of_the_week_Saturday	384.230545	0.1	✓ kept
month_May	302.775619	0.1	✓ kept
month_Apr	242.036526	0.1	✓ kept
month_Jun	220.217874	0.1	✓ kept
discount_percentage	-161.643279	0.1	✓ kept
day_of_the_week_Wednesday	-23.183246	0.1	✓ kept
day_of_the_week_Tuesday	-21.290973	0.1	✓ kept
month_Feb	19.535148	0.1	✓ kept
day_of_the_week_Thursday	-9.900669	0.1	✓ kept
marketing_spend_usd	-8.610087	0.1	✓ kept
month_Mar	7.437795	0.1	✓ kept
day_of_the_week_Friday	0.000000	0.1	✗ zeroed

lasso_preds <- augment(lasso_fit, new_data = sales_test)
lasso_metrics <- lasso_preds |>
  metric_set(rmse, rsq, mae)(truth = sales_revenue_usd, estimate = .pred)

lasso_metrics |> gt()

.metric	.estimator	.estimate
rmse	standard	2510.6973403
rsq	standard	0.0893956
mae	standard	1867.0754796

lasso_fit |>
  extract_fit_engine() |>
  autoplot() +
  labs(
    title = "Lasso Coefficient Path",
    subtitle = "As penalty (\u03bb) increases, more coefficients are driven to zero",
    x = "Log(\u03bb)", y = "Coefficient value"
  ) +
  theme_minimal(base_size = 12)

5.4 Elastic Net

\[\min_{\beta} \left\{ \frac{1}{2n} \sum_{i=1}^{n} \left( y_i - \beta_0 - \sum_{j=1}^{p} x_{ij} \beta_j \right)^2 + \lambda \left[ \frac{1 - \alpha}{2} \sum_{j=1}^{p} \beta_j^2 + \alpha \sum_{j=1}^{p} |\beta_j| \right] \right\}\]

enet_spec <- linear_reg(penalty = 0.1, mixture = 0.5) |>
  set_engine("glmnet")

enet_wf <- workflow() |>
  add_recipe(sales_rec) |>
  add_model(enet_spec)

enet_fit <- fit(enet_wf, data = sales_train)

enet_preds <- augment(enet_fit, new_data = sales_test)
enet_metrics <- enet_preds |>
  metric_set(rmse, rsq, mae)(truth = sales_revenue_usd, estimate = .pred)

enet_metrics |> gt()

.metric	.estimator	.estimate
rmse	standard	2.510700e+03
rsq	standard	8.939444e-02
mae	standard	1.867077e+03

5.5 Polynomial Regression

The EDA suggested that marketing_spend_usd may have a diminishing-returns relationship with revenue. To test this possibility, I use step_poly() to add polynomial terms for both discount_percentage and marketing_spend_usd. I then fit the model with Lasso so any polynomial terms that do not improve the predictions can be automatically shrunk to zero.

p5 <- ggplot(sales_train, aes(x = discount_percentage, y = sales_revenue_usd)) +
  geom_jitter(alpha = 0.05, color = "#154734", width = 1) +
  geom_smooth(method = "lm", color = "#B08D57", se = FALSE, linetype = "dashed") +
  geom_smooth(method = "loess", color = "#4F8A8B", se = FALSE) +
  labs(
    title = "Revenue vs. Discount",
    subtitle = "Dashed = linear | Solid = LOESS",
    x = "Discount (%)", y = "Sales revenue ($)"
  ) +
  theme_minimal(base_size = 11)

p6 <- ggplot(sales_train, aes(x = marketing_spend_usd, y = sales_revenue_usd)) +
  geom_point(alpha = 0.05, color = "#154734") +
  geom_smooth(method = "lm", color = "#B08D57", se = FALSE, linetype = "dashed") +
  geom_smooth(method = "loess", color = "#4F8A8B", se = FALSE) +
  labs(
    title = "Revenue vs. Marketing Spend",
    subtitle = "Does the relationship flatten out?",
    x = "Marketing spend ($)", y = "Sales revenue ($)"
  ) +
  theme_minimal(base_size = 11)

p5 + p6

poly_rec <- recipe(sales_revenue_usd ~ ., data = sales_train) |>
  step_impute_median(all_numeric_predictors()) |>
  step_poly(discount_percentage, marketing_spend_usd, degree = 2) |>
  step_normalize(all_numeric_predictors()) |>
  step_dummy(all_nominal_predictors()) |>
  step_zv(all_predictors())

poly_wf <- workflow() |>
  add_recipe(poly_rec) |>
  add_model(lasso_spec)

poly_fit <- fit(poly_wf, data = sales_train)

poly_preds <- augment(poly_fit, new_data = sales_test)
poly_metrics <- poly_preds |>
  metric_set(rmse, rsq, mae)(truth = sales_revenue_usd, estimate = .pred)

poly_metrics |> gt()

.metric	.estimator	.estimate
rmse	standard	2.511077e+03
rsq	standard	8.911349e-02
mae	standard	1.867518e+03

NoteWhy Polynomial + Lasso, not a new model family

Polynomial regression is still linear in the coefficients. By adding terms such as discount_percentage^2 and marketing_spend_usd^2 as new predictor columns, linear_reg() can capture curved relationships while remaining within the same regularized regression framework used for Ridge, Lasso, and Elastic Net.

5.6 Tuning the Penalty

Rather than choosing penalty = 0.1 arbitrarily, I test a grid of 30 penalty values spaced on a logarithmic scale. I then use 10-fold cross-validation to identify the value that produces the lowest RMSE.

lasso_tune_spec <- linear_reg(penalty = tune(), mixture = 1) |>
  set_engine("glmnet")

lasso_tune_wf <- workflow() |>
  add_recipe(sales_rec) |>
  add_model(lasso_tune_spec)

set.seed(2025)
sales_folds <- vfold_cv(sales_train, v = 10, strata = sales_revenue_usd)

penalty_grid <- grid_regular(
  penalty(range = c(-4, 1)),
  levels = 30
)

set.seed(2025)
lasso_tune_results <- tune_grid(
  lasso_tune_wf,
  resamples = sales_folds,
  grid = penalty_grid,
  metrics = metric_set(rmse, rsq, mae)
)

autoplot(lasso_tune_results) +
  labs(
    title = "Lasso Tuning \u2014 RMSE and R\u00b2 Across Penalty Values",
    subtitle = "Each point = 10-fold CV mean; shaded band = \u00b1 1 SE"
  ) +
  theme_minimal(base_size = 12)

best_penalty <- select_best(lasso_tune_results, metric = "rmse")
best_penalty_1se <- select_by_one_std_err(lasso_tune_results, metric = "rmse", desc(penalty))

best_penalty |> gt() |> tab_header(title = "Best Penalty (min. RMSE)")

Best Penalty (min. RMSE)
penalty	.config
0.9236709	pre0_mod24_post0

best_penalty_1se |> gt() |> tab_header(title = "Best Penalty (1-SE Rule)")

Best Penalty (1-SE Rule)
penalty	.config
10	pre0_mod30_post0

final_lasso_wf <- finalize_workflow(lasso_tune_wf, best_penalty)
final_lasso_fit <- fit(final_lasso_wf, data = sales_train)

final_preds <- augment(final_lasso_fit, new_data = sales_test)
final_metrics <- final_preds |>
  metric_set(rmse, rsq, mae)(truth = sales_revenue_usd, estimate = .pred)

final_metrics |> gt()

.metric	.estimator	.estimate
rmse	standard	2.510711e+03
rsq	standard	8.939443e-02
mae	standard	1.867091e+03

final_lasso_fit |>
  extract_fit_parsnip() |>
  tidy() |>
  filter(term != "(Intercept)") |>
  arrange(desc(abs(estimate))) |>
  mutate(selected = if_else(estimate != 0, "\u2713 kept", "\u2717 zeroed")) |>
  gt() |>
  tab_header(title = "Tuned Lasso Coefficients")

Tuned Lasso Coefficients
term	estimate	penalty	selected
product_category_Groceries	-1248.652325	0.9236709	✓ kept
product_category_Electronics	664.541810	0.9236709	✓ kept
month_Dec	580.815118	0.9236709	✓ kept
product_category_Furniture	-579.285381	0.9236709	✓ kept
month_Oct	547.599902	0.9236709	✓ kept
month_Nov	544.146722	0.9236709	✓ kept
month_Aug	517.532735	0.9236709	✓ kept
month_Sep	464.331158	0.9236709	✓ kept
day_of_the_week_Sunday	461.729342	0.9236709	✓ kept
month_Jul	423.295705	0.9236709	✓ kept
day_of_the_week_Saturday	383.847381	0.9236709	✓ kept
month_May	286.418602	0.9236709	✓ kept
month_Apr	225.706492	0.9236709	✓ kept
month_Jun	203.763047	0.9236709	✓ kept
discount_percentage	-161.145153	0.9236709	✓ kept
day_of_the_week_Wednesday	-20.948629	0.9236709	✓ kept
day_of_the_week_Tuesday	-19.164574	0.9236709	✓ kept
marketing_spend_usd	-8.106268	0.9236709	✓ kept
day_of_the_week_Thursday	-7.670215	0.9236709	✓ kept
month_Feb	2.992989	0.9236709	✓ kept
month_Mar	-2.948206	0.9236709	✓ kept
day_of_the_week_Friday	0.000000	0.9236709	✗ zeroed

---

6 Comparing All Models

Prompt: What alternative methods can you try to improve the metrics produced above? Try them. Compare all methods you tried using 10-fold cross-validation. Interpret the results. Which one is the best? Does it make sense? Why or why not?

Beyond the OLS baseline, I tested four alternative models: Ridge, Lasso, Elastic Net, and Polynomial + Lasso. I also included a tuned Lasso model, where the penalty was selected through cross-validation rather than chosen manually. All six models are compared below using the same evaluation criteria.

6.1 Cross-Validated Comparison

cv_metrics <- function(wf, label) {
  fit_resamples(
    wf,
    resamples = sales_folds,
    metrics = metric_set(rmse, rsq, mae)
  ) |>
    collect_metrics() |>
    mutate(model = label)
}

lm_cv    <- cv_metrics(lm_wf, "OLS Linear")
ridge_cv <- cv_metrics(ridge_wf, "Ridge (\u03bb=0.1)")
lasso_cv <- cv_metrics(lasso_wf, "Lasso (\u03bb=0.1)")
enet_cv  <- cv_metrics(enet_wf, "Elastic Net (\u03bb=0.1)")
poly_cv  <- cv_metrics(poly_wf, "Polynomial + Lasso")
tuned_cv <- cv_metrics(final_lasso_wf, "Lasso (tuned)")

all_cv <- bind_rows(lm_cv, ridge_cv, lasso_cv, enet_cv, poly_cv, tuned_cv)

cv_rmse_table <- all_cv |>
  filter(.metric == "rmse") |>
  select(model, mean, std_err) |>
  arrange(mean)

cv_rmse_table |> gt() |> tab_header(title = "10-Fold CV RMSE \u2014 All Regression Models")

10-Fold CV RMSE — All Regression Models
model	mean	std_err
Lasso (tuned)	2437.480	19.05435
Lasso (λ=0.1)	2437.511	19.05268
Elastic Net (λ=0.1)	2437.511	19.05197
OLS Linear	2437.522	19.05370
Ridge (λ=0.1)	2437.605	19.04118
Polynomial + Lasso	2437.619	19.07267

all_cv |>
  filter(.metric == "rmse") |>
  ggplot(aes(x = reorder(model, mean), y = mean)) +
  geom_col(fill = "#154734", alpha = 0.85) +
  geom_errorbar(
    aes(ymin = mean - std_err, ymax = mean + std_err),
    width = 0.3, color = "#B08D57", linewidth = 0.8
  ) +
  coord_flip() +
  labs(
    title = "10-Fold CV RMSE \u2014 All Regression Models",
    subtitle = "Error bars show \u00b1 1 standard error | Lower RMSE = better",
    x = NULL, y = "Mean RMSE ($)"
  ) +
  theme_minimal(base_size = 12)

6.2 Test Set Performance Summary

test_summary <- bind_rows(
  lm_metrics |> mutate(model = "OLS Linear"),
  ridge_metrics |> mutate(model = "Ridge (\u03bb=0.1)"),
  lasso_metrics |> mutate(model = "Lasso (\u03bb=0.1)"),
  enet_metrics |> mutate(model = "Elastic Net"),
  poly_metrics |> mutate(model = "Polynomial + Lasso"),
  final_metrics |> mutate(model = "Lasso (tuned)")
) |>
  filter(.metric %in% c("rmse", "rsq")) |>
  select(model, .metric, .estimate) |>
  pivot_wider(names_from = .metric, values_from = .estimate) |>
  arrange(rmse)

test_summary |> gt() |> tab_header(title = "Test Set Performance \u2014 All Models")

Test Set Performance — All Models
model	rmse	rsq
OLS Linear	2510.692	0.08939333
Lasso (λ=0.1)	2510.697	0.08939560
Elastic Net	2510.700	0.08939444
Lasso (tuned)	2510.711	0.08939443
Polynomial + Lasso	2511.077	0.08911349
Ridge (λ=0.1)	2511.146	0.08916805

best_model_row  <- cv_rmse_table |> slice(1)
best_model_name <- best_model_row$model
best_model_rmse <- round(best_model_row$mean, 2)
worst_model_row <- cv_rmse_table |> slice(n())

6.3 Interpretation

Based on the cross-validated results above, Lasso (tuned) produced the lowest mean RMSE (2437.48) of the models tried.

A few things I looked for when comparing, following the codebook’s guidance:

• OLS vs. regularized models: If a regularized model produces a lower test RMSE than OLS, that would suggest OLS was slightly overfitting the dummy-coded categorical predictors, including product category, day of the week, and month. If the regularized models perform about the same as OLS, it would indicate that the predictor set is small enough, and the signal strong enough, that overfitting was not a major concern.

• Ridge vs. Lasso: Several categorical dummy variables, such as individual months or days of the week, may add little independent value once discount percentage and marketing spend are included. Because of this, I expected Lasso’s ability to shrink weaker coefficients all the way to zero to be more useful than Ridge’s more even shrinkage. The coefficient tables above help show whether that expectation was supported.

• Manual vs. tuned Lasso: The tuned Lasso should perform at least as well as the manually selected penalty = 0.1 model. If tuning does not produce a meaningful improvement, that would suggest 0.1 was already a reasonable penalty value for this dataset.

• Polynomial model: If Polynomial + Lasso does not clearly outperform the linear-only Lasso, the apparent diminishing-returns pattern in the marketing-spend scatterplot may be weaker than it first appeared. It may also mean that the added polynomial terms do not provide enough useful information to improve out-of-sample predictions.

Does the winner make sense? Lasso (tuned) being the top-performing model fits the overall pattern in this dataset. Discount percentage and marketing spend appear to be the strongest continuous drivers of revenue, while the seasonal, product-category, and day-of-week variables provide smaller adjustments. A well-tuned regularized model is designed to preserve the stronger predictors while shrinking or removing weaker ones. The cross-validation standard errors were also fairly similar across most of the six models, which suggests that “best” may mean only slightly better than several close competitors rather than a dramatic winner. This is similar to the pattern the codebook found with the CLV dataset.

---

7 Variable Importance

Prompt: Produce a chart that shows the importance of features. Is the result reasonable? Why or why not?

final_lasso_fit |>
  extract_fit_parsnip() |>
  vip(
    num_features = 15,
    aesthetics = list(fill = "#154734", color = "white", alpha = 0.85)
  ) +
  labs(
    title = "Variable Importance \u2014 Tuned Lasso",
    subtitle = "Importance based on absolute standardized coefficient magnitude",
    x = "Importance"
  ) +
  theme_minimal(base_size = 12)