# High Dimensional Fixed Effects with Rossman Sales Data

Intro

This tutorial demonstrates how to create models with high dimensional fixed effects using glum. Using tabmat, we can pass categorical variables with a large range of values. glum and tabmat will handle the creation of the one-hot-encoded design matrix.

In some real-world problems, we have used millions of categories. This would be impossible with a dense matrix. General-purpose sparse matrices like compressed sparse row (CSR) matrices help but still leave a lot on the table. For a categorical matrix, we know that each row has only a single non-zero value and that value is 1. These optimizations are implemented in tabmat.CategoricalMatrix.

Background

For this tutorial, we will be predicting sales for the European drug store chain Rossman. Specifically, we are tasked with predicting daily sales for future dates. Ideally, we want a model that can capture the many factors that influence stores sales – promotions, competition, school, holidays, seasonality, etc. As a baseline, we will start with a simple model that only uses a few basic predictors. Then, we will fit a model with a large number of fixed effects. For both models, we will use OLS with L2 regularization.

We will use a gamma distribution for our model. This choice is motivated by two main factors. First, our target variable, sales, is a positive real number, which matches the support of the gamma distribution. Second, it is expected that factors influencing sales are multiplicative rather than additive, which is better captured with a gamma regression than say, OLS.

Note: a few parts of this tutorial utilize local helper functions outside this notebook. If you wish to run the notebook on your own, you can find the rest of the code here.

:

import os
from pathlib import Path

import altair as alt
import matplotlib.pyplot as plt
import numpy as np
import pandas as pd

from glum import GeneralizedLinearRegressor
from sklearn.pipeline import Pipeline

from feature_engineering import apply_all_transformations

import sys
sys.path.append("../")
from metrics import root_mean_squared_percentage_error

pd.set_option("display.float_format", lambda x: "%.3f" % x)
pd.set_option('display.max_columns', None)
alt.data_transformers.enable("json")  # to allow for large plots

:

DataTransformerRegistry.enable('json')


We start by loading in the raw data. If you have not yet processed the raw data, it will be done below. (Initial processing consists of some basic cleaning and renaming of columns.

Note: if you wish to run this notebook on your own, and have not done so already, please download the data from the Rossman Kaggle Challenge. This tutorial expects that it in a folder named “raw_data” under the same directory as the notebook.

:

if not all(Path(p).exists() for p in ["raw_data/train.csv", "raw_data/test.csv", "raw_data/store.csv"]):

if not all(Path(p).exists() for p in ["processed_data/train.parquet", "processed_data/test.parquet"]):
process_data()
"Done"

df = df.iloc[:int(.1*len(df))]

:

store day_of_week date sales customers open promo state_holiday school_holiday year month store_type assortment competition_distance competition_open_since_month competition_open_since_year promo2 promo2_since_week promo2_since_year promo_interval
1016095 1 2 2013-01-01 0 0 False 0 a 1 2013 1 c a 1270.000 9.000 2008.000 0 NaN NaN None
1014980 1 3 2013-01-02 5530 668 True 0 0 1 2013 1 c a 1270.000 9.000 2008.000 0 NaN NaN None
1013865 1 4 2013-01-03 4327 578 True 0 0 1 2013 1 c a 1270.000 9.000 2008.000 0 NaN NaN None
1012750 1 5 2013-01-04 4486 619 True 0 0 1 2013 1 c a 1270.000 9.000 2008.000 0 NaN NaN None
1011635 1 6 2013-01-05 4997 635 True 0 0 1 2013 1 c a 1270.000 9.000 2008.000 0 NaN NaN None

### 1.2 Feature engineering

As mentioned earlier, we want our model to incorporate many factors that could influence store sales. We create a number of fixed effects to capture this information. These include fixed effects for:

• A certain number days before a school or state holiday

• A certain number days after a school or state holiday

• A certain number days before a promo

• A certain number days after a promo

• A certain number days before the store is open or closed

• A certain number days after the store is open or closed

• Each month for each store

• Each year for each store

• Each day of the week for each store

We also do several other transformations like computing the z score to eliminate outliers (in the next step)

:

df = apply_all_transformations(df)

:

1016095 1 2 2013-01-01 0 0 False 0 a 1 2013 1 c a 1270.000 9.000 2008.000 0 NaN NaN None -1 1.000 0 1.0 1.0 1.0 True True True 0.000 0.000 0.000 0.000 0.000 0.000 0.000 0.000 0.000 1.000 1.000 1.000 0 0 0 0 0 0 1_2 1_1 1_1 1_True 1_2013 NaN
1014980 1 3 2013-01-02 5530 668 True 0 0 1 2013 1 c a 1270.000 9.000 2008.000 0 NaN NaN None -1 1.000 1 False 1.0 1.0 True True True 0.000 0.000 0.000 0.000 0.000 0.000 1.000 0.000 0.000 1.000 1.000 1.000 a 0 0 0 0 0 1_3 1_1 1_1 1_False 1_2013 NaN
1013865 1 4 2013-01-03 4327 578 True 0 0 1 2013 1 c a 1270.000 9.000 2008.000 0 NaN NaN None -1 1.000 2 True False 1.0 True True False 0.000 0.000 0.000 0.000 0.000 0.000 1.000 1.000 0.000 1.000 1.000 1.000 0 a 0 0 0 0 1_4 1_1 1_1 1_False 1_2013 NaN
1012750 1 5 2013-01-04 4486 619 True 0 0 1 2013 1 c a 1270.000 9.000 2008.000 0 NaN NaN None -1 1.000 3 True True False True False True 0.000 0.000 0.000 0.000 0.000 1.000 1.000 1.000 1.000 1.000 1.000 1.000 0 0 a 0 0 0 1_5 1_1 1_1 1_False 1_2013 NaN
1011635 1 6 2013-01-05 4997 635 True 0 0 1 2013 1 c a 1270.000 9.000 2008.000 0 NaN NaN None -1 1.000 4 True True True False True True 0.000 0.000 0.000 0.000 1.000 1.000 1.000 1.000 1.000 1.000 1.000 1.000 0 0 0 0 0 0 1_6 1_1 1_1 1_False 1_2013 NaN

### 1.3 Train vs. validation selection

Lastly, we split our data into training and validation sets. Kaggle provides a test set for the Rossman challenge, but it does not directly include outcome data (sales), so we do not use it for our tutorial. Instead, we simulate predicting future sales by taking the last 5 months of our training data as our validation set.

:

validation_window = [pd.to_datetime("2015-03-15"), pd.to_datetime("2015-07-31")]
select_train = (df["sales"].gt(0) & df["date"].lt(validation_window) & df["zscore"].abs().lt(5)).to_numpy()

select_val = (
df["sales"].gt(0)
& df["date"].ge(validation_window)
& df["date"].lt(validation_window)
).to_numpy()

(select_train.sum(), select_val.sum())

:

(57876, 12502)


## 2. Fit baseline GLM

We start with a simple model that uses only year, month, day of the week, and store as predictors. Even with these variables alone, we should still be able to capture a lot of valuable information. Year can capture overall sales trends, month can capture seasonality, week day can capture the variation in sales across the week, and store can capture locality. We will treat these all as categorical variables.

With the GeneralizedLinearRegressor() class, we can pass in pandas.Categorical variables directly without having to encode them ourselves. This is convenient, especially when we start adding more fixed effects. But it is very important that the categories are aligned between calls to fit and predict. One way of achieving this alignment is with a dask_ml.preprocessing.Categorizer. Note, however, that the Categorizer class fails to enforce category alignment if the input column is already a categorical data type.

You can reference the pandas documentation on Categoricals to learn more about how these data types work.

:

baseline_features = ["year", "month", "day_of_week", "store"]
baseline_categorizer = Categorizer(columns=baseline_features)
baseline_glm = GeneralizedLinearRegressor(
family="gamma",
scale_predictors=True,
l1_ratio=0.0,
alphas=1e-1,
)


Fit the model making sure to process the data frame with the Categorizer first and inspect the coefficients.

:

baseline_glm.fit(
baseline_categorizer.fit_transform(df[select_train][baseline_features]),
df.loc[select_train, "sales"]
)

pd.DataFrame(
{'coefficient': np.concatenate(([baseline_glm.intercept_], baseline_glm.coef_))},
index=['intercept'] + baseline_glm.feature_names_
).T

:

intercept year__2013 year__2014 year__2015 month__1 month__2 month__3 month__4 month__5 month__6 month__7 month__8 month__9 month__10 month__11 month__12 day_of_week__1 day_of_week__2 day_of_week__3 day_of_week__4 day_of_week__5 day_of_week__6 day_of_week__7 store__1 store__2 store__3 store__4 store__5 store__6 store__7 store__8 store__9 store__10 store__11 store__12 store__13 store__14 store__15 store__16 store__17 store__18 store__19 store__20 store__21 store__22 store__23 store__24 store__25 store__26 store__27 store__28 store__29 store__30 store__31 store__32 store__33 store__34 store__35 store__36 store__37 store__38 store__39 store__40 store__41 store__42 store__43 store__44 store__45 store__46 store__47 store__48 store__49 store__50 store__51 store__52 store__53 store__54 store__55 store__56 store__57 store__58 store__59 store__60 store__61 store__62 store__63 store__64 store__65 store__66 store__67 store__68 store__69 store__70 store__71 store__72 store__73 store__74 store__75 store__76 store__77 store__78 store__79 store__80 store__81 store__82 store__83 store__84 store__85 store__86 store__87 store__88 store__89 store__90 store__91 store__92 store__93 store__94 store__95 store__96 store__97 store__98 store__99 store__100 store__101 store__102 store__103 store__104 store__105 store__106 store__107 store__108 store__109 store__110 store__111 store__112
coefficient 8.781 -0.010 0.008 0.003 -0.015 -0.021 -0.019 0.020 0.008 -0.006 -0.002 -0.023 -0.034 -0.030 0.022 0.116 0.098 0.010 -0.014 -0.014 0.017 -0.100 0.285 -0.156 -0.136 0.024 0.202 -0.161 -0.087 0.158 -0.094 0.002 -0.081 0.110 0.074 -0.126 -0.088 0.009 0.082 -0.020 0.001 -0.008 0.085 -0.088 -0.178 -0.088 0.183 0.274 0.008 0.192 -0.098 0.059 -0.101 -0.051 -0.225 0.125 0.102 0.198 0.185 0.046 -0.048 -0.153 -0.128 -0.085 0.234 0.017 -0.086 -0.101 -0.094 0.039 -0.243 0.058 -0.208 0.012 0.079 -0.109 0.099 -0.180 0.047 0.259 -0.017 -0.091 0.102 -0.149 -0.009 0.034 0.251 -0.122 -0.035 0.086 0.081 0.190 0.002 0.171 -0.189 -0.173 -0.023 -0.038 0.172 0.077 -0.266 -0.091 0.077 0.038 0.163 -0.248 0.369 0.019 -0.155 -0.060 -0.048 -0.047 0.130 -0.059 -0.047 -0.052 0.038 0.084 -0.121 0.007 -0.119 -0.111 0.108 0.069 0.038 -0.171 0.233 -0.175 0.136 0.058 0.284 -0.006 -0.174 0.018 -0.039

And let’s predict for our test set with the caveat that we will predict 0 for days when the stores are closed!

:

df.loc[lambda x: x["open"], "predicted_sales_baseline"] = baseline_glm.predict(
baseline_categorizer.fit_transform(df.loc[lambda x: x["open"]][baseline_features])
)

df["predicted_sales_baseline"] = df["predicted_sales_baseline"].fillna(0)
df["predicted_sales_baseline"] = df["predicted_sales_baseline"]


We use root mean squared percentage error (RMSPE) as our performance metric. (Useful for thinking about error relative to total sales of each store).

:

train_err = root_mean_squared_percentage_error(
df.loc[select_train, "sales"], df.loc[select_train, "predicted_sales_baseline"]
)
val_err = root_mean_squared_percentage_error(
df.loc[select_val, "sales"], df.loc[select_val, "predicted_sales_baseline"]
)
print(f'Training Error: {round(train_err, 2)}%')
print(f'Validation Error: {round(val_err, 2)}%')

Training Error: 27.96%
Validation Error: 29.57%


The results aren’t bad for a start, but we can do better :)

## 3. GLM with high dimensional fixed effects

Now, we repeat a similar process to above, but, this time, we take advantage of the full range of categoricals we created in our data transformation step. Since we will create a very large number of fixed effects, we may run into cases where our validation data has categorical values not seen in our training data. In these cases, Dask-ML’s Categorizer will output null values when transforming the validation columns to the categoricals that were created on the training set. To fix this, we add Dask-ML’s SimpleImputer to our pipeline.

:

highdim_features = [
"age_quantile",
"competition_open",
"open_lag_1",
"open_lag_2",
"open_lag_3",
"promo_lag_1",
"promo_lag_2",
"promo_lag_3",
"promo",
"school_holiday_lag_1",
"school_holiday_lag_2",
"school_holiday_lag_3",
"school_holiday",
"state_holiday_lag_1",
"state_holiday_lag_2",
"state_holiday_lag_3",
"state_holiday",
"store_day_of_week",
"store_month",
"store_school_holiday",
"store_state_holiday",
"store_year",
]
highdim_categorizer = Pipeline([
("categorize", Categorizer(columns=highdim_features)),
("impute", SimpleImputer(strategy="most_frequent"))
])
highdim_glm = GeneralizedLinearRegressor(
family="gamma",
scale_predictors=True,
l1_ratio=0.0, # only ridge
alpha=1e-1,
)


For reference, we output the total number of predictors after fitting the model. We can see that the number getting a bit larger, so we don’t print out the coefficients this time.

:

highdim_glm.fit(
highdim_categorizer.fit_transform(df[select_train][highdim_features]),
df.loc[select_train, "sales"]
)

print(f"Number of predictors: {len(highdim_glm.feature_names_)}")

Number of predictors: 2771

:

df.loc[lambda x: x["open"], "predicted_sales_highdim"] = highdim_glm.predict(
highdim_categorizer.transform(df.loc[lambda x: x["open"]][highdim_features]),
)

df["predicted_sales_highdim"] = df["predicted_sales_highdim"].fillna(0)
df["predicted_sales_highdim"] = df["predicted_sales_highdim"]

train_err = root_mean_squared_percentage_error(
df.loc[select_train, "sales"], df.loc[select_train, "predicted_sales_highdim"]
)
val_err = root_mean_squared_percentage_error(
df.loc[select_val, "sales"], df.loc[select_val, "predicted_sales_highdim"]
)
print(f'Training Error: {round(train_err, 2)}%')
print(f'Validation Error: {round(val_err, 2)}%')

Training Error: 12.54%
Validation Error: 13.28%


From just the RMSPE, we can see a clear improvement from our baseline model.

## 4. Plot results

Finally, to get a better look at our results, we make some plots.

:

sales_cols = ["sales", "predicted_sales_highdim", "predicted_sales_baseline"]


First, we plot true sales and the sales predictions from each model aggregated over month:

:

_, axs = plt.subplots(2, 1, figsize=(16, 16))

for i, select in enumerate([select_train, select_val]):
ax = axs[i]
df_plot_date = df[select].groupby(
["year", "month"]
).agg("sum")[sales_cols].reset_index()

year_month_date = df_plot_date['month'].map(str)+ '-' + df_plot_date['year'].map(str)
df_plot_date['year_month'] = pd.to_datetime(year_month_date, format='%m-%Y').dt.strftime('%m-%Y')
df_plot_date.drop(columns= ["year", "month"], inplace=True)

df_plot_date.plot(x="year_month", ax=ax)
ax.set_xticks(range(len(df_plot_date)))
ax.set_xticklabels(df_plot_date.year_month, rotation=45)
ax.set_xlabel("date")
ax.set_ylabel("Total sales")
ax.grid(True, linestyle='-.')

axs.set_title("Training Results: Total Sales per Month")
axs.set_title("Validation Results: Total Sales per Month")
plt.show() We can also look at aggregate sales for a subset of stores. We select the first 20 stores and plot in order of increasing sales.

:

_, axs = plt.subplots(2, 1, figsize=(14, 12))
for i, select in enumerate([select_train, select_val]):
ax = axs[i]
df_plot_store = df[select].groupby(
["store"]
).agg("sum")[sales_cols].reset_index()[:20].sort_values(by="sales")

df_plot_store.plot.bar(x="store", ax=ax)
ax.set_xlabel("Store")
ax.set_ylabel("Total sales")

axs.set_title("Training Results: Total Sales by Store")
axs.set_title("Validation Results: Total Sales by Store")
plt.show() We can see that the high dimensional model is much better at capturing the variation between months and individual stores!