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Probabilistic forecasting of seasonal time series
Combining clustering and classification for forecasting
Colin Leverger1,2, Thomas Guyet3, Simon Malinowski4, Vincent Lemaire2,
Alexis Bondu2, Laurence Roz´e5, Alexandre Termier4, and R´egis Marguerie2
1IRISA
F-35042 Rennes Cedex, France
2Orange Labs
3Institut Agro/IRISA
4Universit´e Rennes 1/Inria/IRISA
5INSA/Inria/IRISA
Abstract. In this article, we propose a framework for seasonal time se-
ries probabilistic forecasting. It aims at forecasting (in a probabilistic
way) the whole next season of a time series, rather than only the next
value. Probabilistic forecasting consists in forecasting a probability distri-
bution function for each future position. The proposed framework is im-
plemented combining several machine learning techniques 1) to identify
typical seasons and 2) to forecast a probability distribution of the next
season. This framework is evaluated using a wide range of real seasonal
time series. On the one side, we intensively study the alternative com-
binations of the algorithms composing our framework (clustering, classi-
fication), and on the other side, we evaluate the framework forecasting
accuracy. As demonstrated by our experiences, the proposed framework
outperforms competing approaches by achieving lower forecasting errors.
Keywords: Time series, Probabilistic forecasting, Seasonality
1 Introduction
Forecasting the evolution of a temporal process is a critical research topic, with
many challenging applications. In this work, we focus on time series forecasting
and on data-driven forecasting models. A time series is a timestamped sequence
of numerical values, and the goal of forecasting is, at a given point of time, to
predict the next values of the time series based on previously observed values
and possibly on other linked exogenous observations. Data-driven algorithms
are used to predict future time series values from past data, with models that
are able to adapt automatically to any type of incoming data. The data science
challenge is to learn accurate and reliable forecasting models with as few hu-
man interventions as possible. Time series forecasting has many applications in
medicine (for instance, to forecast blood glucose of a patient [1]), in economy
(for instance, to forecast macroeconomic variable changes [2]), in the financial
domain (forecasting financial time series [3]), in electricity load [4] or in industry
(for instance, to forecast the server load [5,6]).
2 Leverger et al.
Time series forecasting algorithms provide information about possible situa-
tions in the future, and can be used to anticipate crucial decisions. Taking correct
decisions requires anticipation and accurate forecasts. Unfortunately, these ob-
jectives are often contradictory. Indeed, the larger the forecasting horizon, the
wider the range of expectable situations. In such case, a probabilistic forecasting
algorithm is a powerful decision support tool, because it handles the uncertainty
of the predictions. Probabilistic or density forecasting is a class of forecasting
that provides intervals or probability distributions as outcomes of the forecast-
ing. It is claimed in [7] that, in recent years, probabilistic forecasts have become
widely used. For instance, fan charts [8], highest density regions [9] or functional
data analysis [10] enable to forecast ranges for possible values of future data.
We are particularly interested in time series that have some periodic regular-
ities in their values. This kind of time series is said to be seasonal. For instance,
time series related to human activities or natural phenomena are often seasonal,
because they often exhibit daily regularities (also known as the circadian cycle).
Knowing that a time series is seasonal is a valuable information that can help for
forecasting. More specifically, learning the seasonal structures can help to gen-
erate longer-term predictions as it provides information about several seasons
ahead.
Furthermore, seasonality of a time series gives a natural midterm forecasting
horizon. Classical forecasting models (e.g., SARIMA [11]) predict the future
values of a given time series stepwise. The predicted values are used by further
steps. At each step, there is then a risk of the error to be accumulated due to
the recursive nature of the forecasts. The prediction of a whole season at once
aims at spreading the forecasting error all along the season. Thus, we expect
to forecast more accurately the salient part of a season that may lie in the
middle of the season. More practically, the prediction of a whole season at once
allows applications where such prediction is required to plan actions (e.g., to plan
electricity production a day ahead, it is necessary to predict the consumption
for the next 24 hours).
A second limitation of usual seasonal forecasting methods is the assumption
that the seasons have the same shape, i.e., the values evolve in the same way over
the season. The differences are with each other are due to noise and an additive
constant. Nevertheless, most of the real seasonal time series often contain more
than just one periodic pattern. For instance, daily connections to a given website
exhibit different patterns for a weekday or for a Sunday for instance. This kind
of structure cannot be well captured by classical forecasting methods.
In this article, we propose a generic framework called P-F2C (which stands
for “Probabilistic Forecasting with Clustering and Classification”) for seasonal
time series forecasting. This approach extends the F2C framework [6] (which
stands for “Forecasting with Clustering and Classification”). P-F2C predicts fu-
ture values for a complete season ahead at once, and this in a probabilistic man-
ner. The P-F2C predictions may be used for supporting decision-making about
the next season, handling the uncertainty in the future through the probabilistic
presentation of the result.
Probabilistic forecasting of seasonal time series 3
2 Probabilistic seasonal time series forecasting
In this section, we introduce the notations and the problem of seasonal time
series forecasting.
2.1 Seasonal time series
A time series Yis an ordered sequence of values y0:n=y0, . . . , yn−1, where
∀i∈[0, n −1], yi∈R(univariate time series). ndenotes the length of the
observed time series.
Yis said to be (ideally) seasonal with season length sif there exists S=
{S1, . . . , Sp}a finite collection of psub-series (of length s) called typical seasons
such that
∀i∈[0, m −1] , y(s×i):s×(i+1) =
p
X
j=1
σi,j Sj+εi(1)
where mis the number of seasons in the time series, εi∈Rsrepresents a
white noise and Pjσi,j = 1 for all j. In other words, it means that for a seasonal
time series Y, every season in Yis a weighted linear combination of typical
seasons. Intuitively, this modelling of a typical season corresponds to additive
measurements (e.g., consumption or traffic) for which the observed measure
at time tis the sum of individual behaviours. In this case, a typical season
corresponds to a typical behaviour of individuals, and the σ,j represents the
proportion of individuals of type jcontributing to the observed measure.
In the following yi=y(s×i):s×(i+1) ∈Rsdenotes the i-th season of Y.
2.2 Seasonal probabilistic forecasting
Let Y=y0, . . . , yn−1be a seasonal time series, an sbe its season length.
Note that the season length of a time series (s) is estimated using Fisher’s
g-statistics [12]. Without loss of generality, we assume that the length of a time
series is a multiple of the season length, i.e.,n=m×s.mdenotes the number of
seasons in the observed time series. The goal of seasonal probabilistic forecasting
is to estimate
Pr(y∗
n:n+s|y(n−γ×s):n) = Pr(y∗
m|y(m−γ):m) (2)
where y∗
m=y∗
n:n+sare the forecasts of the snext values (next season) of the
observed time series, and y(m−γ):m=y(n−γ×s):nare the observed values of the
last γseasons. γis a parameter given by the user.
We now propose an equivalent formulation of this problem considering our
hypothesis on seasonal time series and we denote S={S1, . . . , Sp}the set of p
typical seasons. Thus, Equation 2 can be rewritten as follows:
Pr(y∗
m|y(m−γ):m) = X
S∈S
Pr(y∗
m|S).Pr(S|y(m−γ):m) (3)
4 Leverger et al.
COMPLETE MATCH
SUBMATCH 1 SUBMATCH 2
Fig. 1: Difference between clustering (on the left), which matches the entire time
series, with coclustering (on the right), which is able to match subintervals of
the time series of various other time series.
where Pr(y∗
m|S) is the probability of having y∗
mgiven that the type of the
next season and Pr(S|y(m−γ):m) is the probability that the next season is of
type Sgiven past observations.
The problem formulation given by Eq. 3 turns the difficult problem of Eq. 2
into two well-known tasks in time series analysis:
–estimating the first term, Pr(y∗
m|S) leads to a problem of time series clus-
tering. The problem is to both define the typical seasons, S, and to have
the distributions of the season values. A clustering of the seasons (yi)i=0:m
of the observed time series identifies the typical seasons (clusters) and gives
the required empirical distributions ˆ
Pr(y, S).
–estimating the second Pr(S|ym−γ:m) is a probabilistic time series classi-
fication problem. This distribution can be empirically learnt from the past
observations (yi−γ:i, S∗
i+1)i=γ:mwhere S∗
idenotes the empirical type of the
i-th season obtained from the clustering assignment above.
This problem formulation and remarks sketch the principles of a probabilistic
seasonal time series forecasting. P-F2C is an implementation of these principles
with a specific time series clustering.
3 The P-F2C forecaster
P-F2C is composed of a clusterer that models the latent typical seasons and a
classifier that predicts the next season type given the recent data. The forecaster
is fit on the historical data of a time series. Then, the forecaster can be applied
on the time series to predict the next season(s).
P-F2C clusterer is based on a probabilistic co-clustering model that is pre-
sented in the next section. In Section 3.2, we present how to use classical classi-
fiers to predict the next seasons.
Probabilistic forecasting of seasonal time series 5
3.1 Coclustering of time series: a probabilistic model
Coclustering is a particular type of unsupervised algorithm which differs from
regular clustering approaches by creating co-clusters. The objective of cocluster-
ing approaches consists in simultaneously partitioning the lines and the columns
of an input data table. Thus, a co-cluster is defined as a set of examples belong-
ing to both a group of rows and a group of columns. In [13], Boull´e proposed
an extension of co-clustering to tri-clustering in order to cluster time series.
In this approach, a time series with an identifier Cis seen as a set of couples
(T, V ), where Tis a timestamp and Va value of a measurement. Thus, the
whole set of time series is a large set of points represented by triples (C, T, V ).
The tri-clustering approach handles the three variables (Cis categorical and
T,Vare numerical) to create homogeneous groups. A co-cluster gathers time
series (group of identifiers) that have similar values during a certain interval of
time. Contrary to the classical clustering approaches (e.g., KMeans, K-shape,
GAK) [14] that are based on the entire time series, the coclustering approach
uses a local criterion. This difference is illustrated in Figure 1: A distance based
clustering (on the left) evaluates the distance between whole time series, in the
co-clustering approaches, the distance is based on subintervals of the seasons.
This enables to identify which parts of the season are the most discriminant.
Besides, tri-clustering is robust to missing values in time series.
The tri-clustering approach of Boull´e is based on the MODL framework
[10]. The MODL framework makes a constant piecewise assumption to estimate
the joint distribution Pr(C, T, V ) by jointly discretising the variables T,Vand
grouping the time series identifiers of the variable C. The resulting model consists
of the Cartesian product6of the three partitions of the variables C,T,V. This
model can be represented as a 3D grid (see Figure 2, on the left). In this 3D grid,
if one considers a given group of time series (i.e., a given group of C), the model
provides a bivariate discretisation which estimates Pr(T, V |C) = Pr(C,T ,V )
Pr(C)as a
2D grid (see Figure 2, on the right). This 2D grid gives the probability to have
a given range of values during a given interval of time. Therefore knowing that
a time series belongs to a given cluster the corresponding 2D grid may then be
used for crafting forecasts (see next section).
In the MODL approach, finding the most probable tri-clustering model is
turning into a model selection problem. To do so, a Bayesian approach called
Maximum A Posteriori (MAP) is used to select the most probable model given
the data. Details about how this 3D grid model is learned may be found in
[13,15]. The main idea could be summarised as finding the grid which maximises
the contrast compared to a grid based on the assumption that T, V and Care
independent (i.e., Pr(V, T , C) compared to Pr(V) Pr(T) Pr(C)). Therefore the
estimation of this MAP model outputs: (i) νintervals of values Vi= [vl
i, vu
i] for
i= 1, . . . , ν, (ii) τintervals of times Ti= [tl
i, tu
i] for i= 1, . . . , τ , (iii) groups
of time series. These groups of time series corresponds to the typical seasons,
6The Cartesian product of the three partitions is used as a constant piecewise esti-
mator – i.e., a 3D histogram.
6 Leverger et al.
Fig. 2: Illustration of a trivariate coclustering model where a slice referred to
forecasting “grid” is extracted.
denoted Sin the above model. |S| is the number of clusters at the finer level
that is optimal in the sense of the MODL framework.
In the time series forecasting approach proposed in this paper, the right
number of (tri-)clusters is optimised regarding to the forecasting task. More
precisely, this number is optimised according to the performance of the model
at prediction time, using the validation ensemble. This value could differ from
|S|. Therefore the MODL coclustering approach allows applying a hierarchical
clustering to the finer level to have a coarse level with a lower number of clusters
called C∗,C∗<|S|. A grid search selects the C∗value based on the forecast
accuracy on the valid dataset.
Let us now come back to the formalisation of probabilistic time series fore-
casting: ˆ
Pr(y∗
m|S) is estimated by the MODL model from the conditional
probabilities Pr(V, T |C=S) where Sdenotes one of the time series groups,
i.e. a typical season. In practice, the grid is used to estimate the distribution
of values at each time point of a season. With MODL, the distribution is a
piecewise constant function.
3.2 Predict the next type of seasons
The problem is here to estimate empirically Pr(Si+1 |y(i−γ):i) the probability of
having a type of season Si+1 ∈ S for the (i+ 1)-th season given the observations
over the γpast seasons. We consider two different sets of features to represent
the γprevious seasons. The first approach consists in having only the time series
values y(i−γ):ias features. The second approach uses the time series values and
the types of the previous seasons as features.
Then, the next season prediction problem consists in learning a probabilis-
tic classifier (Naive-Bayes classifier, logistic regression, decision tree or random
forests) or a time series classifiers (TSForest [16], Rocket [17]). Note that time
series classifiers can use only the time series values.
3.3 Select the best parameters (Portfolio)
The P-F2C forecaster is parameterised by the number of seasons in the past (γ)
used for learning next season type, a maximum number of typical seasons to
Probabilistic forecasting of seasonal time series 7
Fig. 3: At the top: typical seasons of length 10 used for generating the time series,
at the bottom: examples of generated time series with white noise (7 seasons).
detect in a non-supervised way, and the type of classifier. The γparameter is
introduced in the problem definition and its choice is left to the user who specifies
what is the forecasting task. On the other hand, the other parameters may be
difficult to be set by the user, and we do not think that one of the classifiers
will outperform the others for all the time series. For these reasons, the portfolio
approach (denoted PP-F2C) implements a grid search for the best parameters
by splitting the dataset into a training (75%) and a validation dataset (25%) to
identify the best value of the parameters. Once the best values have been set,
the clusterer and the classifier are fitted on the entire dataset.
4 Illustration on a synthetic dataset
This section shows results with synthetic data. The goal is to illustrate the prob-
abilistic grid used in P-F2C method, and to give intuitions behind probabilis-
tic forecasting that are provided by P-F2C. We compare the output of P-F2C
against the output of DeepAR [18], a state-of-the-art probabilistic time series
forecaster.
4.1 The data generated
Generating data is a good strategy for checking assumptions before launching ex-
periments at scale. Indeed, the shape of the generated data is often simpler, and
completely controlled. Experiments may be executed with various parameters,
to plot understandable results and to validate basic expectations.
8 Leverger et al.
The seasonal data generated for this section follows some well-established
seasonal sequences. Three different time series patterns are defined for three
different latent types of season of length 10. In the Figure 3, one type of season
(s1in orange) with always increasing values is observed, one type of season (s2in
green) with two peaks is observed, etc. Those three different types of season are
then repeated 50 times in a defined order (s1, s1, s0, s2, s1, s1, s2, s0, as observed
in Figure 3, on the right, which shows the entire sequence that is being repeated),
and noise is added to the final time series to make the forecasting process less
straightforward.
4.2 Grid probabilistic forecasts
Once trained, we apply the P-F2C forecaster at the end of the time series illus-
trated in Figure 3 on the right. Knowing the sequence of patterns, we can guess
that a season of type s2is coming ahead. Indeed, the last three patterns seems
to follow the sequence [s1, s1, s0].
The Figure 4 shows two examples of forecasts with different values of γ.
The real values of the predicted time series are in blue (noisy version of the
s2pattern). The probabilistic forecasts are shown in a red overlay. It is a set of
rectangles that visualise the homogeneous regions that have been identified by
MODL coclustering. The darker the red, the more probable next season ahead
lay in this (T,V) interval.
The Figure 4 on the left is the forecast obtained with γ= 1. It illustrates a
probabilistic forecast with a lot of uncertainty. Indeed, light red cells are observed
in the figure where the data are predicted to lay (with a low probability). In this
case, the classifier is unable to predict accurately the next type of season. With
γ= 1 the classifier has only the information of the preceding season (of type s0).
In this case, the forecaster encountered two types of season after a s0season:
s1or s2with the same probability. Then, the predicted grid is a mixture of the
two types of grids. For the first half of the season, the forecast is confident in
predicting the linear increase of the value (darker red cells), but for the second
half, the forecast suggests two possible behaviours: continue the linear increase
(s1) or a decrease (s2). Note that the grids of all typical seasons share the same
squaring. MODL necessarily creates the same cuttings of a dimension (Vor T)
along the others (C).
The Figure 4 on the middle is the forecast obtained with γ= 3. It illustrates
a good probabilistic forecast. The real values (in blue) often appear in the red
boxes where the red is very dark. It means that the season type was both well
described by MODL and well predicted by the classifier. In this case, a larger
memory of the forecaster disentangles the two possible choices it had above.
After a [s1, s1, s0], the forecaster always observed seasons of type s2. Thus, the
grid of this pattern is predicted.
It is worth noting that, for γ= 1, the use of the MODL probabilistic grid
suggests two distinct possible evolution of the time series, but there is an un-
certainty on which evolution will actually occur. In the classical probabilistic
Probabilistic forecasting of seasonal time series 9
Fig. 4: One season ahead grid forecasts for the generated time series with γ= 1
at the top left and γ= 3 at the top right, and DeepAR at the bottom.
forecasts, probabilities are distributed around a mean time series. This is illus-
trated on the Figure 4 on the right with DeepAR using the 7 seasons in the
past to predict the next season. On the second half of the season, the predicted
probabilistic distribution suggests a behaviour in between s1and s2with a larger
uncertainty. Such model makes confusion between uncertainty of behaviour and
imprecise forecast. In the case of seasonal time series with different types of
season, the mean time series has no meaning for an analyst.
5 Experiments
This section presents experiments to assess the accuracy of P-F2C. We start by
introducing the experimental settings, then we investigate some parameters of
our model and finally we present the result of an intensive comparison of P-F2C
to competitors.
5.1 Experimental protocol
The framework has been developed in Python 3.5. The MODL coclustering is
performed by the Khiops tool [19]. The classification algorithms are borrowed
from the sklearn library [20].
In our experiments, we used 36 datasets7, from various sources and nature:
technical devices, human activities, electricity consumption, natural processes,
7Datasets details can be downloaded here: https://tinyurl.com/4kffdwhc (tempo-
rary link). It includes the sources of time series.
10 Leverger et al.
123
CD
gamma = 3
gamma = 2
gamma = 1
234
CD
TimeSeriesForestClassifier
LogisticRegression
RandomForestClassifier
DecisionTreeClassifier
GaussianNB
Fig. 5: Critical diagrams used to find the best parameters for the P-F2C imple-
mentation.
etc. All these datasets have been selected because seasonality was identified and
validated with a Fisher g-test [12]. Each time series is normalised using a z-
normalisation prior to data splitting, in order to have comparable results. For
the experiments, 90% of the time series are used to train the forecaster (this train
test is internally split in training and valid datasets) and 10% of the original time
series are used to evaluate the accuracy.
P-F2C and PP-F2C are compared with classical deterministic time-series
forecasters (AR, ARIMA, SARIMA, HoltWinters), with LSTM [21], Prophet [22]
and with the F2C method [6] which uses the principles as P-F2C but with K-
means clustering algorithm and random forest classifiers to learn the structure in
the season sequence. P-F2C being a probabilistic methodology, we also compare
it with DeepAR [18].
We use Mean Absolute Error (MAE) and Continuous Ranked Probability
Score (CRPS) to compare the forecasts to the real time series. The MAE is
dedicated to deterministic forecasts while CRPS is to probabilistic ones. It is
worth noting that the CRPS is analogous to MAE for deterministic forecasts.
Therefore, comparing MAE measure for deterministic forecasts against CRPS
values for probabilistic forecasts is technically sound [23]. The CRPS is used
for DeepAR and P-F2C. All the other approaches forecast crisp time series and
their accuracy is evaluated through MAE. For each experiment, we illustrate the
results with critical difference diagrams. A critical difference diagram represents
the mean rank of the methods that have been obtained on the set of the 36 times
series. The lower the better. In addition, the representation shows horizontal bars
that group some methods. In a same group, the methods are not statistically
different according to the Nemenyi test.
5.2 Parameters sensitivity
In this section, an analysis of the alternative settings of the P-F2C methodology
is conducted. We investigate the effect of two choices: the choice of the γvalue,
i.e., the number of seasons to consider in the history; and the choice of the
classifier to predict the next type of season in case we do not use the portfolio
optimisation.
Probabilistic forecasting of seasonal time series 11
2345678
CD
F2C
PP_F2C
P_F2C
LSTM
SARIMA
HOLTWINTERS
PROPHET
DeepAR
ARIMA
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DeepAR win: 5
PP_F2C win: 27
p−value=4.597e−06
0.00
0.25
0.50
0.75
1.00
0.0 0.3 0.6 0.9
DeepAR
PP_F2C
DeepAR vs PP_F2C in terms of CRPS error.
Fig. 6: At the top: Critical diagram of the comparison between different predic-
tion approaches (acronyms of method are detailed in the text). At the bottom:
Win-Tie-Loose graph between PP-F2C and DeepAR.
Figure 5 on the left shows a critical diagram that compares the ranking of
P-F2C with different values of γ(1, 2 or 3). For this experiment, the classi-
fier is the RandomForestClassifier (and we had the same results with the other
classifiers). We notice that the larger γ, the lower the error. Indeed, as seen in
Section 4, larger γimproves the accuracy of the forecast of the next season type.
Nonetheless, we observed that for some time series, lower γmay be better. We
explain this counter-intuitive results by the small length of some of the time
series. In the cases, the number of seasons in the training set is too small to fit
the numerous parameters of a classifier with γ×sfeatures.
Figure 5 on the right shows a critical diagram that compares the classifiers
used to predict the next type of season. It shows that time series forest classifier
[16] is on average in first position. This classifier has been designed specifically for
time series classification, it explains why it outperforms the other approaches.
Nonetheless, the differences with Logistic Regression and Random Forest are
not statistically significant. Their capability to use extra-information, such as
the type of seasons, may be an interesting advantage to improve performances.
5.3 P-F2C and PP-F2C vs opponents
The critical diagram of Figure 6 compares the performances of the methods.
P-F2C denotes our approach configured with the best parameters on average
12 Leverger et al.
found in Section 5.2. PP-F2C denotes P-F2C that is optimised on the valid test
for each dataset (portfolio). It shows that rank-wise, the seasonal forecaster F2C,
P-F2C and PP-F2C are performing better than the others. We can first notice
that the portfolio actually improve the performances of P-F2C. Nonetheless, the
non-probabilistic approach outperform PP-F2C.
We also notice that F2C outperforms PP-F2C. Even if a PP-F2C forecast fits
the time series (see Figure 4), the piece-wise approximation generates a spread of
the probabilistic distribution that penalises the CRPS. Nonetheless, it is worth
noting that the rank difference with F2C is not statistically significant, and that
probabilistic forecast convey meaningful information to trust the forecasts.
Then, we compared PP-F2C with another probabilistic forecaster, i.e. DeepAR.
The critical diagram of Figure 6 shows that PP-F2C outperforms DeepAR signif-
icantly (p < 10−6). The win/tie/lose graph on the right shows how many times
PF2C won against DeepAR (points below the diagonal) and the relative val-
ues of CRPS. The point positions illustrate that PP-F2C outperforms DeepAR
significantly on most of the datasets.
6 Conclusion
P-F2C is a probabilistic forecaster for seasonal time series. It assumes that sea-
sons are a mixture of typical seasons to transform the forecasting problem into
both a clustering and a classification of time series. The P-F2C applies param-
eterless coclustering approach that generates grid forecasts, each typical grid
being a typical seasonal behaviour. In addition we proposed PP-F2C that adjust
P-F2C parameters for each time series. PP-F2C outperforms on average the com-
petitors except F2C on various seasonal time series. F2C is based on the same
principle as PP-F2C but is not probabilistic and parameterless. Nonetheless,
we illustrated the interest of probabilistic grid forecasting to give information
about uncertain distinct mean behaviours. Indeed, the probabilistic grid mixture
is more interpretable than combining probabilistic distribution around a mean.
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