Forecasting time series with missing values¶

In many real use cases of forecasting, although historical data are available, it is common for the time series to be incomplete. The presence of missing values in the data is a major problem since most forecasting algorithms require the time series to be complete in order to train a model.

A commonly employed strategy to overcome this problem is to impute missing values before training the model, for example, using a moving average. However, the quality of the imputations may not be good, impairing the training of the model. One way to improve the imputation strategy is to combine it with weighted time series forecasting. The latter consists of reducing the weight of the imputed observations and thus their influence during model training.

This document shows two examples of how skforecast makes it easy to apply this strategy.

Libraries¶

In [1]:

# Libraries
# ==============================================================================
import pandas as pd
import numpy as np
import matplotlib.pyplot as plt

plt.style.use('fivethirtyeight')
dark_style = {
    'figure.facecolor': '#212946',
    'axes.facecolor': '#212946',
    'savefig.facecolor':'#212946',
    'axes.grid': True,
    'axes.grid.which': 'both',
    'axes.spines.left': False,
    'axes.spines.right': False,
    'axes.spines.top': False,
    'axes.spines.bottom': False,
    'grid.color': '#2A3459',
    'grid.linewidth': '1',
    'text.color': '0.9',
    'axes.labelcolor': '0.9',
    'xtick.color': '0.9',
    'ytick.color': '0.9',
    'font.size': 12,
    'lines.linewidth': 1.5
}
plt.rcParams.update(dark_style)

from sklearn.linear_model import Ridge
from lightgbm import LGBMRegressor
from sklearn.metrics import mean_absolute_error
from skforecast.ForecasterAutoreg import ForecasterAutoreg
from skforecast.model_selection import backtesting_forecaster

# Libraries # ============================================================================== import pandas as pd import numpy as np import matplotlib.pyplot as plt plt.style.use('fivethirtyeight') dark_style = { 'figure.facecolor': '#212946', 'axes.facecolor': '#212946', 'savefig.facecolor':'#212946', 'axes.grid': True, 'axes.grid.which': 'both', 'axes.spines.left': False, 'axes.spines.right': False, 'axes.spines.top': False, 'axes.spines.bottom': False, 'grid.color': '#2A3459', 'grid.linewidth': '1', 'text.color': '0.9', 'axes.labelcolor': '0.9', 'xtick.color': '0.9', 'ytick.color': '0.9', 'font.size': 12, 'lines.linewidth': 1.5 } plt.rcParams.update(dark_style) from sklearn.linear_model import Ridge from lightgbm import LGBMRegressor from sklearn.metrics import mean_absolute_error from skforecast.ForecasterAutoreg import ForecasterAutoreg from skforecast.model_selection import backtesting_forecaster

  Note

In this document, a forecaster of type ForecasterAutoreg is used. The same strategy can be applied with any forecaster from skforecast.

Data¶

In [2]:

# Data download
# ==============================================================================
url = (
    'https://raw.githubusercontent.com/JoaquinAmatRodrigo/'
    'Estadistica-machine-learning-python/master/data/usuarios_diarios_bicimad.csv'
)
data = pd.read_csv(url, sep=',')

# Data preprocessing
# ==============================================================================
data['fecha'] = pd.to_datetime(data['fecha'], format='%Y-%m-%d')
data = data[['fecha', 'Usos bicis total día']]
data.columns = ['date', 'users']
data = data.set_index('date')
data = data.asfreq('D')
data = data.sort_index()
data.head(3)

# Data download # ============================================================================== url = ( 'https://raw.githubusercontent.com/JoaquinAmatRodrigo/' 'Estadistica-machine-learning-python/master/data/usuarios_diarios_bicimad.csv' ) data = pd.read_csv(url, sep=',') # Data preprocessing # ============================================================================== data['fecha'] = pd.to_datetime(data['fecha'], format='%Y-%m-%d') data = data[['fecha', 'Usos bicis total día']] data.columns = ['date', 'users'] data = data.set_index('date') data = data.asfreq('D') data = data.sort_index() data.head(3)

Out[2]:

	users
date
2014-06-23	99
2014-06-24	72
2014-06-25	119

In [3]:

# Generating gaps with missing values
# ==============================================================================
gaps = [
    ['2020-09-01', '2020-10-10'],
    ['2020-11-08', '2020-12-15'],
]

for gap in gaps:
    data.loc[gap[0]:gap[1]] = np.nan

# Generating gaps with missing values # ============================================================================== gaps = [ ['2020-09-01', '2020-10-10'], ['2020-11-08', '2020-12-15'], ] for gap in gaps: data.loc[gap[0]:gap[1]] = np.nan

In [4]:

# Split data into train-test
# ==============================================================================
data = data.loc['2020-06-01': '2021-06-01']
end_train = '2021-03-01'
data_train = data.loc[: end_train, :]
data_test  = data.loc[end_train:, :]

print(f"Dates train : {data_train.index.min()} --- {data_train.index.max()}  (n={len(data_train)})")
print(f"Dates test  : {data_test.index.min()} --- {data_test.index.max()}  (n={len(data_test)})")

# Split data into train-test # ============================================================================== data = data.loc['2020-06-01': '2021-06-01'] end_train = '2021-03-01' data_train = data.loc[: end_train, :] data_test = data.loc[end_train:, :] print(f"Dates train : {data_train.index.min()} --- {data_train.index.max()} (n={len(data_train)})") print(f"Dates test : {data_test.index.min()} --- {data_test.index.max()} (n={len(data_test)})")

Dates train : 2020-06-01 00:00:00 --- 2021-03-01 00:00:00  (n=274)
Dates test  : 2021-03-01 00:00:00 --- 2021-06-01 00:00:00  (n=93)

In [5]:

# Time series plot
# ==============================================================================
fig, ax = plt.subplots(figsize=(12, 4))
data_train.users.plot(ax=ax, label='train', linewidth=1)
data_test.users.plot(ax=ax, label='test', linewidth=1)

for gap in gaps:
    ax.plot(
        [pd.to_datetime(gap[0]), pd.to_datetime(gap[1])],
        [data.users[pd.to_datetime(gap[0]) - pd.Timedelta(days=1)],
         data.users[pd.to_datetime(gap[1]) + pd.Timedelta(days=1)]],
        color = 'red',
        linestyle = '--',
        label = 'gap'
        )

ax.set_title('Number of users BiciMAD')
handles, labels = plt.gca().get_legend_handles_labels()
by_label = dict(zip(labels, handles))
ax.legend(by_label.values(), by_label.keys(), loc='lower right');

# Time series plot # ============================================================================== fig, ax = plt.subplots(figsize=(12, 4)) data_train.users.plot(ax=ax, label='train', linewidth=1) data_test.users.plot(ax=ax, label='test', linewidth=1) for gap in gaps: ax.plot( [pd.to_datetime(gap[0]), pd.to_datetime(gap[1])], [data.users[pd.to_datetime(gap[0]) - pd.Timedelta(days=1)], data.users[pd.to_datetime(gap[1]) + pd.Timedelta(days=1)]], color = 'red', linestyle = '--', label = 'gap' ) ax.set_title('Number of users BiciMAD') handles, labels = plt.gca().get_legend_handles_labels() by_label = dict(zip(labels, handles)) ax.legend(by_label.values(), by_label.keys(), loc='lower right');

Impute missing values¶

In [6]:

# Value imputation using linear interpolation
# ======================================================================================
data['users_imputed'] = data['users'].interpolate(method='linear')
data_train = data.loc[: end_train, :]
data_test  = data.loc[end_train:, :]

# Value imputation using linear interpolation # ====================================================================================== data['users_imputed'] = data['users'].interpolate(method='linear') data_train = data.loc[: end_train, :] data_test = data.loc[end_train:, :]

Give weight of zero to imputed values¶

To minimize the influence on the model of imputed values, a custom function is defined to create weights following the rules:

• Weight of 0 if the index date has been imputed or is within 14 days ahead of an imputed day.

• Weight of 1 otherwise.

If an observation has a weight of 0, it has no influence at all during model training.

  Note

Imputed values should neither participate in the training process as a target nor as a predictor (lag). Therefore, values within a window size as large as the lags used should also be excluded.

In [7]:

# Custom function to create weights
# ==============================================================================
def custom_weights(index):
    """
    Return 0 if index is in any gap.
    """
    gaps = [
        ['2020-09-01', '2020-10-10'],
        ['2020-11-08', '2020-12-15'],
    ]
    
    missing_dates = [pd.date_range(
                        start = pd.to_datetime(gap[0]) + pd.Timedelta('14d'),
                        end   = pd.to_datetime(gap[1]) + pd.Timedelta('14d'),
                        freq  = 'D'
                    ) for gap in gaps]
    missing_dates = pd.DatetimeIndex(np.concatenate(missing_dates))   
    weights = np.where(index.isin(missing_dates), 0, 1)

    return weights

# Custom function to create weights # ============================================================================== def custom_weights(index): """ Return 0 if index is in any gap. """ gaps = [ ['2020-09-01', '2020-10-10'], ['2020-11-08', '2020-12-15'], ] missing_dates = [pd.date_range( start = pd.to_datetime(gap[0]) + pd.Timedelta('14d'), end = pd.to_datetime(gap[1]) + pd.Timedelta('14d'), freq = 'D' ) for gap in gaps] missing_dates = pd.DatetimeIndex(np.concatenate(missing_dates)) weights = np.where(index.isin(missing_dates), 0, 1) return weights

ForecasterAutoreg is trained including the custom_weights function.

In [8]:

# Create a recursive multi-step forecaster (ForecasterAutoreg)
# ==============================================================================
forecaster = ForecasterAutoreg(
                 regressor   = LGBMRegressor(random_state=123),
                 lags        = 14,
                 weight_func = custom_weights
             )

# Backtesting: predict the next 7 days at a time.
# ==============================================================================
metric, predictions = backtesting_forecaster(
                            forecaster         = forecaster,
                            y                  = data.users_imputed,
                            initial_train_size = len(data.loc[:end_train]),
                            fixed_train_size   = False,
                            steps              = 7,
                            metric             = 'mean_absolute_error',
                            refit              = True,
                            verbose            = False
                        )
print(f"Backtesting metric (mean_absolute_error): {metric:.2f}")
predictions.head(4)

# Create a recursive multi-step forecaster (ForecasterAutoreg) # ============================================================================== forecaster = ForecasterAutoreg( regressor = LGBMRegressor(random_state=123), lags = 14, weight_func = custom_weights ) # Backtesting: predict the next 7 days at a time. # ============================================================================== metric, predictions = backtesting_forecaster( forecaster = forecaster, y = data.users_imputed, initial_train_size = len(data.loc[:end_train]), fixed_train_size = False, steps = 7, metric = 'mean_absolute_error', refit = True, verbose = False ) print(f"Backtesting metric (mean_absolute_error): {metric:.2f}") predictions.head(4)

100%|██████████| 14/14 [00:00<00:00, 26.19it/s]

Backtesting metric (mean_absolute_error): 1904.83

Out[8]:

	pred
2021-03-02	10524.159747
2021-03-03	10087.283682
2021-03-04	8882.926166
2021-03-05	9474.810215

In [9]:

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