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Introduction

When faced with scenarios involving the prediction of hundreds or thousands of time series, a crucial decision arises: should one develop individual models for each series, or should one use a unified model to handle them all at once?

In Single-Series Modeling (Local Forecasting Model), a separate predictive model is created for each time series. While this method provides a comprehensive understanding of each series, its scalability can be challenged by the need to create and maintain hundreds or thousands of models.

Multi-Series Modeling (Global Forecasting Model) involves building a single predictive model that considers all time series simultaneously. It attempts to capture the core patterns that govern the series, thereby mitigating the potential noise that each series might introduce. This approach is computationally efficient, easy to maintain, and can yield more robust generalizations across time series, albeit potentially at the cost of sacrificing some individual insights.

To help understand the benefits of each forecasting strategy, this document focuses on examining and comparing the results obtained when predicting the energy consumption of over a thousand buildings using the ASHRAE - Great Energy Predictor III dataset available on Kaggle. Efficient management of energy consumption in residential buildings is essential for sustainable urban development, therefore accurate forecasting of energy consumption can play a crucial role in optimizing resource allocation and promoting energy conservation initiatives.

Global forecasting modeling assumes that series that behave similarly can benefit from being modeled together. Although the primary use of each building is available in the dataset, it may not reflect groups with similar patterns of energy use, so additional groups are created using clustering methods. A total of 5 experiments are performed:

  • Modelling each building individually.

  • Modelling all buildings together with a unique global model strategy.

  • Modelling groups of buildings based on their primary use (a global model per primary use).

  • Modelling groups of buildings based on time series feature clustering (a global model per cluster).

  • Modelling groups of buildings based on Dynamic Time Warping (DTW) clustering (a global model per cluster).

Energy consumption is predicted for each building on a daily basis using a 7-day-ahead recursive backtesting scheme (without refitting), evaluated over the last five months of the year (August-December 2016, 22 weekly folds) following each strategy. The effectiveness of each approach is evaluated using several performance metrics, including Mean Absolute Error (MAE), Absolute Error, and Bias. The goal of the study is to identify the most effective approach both for overall predictions and for a specific group of buildings.

✏️ Note

Read the Conclusions First?

If you prefer a quick overview before diving into the details, consider starting with the Conclusions section. This approach allows you to tailor your reading to your interests and time constraints, and summarizes our findings and insights. After reading the conclusions, you may find certain sections particularly relevant or interesting. Feel free to navigate directly to those parts of the article for a deeper understanding.

Libraries

Libraries used in this document.

# Data management
# ==============================================================================
import numpy as np
import pandas as pd
from datetime import datetime
from skforecast.datasets import fetch_dataset
from tqdm.auto import tqdm

# Plots
# ==============================================================================
import matplotlib.pyplot as plt
import matplotlib.ticker as ticker
from skforecast.plot import set_dark_theme

# Forecasting
# ==============================================================================
import lightgbm
from lightgbm import LGBMRegressor
import skforecast
from skforecast.recursive import ForecasterRecursive, ForecasterRecursiveMultiSeries
from skforecast.model_selection import (
    backtesting_forecaster,
    TimeSeriesFold,
    backtesting_forecaster_multiseries
)
from skforecast.preprocessing import (
    CalendarFeatures,
    reshape_series_long_to_dict,
    reshape_exog_long_to_dict
)

# Feature extraction
# ==============================================================================
import tsfresh
from tsfresh import extract_features, select_features
from tsfresh.feature_extraction.settings import ComprehensiveFCParameters, from_columns

# Clustering
# ==============================================================================
from sklearn.preprocessing import StandardScaler
from sklearn.decomposition import PCA
from sklearn.cluster import KMeans
from sktime.clustering.k_means import TimeSeriesKMeans

# Warnings configuration
# ==============================================================================
import warnings
warnings.filterwarnings('once')
warnings.simplefilter('ignore', category=skforecast.exceptions.InputTypeWarning)

color = '\033[1m\033[38;5;208m' 
print(f"{color}Version skforecast: {skforecast.__version__}")
print(f"{color}Version lightgbm: {lightgbm.__version__}")
Version skforecast: 0.25.0
Version lightgbm: 4.7.0

Data

Data used in this document has been obtained from the Kaggle competition Addison Howard, Chris Balbach, Clayton Miller, Jeff Haberl, Krishnan Gowri, Sohier Dane. (2019). ASHRAE - Great Energy Predictor III. Kaggle.

Three files are used to create the modeling data set:

  • weather_train.csv and weather_test.csv: These files contain weather-related data for each building, including outdoor air temperature, dew temperature, relative humidity, and other weather parameters. The weather data is crucial for understanding the impact of external conditions on building energy usage.

  • building_metadata.csv: This file provides metadata for each building in the dataset, such as building type, primary use, square footage, floor count, and year built. This information helps in understanding the characteristics of the buildings and their potential influence on energy consumption patterns.

  • train.csv: the train dataset contains the target variable, i.e., the energy consumption data for each building, along with the timestamps for the energy consumption readings. It also includes the corresponding building and weather IDs to link the information across different datasets.

The three files have been preprocessed to remove buildings with less than 85% of values not equal to NaN or zero, to use only the electricity meter, and to aggregate the data to a daily frequency.

# Load preprocessed data
# ==============================================================================
data = fetch_dataset('ashrae_daily')
data.head(3)
╭────────────────────────────────── ashrae_daily ──────────────────────────────────╮
│ Description:                                                                     │
│ Daily energy consumption data from the ASHRAE competition with building metadata │
│ and weather data.                                                                │
│                                                                                  │
│ Source:                                                                          │
│ Kaggle competition Addison Howard, Chris Balbach, Clayton Miller, Jeff Haberl,   │
│ Krishnan Gowri, Sohier Dane. (2019). ASHRAE - Great Energy Predictor III.        │
│ Kaggle. https://www.kaggle.com/c/ashrae-energy-prediction/overview               │
│                                                                                  │
│ URL:                                                                             │
│ https://huggingface.co/datasets/skforecast/ashrae_daily/resolve/main/ashrae_dail │
│ y.parquet                                                                        │
│                                                                                  │
│ Shape: 444324 rows x 10 columns                                                  │
╰──────────────────────────────────────────────────────────────────────────────────╯
building_id meter_reading site_id primary_use square_feet air_temperature dew_temperature sea_level_pressure wind_direction wind_speed
timestamp
2016-01-01 id_105 1102.766933 1 Education 50623 3.8 2.4 1020.9 240.0 3.1
2016-01-01 id_106 17.606200 1 Education 5374 3.8 2.4 1020.9 240.0 3.1
2016-01-01 id_107 8233.625107 1 Education 97532 3.8 2.4 1020.9 240.0 3.1
# Ensure all time series indexes are complete without gaps
# ==============================================================================
data = (
    data
    .groupby('building_id')
    .apply(lambda group: group.asfreq('D', fill_value=np.nan), include_groups=False)
    .reset_index()
    .set_index('timestamp')
)
# Fill missing values of air_temperature and wind_speed using fowrard and backward fill
# ==============================================================================
# Imputation must be done separately for each building
data = data.sort_values(by=['building_id', 'timestamp'])
data['air_temperature'] = data.groupby('building_id')['air_temperature'].ffill().bfill()
data['wind_speed'] = data.groupby('building_id')['wind_speed'].ffill().bfill()
data = data.sort_index()

print(
    f"Range of dates available : {data.index.min()} --- {data.index.max()}  "
    f"(n_days={(data.index.max() - data.index.min()).days})"
)
Range of dates available : 2016-01-01 00:00:00 --- 2016-12-31 00:00:00  (n_days=365)

Exploratory data analysis

Building primary use

One of the key attributes associated with each building is its designated use. This feature may play a crucial role in influencing the energy consumption pattern, as distinct uses can significantly impact both the quantity and timing of energy consumption.

# Number of buildings and type of buildings based on primary use
# ==============================================================================
n_building = data['building_id'].nunique()
n_type_building = data['primary_use'].nunique()

print(f"Number of buildings: {n_building}")
print(f"Number of building types: {n_type_building}")
display(data.drop_duplicates(subset=['building_id'])['primary_use'].value_counts())
Number of buildings: 1214
Number of building types: 16
primary_use
Education                        463
Office                           228
Entertainment/public assembly    157
Public services                  150
Lodging/residential              112
Other                             19
Healthcare                        19
Parking                           13
Warehouse/storage                 12
Manufacturing/industrial          10
Services                           9
Food sales and service             5
Technology/science                 5
Retail                             5
Utility                            4
Religious worship                  3
Name: count, dtype: int64

For certain primary use categories, there is a limited number of buildings within the dataset. To streamline the analysis, categories with fewer than 100 buildings are grouped into the "Other" category.

# Types of buildings (primary use) with less than 100 samples are grouped as "Other".
# ==============================================================================
infrequent_categories = (
    data
    .drop_duplicates(subset=['building_id'])['primary_use']
    .value_counts()
    .loc[lambda x: x < 100]
    .index
    .tolist()
)
print("Infrequent categories:")
print("=====================")
print('\n'.join(infrequent_categories))

data['primary_use'] = np.where(
    data['primary_use'].isin(infrequent_categories),
    'Other',
    data['primary_use']
)
Infrequent categories:
=====================
Other
Healthcare
Parking
Warehouse/storage
Manufacturing/industrial
Services
Food sales and service
Technology/science
Retail
Utility
Religious worship

Next, a plot is created showing the energy consumption for a randomly selected building within each respective category, and a plot of all available time series for each category.

# Time series for 1 random selected building per group
# ==============================================================================
set_dark_theme()
fig, axs = plt.subplots(nrows=3, ncols=2, figsize=(8, 5.5), sharex=True, sharey=False)
sample_ids = (
    data
    .groupby('primary_use')['building_id']
    .apply(lambda x: x.sample(1, random_state=333))
    .tolist()
)
axs = axs.flatten()

for i, building_id in enumerate(sample_ids):
    data_sample = data[data['building_id'] == building_id]
    building_type = data_sample['primary_use'].unique()[0]
    data_sample.plot(
        y        = 'meter_reading',
        ax       = axs[i],
        legend   = False,
        title    = f"Building: {building_id}, type: {building_type}",
        fontsize = 8
    )
    axs[i].set_xlabel("")
    axs[i].set_ylabel("")
    # Scientific notation for y axis
    axs[i].ticklabel_format(axis='y', style='sci', scilimits=(0, 0))
    axs[i].title.set_size(9)

fig.suptitle('Energy consumption for 6 random buildings', fontsize=12)
fig.tight_layout()
plt.show()

⚠️ Warning

Although 1214 buildings are available, to keep model training within a reasonable time frame a subset of, for example, 600 randomly selected buildings can be used. The reader is encouraged to adapt the number of buildings if necessary and check whether the conclusions hold.

# Sample 600 buildings
# ==============================================================================
rng = np.random.default_rng(12345)
buildings = data['building_id'].unique()
buildings_selected = rng.choice(
                         buildings,
                         size    = 600,
                         replace = False
                     )

data = data.query("building_id in @buildings_selected")

Clustering by energy consumption patterns

The idea behind modeling multiple series at the same time is to be able to capture the main patterns that govern the series, thereby reducing the impact of the potential noise that each series may have. This means that series that behave similarly may benefit from being modeled together. One way to identify potential groups of series is to perform a clustering study prior to modeling. If clear groups are identified as a result of clustering, it is appropriate to model each of them separately.

Clustering is an unsupervised analysis technique that groups a set of observations into clusters that contain observations that are considered homogeneous, while observations in different clusters are considered heterogeneous. Algorithms that cluster time series can be divided into two groups: those that use a transformation to create features prior to clustering (feature-driven time series clustering), and those that work directly on the time series (elastic distance measures).

  • Feature-driven time series clustering: Features describing structural characteristics are extracted from each time series, then these features are fed into arbitrary clustering algorithms. This features are obtained by applying statistical operations that best capture the underlying characteristics: trend, seasonality, periodicity, serial correlation, skewness, kurtosis, chaos, nonlinearity, and self-similarity.

  • Elastic distance measures: This approach works directly on the time series, adjusting or "realigning" the series in comparison to each other. The best known of this family of measures is Dynamic Time Warping (DTW).

For a more detailed review of time series clustering, see A review and evaluation of elastic distance functions for time series clustering.

⚠️ Warning

Both clustering approaches used below (time series features and DTW) are computed using the entire observed series for each building, i.e. the full year, including the August-December period that is later used as the backtesting test window. This is a simplification for benchmarking purposes: the resulting cluster assignments, used to decide which buildings are modeled together, are informed by data that would not yet be available at the time of a real forecast. In a production pipeline, clusters would need to be formed using only the data available up to the training cutoff to avoid this kind of lookahead bias.

Clustering based on time series features

Feature creation

tsfresh is a powerful Python library for feature engineering from time-series and sequential data, including statistical measures, Fourier coefficients, and various other time-domain and frequency-domain features. It provides a systematic approach to automate the calculation of features and select the most informative ones.

To start, the default configuration of tsfresh is used, which calculates all available features. The easiest way to access this configuration is to use the functions provided by the ComprehensiveFCParameters class. It creates a dictionary which is a mapping from string (the name of each feature) to a list of dictionary parameters to be used when the function with that name is called.

# Default features and settings created by tsfresh
# ==============================================================================
default_features = ComprehensiveFCParameters()
print("Name of the features extracted by tsfresh:")
print("=========================================")
print('\n'.join(list(default_features)))
Name of the features extracted by tsfresh:
=========================================
variance_larger_than_standard_deviation
has_duplicate_max
has_duplicate_min
has_duplicate
sum_values
abs_energy
mean_abs_change
mean_change
mean_second_derivative_central
median
mean
length
standard_deviation
variation_coefficient
variance
skewness
kurtosis
root_mean_square
absolute_sum_of_changes
longest_strike_below_mean
longest_strike_above_mean
count_above_mean
count_below_mean
last_location_of_maximum
first_location_of_maximum
last_location_of_minimum
first_location_of_minimum
percentage_of_reoccurring_values_to_all_values
percentage_of_reoccurring_datapoints_to_all_datapoints
sum_of_reoccurring_values
sum_of_reoccurring_data_points
ratio_value_number_to_time_series_length
sample_entropy
maximum
absolute_maximum
minimum
benford_correlation
time_reversal_asymmetry_statistic
c3
cid_ce
symmetry_looking
large_standard_deviation
quantile
autocorrelation
agg_autocorrelation
partial_autocorrelation
number_cwt_peaks
number_peaks
binned_entropy
index_mass_quantile
cwt_coefficients
spkt_welch_density
ar_coefficient
change_quantiles
fft_coefficient
fft_aggregated
value_count
range_count
approximate_entropy
friedrich_coefficients
max_langevin_fixed_point
linear_trend
agg_linear_trend
augmented_dickey_fuller
number_crossing_m
energy_ratio_by_chunks
ratio_beyond_r_sigma
linear_trend_timewise
count_above
count_below
lempel_ziv_complexity
fourier_entropy
permutation_entropy
query_similarity_count
mean_n_absolute_max

Many of these features are calculated using different values for their arguments.

# Default configuration for feature "partial_autocorrelation"
# ==============================================================================
default_features['partial_autocorrelation']
[{'lag': 0},
 {'lag': 1},
 {'lag': 2},
 {'lag': 3},
 {'lag': 4},
 {'lag': 5},
 {'lag': 6},
 {'lag': 7},
 {'lag': 8},
 {'lag': 9}]

To access the detailed view of each feature and the parameter values included in the default configuration, use the following code:

# Default configuration for all extracted feature
# ==============================================================================
# from pprint import pprint
# pprint(default_features)

Once the configuration of features has been defined, the next step is to extract them from the time series. The function extract_features() of tsfresh is used for this purpose. This function receives as input the time series and the configuration of the features to be extracted. The output is a dataframe with the extracted features.

# Feature extraction
# ==============================================================================
ts_features = extract_features(
    timeseries_container  = data[['building_id', 'meter_reading']].reset_index(),
    column_id             = "building_id",
    column_sort           = "timestamp",
    column_value          = "meter_reading",
    default_fc_parameters = default_features,
    impute_function       = tsfresh.utilities.dataframe_functions.impute,
    n_jobs                = 4
)

print("Shape of ts_features:", ts_features.shape)
ts_features.head(3)
Shape of ts_features: (600, 783)
meter_reading__variance_larger_than_standard_deviation meter_reading__has_duplicate_max meter_reading__has_duplicate_min meter_reading__has_duplicate meter_reading__sum_values meter_reading__abs_energy meter_reading__mean_abs_change meter_reading__mean_change meter_reading__mean_second_derivative_central meter_reading__median ... meter_reading__fourier_entropy__bins_5 meter_reading__fourier_entropy__bins_10 meter_reading__fourier_entropy__bins_100 meter_reading__permutation_entropy__dimension_3__tau_1 meter_reading__permutation_entropy__dimension_4__tau_1 meter_reading__permutation_entropy__dimension_5__tau_1 meter_reading__permutation_entropy__dimension_6__tau_1 meter_reading__permutation_entropy__dimension_7__tau_1 meter_reading__query_similarity_count__query_None__threshold_0.0 meter_reading__mean_n_absolute_max__number_of_maxima_7
id_1001 1.0 0.0 0.0 1.0 2.543133e+05 4.114736e+08 212.298183 -0.049314 0.026097 304.500150 ... 0.090729 0.090729 0.700821 1.752526 3.038890 4.327469 5.278212 5.735413 0.0 2871.142857
id_1003 1.0 0.0 0.0 0.0 7.423270e+05 1.561927e+09 218.155580 -0.904797 0.782968 1996.688070 ... 0.235155 0.446547 1.719214 1.636071 2.696952 3.780121 4.766919 5.454277 0.0 3171.445826
id_1004 1.0 0.0 1.0 1.0 3.275762e+06 3.185436e+10 1435.418191 0.328759 -2.879118 9054.500992 ... 0.136002 0.245901 0.771605 1.637745 2.735159 3.797395 4.713989 5.339062 0.0 13217.571568

3 rows × 783 columns

As a result of the extraction process, 783 features have been created for each time series (building_id in this use case). The returned dataframe has as its index the column specified in the column_id argument of extract_features.

The default extraction of tsfresh produces a huge number of features. However, only a few of these may be of interest in each use case. To select the most relevant ones, tsfresh includes an automated selection process based on hypothesis tests (FeatuRE Extraction based on Scalable Hypothesis tests). In this process, the features are individually and independently evaluated for their significance in predicting the target under investigation.

⚠️ Warning

The selection process used by tsfresh is based on the significance of each feature in accurately predicting the target variable. To perform this process, a target variable is required – for example, the type of building associated with a given time series. However, there are cases where a target variable is not readily available. In such cases, alternative strategies can be used:

  • Instead of calculating all the standard features, focus only on those that are likely to be relevant to the specific application, based on expert knowledge.
  • Exclude features based on criteria such as low variance and high correlation. This helps refine the set of features to consider, focusing on those that provide the most meaningful information for analysis.
  • Use techniques such as PCA, t-SNE, or auto-encoders to reduce dimensionality.

As feature selection is a critical step that affects the information available for the next steps of the analysis, it is highly recommended to understand the parameters that control the behavior of select_features().

⚠️ Warning

The order of the index returned in the features DataFrame is not the same as the order of the columns in the original DataFrame. Therefore, the data passed to the argument y in select_features must be sorted to ensure the correct association between the features and the target variable.

# Select relevant features
# ==============================================================================
target = (
    data[['building_id', 'primary_use']]
    .drop_duplicates()
    .set_index('building_id')
    .loc[ts_features.index, :]
    ['primary_use']
)

assert ts_features.index.equals(target.index)

ts_features_selected = select_features(
    X = ts_features,
    y = target,
    fdr_level = 0.001 # Very restrictive filter
)

ts_features_selected.index.name = 'building_id'
print(f"Number of features before selection: {ts_features.shape[1]}")
print(f"Number of features after selection: {ts_features_selected.shape[1]}")
ts_features_selected.head()
Number of features before selection: 783
Number of features after selection: 262
meter_reading__cwt_coefficients__coeff_13__w_2__widths_(2, 5, 10, 20) meter_reading__time_reversal_asymmetry_statistic__lag_3 meter_reading__fft_coefficient__attr_"imag"__coeff_14 meter_reading__cwt_coefficients__coeff_12__w_2__widths_(2, 5, 10, 20) meter_reading__fft_coefficient__attr_"real"__coeff_52 meter_reading__change_quantiles__f_agg_"mean"__isabs_False__qh_0.4__ql_0.2 meter_reading__cwt_coefficients__coeff_9__w_2__widths_(2, 5, 10, 20) meter_reading__cwt_coefficients__coeff_8__w_2__widths_(2, 5, 10, 20) meter_reading__cwt_coefficients__coeff_5__w_2__widths_(2, 5, 10, 20) meter_reading__fft_coefficient__attr_"abs"__coeff_52 ... meter_reading__number_peaks__n_3 meter_reading__ar_coefficient__coeff_1__k_10 meter_reading__lempel_ziv_complexity__bins_5 meter_reading__augmented_dickey_fuller__attr_"usedlag"__autolag_"AIC" meter_reading__approximate_entropy__m_2__r_0.3 meter_reading__ar_coefficient__coeff_2__k_10 meter_reading__sample_entropy meter_reading__lempel_ziv_complexity__bins_10 meter_reading__mean_second_derivative_central meter_reading__fft_coefficient__attr_"imag"__coeff_66
building_id
id_1001 6.826138 3.442917e+07 -13196.753777 12.030563 -7628.816978 -0.129015 -3.912888 -9.874489 9.502322 15873.642987 ... 39.0 0.800818 0.215847 13.0 0.569996 -0.042719 0.135011 0.284153 0.026097 -12066.902088
id_1003 532.475812 -1.685504e+07 -9842.688817 580.669414 -30259.389852 -57.335155 -103.029346 -85.134903 523.822698 30879.572926 ... 51.0 0.631371 0.251366 15.0 1.075901 -0.133314 1.373685 0.336066 0.782968 -480.328641
id_1004 3842.017430 -2.048682e+09 -16987.081591 3477.250740 -251461.502332 -32.344863 -4909.050477 -2610.620993 4888.722487 267714.847953 ... 48.0 0.536810 0.262295 16.0 0.806466 -0.222787 0.909909 0.338798 -2.879118 -15323.742941
id_1010 35.982333 -5.002756e+03 -197.270808 40.074998 -2086.324646 0.384581 -25.906483 -26.748608 29.887855 2393.603859 ... 47.0 0.615801 0.270492 14.0 1.177241 -0.080681 1.809459 0.355191 -0.006868 -0.316418
id_1014 428.452168 -2.029309e+07 -895.160202 465.314991 -51416.366795 39.945500 -286.781950 -317.905714 323.499854 53759.415208 ... 45.0 0.671580 0.273224 15.0 0.778352 -0.257865 0.826844 0.352459 -0.509944 3568.761154

5 rows × 262 columns

Some created features may have missing values for some of the time series. Since most clustering algorithms do not allow for missing values, these are excluded.

# Remove features with missing values
# ==============================================================================
ts_features_selected = ts_features_selected.dropna(axis=1, how='any')
print(f"Number of features after selection and removing missing values: {ts_features_selected.shape[1]}")
Number of features after selection and removing missing values: 262

Once the final feature matrix is ready, it may be useful to create a new dictionary to store the selected features and the parameters used to calculate them. This can be easily done using the from_columns function.

# Dictionary with selected features and their configuration
# ==============================================================================
selected_features_info = from_columns(ts_features_selected)
# pprint(selected_features_info['meter_reading'])

K-means clustering

A K-means clustering method is used to group the buildings. Since clustering is known to be negatively affected by high dimensionality, and since several hundred features have been created for each building, a PCA is used to reduce the dimensionality of the data before the k-means is applied.

# Scale features so all of them have mean 0 and std 1
# ==============================================================================
scaler = StandardScaler().set_output(transform="pandas")
ts_features_selected_scaled = scaler.fit_transform(ts_features_selected)
ts_features_selected_scaled.head(2)
meter_reading__cwt_coefficients__coeff_13__w_2__widths_(2, 5, 10, 20) meter_reading__time_reversal_asymmetry_statistic__lag_3 meter_reading__fft_coefficient__attr_"imag"__coeff_14 meter_reading__cwt_coefficients__coeff_12__w_2__widths_(2, 5, 10, 20) meter_reading__fft_coefficient__attr_"real"__coeff_52 meter_reading__change_quantiles__f_agg_"mean"__isabs_False__qh_0.4__ql_0.2 meter_reading__cwt_coefficients__coeff_9__w_2__widths_(2, 5, 10, 20) meter_reading__cwt_coefficients__coeff_8__w_2__widths_(2, 5, 10, 20) meter_reading__cwt_coefficients__coeff_5__w_2__widths_(2, 5, 10, 20) meter_reading__fft_coefficient__attr_"abs"__coeff_52 ... meter_reading__number_peaks__n_3 meter_reading__ar_coefficient__coeff_1__k_10 meter_reading__lempel_ziv_complexity__bins_5 meter_reading__augmented_dickey_fuller__attr_"usedlag"__autolag_"AIC" meter_reading__approximate_entropy__m_2__r_0.3 meter_reading__ar_coefficient__coeff_2__k_10 meter_reading__sample_entropy meter_reading__lempel_ziv_complexity__bins_10 meter_reading__mean_second_derivative_central meter_reading__fft_coefficient__attr_"imag"__coeff_66
building_id
id_1001 -0.291819 0.036635 -0.399262 -0.310218 0.256123 0.114873 0.264589 0.261563 -0.348869 -0.234496 ... -1.322932 0.336083 -0.436683 0.282832 -0.912063 0.488825 -2.087332 -0.576604 0.244556 -0.880899
id_1003 -0.058363 0.034091 -0.256339 -0.060110 0.118053 -0.177152 0.225910 0.215803 -0.207669 -0.152704 ... 1.171205 -0.316247 0.425191 0.720766 1.172814 -0.073738 1.132361 0.588872 0.536417 0.016526

2 rows × 262 columns

# Plot variance reduction as a function of the number of PCA components
# ==============================================================================
pca = PCA()
pca.fit(ts_features_selected_scaled)
n_components = np.argmax(np.cumsum(pca.explained_variance_ratio_) > 0.85) + 1

fig, ax = plt.subplots(nrows=1, ncols=1, figsize=(6, 2.5))
ax.plot(np.cumsum(pca.explained_variance_ratio_))
ax.set_title('Variance explained by PCA components')
ax.set_xlabel('Number of components')
ax.set_ylabel('Cumulative explained variance')
ax.axhline(y=0.85, color='red', linestyle='--')
ax.axvline(x=n_components, color='red', linestyle='--')
ax.text(
    x=n_components+10,
    y=0.4,
    s=f"n_components = {n_components}",
    color='red',
    verticalalignment='center'
)

plt.show();

The dimensionality of the original 262 features is reduced by using the first 16 principal components (explaining 85% of the variance).

# PCA with as many components as necessary to explain 85% of the variance
# ==============================================================================
pca = PCA(n_components=0.85)
pca_projections = pca.fit_transform(ts_features_selected_scaled)

# Create a data frame with the projections. Each column is a principal component
# and each row is the id of the building.
pca_projections = pd.DataFrame(
    pca_projections,
    index   = ts_features_selected_scaled.index,
    columns = [f"PC{i}" for i in range(1, pca.n_components_ + 1)]
)

print(f"Number of components selected: {pca.n_components_}")
print(f"Explained variance: {pca.explained_variance_ratio_.sum():.3f}")
pca_projections.head(3)
Number of components selected: 16
Explained variance: 0.852
PC1 PC2 PC3 PC4 PC5 PC6 PC7 PC8 PC9 PC10 PC11 PC12 PC13 PC14 PC15 PC16
building_id
id_1001 -3.953822 4.591653 2.225976 2.705873 2.598711 1.528322 -2.501696 0.186481 -3.489617 -0.586468 0.899327 1.244604 -2.319459 3.545177 3.871482 -1.581641
id_1003 -2.111044 -3.083024 -1.824578 0.121061 -1.716628 0.034439 0.221577 -0.395868 -0.155805 0.284908 -0.612728 -0.313265 0.357750 -0.209675 0.709589 1.054758
id_1004 9.973499 -2.132875 -0.612174 1.788844 -2.797131 -0.123844 -3.009209 1.008297 -2.330331 -1.814039 1.339265 -1.029985 -1.903606 -4.379159 -0.572582 0.913577

One of the inherent challenges of clustering is determining the optimal number of clusters, since there is no ground truth or predefined number of clusters in unsupervised learning. Several heuristics have been proposed to guide this selection, and in this case, the elbow method is used.

The elbow method involves plotting the total within-cluster sum of squares (WSS) against the number of clusters (k). The optimal number of clusters is typically identified at the "elbow" of the curve, where the rate of decrease in WSS begins to decrease significantly. This suggests that adding more clusters beyond this point provides little improvement in clustering performance.

# Optimal number of clusters
# ==============================================================================
# Identify the optimal number of clusters using the elbow method
range_n_clusters = range(1, 20)
inertias = []
size_of_clusters = []

for n_clusters in range_n_clusters:
    kmeans = KMeans(
                 n_clusters   = n_clusters, 
                 n_init       = 20, 
                 random_state = 963852
             )
    kmeans.fit(pca_projections)
    inertias.append(kmeans.inertia_)
    sizes = [int(i) for i in np.unique(kmeans.labels_, return_counts=True)[1]]
    size_of_clusters.append(sizes)

inertias = pd.Series(inertias, index=range_n_clusters)
dicrease_in_inertia = inertias.pct_change() * -100

fig, axs = plt.subplots(1, 2, figsize=(9, 3.5))
inertias.plot(marker='o', ax=axs[0])
axs[0].xaxis.set_major_locator(ticker.MaxNLocator(integer=True))
axs[0].set_title("Cluster inertia vs number of clusters", fontsize=10)
axs[0].set_xlabel('Number of clusters')
axs[0].set_ylabel('Intra-cluster variance (inertia)')

dicrease_in_inertia.plot(kind='bar', ax=axs[1])
axs[1].set_title("Decrease in inertia", fontsize=10)
axs[1].set_xlabel('Number of clusters')
axs[1].set_ylabel('Decrease in inertia (%)')
fig.tight_layout()
plt.show();

The chart shows that after 9 clusters the decrease in inertia slows down, therefore 9 clusters may be a good choice.

In addition to analyzing the evolution of the inertia (intra-variance), it is important to check the size of the clusters being created. The presence of small clusters may indicate overfitting or anomalous samples that do not fit well into any of the groups.

# Distribution of cluster sizes
# ==============================================================================
for n_clusters, sizes in zip(range_n_clusters, size_of_clusters):
    print(f"Size of clusters (n = {n_clusters}): {sizes}")
Size of clusters (n = 1): [600]
Size of clusters (n = 2): [25, 575]
Size of clusters (n = 3): [574, 3, 23]
Size of clusters (n = 4): [304, 3, 269, 24]
Size of clusters (n = 5): [274, 241, 3, 62, 20]
Size of clusters (n = 6): [66, 3, 237, 18, 4, 272]
Size of clusters (n = 7): [66, 21, 2, 1, 1, 272, 237]
Size of clusters (n = 8): [274, 2, 238, 1, 15, 5, 1, 64]
Size of clusters (n = 9): [228, 2, 179, 3, 17, 110, 1, 59, 1]
Size of clusters (n = 10): [179, 6, 1, 226, 111, 1, 14, 1, 60, 1]
Size of clusters (n = 11): [111, 1, 58, 1, 1, 228, 6, 1, 179, 13, 1]
Size of clusters (n = 12): [104, 171, 2, 7, 177, 56, 1, 1, 7, 2, 71, 1]
Size of clusters (n = 13): [86, 1, 54, 3, 1, 176, 23, 183, 1, 5, 6, 1, 60]
Size of clusters (n = 14): [115, 3, 57, 1, 45, 1, 1, 1, 138, 1, 133, 12, 2, 90]
Size of clusters (n = 15): [75, 1, 6, 126, 22, 53, 1, 59, 1, 136, 3, 5, 1, 110, 1]
Size of clusters (n = 16): [113, 1, 77, 32, 2, 3, 1, 135, 38, 54, 126, 11, 1, 1, 3, 2]
Size of clusters (n = 17): [130, 3, 1, 87, 134, 1, 16, 1, 111, 2, 1, 47, 4, 56, 1, 2, 3]
Size of clusters (n = 18): [88, 1, 23, 56, 112, 1, 4, 1, 2, 75, 1, 1, 32, 1, 126, 5, 69, 2]
Size of clusters (n = 19): [104, 1, 22, 129, 1, 2, 113, 1, 3, 84, 3, 75, 1, 52, 1, 2, 2, 3, 1]

When 9 clusters are created, the size of the clusters is not well balanced. Before modeling the time series, the smaller clusters (less than 20 observations) are combined into a single cluster named "Other".

# Train clustering model with 9 clusters and assign each building to a cluster
# ==============================================================================
kmeans = KMeans(n_clusters=9, n_init=20, random_state=963852)
kmeans.fit(pca_projections)
clusters = kmeans.predict(pca_projections)
clusters = pd.DataFrame({
    'building_id': pca_projections.index,
    'cluster_base_on_features': clusters.astype(str)
})

# Merge cluster minor clusters into a single cluster
threshold = 20
cluster_size = clusters['cluster_base_on_features'].value_counts()
cluster_size = cluster_size[cluster_size < threshold].index.tolist()
clusters['cluster_base_on_features'] = np.where(
    clusters['cluster_base_on_features'].isin(cluster_size),
    'Other',
    clusters['cluster_base_on_features']
)

clusters['cluster_base_on_features'].value_counts()
cluster_base_on_features
0        228
2        179
5        110
7         59
Other     24
Name: count, dtype: int64
# Add the predicted cluster to the data frame with the building information
# ==============================================================================
data = pd.merge(
           data.reset_index(), # To avoid losing the index
           clusters,
           on       ='building_id',
           how      ='left',
           validate = 'm:1'
       ).set_index('timestamp')

data.head(3)
building_id meter_reading site_id primary_use square_feet air_temperature dew_temperature sea_level_pressure wind_direction wind_speed cluster_base_on_features
timestamp
2016-01-01 id_970 0.000000 9 Other 346056 7.8 NaN NaN NaN 3.1 5
2016-01-01 id_1066 639.270004 12 Education 55800 1.9 -1.2 1016.2 200.0 5.0 5
2016-01-01 id_862 483.950409 8 Other 27640 25.0 20.0 1019.7 0.0 0.0 5

Once each building has been assigned to a cluster, it is useful to examine the characteristics of the grouped buildings. For example, the distribution of the primary use of the buildings.

# Percentaje of building types per cluster
# ==============================================================================
primary_usage_per_cluster = ( 
    data
    .groupby('cluster_base_on_features')['primary_use']
    .value_counts(normalize=True)
    .unstack()
    .fillna(0)
)

primary_usage_per_cluster = (100 * primary_usage_per_cluster).round(1)
primary_usage_per_cluster
primary_use Education Entertainment/public assembly Lodging/residential Office Other Public services
cluster_base_on_features
0 23.7 14.9 21.5 10.5 13.6 15.8
2 54.2 7.3 0.6 27.9 6.1 3.9
5 30.0 22.7 0.9 16.4 13.6 16.4
7 55.9 13.6 1.7 15.3 6.8 6.8
Other 50.0 8.3 0.0 29.2 4.2 8.3

The results suggest (the table should be read horizontally) that the clustering process based on features extracted from the time series generates groups that differ from those formed by the main purpose of the building.

Clustering based on elastic distance (DTW)

DTW is a technique that measures the similarity between two temporal sequences, which may vary in speed. In essence, it's an elastic distance measure that allows the time series to be stretched or compressed to align with each other optimally. Clustering with DTW involves grouping time series data based on their dynamic time-warped distances, ensuring that time series within the same cluster have similar shapes and patterns, even if they are out of phase in time or have different lengths. This method is particularly useful in various applications like speech recognition, data mining, and financial analysis, where the alignment of sequences in time is crucial for accurate pattern recognition and analysis.

The class TimeSeriesKMeans class from the Sktime library enables the application of K-means clustering with a variety of distance metrics, including DTW, Euclidean, ERP, EDR, LCSS, squared, DDTW, WDTW, and WDDTW. Many of these metrics are elastic distances, making the method well-suited for time series data.

Sktime requires time series to be structured in a long format with a multi-index. The outermost index level represents the series ID, while the innermost level corresponds to the datetime.

# Convert data to long format with a multiindex: (building_id, timestamp)
# ==============================================================================
data_long = (
    data
    .reset_index()
    .loc[:, ['timestamp', 'building_id', 'meter_reading']]
    .set_index(['building_id', 'timestamp'])
    .sort_index(ascending=True)
)
data_long
meter_reading
building_id timestamp
id_1001 2016-01-01 142.999700
2016-01-02 141.000801
2016-01-03 137.000300
2016-01-04 133.000100
2016-01-05 127.000300
... ... ...
id_999 2016-12-27 2627.250000
2016-12-28 2667.250000
2016-12-29 2495.000000
2016-12-30 2059.500000
2016-12-31 1899.500000

219600 rows × 1 columns

⚠️ Warning

The next cell has a run time of approximately 5 minutes.

# Fit clustering model
# ==============================================================================
model = TimeSeriesKMeans(
            n_clusters   = 4,
            metric       = "dtw",
            max_iter     = 10,
            random_state = 123456
        )

model.fit(data_long)
TimeSeriesKMeans(max_iter=10, n_clusters=4, random_state=123456)
Please rerun this cell to show the HTML repr or trust the notebook.
# Cluster prediction
# ==============================================================================
clusters = model.predict(data_long)
clusters = pd.DataFrame({
               'building_id': data_long.index.get_level_values('building_id').drop_duplicates(),
               'cluster_base_on_dtw': clusters.astype(str),
           })

# Size of each cluster
clusters['cluster_base_on_dtw'].value_counts().sort_index().to_frame(name='Number of series')
Number of series
cluster_base_on_dtw
0 179
1 14
2 342
3 65
# Add the predicted cluster to the data frame with the building information
# ==============================================================================
data = pd.merge(
           data.reset_index(),  # To avoid losing the index
           clusters,
           on       = 'building_id',
           how      = 'left',
           validate = 'm:1'
       ).set_index('timestamp')

data.head(3)
building_id meter_reading site_id primary_use square_feet air_temperature dew_temperature sea_level_pressure wind_direction wind_speed cluster_base_on_features cluster_base_on_dtw
timestamp
2016-01-01 id_970 0.000000 9 Other 346056 7.8 NaN NaN NaN 3.1 5 2
2016-01-01 id_1066 639.270004 12 Education 55800 1.9 -1.2 1016.2 200.0 5.0 5 2
2016-01-01 id_862 483.950409 8 Other 27640 25.0 20.0 1019.7 0.0 0.0 5 2

✏️ Note

Sktime provides additional clustering algorithms, such as TimeSeriesKMeansTslearn, TimeSeriesKMedoids and TimeSeriesKShapes that worth exploring. Other great libraries for clustering time series are: DTAIDistance, Tslearn and AEON.

# Save data for modelling to avoid repeating the previous steps
# ==============================================================================
data.to_parquet('data_modelling.parquet')

Modelling and forecasting

After establishing the three grouping criteria (building usage, clustering based on time series features, and clustering based on dynamic time warping), both single and multi-series global models are trained. The evaluation that follows concentrates on determining how effectively these models can forecast daily data over the last five months of the year (August-December 2016), using a 7-day-ahead backtesting scheme without refitting (22 weekly folds). During this assessment, three distinct metrics are used:

  • The average value of the Mean Absolute Error (MAE) for all buildings.

  • The absolute sum of the errors, i.e. the absolute deviation between the predicted value and the actual consumption for all buildings.

  • The bias for the predictions summed over all buildings.

In addition to the lagged values of the own time series, the day of the week (sine-cosine encoded), the outdoor temperature and the wind speed are included as exogenous variables.

⚠️ Warning

The air_temperature and wind_speed exogenous variables used during backtesting correspond to the actual historical observations for the test period, not forecasts made ahead of time. This is a common simplification for benchmarking purposes, but in a real deployment future weather values are not known in advance and would need to be replaced by weather forecasts (e.g. from an external weather-forecast provider) or by features that are genuinely available at prediction time.

✎ Note

For a more detailed explanation of time series model validation, readers are encouraged to consult the Backtesting user guide. For more information about calendar features and cyclical encoding visit Calendar features and Cyclical features in time series.
# Load data for modelling
# ==============================================================================
data = pd.read_parquet('data_modelling.parquet')

The data is reshaped from a long data frame into a dictionary of series. Although this is not strictly required, since skforecast allows multiple input formats, it is the recommended format for multi-series models, since it allows series of different lengths and with different subsets of exogenous features to be combined easily.

# Transform series and exog to dictionaries
# ==============================================================================
series_dict = reshape_series_long_to_dict(
    data      = data.reset_index(),
    series_id = 'building_id',
    index     = 'timestamp',
    values    = 'meter_reading',
    freq      = 'D'
)

exog_dict = reshape_exog_long_to_dict(
    data      = data[['building_id', 'primary_use', 'air_temperature', 'wind_speed']].reset_index(),
    series_id = 'building_id',
    index     = 'timestamp',
    freq      = 'D'
)
# Partition data in train and test
# ==============================================================================
# The train/test split used for modelling is handled internally by TimeSeriesFold
# (see `cv` below), which receives the full, undivided series_dict/exog_dict.
# The splits are created in case they need to be used for additional analysis.
end_train = '2016-07-31 23:59:00'
series_dict_train = {k: v.loc[: end_train,] for k, v in series_dict.items()}
exog_dict_train   = {k: v.loc[: end_train,] for k, v in exog_dict.items()}
series_dict_test  = {k: v.loc[end_train:,] for k, v in series_dict.items()}
exog_dict_test    = {k: v.loc[end_train:,] for k, v in exog_dict.items()}
# Forecaster definitions for single and global models
# ==============================================================================
calendar_features = CalendarFeatures(features = ['day_of_week'],  encoding='cyclical')

params_lgbm_single = {
    'n_estimators': 500,
    'learning_rate': 0.01,
    'max_depth': 4,
    'random_state': 8520,
    'verbose': -1
}
forecaster_single = ForecasterRecursive(
                        estimator = LGBMRegressor(**params_lgbm_single),
                        lags      = 31,
                    )

params_lgbm_global = {
    'n_estimators': 500,
    'learning_rate': 0.01,
    'max_depth': 10,
    'random_state': 8520,
    'verbose': -1
}
forecaster_global = ForecasterRecursiveMultiSeries(
                        estimator          = LGBMRegressor(**params_lgbm_global),
                        lags               = 31,
                        calendar_features  = calendar_features,
                        encoding           = "ordinal_category"
                    )
# Backtesting definition
# ==============================================================================
cv = TimeSeriesFold(
        steps              = 7,
        initial_train_size = end_train,
        refit              = False
     )
# Exogenous variables included in the model
# ==============================================================================
exog_features = ['primary_use', 'air_temperature', 'wind_speed']
# Table of results for all models
# ==============================================================================
table_results = pd.DataFrame(columns=['model', 'mae', 'abs_error', 'bias', 'elapsed_time'])
table_results = table_results.set_index('model')
table_results = table_results.astype({'mae': float, 'abs_error': float, 'bias': float, 'elapsed_time': object})

Single model for each building

An individual forecasting model is trained and evaluated for each building.

# Train and test a model for each building
# ==============================================================================
predictions_all_buildings = {}
metrics_all_buildings = {}
errors_all_buildings = {}

# Train and predict for each building
start = datetime.now()

for building in tqdm(data['building_id'].unique(), desc='Modelling buildings'):

    # Get data for the building
    data_building = data[data['building_id'] == building]
    data_building = data_building.asfreq('D').sort_index()

    # Backtesting
    try:
        metric, predictions = backtesting_forecaster(
                                  forecaster    = forecaster_single,
                                  y             = data_building['meter_reading'],
                                  exog          = data_building[exog_features],
                                  cv            = cv,
                                  metric        = 'mean_absolute_error',
                                  show_progress = False
                              )
        predictions_all_buildings[building] = predictions['pred']
        metrics_all_buildings[building] = metric.at[0, 'mean_absolute_error']
        errors_all_buildings[building] = (
            predictions['pred'] - data_building.loc[predictions.index, 'meter_reading']
        )
    except Exception as e:
        print(f"Error modelling building {building}: {e}")

end = datetime.now()

predictions_all_buildings = pd.DataFrame(predictions_all_buildings)
errors_all_buildings = pd.DataFrame(errors_all_buildings)
mean_metric_all_buildings = pd.Series(metrics_all_buildings).mean()
sum_abs_errors_all_buildings = errors_all_buildings.abs().sum(axis=1).sum()
sum_bias_all_buildings = errors_all_buildings.sum().sum()

table_results.loc['One model per building', ['mae', 'abs_error', 'bias', 'elapsed_time']] = [
    mean_metric_all_buildings,
    sum_abs_errors_all_buildings,
    sum_bias_all_buildings,
    end - start
]

print(
    f"\nAverage mean absolute error for all buildings: {mean_metric_all_buildings:.0f}\n"
    f"Sum of absolute errors for all buildings (x 10,000): {sum_abs_errors_all_buildings/10000:.0f}\n"
    f"Bias (x 10,000): {sum_bias_all_buildings/10000:.0f}"
)
Average mean absolute error for all buildings: 503
Sum of absolute errors for all buildings (x 10,000): 4619
Bias (x 10,000): -31
# Plot predictions vs real value for 2 random buildings
# ==============================================================================
rng = np.random.default_rng(147)
selected_buildings = rng.choice(data['building_id'].unique(), size=2, replace=False)

fig, axs = plt.subplots(2, 1, figsize=(6, 4), sharex=True)
axs = axs.flatten()

for i, building in enumerate(selected_buildings):
    series_dict_test[building].plot(ax=axs[i], label='test')
    predictions_all_buildings[building].plot(ax=axs[i], label='One model per building')
    axs[i].set_title(f"Building {building}", fontsize=10)
    axs[i].set_xlabel("")
    axs[i].legend()

fig.tight_layout()
plt.show();

Global multi-series model for all buildings

A global model for all buildings is trained and tested using the skforecast class ForecasterRecursiveMultiSeries. This forecaster allows the user to pass the time series in several formats, including a dataFrame with the time series organized as columns.

For more information about using series of different lengths, or different exogenous variables for each series, see Global Forecasting Models.

# Forecaster multi-series to model all buildings at the same time
# ==============================================================================
start = datetime.now()

metric, predictions = backtesting_forecaster_multiseries(
                          forecaster            = forecaster_global,
                          series                = series_dict,
                          exog                  = exog_dict,
                          cv                    = cv,
                          metric                = 'mean_absolute_error',
                          add_aggregated_metric = False,
                          verbose               = False,
                          show_progress         = True
                      )

end = datetime.now()

# Combine predictions with real values
results = predictions.reset_index(names=['timestamp']).merge(
    data[['building_id', 'meter_reading']].reset_index(names=['timestamp']),
    left_on=['timestamp', 'level'],
    right_on=['timestamp', 'building_id'],
    how='inner'
)
results['error'] = results['pred'] - results['meter_reading']
mean_metric_all_buildings = metric['mean_absolute_error'].mean()
sum_abs_errors_all_buildings = results['error'].abs().sum()
sum_bias_all_buildings = results['error'].sum()
table_results.loc['Global model', ['mae', 'abs_error', 'bias', 'elapsed_time']] = [
    mean_metric_all_buildings,
    sum_abs_errors_all_buildings,
    sum_bias_all_buildings,
    end - start
]

print(
    f"\nAverage mean absolute error for all buildings: {mean_metric_all_buildings:.0f}\n"
    f"Sum of absolute errors for all buildings (x 10,000): {sum_abs_errors_all_buildings/10000:.0f}\n"
    f"Bias (x 10,000): {sum_bias_all_buildings/10000:.0f}"
)
Average mean absolute error for all buildings: 456
Sum of absolute errors for all buildings (x 10,000): 4182
Bias (x 10,000): 881
# Add predictions to the already existing plot (not showing the plot yet)
# ==============================================================================
for i, building in enumerate(selected_buildings):
    predictions.query("level==@building")['pred'].plot(ax=axs[i], label='Global model')
    axs[i].legend()

Global multi-series model by building usage

# Forecaster multi-series models for buildings grouped by primary usage
# ==============================================================================
predictions_all_buildings = []
metrics_all_buildings = []
start = datetime.now()

for primary_usage in data['primary_use'].unique():

    print(
        f"Training and testing model for primary usage: {primary_usage} "
        f"(n = {data[data['primary_use'] == primary_usage]['building_id'].nunique()})"
    )

    # Create a subset based on primary use clusters
    building_id_subset = set(data.loc[data['primary_use'] == primary_usage, 'building_id'].unique())
    series_dict_subset = {k: v for k, v in series_dict.items() if k in building_id_subset}
    exog_dict_subset   = {k: v for k, v in exog_dict.items() if k in building_id_subset}

    metric, predictions = backtesting_forecaster_multiseries(
                            forecaster            = forecaster_global,
                            series                = series_dict_subset,
                            exog                  = exog_dict_subset,
                            cv                    = cv,
                            metric                = 'mean_absolute_error',
                            add_aggregated_metric = False,
                            verbose               = False,
                            show_progress         = True,
                            suppress_warnings     = True
                        )
    predictions_all_buildings.append(predictions)
    metrics_all_buildings.append(metric)

end = datetime.now()

predictions_all_buildings = pd.concat(predictions_all_buildings, axis=0)
metrics_all_buildings = pd.concat(metrics_all_buildings, axis=0)
results = predictions_all_buildings.reset_index(names=['timestamp']).merge(
    data[['building_id', 'meter_reading']].reset_index(names=['timestamp']),
    left_on=['timestamp', 'level'],
    right_on=['timestamp', 'building_id'],
    how='inner'
)
results['error'] = results['pred'] - results['meter_reading']

mean_metric_all_buildings = metrics_all_buildings['mean_absolute_error'].mean()
sum_abs_errors_all_buildings = results['error'].abs().sum()
sum_bias_all_buildings = results['error'].sum()

table_results.loc['Global model per primary usage', ['mae', 'abs_error', 'bias', 'elapsed_time']] = [
    mean_metric_all_buildings,
    sum_abs_errors_all_buildings,
    sum_bias_all_buildings,
    end - start
]

print(
    f"\nAverage mean absolute error for all buildings: {mean_metric_all_buildings:.0f}\n"
    f"Sum of absolute errors for all buildings (x 10,000): {sum_abs_errors_all_buildings/10000:.0f}\n"
    f"Bias (x 10,000): {sum_bias_all_buildings/10000:.0f}"
)
Training and testing model for primary usage: Other (n = 62)
Training and testing model for primary usage: Education (n = 229)
Training and testing model for primary usage: Entertainment/public assembly (n = 82)
Training and testing model for primary usage: Office (n = 108)
Training and testing model for primary usage: Public services (n = 67)
Training and testing model for primary usage: Lodging/residential (n = 52)
Average mean absolute error for all buildings: 457
Sum of absolute errors for all buildings (x 10,000): 4193
Bias (x 10,000): 608
# Add predictions to the already existing plot (not showing the plot yet)
# ==============================================================================
for i, building in enumerate(selected_buildings):
    predictions_all_buildings.query("level==@building")['pred'].plot(ax=axs[i], label='Global model per primary usage')
    axs[i].legend()

Global multi-series model by cluster (features)

A global model for each cluster based on time series features is trained and tested.

# Forecaster multi-series models for buildings grouped by time series features
# ==============================================================================
predictions_all_buildings = []
metrics_all_buildings = []
start = datetime.now()

for cluster in data['cluster_base_on_features'].unique():

    print(
        f"Training and testing model for cluster: {cluster} "
        f"(n = {data[data['cluster_base_on_features'] == cluster]['building_id'].nunique()})"
    )

    # Create subset based on DTW clusters
    building_id_subset = set(data.loc[data['cluster_base_on_features'] == cluster, 'building_id'].unique())
    series_dict_subset = {k: v for k, v in series_dict.items() if k in building_id_subset}
    exog_dict_subset   = {k: v for k, v in exog_dict.items() if k in building_id_subset}

    metric, predictions = backtesting_forecaster_multiseries(
                            forecaster            = forecaster_global,
                            series                = series_dict_subset,
                            exog                  = exog_dict_subset,
                            cv                    = cv,
                            metric                = 'mean_absolute_error',
                            add_aggregated_metric = False,
                            verbose               = False,
                            show_progress         = True,
                            suppress_warnings     = True
                        )
    predictions_all_buildings.append(predictions)
    metrics_all_buildings.append(metric)

end = datetime.now()

predictions_all_buildings = pd.concat(predictions_all_buildings, axis=0)
metrics_all_buildings = pd.concat(metrics_all_buildings, axis=0)
results = predictions_all_buildings.reset_index(names=['timestamp']).merge(
    data[['building_id', 'meter_reading']].reset_index(names=['timestamp']),
    left_on=['timestamp', 'level'],
    right_on=['timestamp', 'building_id'],
    how='inner'
)
results['error'] = results['pred'] - results['meter_reading']

mean_metric_all_buildings = metrics_all_buildings['mean_absolute_error'].mean()
sum_abs_errors_all_buildings = results['error'].abs().sum()
sum_bias_all_buildings = results['error'].sum()

table_results.loc['Global model per cluster (features)', ['mae', 'abs_error', 'bias', 'elapsed_time']] = [
    mean_metric_all_buildings,
    sum_abs_errors_all_buildings,
    sum_bias_all_buildings,
    end - start
]

print(
    f"\nAverage mean absolute error for all buildings: {mean_metric_all_buildings:.0f}\n"
    f"Sum of absolute errors for all buildings (x 10,000): {sum_abs_errors_all_buildings/10000:.0f}\n"
    f"Bias (x 10,000): {sum_bias_all_buildings/10000:.0f}"
)
Training and testing model for cluster: 5 (n = 110)
Training and testing model for cluster: 7 (n = 59)
Training and testing model for cluster: 2 (n = 179)
Training and testing model for cluster: 0 (n = 228)
Training and testing model for cluster: Other (n = 24)
Average mean absolute error for all buildings: 420
Sum of absolute errors for all buildings (x 10,000): 3856
Bias (x 10,000): 557
# Add predictions to the already existing plot (not showing the plot yet)
# ==============================================================================
for i, building in enumerate(selected_buildings):
    predictions_all_buildings.query("level==@building")['pred'].plot(ax=axs[i], label='Global model per cluster (features)')
    axs[i].legend()

Global multi-series model by cluster (elastic distance DTW)

A global model for each cluster based on elastic distance (DTW) is trained and tested.

# Forecaster multi-series models for buildings grouped by DTW
# ==============================================================================
predictions_all_buildings = []
metrics_all_buildings = []
start = datetime.now()

for cluster in data['cluster_base_on_dtw'].unique():

    print(
        f"Training and testing model for cluster: {cluster} "
        f"(n = {data[data['cluster_base_on_dtw'] == cluster]['building_id'].nunique()})"
    )

    # Create subset based on DTW clusters
    building_id_subset = set(data.loc[data['cluster_base_on_dtw'] == cluster, 'building_id'].unique())
    series_dict_subset = {k: v for k, v in series_dict.items() if k in building_id_subset}
    exog_dict_subset   = {k: v for k, v in exog_dict.items() if k in building_id_subset}

    metric, predictions = backtesting_forecaster_multiseries(
                            forecaster            = forecaster_global,
                            series                = series_dict_subset,
                            exog                  = exog_dict_subset,
                            cv                    = cv,
                            metric                = 'mean_absolute_error',
                            add_aggregated_metric = False,
                            verbose               = False,
                            show_progress         = True,
                            suppress_warnings     = True
                        )
    predictions_all_buildings.append(predictions)
    metrics_all_buildings.append(metric)

end = datetime.now()

predictions_all_buildings = pd.concat(predictions_all_buildings, axis=0)
metrics_all_buildings = pd.concat(metrics_all_buildings, axis=0)
results = predictions_all_buildings.reset_index(names=['timestamp']).merge(
    data[['building_id', 'meter_reading']].reset_index(names=['timestamp']),
    left_on=['timestamp', 'level'],
    right_on=['timestamp', 'building_id'],
    how='inner'
)
results['error'] = results['pred'] - results['meter_reading']

mean_metric_all_buildings = metrics_all_buildings['mean_absolute_error'].mean()
sum_abs_errors_all_buildings = results['error'].abs().sum()
sum_bias_all_buildings = results['error'].sum()

table_results.loc['Global model per cluster (DTW)', ['mae', 'abs_error', 'bias', 'elapsed_time']] = [
    mean_metric_all_buildings,
    sum_abs_errors_all_buildings,
    sum_bias_all_buildings,
    end - start
]

print(
    f"\nAverage mean absolute error for all buildings: {mean_metric_all_buildings:.0f}\n"
    f"Sum of absolute errors for all buildings (x 10,000): {sum_abs_errors_all_buildings/10000:.0f}\n"
    f"Bias (x 10,000): {sum_bias_all_buildings/10000:.0f}"
)
Training and testing model for cluster: 2 (n = 342)
Training and testing model for cluster: 3 (n = 65)
Training and testing model for cluster: 0 (n = 179)
Training and testing model for cluster: 1 (n = 14)
Average mean absolute error for all buildings: 421
Sum of absolute errors for all buildings (x 10,000): 3868
Bias (x 10,000): 601
# Add predictions to the already existing plot (not showing the plot yet)
# ==============================================================================
for i, building in enumerate(selected_buildings):
    predictions_all_buildings.query("level==@building")['pred'].plot(ax=axs[i], label='Global model per cluster (DTW)')
    axs[i].legend()

Results

# Table of results
# ==============================================================================
def highlight_best(column):
    # Bias is best when closest to zero, not when most negative;
    # mae and abs_error are best when smallest.
    best_idx = column.abs().idxmin() if column.name == 'bias' else column.idxmin()
    return ['background-color: green' if idx == best_idx else '' for idx in column.index]

table_results['elapsed_time'] = table_results['elapsed_time'].astype(str).str[:7]
table_results.style.apply(highlight_best, subset=['mae', 'abs_error', 'bias'], axis=0).format(precision=0)
  mae abs_error bias elapsed_time
model        
One model per building 503 46194541 -312364 0:01:27
Global model 456 41816769 8808073 0:00:12
Global model per primary usage 457 41930577 6084150 0:00:15
Global model per cluster (features) 420 38555819 5572232 0:00:15
Global model per cluster (DTW) 421 38679398 6013944 0:00:15
# Plot predictions vs real value for 2 random buildings for all models
# ==============================================================================
handles, labels = axs[0].get_legend_handles_labels()
fig.legend(handles, labels, loc='upper center', bbox_to_anchor=(0.5, 0.05), ncol=2)
for ax in axs:
    ax.legend().remove()
fig

Conclusion

The decision between using individual models for each building or a global model for all can greatly impact the outcome. Our examination compares these two strategies, highlighting the trade-offs between the forecasting capabilities and computational efficiency.

  • Predictive Capability: Global models outperform individual models in terms of mean absolute error, suggesting that they better capture the common patterns across series, leading to more accurate forecasts.

  • Computational Efficiency: Global models prove to be more efficient, requiring less time than running individual models for each building. This highlights the advantage of the global model in time-sensitive applications where rapid training and prediction are valued.

  • Best Model: For this use case, using multiple global models, one per cluster, achieves the best overall performance, with the lowest mean absolute error and absolute error compared to using a single global model.

  • Bias Trade-off: single-building models have by far the smallest absolute bias (-31, in units of 10,000), while every global-model variant shows a larger positive bias (557-881, in the same units). The lower mean absolute error and absolute error achieved by the global approaches therefore come at the cost of a less centered bias.

Further research

This analysis has provided interesting insights into the effectiveness of global versus individual models in time series forecasting. Possible next steps could include:

  • Manually review buildings with high error. Identify if there is a group for which the model is not performing well.

  • Add more exogenous features: see skforecast's user guide Calendar Features for calendar and sunlight features that typically affect energy consumption.

  • Optimize lags and hyperparameters: Use Grid, Random or Bayesian search to find the best model configuration.

  • Try other machine learning algorithms.

Session information

import session_info
session_info.show(html=False)
-----
lightgbm            4.7.0
matplotlib          3.10.9
numpy               2.4.6
pandas              2.3.3
session_info        v1.0.1
skforecast          0.25.0
sklearn             1.7.2
sktime              1.1.0
tqdm                4.67.3
tsfresh             0.21.2
-----
IPython             9.15.0
jupyter_client      8.9.1
jupyter_core        5.9.1
-----
Python 3.13.14 | packaged by conda-forge | (main, Jun 12 2026, 09:44:26) [MSC v.1944 64 bit (AMD64)]
Windows-11-10.0.26200-SP0
-----
Session information updated at 2026-09-24 18:36

Citation

How to cite this document

If you use this document or any part of it, please acknowledge the source, thank you!

Global Forecasting Models: Comparative Analysis of Single and Multi-Series Forecasting Modeling by Joaquín Amat Rodrigo and Javier Escobar Ortiz, available under a CC BY-NC-SA 4.0 at https://www.cienciadedatos.net/documentos/py53-global-forecasting-models.html

How to cite skforecast

If you use skforecast for a publication, we would appreciate it if you cite the published software.

Zenodo:

Amat Rodrigo, Joaquin, & Escobar Ortiz, Javier. (2026). skforecast (v0.25.0). Zenodo. https://doi.org/10.5281/zenodo.8382788

APA:

Amat Rodrigo, J., & Escobar Ortiz, J. (2026). skforecast (Version 0.25.0) [Computer software]. https://doi.org/10.5281/zenodo.8382788

BibTeX:

@software{skforecast, author = {Amat Rodrigo, Joaquin and Escobar Ortiz, Javier}, title = {skforecast}, version = {0.25.0}, month = {09}, year = {2026}, license = {BSD-3-Clause}, url = {https://skforecast.org/}, doi = {10.5281/zenodo.8382788} }


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