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Bike-Sharing Dataset

The Bike-Sharing Dataset contains the hourly and daily count of rental bikes between years 2011 and 2012 in Capital bikeshare system with the corresponding weather and seasonal information. The dataset contains 14 features with information about the day-type, e.g., month, hour, which day of the week, whether it is working-day, and the weather conditions, e.g., temperature, humidity, wind speed, etc. The target variable is the number of bike rentals per hour. The dataset contains 17,379 instances. 

import effector
import numpy as np
import tensorflow as tf
from tensorflow import keras
import random

np.random.seed(42)
tf.random.set_seed(42)
random.seed(42)

2025-02-26 11:04:59.976580: I tensorflow/core/platform/cpu_feature_guard.cc:210] This TensorFlow binary is optimized to use available CPU instructions in performance-critical operations.
To enable the following instructions: AVX2 FMA, in other operations, rebuild TensorFlow with the appropriate compiler flags.

Preprocess the data

from ucimlrepo import fetch_ucirepo 

bike_sharing_dataset = fetch_ucirepo(id=275) 
X = bike_sharing_dataset.data.features 
y = bike_sharing_dataset.data.targets 

X = X.drop(["dteday", "atemp"], axis=1)

X.head()
season yr mnth hr holiday weekday workingday weathersit temp hum windspeed
0 1 0 1 0 0 6 0 1 0.24 0.81 0.0
1 1 0 1 1 0 6 0 1 0.22 0.80 0.0
2 1 0 1 2 0 6 0 1 0.22 0.80 0.0
3 1 0 1 3 0 6 0 1 0.24 0.75 0.0
4 1 0 1 4 0 6 0 1 0.24 0.75 0.0 
print("Design matrix shape: {}".format(X.shape))
print("---------------------------------")
for col_name in X.columns:
    print("Feature: {:15}, unique: {:4d}, Mean: {:6.2f}, Std: {:6.2f}, Min: {:6.2f}, Max: {:6.2f}".format(col_name, len(X[col_name].unique()), X[col_name].mean(), X[col_name].std(), X[col_name].min(), X[col_name].max()))

print("\nTarget shape: {}".format(y.shape))
print("---------------------------------")
for col_name in y.columns:
    print("Target: {:15}, unique: {:4d}, Mean: {:6.2f}, Std: {:6.2f}, Min: {:6.2f}, Max: {:6.2f}".format(col_name, len(y[col_name].unique()), y[col_name].mean(), y[col_name].std(), y[col_name].min(), y[col_name].max()))

Design matrix shape: (17379, 11)
---------------------------------
Feature: season         , unique:    4, Mean:   2.50, Std:   1.11, Min:   1.00, Max:   4.00
Feature: yr             , unique:    2, Mean:   0.50, Std:   0.50, Min:   0.00, Max:   1.00
Feature: mnth           , unique:   12, Mean:   6.54, Std:   3.44, Min:   1.00, Max:  12.00
Feature: hr             , unique:   24, Mean:  11.55, Std:   6.91, Min:   0.00, Max:  23.00
Feature: holiday        , unique:    2, Mean:   0.03, Std:   0.17, Min:   0.00, Max:   1.00
Feature: weekday        , unique:    7, Mean:   3.00, Std:   2.01, Min:   0.00, Max:   6.00
Feature: workingday     , unique:    2, Mean:   0.68, Std:   0.47, Min:   0.00, Max:   1.00
Feature: weathersit     , unique:    4, Mean:   1.43, Std:   0.64, Min:   1.00, Max:   4.00
Feature: temp           , unique:   50, Mean:   0.50, Std:   0.19, Min:   0.02, Max:   1.00
Feature: hum            , unique:   89, Mean:   0.63, Std:   0.19, Min:   0.00, Max:   1.00
Feature: windspeed      , unique:   30, Mean:   0.19, Std:   0.12, Min:   0.00, Max:   0.85

Target shape: (17379, 1)
---------------------------------
Target: cnt            , unique:  869, Mean: 189.46, Std: 181.39, Min:   1.00, Max: 977.00

def preprocess(X, y):
    # Standarize X
    X_df = X
    x_mean = X_df.mean()
    x_std = X_df.std()
    X_df = (X_df - X_df.mean()) / X_df.std()

    # Standarize Y
    Y_df = y
    y_mean = Y_df.mean()
    y_std = Y_df.std()
    Y_df = (Y_df - Y_df.mean()) / Y_df.std()
    return X_df, Y_df, x_mean, x_std, y_mean, y_std

# shuffle and standarize all features
X_df, Y_df, x_mean, x_std, y_mean, y_std = preprocess(X, y)

def split(X_df, Y_df):
    # shuffle indices
    indices = X_df.index.tolist()
    np.random.shuffle(indices)

    # data split
    train_size = int(0.8 * len(X_df))

    X_train = X_df.iloc[indices[:train_size]]
    Y_train = Y_df.iloc[indices[:train_size]]
    X_test = X_df.iloc[indices[train_size:]]
    Y_test = Y_df.iloc[indices[train_size:]]

    return X_train, Y_train, X_test, Y_test

# train/test split
X_train, Y_train, X_test, Y_test = split(X_df, Y_df)

Fit a Neural Network

# Train - Evaluate - Explain a neural network
model = keras.Sequential([
    keras.layers.Dense(1024, activation="relu"),
    keras.layers.Dense(512, activation="relu"),
    keras.layers.Dense(256, activation="relu"),
    keras.layers.Dense(1)
])

optimizer = keras.optimizers.Adam(learning_rate=0.001)
model.compile(optimizer=optimizer, loss="mse", metrics=["mae", keras.metrics.RootMeanSquaredError()])
model.fit(X_train, Y_train, batch_size=512, epochs=20, verbose=1)
model.evaluate(X_train, Y_train, verbose=1)
model.evaluate(X_test, Y_test, verbose=1)

Epoch 1/20
28/28 ━━━━━━━━━━━━━━━━━━━━ 1s 12ms/step - loss: 0.6231 - mae: 0.5745 - root_mean_squared_error: 0.7853
Epoch 2/20
28/28 ━━━━━━━━━━━━━━━━━━━━ 0s 13ms/step - loss: 0.3870 - mae: 0.4506 - root_mean_squared_error: 0.6219
Epoch 3/20
28/28 ━━━━━━━━━━━━━━━━━━━━ 0s 12ms/step - loss: 0.2976 - mae: 0.3851 - root_mean_squared_error: 0.5454
Epoch 4/20
28/28 ━━━━━━━━━━━━━━━━━━━━ 0s 13ms/step - loss: 0.2237 - mae: 0.3326 - root_mean_squared_error: 0.4728
Epoch 5/20
28/28 ━━━━━━━━━━━━━━━━━━━━ 0s 14ms/step - loss: 0.1619 - mae: 0.2836 - root_mean_squared_error: 0.4023
Epoch 6/20
28/28 ━━━━━━━━━━━━━━━━━━━━ 0s 13ms/step - loss: 0.1193 - mae: 0.2386 - root_mean_squared_error: 0.3451
Epoch 7/20
28/28 ━━━━━━━━━━━━━━━━━━━━ 0s 11ms/step - loss: 0.0906 - mae: 0.2075 - root_mean_squared_error: 0.3009
Epoch 8/20
28/28 ━━━━━━━━━━━━━━━━━━━━ 0s 12ms/step - loss: 0.0753 - mae: 0.1895 - root_mean_squared_error: 0.2745
Epoch 9/20
28/28 ━━━━━━━━━━━━━━━━━━━━ 0s 12ms/step - loss: 0.0669 - mae: 0.1784 - root_mean_squared_error: 0.2586
Epoch 10/20
28/28 ━━━━━━━━━━━━━━━━━━━━ 0s 12ms/step - loss: 0.0610 - mae: 0.1703 - root_mean_squared_error: 0.2469
Epoch 11/20
28/28 ━━━━━━━━━━━━━━━━━━━━ 0s 13ms/step - loss: 0.0554 - mae: 0.1614 - root_mean_squared_error: 0.2353
Epoch 12/20
28/28 ━━━━━━━━━━━━━━━━━━━━ 0s 13ms/step - loss: 0.0500 - mae: 0.1524 - root_mean_squared_error: 0.2235
Epoch 13/20
28/28 ━━━━━━━━━━━━━━━━━━━━ 0s 12ms/step - loss: 0.0462 - mae: 0.1459 - root_mean_squared_error: 0.2149
Epoch 14/20
28/28 ━━━━━━━━━━━━━━━━━━━━ 0s 12ms/step - loss: 0.0431 - mae: 0.1407 - root_mean_squared_error: 0.2075
Epoch 15/20
28/28 ━━━━━━━━━━━━━━━━━━━━ 0s 12ms/step - loss: 0.0410 - mae: 0.1372 - root_mean_squared_error: 0.2026
Epoch 16/20
28/28 ━━━━━━━━━━━━━━━━━━━━ 0s 12ms/step - loss: 0.0399 - mae: 0.1360 - root_mean_squared_error: 0.1996
Epoch 17/20
28/28 ━━━━━━━━━━━━━━━━━━━━ 0s 12ms/step - loss: 0.0381 - mae: 0.1325 - root_mean_squared_error: 0.1952
Epoch 18/20
28/28 ━━━━━━━━━━━━━━━━━━━━ 0s 12ms/step - loss: 0.0378 - mae: 0.1323 - root_mean_squared_error: 0.1945
Epoch 19/20
28/28 ━━━━━━━━━━━━━━━━━━━━ 0s 12ms/step - loss: 0.0376 - mae: 0.1325 - root_mean_squared_error: 0.1940
Epoch 20/20
28/28 ━━━━━━━━━━━━━━━━━━━━ 0s 12ms/step - loss: 0.0378 - mae: 0.1331 - root_mean_squared_error: 0.1944
435/435 ━━━━━━━━━━━━━━━━━━━━ 1s 1ms/step - loss: 0.0420 - mae: 0.1422 - root_mean_squared_error: 0.2048
109/109 ━━━━━━━━━━━━━━━━━━━━ 0s 1ms/step - loss: 0.0701 - mae: 0.1725 - root_mean_squared_error: 0.2633





[0.06317276507616043, 0.16644978523254395, 0.25134193897247314]

We train a deep fully-connected Neural Network with 3 hidden layers for \(20\) epochs. The model achieves a root mean squared error on the test of about \(0.24\) units, that corresponds to approximately \(0.26 * 181 = 47\) counts.

Explain

We will focus on the feature temp (temperature) because its global effect is quite heterogeneous and the heterogeneity can be further explained using regional effects.

def model_jac(x):
    x_tensor = tf.convert_to_tensor(x, dtype=tf.float32)
    with tf.GradientTape() as t:
        t.watch(x_tensor)
        pred = model(x_tensor)
        grads = t.gradient(pred, x_tensor)
    return grads.numpy()

def model_forward(x):
    return model(x).numpy().squeeze()

scale_y = {"mean": y_mean.iloc[0], "std": y_std.iloc[0]}
scale_x_list =[{"mean": x_mean.iloc[i], "std": x_std.iloc[i]} for i in range(len(x_mean))]
scale_x = scale_x_list[3]
feature_names = X_df.columns.to_list()
target_name = "bike-rentals"
y_limits=[-200, 800]
dy_limits = [-300, 300]

scale_x_list[8]["mean"] += 8
scale_x_list[8]["std"] *= 47

scale_x_list[9]["std"] *= 100
scale_x_list[10]["std"] *= 67

Global Effect

Overview of All Features

We start by examining all relevant features. Feature effect methods are generally more meaningful for numerical features. To compare effects across features more easily, we standardize the y_limits.

Relevant features:

  • month
  • hr
  • weekday
  • workingday
  • temp
  • humidity
  • windspeed

We observe that features: hour, temperature and humidity have an intersting structure. Out of them hour has by far the most influence on the output, so it makes sensce to fucse on it further.

pdp = effector.PDP(data=X_train.to_numpy(), model=model_forward, feature_names=feature_names, target_name=target_name, nof_instances=2000)
for i in [2, 3, 8, 9, 10]:
    pdp.plot(feature=i, centering=True, scale_x=scale_x, scale_y=scale_y, show_avg_output=True, nof_ice=200, y_limits=y_limits)

2025-02-26 11:05:19.691215: W external/local_xla/xla/tsl/framework/cpu_allocator_impl.cc:83] Allocation of 245760000 exceeds 10% of free system memory.
2025-02-26 11:05:19.739214: W external/local_xla/xla/tsl/framework/cpu_allocator_impl.cc:83] Allocation of 245760000 exceeds 10% of free system memory.
2025-02-26 11:05:19.779244: W external/local_xla/xla/tsl/framework/cpu_allocator_impl.cc:83] Allocation of 245760000 exceeds 10% of free system memory.
2025-02-26 11:05:20.164766: W external/local_xla/xla/tsl/framework/cpu_allocator_impl.cc:83] Allocation of 245760000 exceeds 10% of free system memory.
2025-02-26 11:05:20.211132: W external/local_xla/xla/tsl/framework/cpu_allocator_impl.cc:83] Allocation of 245760000 exceeds 10% of free system memory.

png

png

png

png

png

PDP

pdp = effector.PDP(data=X_train.to_numpy(), model=model_forward, feature_names=feature_names, target_name=target_name, nof_instances=5000)
pdp.plot(feature=3, centering=True, scale_x=scale_x, scale_y=scale_y, show_avg_output=True, nof_ice=200)

png

RHALE

rhale = effector.RHALE(data=X_train.to_numpy(), model=model_forward, model_jac=model_jac, feature_names=feature_names, target_name=target_name)
rhale.plot(feature=3, heterogeneity="std", centering=True, scale_x=scale_x, scale_y=scale_y, show_avg_output=True)

png

shap_dp = effector.ShapDP(data=X_train.to_numpy(), model=model_forward, feature_names=feature_names, target_name=target_name, nof_instances=500)
shap_dp.plot(feature=3, centering=True, scale_x=scale_x, scale_y=scale_y, show_avg_output=True)

PermutationExplainer explainer: 501it [02:44,  2.91it/s]

png

Conclusion

All methods agree that hour affects bike-rentals with two peaks, around 8:00 and 17:00, likely reflecting commute hours. However, the effect varies significantly, so regional effects may help in understanding the origin of this heterogeneity.

Regional Effect

RegionalPDP

regional_pdp = effector.RegionalPDP(data=X_train.to_numpy(), model=model_forward, feature_names=feature_names, nof_instances=5_000)
regional_pdp.summary(features=3, scale_x_list=scale_x_list)

100%|██████████| 1/1 [00:01<00:00,  1.15s/it]



Feature 3 - Full partition tree:
🌳 Full Tree Structure:
───────────────────────
hr 🔹 [id: 0 | heter: 0.44 | inst: 5000 | w: 1.00]
    workingday = 0.00 🔹 [id: 1 | heter: 0.38 | inst: 1535 | w: 0.31]
        temp ≤ 6.81 🔹 [id: 3 | heter: 0.20 | inst: 774 | w: 0.15]
        temp > 6.81 🔹 [id: 4 | heter: 0.22 | inst: 761 | w: 0.15]
    workingday ≠ 0.00 🔹 [id: 2 | heter: 0.30 | inst: 3465 | w: 0.69]
        temp ≤ 6.81 🔹 [id: 5 | heter: 0.21 | inst: 1547 | w: 0.31]
        temp > 6.81 🔹 [id: 6 | heter: 0.21 | inst: 1918 | w: 0.38]
--------------------------------------------------
Feature 3 - Statistics per tree level:
🌳 Tree Summary:
─────────────────
Level 0🔹heter: 0.44
    Level 1🔹heter: 0.32 | 🔻0.12 (26.86%)
        Level 2🔹heter: 0.21 | 🔻0.12 (35.85%)

for node_idx in [1, 2]:
    regional_pdp.plot(feature=3, node_idx=node_idx, centering=True, scale_x_list=scale_x_list, scale_y=scale_y, y_limits=y_limits)

png

png

for node_idx in [3, 4, 5, 6]:
    regional_pdp.plot(feature=3, node_idx=node_idx, centering=True, scale_x_list=scale_x_list, scale_y=scale_y, y_limits=y_limits)

png

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png

png

RegionalRHALE

regional_rhale = effector.RegionalRHALE(data=X_train.to_numpy(), model=model_forward, model_jac=model_jac, feature_names=feature_names, target_name=target_name)
regional_rhale.summary(features=3, scale_x_list=scale_x_list)

100%|██████████| 1/1 [00:03<00:00,  3.19s/it]



Feature 3 - Full partition tree:
🌳 Full Tree Structure:
───────────────────────
hr 🔹 [id: 0 | heter: 5.68 | inst: 13903 | w: 1.00]
    workingday = 0.00 🔹 [id: 1 | heter: 0.75 | inst: 4385 | w: 0.32]
    workingday ≠ 0.00 🔹 [id: 2 | heter: 5.44 | inst: 9518 | w: 0.68]
--------------------------------------------------
Feature 3 - Statistics per tree level:
🌳 Tree Summary:
─────────────────
Level 0🔹heter: 5.68
    Level 1🔹heter: 3.96 | 🔻1.71 (30.22%)

regional_rhale.plot(feature=3, node_idx=1, centering=True, scale_x_list=scale_x_list, scale_y=scale_y)
regional_rhale.plot(feature=3, node_idx=2, centering=True, scale_x_list=scale_x_list, scale_y=scale_y)

png

png

Regional SHAP-DP

regional_shap_dp = effector.RegionalShapDP(data=X_train.to_numpy(), model=model_forward, feature_names=feature_names, nof_instances=500)

regional_shap_dp.summary(features=3, scale_x_list=scale_x_list)

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Feature 3 - Full partition tree:
🌳 Full Tree Structure:
───────────────────────
hr 🔹 [id: 0 | heter: 0.06 | inst: 500 | w: 1.00]
    workingday = 0.00 🔹 [id: 1 | heter: 0.03 | inst: 150 | w: 0.30]
        temp ≤ 6.81 🔹 [id: 3 | heter: 0.01 | inst: 71 | w: 0.14]
        temp > 6.81 🔹 [id: 4 | heter: 0.02 | inst: 79 | w: 0.16]
    workingday ≠ 0.00 🔹 [id: 2 | heter: 0.04 | inst: 350 | w: 0.70]
        temp ≤ 6.81 🔹 [id: 5 | heter: 0.02 | inst: 150 | w: 0.30]
        temp > 6.81 🔹 [id: 6 | heter: 0.02 | inst: 200 | w: 0.40]
--------------------------------------------------
Feature 3 - Statistics per tree level:
🌳 Tree Summary:
─────────────────
Level 0🔹heter: 0.06
    Level 1🔹heter: 0.03 | 🔻0.03 (43.39%)
        Level 2🔹heter: 0.02 | 🔻0.02 (48.69%)

for node_idx in [1, 2]:
    regional_shap_dp.plot(feature=3, node_idx=node_idx, centering=True, scale_x_list=scale_x_list, scale_y=scale_y)

png

png

for node_idx in [3, 4, 5, 6]:
    regional_shap_dp.plot(feature=3, node_idx=node_idx, centering=True, scale_x_list=scale_x_list, scale_y=scale_y, y_limits=y_limits)

png

png

png

png

Conclusion

Regional methods reveal two distinct patterns: one for working days and another for non-working days.

  • On working days, the effect mirrors the global trend, with peaks around 8:00 and 17:00, likely due to commuting.
  • On non-working days, the pattern shifts to a single peak around 13:00, likely driven by sightseeing and leisure.

PDP and ShapDP push it more; that on non-working days, hour interacts with temperature — peaks between 12:00 and 14:00 are more pronounced in warmer conditions. Makes sense, right? If you go for sightseeing, you probably