HERMES: Hybrid Error-corrector Model with inclusion of External Signals for nonstationary fashion time series
Authors: Etienne David
Jean Bellot
Sylvain Le Corff
Venue: Accepted by TMLR ()

Abstract: Developing models and algorithms to predict nonstationary time series is a long standing statistical problem. It is crucial for many applications, in particular for fashion or retail industries, to make optimal inventory decisions and avoid massive wastes. By tracking thousands of fashion trends on social media with state-of-the-art computer vision approaches, we propose a new model for fashion time series forecasting. Our contribution is  twofold. We first provide publicly a dataset gathering 10000 weekly fashion time series. As influence dynamics are the key of emerging trend detection, we associate with each time series an external weak signal representing behaviours of influencers. Secondly, to leverage such a dataset, we propose a new hybrid forecasting mode. Our approach combines per-time-series parametric models with seasonal components and a global recurrent neural network to include sporadic external signals. This hybrid model provides state-of-the-art results on the proposed fashion dataset, on the weekly time series of the M4 competition, and illustrates the benefit of the contribution of external weak signals.

Published in Transactions on Machine Learning Research (09/2023)
HERMES: Hybrid Error-corrector Model with inclusion of External Signals for nonstationary fashion time series
Étienne David
etienne. Cicenne.darisd@heeuitecc.cmmm
SAMOVAR, Télcom SudParis,
Institut Polylechnique de Paris, 91120 Palaiseau, France
Jean Bellot
jean. won..bellot@hheuriecc..com
Heuritech, 6 Rue de Braque, 75003 Paris
Sylvain Le Corff
sylvain. le corf@Bsortomm---unberrsss..
LPSM,
Sorbonne Université, UMR CNRS 8001, 75005, Parts
Reviewed on OpenReview: https: openreview. net/ http://www.....oooooo//o//oaaaa.
Abstract
Developing models and algorithms to predict nonstationary time series is a long stand- ing statistical problem. It crucial for many applications, in particular for fashion or retail industries, to make optimal inventory decisions and avoid massive wastes. By tracking thou- sands of fashion trends on social media with soate--o-thheartt computer vision approaches, we propose new model for fashion time series forecasting. Our contribution twofold. We first provide publicly dataset gathering 10000 weekly fashion time series. As influence dynamics are the key of emerging trend detection, associate with each time series external weak signal representing behaviours of influencers. Secondly, to leverage such dataset, we propose a new hybrid forecasting mode!" Our approach combines per-time- scries paramctric models with scasonal components and a global recurrent noural network to include sporadic external signals. This hybrid model provides state-of-the-arl results on the proposed fashion dataset, on the weekly time series of the M4 competition (Makridakis et al. 2018) and illustrates the benefit of the contribution of external weak signals.
1 Introduction
Multivariate time series forecasting is a widespread statistical problem with many applications, see for instance Särkkä (0133: Douc et al. (2014); Zucchini et al. (2017) and the numerous references therein. Parametric generative models provide explainable predictions with statistical guarantees owing to a precise modeling of the predictive distributions of new data based on a record of past obscrvations. Calibrating these models, for instance using maximum likelihood inference, often requires fair amount of tuning to design time-series-specific models able to provide accurate forecasts and sharp confidence intervals. Depending on the case, statistical properties of the signal and the available data, many families of models have been proposed for time series. The exponential smoothing model (Brown & Meyer 1961 the Trigonometric Box-Cox transform, ARMA errors, Trend, and Seasonal components model (TBATS) (Livera et al. 2011), or the ARIMA with the Box-Jenkins approach (Box et al. 2015) are for instance very popular parametric generative models. Hidden Markov models (HMM) are also widespread and presuppose that available observations are defined using missing data describing the dynamical system. This hidden state is assumed to be Markov chain such that at cach time step the reccived observation is random
https://github. com/ coom/etiin itteea/ERadSs

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function of the corresponding latent data. Although hidden states are modeled Markov chain, the observations arising therefrom have a complex statistical structure. In various applications where signals exhibit non-stationarities such as trends and seasonality, classical HMM are not adapted. However, Touron (2017) recently proposed seasonal HMM, assuming that transition probabilitios between the states, as well as the emission distributions, are not constant. time but evolve in periodic manner. Strong consistency results were established in Touron (2019) and Expectation Maximization basod numerical experiments were proposed. Although these works provide promising results, HMM are computationally expensive train and are not yet well studied for seasonal sequences with thousands of components.
In many fields, single or few time scries have become thousands of sequences with various statistical proper- ties. In this new context, classical time-series-specifie statistical models show limitations when dealing with mumerous heterogeneous data. By constrast, recurrent neural networks and recent sequence to sequence deep learning architectures offer very appealing numerical alternatives thanks to their capability of leveraging any kind of hetcrogencous multivariate data, seo for instance Hochreiter & Schmidhuber (1997): Vaswani et al.. 20177; Siami-Namini al. (2018);
|; Li et al. (2019); Lim el. al. (2019); Salinas et al. (2020). The DeepAR model proposed in (Sainaas et al. (2020) provides global model from many time series based on a multi-layer recurrent neural network with LSTM cells. More recently, applications using the Transformer model have been proposed (Li et al. 2 The Temporal Fusion Transformers (TFT) approach is a direct alternative to the DeepAR R. model (Lim et al. 2019.. Unfortunately, all these solutions suffer from two main weaknesses. Firstly, many of them are black- boxes as the final forecast usually does not come with a statistical guarantee although a few recent works focused on measuring uncertainty in recurrent neural networks, see Martin et al.. p. Secondly, without fine preprocessing and well chosen hyperparameters, these methods may lead to poor results and be outperformed by traditional statistical models, see Makridakis et al. (2018).
In this paper, we consider an emerging time series forecasting application referred to as fashion trends prediction. In fashion and retails industries, accurately anticipating consumers' needs is vital and wrong decisions can lead to massive wastes. With the explosion of social network and the recent advances in image recognition, it possible to translate the visibility of fashion items on social media over time into time series. Consequently, models and algorithms can be trained to accurately anticipate and predict consumer behaviour. In Ma et al. (2020), a dataset is provided using social media pictures and an image recognition framework to detect several clothes: 2000 fashion time series are proposed with weekly scasonality. However, only 3 years of historical data is available (144 data points) that may not be sufficient for some statistical approaches. In Ma et al. (020)) another dataset is presented gathering 8000 fashion sequences with an historical available data increased to 5 years. Nevertheless, only 120 values are available for cach fashion time series and the overall volume remains low for a large part of the sequences resulting in a lot of noise and no clear patterns. In this paper, we propose a new fashion dataset overcoming the weaknesses of the two previous ones. Based on recent image recognition algorithms (Ren et al.] 2015 Chollet, 2017], we built large fashion dataset containing 10000 weekly sequences of fashion trends on social media with 5 years of historical data from 01-01-2015 to 30-12-2019. This dataset has very appealing properties: all time series have the same length (261 data points), there is no missing value and there is no sparse time scries even for niche trends. Concerning fashion dynamics, some of them appear to be really volatile with nonlinear changes of dynamics resulting from the emergence of new tendencies. In this context, understanding early signals of the apparition of a trend one of the key to accurately forecast the future of the fashion. Consequently, the originality of our dataset comes from the fact that additional external weak signals are introduced. With our fashion expertise, we detected several groups of highly influential fashion users. Analyzing their specific behaviours social media, we associate with each time series an external weak signal representing the same fashion trends on this sub-category of users. They are called weak signals because they are often alerts or events that are too sparse, or too incomplete to allow on their own an accurate estimation of their impact on the prediction of the target signal. Exploring this new way of representing fashion, we aim at designing a model able to deal with such a large dataset, leveraging complex external weak signals and finally providing the most accurate forecasts.
Recurrent neural networks are appealing to tackle our forecasting problem due to their capability of leveraging external data. Recently, hybrid models combining deep neural network (DNN) architectures with widespread statistical models to deal with seasonality and trends have been proposed, see for instance Zhang (2003):
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a)
b)
C)
Figure 1: From social media to fashion time series. a) A complete image dataset of 150 millions pictures is collected from social media users localized on strategic markets. b) A visual recognition pipeline is applied on images. Global fashion items are detected with collection of fine- ine-grain attributes. c) Results are aggregated by fashion trend over time and normalized in order to remove social media bias.
Jianwei et al. 2019 Bandara et al. 2020). The approach providing the most striking results was proposed in Smyl 2020 in the context of the M4 forecasting competition (Makridakis et al., 2020). Given a large dataset, a per- ten-time-seres multiplicative exponential smoothing model was introduced to estimate simple but fundamental components for each time series and compute first prediction. Then a global recurrent neural network was trained on the entire dataset to correct errors of the previous exponential smoothing models.
Following this work, we present in this paper HERMES, forecasting with inclusion of external signals. This new architecture is decomposed into two parts: First, per-time-series parametric statistical model is trained on each sequence. Then, global recurrent neural network is trained to evaluate and correct the forecast of the first collection of models. The ability to deal with external signals reveal the real potential of the hybrid approach: a global neural network, able to leverage large amounts of heterogeneous data, deal with any kind of external weak signals, learn context and finally correct weaknesses and errors of parametric models.
hybrid recurrent model for time series
The paper is organized follows. Section 2 presents the new fashion dataset provided with this article. Then, the proposed forecasting approach is presented in Section 3 Section describes the experiments and comparisons with several benchmarks on 2 different use cases: the fashion datasct and the M4 competition weekly dataset. Finally, a general conclusion and some research perspectives are given in Section 5
2 From social media to fashion time series
2.1 Translate fashion to data
An image dataset of 150 millions pictures is collected from different social media such Instagram or Weibo, We targeted 5 strategic markets for the retail industry using posts localisation: the United States, Europe, Japan, Brazil and China. With compliance of privacy and data protection, only public accounts are selected and no potential private information is used, saved or revealed during the whole process. The second step consists in creating a powerful visual recognition framework to be able to detect clothes details on pictures like the type of clothing, the form, the size, the color, the texture, etc. To do so, the following framework is designed.
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Fashion time series
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Influencers weak signal
Dom M.A
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Figure 2: A shoes trend of the fashion dataset. black the main signal and in orange its associated fashion- forward weak signal. The sudden explosion of the influencers signal at the end of 2018 announces the future burst of the trend in the mass market.
First, an object detection model trained to detect the position, the size and the general type of possible multiple fashion items on a picture. This localization model based on the Faster-RCNN architecture introduced in Ren et al. (2015). Starting from weights trained on MS-COCO (Lin al. 2014), the model is fine-tuned with our data with standard setup following the original paper.
Additionally, several visual recognition models are trained classifying rich collection of 350 fashion details. We train one classifier for each category of fashion item: one for pants, another for tops, third for shoes, etc. These models are all based on the Xception architecture introduced in Chollct (2017). So as to trained them, large amount of social media pictures (between 200k and 800k training images depending on the category) have been manually lagged to constitute meaningful training datasets depending on the classification task. Architectures are first initialized with public weights trained on ImageNet (Russakovsky al. 2014] and then fine tuned on the manually labelod dataset corresponding to their task.
At inference time, we first apply the localisation model which predicts boxes of generic fashion items (Lops, pants, shoes, dresses, etc.) for each image. Then, each fashion item is cropped from its full image, resized the classifiers' input size (299 299 px) and fed into the related classifier: top will be fed into the model trained tops, etc. We obtain for each image a set of boxes, associated with a general category and set of fine-grain attributes describing this object. As a final step, fashion experts aggregate those attributes to define relevant trends for the fashion and retails industry. We call them fashion trends. They can be just one attribute detected by a visual recognition model, for example the sneakers fashion trend. They can also be a combination of several attributes, for instance the Mini A-line dress fashion trend which is a combination of several attributes (dress length, shape, catogory....) cach one detected with specific vision recognition model.
The 150 million of social media pictures are analyzed with this visual recognition pipeline. Out of those images, we detected clothes in 96 millions posts making the final datasct used in this paper. We aggregate results by fashion trend definition over the time and thousands of trends are finally translated from social media to time series. We note --s.m the final raw sequence representing the fashion trend j of the cloth type e for the gender g on market m. At cach time t, yf yf.g.mm. represents the number of posted pictures the market m during the week where computer vision algorithms detected the fashion trend j of the cloth type for the gender g. As illustration, example of fashion time series is given in Figure
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2.2 Removing social media bias
Due to the increasing use of social media and continuous changes of users' behaviours, a normalization step is applied to the raw sequences yf...m.. in order to remove bias. The pre-processing of data presented in this section does not constitute a technical contribution. This step is presented in depth to explain how the dataset provided with this paper constructed. Let ges.." be the global sequence of the cloth type c for the gender 9 on market m (e.g the evolution of the skirts in general for female in Europe). With the R package stats, the Seasonal-Trend decomposition using LOESS (Cleveland the scasonal component of fr...m. The resulting descasonalized signal is called graa.".. Finally, for any fashion trend j, the following normalized sequence is defined for all 0 S L T:
al. [990) is used to remove
Dt G.g.m.
Vi
(1)
g.m.m.
where denotes the number of available time steps and i represent a unique identifier summarizing the 4 information e, 9: m and j. The time series yf...m. is divided by the deseasonalized signal y=.. and not fr... in order to avoid removing the seasonality of all the fashion trend sequences. With this normalizing step, most of the social media bias is removed and the final normalized sequences are expressed in share of category. As an illustration, example of the normalization process is displayed in Figure 3 In this example, the raw and normalized time series of the Jersey top trend for females in China (y with c=Top, g=Female, m=China and j=Jersey top) are presented. So as to compute the normalized signal, the raw time series of the Jersey top trend for females in China is divied by the deseasonalized raw time series representing the global Top trend for female in China (7 with c= g-Female and m=China).
2.3 Weak signal
In theoretical fashion dynamics (Rogers 1962), different categorics of adopters follow a trend in succession, resulting in several adoption waves. So as to catch the early signal of the emergence of a trend, 6000 social media influencers were selectioned by hand by fashion exports. Aggregating them, a specific "fashion- oriented* panel is created. With the same methodology as for the main panel described in Section [2.] and Section 2.2 normalized time series representing each fashion trend on this specific population is created. We named foshion-forware this weak signal. For all fashion sequence (vt)SesT, let {vt f.8 higer be the normalized sequence representing the behaviours of influencers regarding the fashion trend i. As we want to detect shifts between the main signal and the fashion forward signal, the following input is computed for the hybrid model: for all E [I....,T} and any fashion trend i,
36
w
Vi
vi
where T denotes the number of available time steps. Values close to 0.5 indicate similar behaviour between the influencers panel and the general panel. For instance, an emerging fashion shoes trend with its fashion- forwards weak signal is represented in Figure? 2
2.4 Fashion dataset
With this paper, we provide publicly sample of 10000 normalized fashion trends for men and women, over 9 different categories and 5 different markets. Each sequence has 261 time steps, from 2015-01-05 to 2019-12-31 with weekly values and no missing values. This collection of 10000 fashion trends was selected in order to represent finely the issues faced by the fashion industry. For instance, some sequences show complex behaviours with sudden changes, referred to as emorging or declining trends. central point of this work is to accurately detect and forecast. such trends. In addition, each fashion time series is linked with its associated normalized fashion forward signal as presented in the section above. An overview of the dataset can be found in Table [] All the trends names are anonymized and for cach trend, only the following macro information are revealed: the geolocalisation, the gender and the cloth type.
https://github. com/etiiav/HERMEs
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raw time series
2000
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2020
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wW
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Figure 3: Example of difference between the raw sequence and the normalized one for the Jersey top fashion trend for females in China. In this example, the Jersey top fashion raw time series for females in China is normalized by the deseasonalized global top raw time series for females China. The final normalized time series finally expresssed as share of the global catagery. (Top) Time series representing the raw signal of the Jersey top fashion trend for females in China. (Bottom) Time series representing the normalized signal of the Jersey top fashion trend for females in China.
Table 1: Fashion time series overview. For each couple geozone/categong the table gives the number of trends Cremale/Male))
Top
Pants
Short
Skirt
Dress
Coat
Shoes
Color
Texture
United States
111/208
149/112
17/22
29/-
20/-
208/151
293/86
38/41
85/81
Europe
409/228
194/114
48/21
28/-
20/-
211/159
303/78
41/42
87/74
Japan
103/218
136/107
19/31
25/-
23/
185/149
S11/78
16/12
92/65
China
424/202
147/114
46,29
27/-
27/-
178/161
310/78
41/47
88/77
Brazil
131/222
194/117
19/27
30/-
28
203/152
811/76
48/41
107/84
Total
2078/1078
700/5664
239/130
142/-
118/-
985/772
1528 /196
214/216
459/381

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3 HERMES: new hybrid model for time series forecasting
We introduce new hybrid approach for time series forecasting composed of two parts: collection of per-time ser-iim--seres parametric models, and global error-corrector neural network train on all time series. Per- time-series parametric models are used in particular to learn local behaviours and to normalize sequences by removing trends and seasonality. Then, recurrent neural network driven by the weak signals is trained to correct these per-time-series models.
Consider N 3 1 time scries. For all E n & N and t & T, let ye be the value of the n-th sequence at time l and y" = {yz HEgeT be all the values of this sequence. The objective of this paper is to propose a model to forecast all time series in given time frame N, i.e. we aim at sampling (17+1T+HhESEN based IEn:N.
3.1 Per-time-series predictors
For all 1 < n & N, we note f (.; predictor) the n-th parametric model of the n-th sequence where 0 predictor are unknown paramcters. Given the sequences ] IEEEN and the estimated paramcters {f IIEEEN the time-series-specifie forecasts WTTIITTTTIEN are, for all (L....,N), for all E [I....,
Vr+ar = ("("V" 6 predictor)
(2)
During the M4 competition, the hybrid model of Smyl 2020 was based on a multiplicative exponential smoothing model as the time-series-specifie predictor. However, on sporadic time series, this choice leads to poor results and instability. In this paper, more general framework able to deal with any kind of per-time- series models is provided. Thus, the choice of the parametric model can be adjusted depending on the nature of the time series. The only limitation is the computational time we aim at forecasting thousands of time scries simultancously. For instance, hidden Markov models provide very interesting framework but inference of such models requires computationally costly iterative procedures such as Expectation Maximization--asee algorithms often combined with Monte Carlo estimates of unknown expectations. Choosing these approaches as per-timeee time-sinssssies predictors would considerably slow the overall training process of the hybrid model.
In Section 4 we introduce three declinations of the hybrid framework using different per-time-sette time-semieess predictors to highlight the adaptability of our approach. The first one is based reference similar to the baseline |myyl (2020), , the second one uses Thetam as per-time-series predictor Hyndman et al. 2020 and the last one uses a TBATS model (Livera et al. 2011
exponential smoothing
For non stationary time series, changes of behaviours are not always predictable using the past of the sequence. In some cases, these changes depend on external variables not considered by univariate parametric models. The difficulty is that the exact influence of external variables on the main signal is mostly unknown. This motivates the introduction of global RNN trained on all time series and able to consider and leverage external signals.
3.2 Error-corrector recurrent model
The second part of the model is a global RNN, trained on all the sequences to correct the weaknesses of the first per-timo-scrics parametric modcls. This task requires a thorough data pro-proccssing as recurrent neural networks training highly sensitive to the scale of the data and requires well-designed inputs.
Let E N be the window size, usually this window is proportional to the forecast horizon x h. The RNN input is defined as the following normalized, descasonalized and rescaled sequence z {27- - for all 1 < n & N and < & w.
7 V wti
WT+
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5 J7
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ground
TBATS pred.
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Mmm
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Figure 4: Hermes forecast example on a time series representing the vertical stipes texture fashion trend for females in Brazil. In green the prediction of the TBATS per-time-series predictors. In red the final forecast of our HERMES hybrid model.
where & - hLi/h] with L-J the floor function. With the numerator part yi- MT+R//>> the per-time- series prediction is included in the RNN input and all the fundamental patterns already learned by this first predictor are removed from the time series. Then the denominator D is use to rescaled all input at the same level as the time series can have different scales. Another option could have been to divide directly vi-wti by Fr+ but with time series hitting O, this option is not valid. Let RNN(,; (corrector) be the recurrent neural network model where Ocorrector are unknown parameters. Given the RNN input sequences [zt HENEN and the global RNN estimated parameters Pcorrector, the correction terms [pcorr.a ( 4 are, for all (l,...,N) N), for all E (4),
DT+IT RNN(Z); 0cor:ector )i 97
Thus, if no external signals are available, the final hermes forecast is, for all 1 < & N and all i E fl....,
VT+ 1rrerr +
UT+H/
aporr.n
(3)
fd (Vll:: Ofredictrrrrro + RNN(2 ceorrector) 77
3.3
Weak signal
In addition to the N target time series, K x N external sequences indexed from 0 to T are now considered. For all < & N, 1 < k < K and < < T, let m.. be the value of the k-th external sequence at time f associated with the sequence y". Let w = [[w) n. heerhhhee be all the values of the external signals. In addition, let w Concatenating 2 and wh, new input for the RNN is defined:
be only the last terms of the external sequences.
- x f {27 W wT m.
K.K WT w-- hiigw
Finally, for all < n < and for all ie (1,..., h.] the final prediction becomes:
Wr+ Spredn +
coorn. UT+4T
(4)
= f (viir; (predictor) )a + RNN(xT;; Ocorrector F7
An illustration of the proposed model is displayed in Figure 5 and first forecast. example is given in Figure

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Eaternal signal
Main time-scries
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Per-time-serica Predictor
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Figure 5: Architecture of the hybrid model with weak signals. The proposed framework can decomposed in steps: provide time series. ii) (a) fit first statistical model with the provided time scries, (b) compute first prediction and (c) preprocess the time series for the Global RNN. iii) If available, external signals be added as part of the RNN input. iv) With pre-trained RNN, compute correction of the first statistical prediction. v) Compute the final forecast by adding the first time series prediction and the RNN correction.

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4 Experimental results
In this section, performance of the hybrid model is assessed and compared with several other approaches. The HERMES framework is evaluated on two different use cases.
A first application is proposed on the fashion dataset. The HERMES model is trained forecasting the fashion time series one year ahead (h=52). This case is mostly guided by the fashion and retail industry where clothes collections are usually prepared one year or more in advance. As additional signals representing influencers behaviours are available, this allows to set the focus on the ability of our framework to leverage these weak signals.
A second application is proposed on the weekly dataset of the M4 competition (Makridakis et al, 2018). Based the competition's rules, the forecasting horizon is set to 13 and external signals are available. Furthermore, the M4 weekly time series come from different sectors and have variable lengths.
4.1 Fashion use case
4.1.1 Training
The fashion dataset is split into three blocks, train, eval and lest sets. The 3 first years are used as the train set, the 4th year kept for the eval set and the test set is made of the last year. The hybrid model is trained to compute a yne-year ahead prediction, ht equal to 52, and the window size w is fixed at 104. Using the two first years of the train set, a first per-time-series parametric model for each time series fitted. With the resulting collection of local models, forecast of the third year is computed for each sequence. Corrector inputs are finally computed and the RNN is trained at correcting this first collection of third-year forecasts. For the eval set, per-time-series predictors are fitted a second time using the three first years and forecasts of the fourth year are computed. The eval set is used during training to control the learning of the RNN model and prevent overfitting. The per-timo-scrics predictors are fitted a last time for the test set using the four first years. The final accuracy measures an example of our split is shown in Figure [ Note that we have just enough historical data to perform the proposed train/eval/tess split. Therefore, the common solution of applying a rolling window to increase the different splits cannot be used. Only one couple input/output is provided for each split per time series.
our models are computed on this lest set. As an illustration,
For the first parametric per-time-series models, existing Python libraries named statsmodels and tbats are used to estimate the different parameters 0 Piredictor. The architecture of the recurrent neural network euror-correcoor model composed of 3 LSTM layers of shape 50 and a final Dense layer to provide the correct output dimension. A classical Adam optimizer used with a learning rate and a batch size set using a grid search. The loss function is defined as follows:
h
1
1 4+/Tl
77
This choice of L1 loss function is motivated by its robustness to outliers which accounts for some time series in the fashion industry with very specific behaviours. The loss and previous parameters are all set with a complete grid search and have to be adapted regarding the use case. See additional results concerning the loss function choice and C.2 for a complete grid search example. The code is developed in Python using the Tensorflow library and publicly available]] It allows the use of GPU to speed up the training process.
4.1.2 Benchmarks, hybrid models and metrics
As benchmarks, several widespread statistical methods and deep learning approaches were selected. Using the R package forecast and the Python packages statsmodels, tbats, for each time series, predictions are computod with the following methods: snaive, ets, stlm, thetam, thats and auto.orima The forecast of
https://github. wwcchhuu..cm/eeeeeeeeeeeeeeeeeee wttp:/taaa/eeeeeeeeeeeeee
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eval
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train output window
wW
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share of category
w
0.08
0.06
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Figure Temporal split for our training process. The three first years define our training set. The fourth year is used as our eval set and the final year is reserved for the test set.
the snaive method only the repetition of the last past period. The els model an additive exponential smoothing with level component and a seasonal component. The stim approach uses multiplicative decomposition and models the seasonally adjusted time series with an exponential smoothing model. The Thetam model decomposes the original signal in O-lines, predicts each one separately and recomposes them to produce the final forecast and thots uses trigonometrical seasonality. Finally, outo.arimo is the R implementation of the ARIMA model with automatic selection of the best parameters. A complete description and references for these models can be found in Hyndman et al. (2020).
We also compare our model with recent deep learning architectures for time series. The Prophet model introduced in Taylor & Letham (017) is proposed and implemented with the available package bearing the same name prophet, and three models based on recurrent neural nelworks called Islm, Isim-ws and deepar are used. Concerning (stm and lstm-ws, they are both composed of 3 LSTM layers of shape 50 and a final Dense layer of shape 52. The first one (Istm) has only accoss to the main signal while the sccond one (Istm- ws) has access to the external signal. The last approach is the recent state-of-the-ard model called DeepAR and introduced in (alinas et al. (2020). The Python package GluonTS (Alexandrov et al. 2020 was used to train DeepAR on the fashion dataset with a gridsearch to tune the main hyperparameters.
A strength of the proposed HERMES framework that it can handle any kind of parametric model. Thus, three versions of HERMES are proposed on the fashion dataset using different local parametric models. The first one uses predictors an additive exponential smoothing model a reference close to [Syyl (2020). The second one is based on the Thetam parametric model (Hyndman et al. 2020). Finally the last relies on the TBATS model of Livera et al. (2011) and achieves the highest accuracy results the fashion datasct. Declinations are called respectively hermes-ets, hermes-tbats and hermes-thetan according to the per-time-series model choice. For each of them, a variation with the inclusion of the weak signals (-ws) is presented.
To compare the different methods, we use the Mean Absolute Scaled Error (MASE) for scasonal time scries. As our sequences have completely different scales, from 10-5 to 10-1 this metric was chosen to compute fair error measure, independent of the scale of the sequence and suited for our seasonal fashion time series,
The MASE metric is defined as follows, with T the length of the time series, the seasonal period and h the horizon:
MASE T- m Σ Lj=1
Yr+j - Pr+sl
h
T- IY, Y:-ml
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Table 2: Results summary on the 10000 time series of the fashion dataset. For each metric, the average on all time series is computed. For approaches using neural networks, 10 models trained with different seeds. The mean and the standard deviation of the 10 results are displayed. Models with in their name have access to the external signal.
MASE L
ACCURACY 1
mean
std
meon
std
snaive thetam arima ets prophet
0.881
0.357
0.845
0.463
0.826
0.464
0.807
0.449
0.786
0.485
stlm
0.770
0.482
hermes-els-un
0.769
0.005
0.501
0.007
sarmes-thetann
0.764
0.003
0.497
0.005
hermes-thetone
0.760
0.004
0.520
0.010
hermes-els
0.758
0.001
0.490
0.006
deepar
0.752
0.018
0.459
0.015
lbats
0.745
0.453
lstm- ws"
0.728
0.004
0.500
0.008
Istm
0.724
0.003
0.498
0.007
hermes-tbats
0.715
0.002
0.488
0.008
hermes-thats-uss
0.712
0.004
0.510
0.005
Detecting emerging and declining trends is a crucial issue for the fashion industry. A correct or incorrect prediction could lead good returns or massive waste due to overstock unsold clothes. In addition the MASE accuracy metric, the different methods are also evaluated differences between methods using weak signals not. In given year, increasing trend is defined as a trend that does more than 5% of growth on average with respect to the previous year. In the same way, a decreasing trend is defined as trend that declines by 5% on average or more. Other trends are classified as flat trends. With this threshold, the proposed fashion dataset is almost balanced on the test set: There are 3087 increasing trends, 3342 decreasing trends and 3571 flat trends. To compare the different methods on this classification task, the accuracy metric, defined the percentage of correct classification, is used.
classification task and especially
4.1.3 Results
10000 Fashion time series global accuracy. For the two metrics and for each model, we compute the average on all sequences in the final year. Results are displayed in Table 2 For methods using neural networks, 10 models are trained with different seeds. The average and the standard deviation of their results are computed and displayed. For the statistical models, TBATS largely dominates the alternatives in terms of MASE. It is one of the main motivations why this model is used on the best HERMES candidate as the predictor model. All HERMES versions show consequent improvements regarding their per-time-series predictors in terms of MASE. hermes-thuee outperforms the two other declinations and all the benchmarks on the fashion dataset. This result highlights two important features of the proposed hybrid model: i) the final accuracy of the model is strongly impacted by the choice of the per-time-series predictor and ii) if the per-time-series predictor is well chosen regarding the nature of the time series, the resulting HERMES model seems to outperforms other state-of-the-ane approaches.
Regarding the impact of the weak signals, Table 2 highlights an interesting improvement of the accuracy metric when weak signals are included. Figure 7 displays some examples of hermes-tbats predictions and illustrates some Tbats weaknesses that can be corrected by the hybrid approach.
10000 Fashion time series classification task. Classification results between the thats model and the hybrid method hermes-thal thals given in Table 3 we note impressive decrease of impactful errors: i.e.
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0.012
ground truth
TEATS pred.
0.010
pred.
share category
0.008
0.006
0.004
0.002
2015
2016
2017
2018
2019
2020
Time (Weekly)
0.045
ground truth
TBAT5 pred.
0.040
HERMES pred.
share category
0.035
0.030
0.025
www
0.020
0.015
2015
2016
2017
2018
2019
2020
Time (Weekly)
hermes-tbats forecast examples. In green the prediction of the per-time-series predictors thats. In red the final forecast of our HERMES hybrid model hermes-tbals. (Top) Time series representing a top fashion trend for females in The United States. (Bottom) Time series representing the horizontal stipes texture fashion trend for females in China.
Figure 7:
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Table tbats, hermes-tbats and hermes-tbats- ws models confusion matrix
thats prod-flat
hermes-Ibats pred-dox pred-flat
pred-doc
pred-ine
pred-ine
true-dee
902
2113
327
true-dec
1261
1960
121
true-flat
351
2920
300
true-flat
549
2823
199
truc-ine
300
2078
true-ine
214
2004
hermes-tbuaie -batswws prod-flat
pred-dox
prod-ine
true-dec
1956
1245
truc-flat
1257
2087
227
truc-inc
358
1620
1100
Table 4: Results sunmary on 1000 time series and 100 time series of the fashion dataset. The MASE average on all the time series is computed. For the two approaches using a neural network, 10 models with different seeds are trained. the mean and the standard deviation of the 10 results are displayed. Models with in their name have accoss to the external signal.
1000 time series fashion dataset
100 time series fashion dataset
MASE J.
ACCURACY 1
MASE
ACCURACY
mean
mean
meam
sld
sonaine
0.871
0.383
smuzine
0.876
0.330
thelam
0.484
thetam
arinen
0.821
0.472
artia
0.814
0,400
ets
0.801
0.460
hermesstthttum
0.812
0.009
0.446
0.031
prophet
0.476
istna
hermee-eess
0,787
0.001
0.482
0.009
Aermees- Meeaamr---
0.810
0,008
0.479
0.021
hermea-thetone
0.796
0.002
0.476
0.009
derpor
0.804
0.024
0.393
0.029
hermes--ee ots-ass
0.004
fusrmmss- ets-
0.010
stim
0.785
0.483
hermes-ets
(1.990
0,001
0.371
0,005
hermes: thetorn-us
0.763
0.005
0.501
0.009
istm- un
0.789
0.010
0.485
0.036
lefo
0.0077
0.487
deepor
0,738
0.017
0.465
0.013
prophet
0.767
0.499
thats
0.734
0.466
thots
0.745
0.440
Isfmu-uss
0.0115
stim
hermes--lol -oota
0,721
0.002
0.487
0.014
Sermess-batts
0.741
0.005
0.462
0.021
hermess-hatts-ee
0.717
0.004
0.500
0.010
hermes- thots-aus
0.737
0.004
0.496
0.027
forecasting increase instead of decrease and vice versa. The hermes-thass model divides by 3 the error rate in comparison to thats with only slight decrease of the number of correct increase increase/decrease decrease predictions. However, with our weak signals, we see that hermes-fbats-wr uss is able to catch twice much as its relative model without weak signals while keeping relatively low number of impactful errors.
Size of the dataset. In addition to the results on the whole fashion dataset, the robustness of the HERMES model analyzed when is trained on smaller datasets. Two experiments are performed on a sub sample of respectively 1000 and 100 randomly selected time series. Results are given in Table The hybrid framework cermes-tbats--e ws achieves the best performance in terms of global accuracy that the accuracy of the full neural network lstm decreases when the dataset size decreases. On the small dataset of 100 time scries, local statistical model like thats or stlm largely outperforms deep-model-bascd approaches such as Isim of deepar. In fact, providing sharp predictions from scratch is a complex lask and high-dimensional recurrent neural networks require large amounts of data to do so. By contrast, the HERMES approach can rely on its first statistical part and consequently needs less data to be trained and obtain interesting performance. We can nevertheless note that the gain brought by the error-corrector recurrent. model decreases the size of the dataset diminishes.
both datasets. We can note
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repeated year
oval
test
train input window
5000
M'M
TM
train output window
M/ ! Wown
4500
share of category
MY
M
4000
in
Mann
M Kanny
3500
3000
2500
2000
2012
2013
2014
2015
2016
2017
Time (Weekly)
Figure 8: One of the shortest sequences of the M4 weekly dataset (93 time steps). In order to fit its predictor, the last complete year of the train set is duplicated in order to reach total length of 300 time steps.
4.2 M4 competition use case
We also assessed the performance HERMES using the M4 weekly dataset (Makridakis el al. 2020 The M4 dataset gathers 359 weekly time series and has 3 main differences compared the proposed fashion dataset. Firstly, sequences do not have the same length with sequences lying between 93 and 2610 time steps. Secondly, the 359 time series come from different sectors such that finance or Industry. Accordingly, they have very distinct scales and dynamics. Thirdly, compared to the previous fashion application, the time horizon of the prediction is set to 13 for the weekly dataset and no additional external signals are provided.
4.2.1 Training
The M4 dataset is preprocessed follows. As some sequences are short (93 time steps), they limit the window size w of the RNN. Consequently, 300 time steps are kept for cach sequence. shorter sequences are duplicated in order to reach the length of 300 and longer sequences are cropped so as to keep the last 300 time steps. An overview of our train, eval, test set split and the resizing of the shortest sequences is given in Figure S. Secondly, several M4 weekly time series have a large volume and a high level of variability. Consequently, Equation 2 of the HERMES framework is changed to:
T+IT = exp (J (log ; (predictor))))
(5)
This simple modification increases significantly the accuracy of the per-time-series predictors tested the M4 weekly dataset while reducing the fitting time. As for the fashion dataset, a complete grid search is done on the M4 weekly dataset to fix hyperparameters of the HERMES architecture. The horizon h is set to 13 and the window size w to 104. For the RNN part, the same architecture as described in Section 4.1.22 is used. The Adam optimizer is used and the MASE is directly used as the loss function. As some the m4 woekly time series have many years of historical data, a rolling window is finally applied on the train set so as to increase the number of examples and improve the training process. The number of slinding windows, the learning rate, the batch size, the RNN architecture and input size set using grid scarch and detailled in
4.2.2 Evaluation
The proposed model is evaluated along with rich collection of benchmarks provided by the M4 competition, encompassing statistical models and neural network approaches. In addition, the hybrid model named Uber of S.Smyl is added. For complete description and references of the benchmark models and the hybrid model
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Table 5: Results summary on the m4 weekly dataset. For each metric, the average on all our time series is computed. For approaches using neural network, 10 models trained with different sceds. The mcan and the standard deviation of the 10 results are displayed.
SMAPE
MASE
OWA
mean
std
mean
std
mean
std
MLP
21.349
13.568
3.608
RNN
15.220
5.132
1.755
snaive
9.161
2.777
1.000
SES
9.012
2.685
0.975
Theta
9.093
2.637
0.971
Holt
9.708
2.420
0.966
Com
8.944
2.432
0.926
Damped
8.866
2.404
0.917
Uber Smyl 2020
7.817
2.356
0.851
tbats
7.409
2.204
0.801
hermes-tbate thats Pawlikowski, et al.
7.383
0.016
2.191
0.010
0.797
0.002
6.919
2.158
0.766
Petropoulos & Svetunkov
6.726
2.133
0.751
Darin & Stellwagen
6.582
2.107
0.739
foorma-hermss
6.614
2.058
0.732
Uber, see Makridakis et al. (020 and Smyl 2020 As HERMES candidate, version using TBATS is proposed and called hermes-ther thats. We propose focus on the top 3 models reaching the highest accuracy on the M4 weekly dataset. These three methods are based on an ensembling and combine various approaches. The first model is presented in Darin & Stellwagen (2020) and called Darin & Stellwagen. The second model is introduced in Petropoulos Svetunkov (2020) and called Petropoulos & Svedunkov. Finally, a description of the third model called Pawlikowski, et al. can be found in Pawlikowski & Chorowska (2020). An ensembling combining 4 HERMES variations is proposed. It is based the FFORMA algorithm introduced in Montero-- Manso et al. (2020) and called fforma-hermes. A complete description of the training process of the proposed ensembllnn is given in B.3 Following the M4 compelition methodology, all the candidates are evaluated according to the MASE, the SMAPE and the OWA measures. A complete definition of these metrics is
proposed in Makridakis et al. (2200) and summarized in B.1 Soc also B.1 for additional information about the M4 weekly dalaset.
4.2.3 Results
The final results for the M4 weekly dataset are displayed in Table 5 The HERMES approach hermes- thats outperforms all the benchmarks. This result is partially induced by the use of TBATS per-time-series predictors which achieve impressively good results on the test set.. Regarding the hybrid model proposed by S.Smyl, its accuracy remains low in comparison to thats and hermes-tbats. For the ensembling methods, the proposed FFORMA model with HERMES variations foorma-hermes reaches the same high level of accuracy the top 3 methods of the competition the weekly dataset. The results provided by bermes-tbat confirm that the HERMES model is well suited for a large collection of forecasting tasks even difficult ones with small datasets, helerogeneous time series and the absence of additional useful external signals. Secondly, the accuracy gap between the proposed hybrid model and the approach proposed in [myl (2020) illustrates the importance of global framework able to leverage any kind of per-time-scries prodictors depending on the use cases. Finally, our model can be easily included as part of an ensembling method improve the final robustness and accuracy of the predictions.
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4.3 Discussion
Results of the two previous forecasting tasks illustrate that the proposed hybrid method is a general frame- work, easily adaptable to different use and able to compete with other state-of-the-art methods. With its neural network component, it is also able to leverage any kind of external signals, which which could become easential in more and more forecasting applications. The experiments highlight some limitations that may be the topic of future works.
In its current form, the use of the neural network prevents the interpretability of the final prediction. The of the external signal can not be easily assessed and no confidence intervals are provided with the hybrid framework. Future works focus designing more intepretable models, using for instance Bayesian neural networks, or combining neural networks with state space models.
The proposed hybrid approach can be computationally intensive, for instance compared to classical neural-network- based models. A solution to improve this limitation is to propose new procedures to train jointly both parts of the model with mini batches of observations.
In the Fashion use case, the inclusion and use of the external signals could be improved the external signals are relevant only in few fashion time series. An ongoing work is to use several neural networks specialized different uses of the external signals. Depending on the past of the time series and the external signals, hidden state could be used to switch between models.
5 Conclusion
In this paper, we propose a new hybrid model for non stationary time series forecasting. By mixing the performance of local parametric models and a global neural network, hermes-tbuls clearly outperforms tra- ditional statistical methods and full neural network models on two forecasting tasks. Furthermore, this new model is totally suited to deal with external signals. With a fine pre-processing and well-designed archi- tocture, the proposed hybrid framework succeeds at leveraging complex extra data and reaches promising accuracy levels. In addition, fashion dataset gathering sample of 10000 lime series and a collection of weak signals is provided. We show that it is possible to represent fashion data with time series and introduce statistical models based on social modia data. By making it publicly available, we hope that it will enhance the diversity of datasets for time series forecasting and pave the way for further explorations. As a possible future work, designing new models for the weak signals would improve their inclusion in the HERMES ar- chitecture. Focusing on the examples with important changes of behaviours, fine analysis of the impact of the collection of weak signals is the topic of ongoing works. In the same way, interesting improvement of the hybrid framework can be to introduce not single but several neural worksrks trained at correcting different kinds of wcaknesses. A perspective is to add a latent discrete label to select dynamically the regime shifts.
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A Fashion dataset: a study of sub samples of time series sharing the same behaviour
A study of subsamples of the fashion dataset is proposed in this section. They are defined as follows.
disrupting time scries: in retail and fashion industrics, strategic issuc is to correctly anticipate new trends that will burst or collapse in the next weeks, months or years. To detect this subsample of trends in the fashion dataset, the following approach is proposed: using the snaive model, a prediction of the last year of data is computed for cach trend with the associated MASE. The disrupting time series are defined as the 1000 time series where the snaive prediction has the highest MASE.
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stable time series: by contrast, a group of stable trends is presented. They are the 1000 time series where the snaive prediction with the lowest MASE.
seasonal time series: compute for each trend the seasonality strength metric introduced in Wang et al. (2006). The seasonal time series group are the 1000 fashion time series with the highest seasonality strength metric.
noisy time series:
Finally, several time series of the fashion dataset represent niche trends only worn and posted on social media by few people. The average volume for these sequences is low resulting in sporadic and noisy time series difficult to forecast. In order to detect them, we use the seasonality and trend strength metrics presented in Wang et al. (2006). For each time series, the mean of these two quantities is computed and we define the time series group as the 1000 sequences with the lowest average.
Table G displays the MASE of each model introduced in Section fo the 4 subsamples of time series. Several relevant results can be highlighted. Firstly, models using the influencer weak signals strongly outperform other candidates on the disruptive time series. This result even more visible with the HERMES approach where models using the weak signals are always better than the ones without. These results illustrato empirically the impact of weak signals final predictions. Secondly, results of hermes-lbals stable time series and seasonal time series, highlight the interest in using hybrid model. This is more visible on scasonal time scries where hybrid approaches largely outperform neural network models presented in the benchmarks. Finally, on noisy time series, we can see that neural network models reach the highest accuracy. Furthermore, for all the HERMES approaches, we can note that the final hybrid predictions and the per-time-series predictions are the same.
B
M4 weekly dataset, Ensembling training and results
B.1 M4 weekly dataset
The M4 weekly dataset collection of 359 time series with contrasting behaviours and sizes. An overview of the dataset is given in Table and some examples of sequences are given in Figure This case is not properly suited for the HERMES approach the dataset is small and there is no clear link between time series. Moreover, no additional external signals are available that could help the RNN part to correct the first errors of the ver-time-series predictors.
B.2 M4 accuracy metrics
The M4 competition proposes 3 metrics to evaluate the different approaches: the mean absolute scaled error (MASE), the symmetric mean absolute percentage error (SMAPE) and the overall weighted average (OWA). MASE and SMAPE are defined as follow:
MASE = T m Σラ YT+j Fr+sl
2 h |Y++ - Yrtsl h Σ Yr + 18r+sl'
SMAPE
h Σl [Y Y,mll
where h is the forecast horizon and m the length of the seasonality. The final OWA is computed by following these steps: i) compute the average MASE and SMAPE of model. ii) Divide the previous results by the MASE and SMAPE computed with the benchmark method snaïve. iii) Compute the OWA as the average of the relative MASE and SMAPE obtained step ii). As an example on the M4 weekly dataset, the method hermes- -tatts has a MASE of 7.383 and a SMAPE of 2.191. The benchmark method snaive has a MASE of 9.161 and SMAPE of 2.777. Thus the OWA of hermes-thats is equal to 0.797.
B.3 FFORMA ensembling with HERMES variations
In this section, a complete description of the proposed ensembling on the M4 weekly dataset provided. In first time, 4 HERMES variations trained using different per-time-series predictors. The first
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Table 6: Results summary on 4 differents subsamples of Fashion time series: i) disrupting time series, ii) stable time series, iii) seasonal time series and iv) noisy time series. Models with a in their name have access to the external signal.
disrupting time series
stable time scries
MASE
MASE
std
std
snaive
1.455
prophet
0.629
thelam
1.314
thelam
0.615
ets
1.27
ets
0.611
arima
1.256
arima
0.565
tbats
1,229
hermes- ceames--ts-us us
0,555
0.007
hermes-thetam
1.209
0.005
snatue
0.536
hermes-er ets
1.202
0.007
deepor
0,531
0.024
sthm
1.198
seemmes-thetam-ss
0.522
0.004
hermes-tbate
L.195
0.01
hermes-els
0.518
0.002
prophet
1.193
stlm
0.513
deepar
1,18
0.08
0,508
0.005
Istm
1.15
0.01
thats
0.501
1.145
0.019
Istm-ws
0.492
0.007
hermes- sts-wss"
1.131
0.024
hermes hermes-thats thats-us wss
0.477
0.008
hermes-tbals-use
1.092
0.007
Istm
0.47
0.004
Istm-ws
1.086
0.009
hermes-tbats
0.451
0.002
seasonal time series
noisy time series
MASE
MASE
moon
std
mean
std
snaive
0.895
snaive
0.842
ets
0.895
hermes-etse etreess-etswsss us
0.739
0.005
prophet
0.851
hermes-de ets
0.726
0.002
deepor
0.836
0.035
0.721
Istm-wv ws*
0.829
0.014
thetam
0.717
thetam
0.826
silm
0.715
Isim
0.823
0.013
prophet
0.698
hermes-thetone
0.815
0.008
arima
0.696
thats
0.81
hermes-theee
0.672
0.003
hermes-cles etsiessssssss
0.809
0.01
hermes-thetam
0.669
0.001
hermes- thetom--ss us
0.808
0.008
deepar
0.661
0.007
arima
0.805
hermes-tluu tbots-ws
0.647
0.006
silm
0.786
thats
0.646
hermes-efa
0.785
0.003
hermes-tlee tbots
0.644
0.003
hermes- thots ws*
0.777
0.012
Istm-ws
0.637
0.004
hermes-tbats
0.772
0.003
Istm
0.636
0.003
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10000
8000
6000
Wp Wpma
4000
2000
2000
2002
2004
2006
2008
2012
2014
2016
2018
Time (Weekly)
10000
8000
6000
4000
2000
1988
1992
1996
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2004
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Time (Weekly)
1850
wwwp
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2013 Time (Weekly)
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Figure 9: Examples of time series from the M4 weekly dataset. From Top to Bottom time series called W10 from the Other category, W20 from the Macro category and W220 from the Finance category.
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Published in Transactions on Machine Learning Research (09/2023)
Table 7: M4 weekly datasct overview. For each category, the number of soquences and the average length are given.
Nb. sequences
Avg. length
Min. length
Demographic
1659
1615
Finance
164
1237
260
Industry
6
356
Macro
1264
522
Micro
112
93
Other
1598
470
14000
thetam pred.
hermes-thetam pred.
13800
13600
13400
13200
13000
12800
2015-10
2016-01
2016-04
2016-07
2016-10
2017-01
2017-04
2017-07
Time (Weekly)
4430
ground truth ets-add pred.
4420
pred.
4410
4400
4390
4380
4370
4360
2015-10
2016-01
2016-04
2016-07
2016-10 Time (Weekly)
2017-01
2017-04
2017-07
Figure 10: forecast examples of HERMES variations 2 time series of the M4 weekly dataset. At the top, the W133 time series is displayed with the prediction of the per-time-series predictor thelam (green) and the final forecast of the HERMES hybrid model termes-thetam (red). At the bottom, the W262 time series is represented with the corresponding prediction of the per-time-series predictors ets-add (green) and the HERMES correction of hermes-els-add (red).
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Published in Transactions on Machine Learning Research (09/2023)
Table 8: Results summary on the m4 weekly dataset of the HERMES variations. For each metric, the average all the time series is computed. For approaches using neural network, 10 models trained with different seeds. The mean and the standard deviation of the 10 results are displayed. For the statistical models ets- add, ets-mul and thetam, the Python package statsmodels is used. The Python package tbats is used for the thats approach.
SMAPE
MASE
OWA
mean
std
mean
std
mean
std
ets-mul sarmes-ets-man
8.933
2.412
0.922
8.889
0.021
2.377
0.016
0.913
0.004
ets-add
8.929
2.410
0.921
hermes-ets-adee
8.880
0.022
2.377
0.016
0.913
0.004
thetam
7.609
2.377
0,843
Cormes-thetane
7.590
0.012
2.359
0.010
0.839
0.002
thats
7.409
2.204
0.801
hermes-thata 7.383
0.016
2.191
0.010
0.797
0.002
called hermes-tbats uses TBATS and is presented in Section 4.2 The second version is called hermes- thetam and use the Thetam method provided with the Python package statsmodels. The two remaining variations use as per-time-series predictors an additive or multiplicative exponential smoothing and are called respectively hermes-ets-adee and hermes-ets-mal. As for Thetam, the Python package statsmodels is used to fit. the different exponential smoothing models. Concerning the HERMES architecture, for simplicity, hyperparameters described in Section 4.2 are used for cach version but grid search could have be run for each of them. 10 models are trained per version with different seeds and the best one based on the eval set is kept for the ensemble model. In second time, the FFORMA ensembling introduced let al. (2020) used to combine the HERMES methods. The R package MAmetalearning containing the FFORMA model directly used without change of the hyperparameters, imported in Python with the library Rpy2 and combined with the HERMES code base.
Mlontero-Manso
B.4 M4 weekly dataset results
In addition of the results provided is Section 4.2 Table g displays the results of all the HERMES variations included in the FFORMA ensembling as well as the accuracy of the per-time-serics prodictors. In cach cases, HERMES approaches always improve the predictors accuracy. These improves can appear slight but are justified regarding the absence of link between time series and the absence of additional useful external signals. Nevertheless, efficient corrections be obtained some examples displayed in Figure 10
C Training parameters and loss
C.1 Loss grid search on the fashion Dataset
Using deep learning models in time series forecasting is an appealing to achieve higher accuracy per- formance. However, it induces two main issues. First, it requires a large enough dataset lo train the model as illustrated in Section ! Second, a dataset can hide contrasting time series in terms of scale, noise and bchaviour. These differences can impact training performance. For the HERMES architocture, some candi- date losses were defined for the training: the Mean Absolute Error (MAE), the Mean Square Error (MSE), the Scaled Mean Absolute Error (SMAE) and the Scaled Mean Square Error (SMSE). The loss functions
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Published in Transactions on Machine Learning Research (09/202))
0.724
Mean Absolute Scaled Error
0.722
0.720
0.718
0.716
0.714
0.712
0.710
MAE
MSE
SMSE
SMAE
Loss function
Figure 11: MASE accuray for the hermes-thats-wt model depending on the loss used during the RNN training. For each loss, 10 models with different seeds have been trained. The mean and the standard deviation are represented with point and vertical line.
are defined as follows:
h MAE
UT++ D7tell,
h iml
MSE 1
Σ
(07++ )
h
i=1
h
SMAF
1V7 Dr+irl.
TT
h on 0T+1r))
SMSE
(UT+:
For each loss, 10 sermesstbals-ure models have been trained with different seeds and the final mean and standard deviation are given Figure II The final Scaled Mean Absolute Error reaches the lowest MASE and was selocted train all the HERMES models on the fashion datasct.
C.2 Parameters grid search on the M4 weekly Dataset
In addition to the loss function, the HERMES model also depends on several hyperparameters to set correctly in order lo reach satisfactory performance. For instance, an overview of the learning rate, batch size and number of windows per time series grid search for the M4 weekly dataset is shown in Figure [2] For each parameter, collection of 10 cermes-tbats models have been trained with different seeds and the final OWA calculated. As in the Figure II the mean and the standard deviation of each group of 10 trainings is computed. For the final hermes-thals model of the M4 weekly dataset, the following set of parameters selected: 3 windows per time series were used as the train set, the batch size was set to 8 and the learning rate was fixed to 0.005.
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Published in Transactions on Machine Learning Research (09/202))
C359
1.16
2752
1300
11
325
0.90
0337
114
0.90
0.703
0 0.00
Figure 12: OWA for the hermes-thats model on the eval set of the M4 weekly dataset. 5 hyperparameters used during the RNN training are lested: the number of moving windows per time series (top left), the learning rate (top right), the batch size (middle), the window size for the RNN input (bottom left) and the dimension of the LSTM layers output (bottom right). For each parameter, 10 models with different seeds have been trained. The mean and the standard deviation of the OWA on the eval set are represented with point and vertical line.
26


Reviews:
[{"review_id": "WgyHWtCzob", "reviewer": "Reviewer_o2ju", "summary": "", "questions": "", "limitations": "", "rating": null, "confidence": null, "soundness": null, "presentation": null, "contribution": null, "strengths": "", "weaknesses": "", "quality": null, "clarity": null, "significance": null, "originality": null, "strengths_and_weaknesses": "Strength\n- Dataset is very interesting and should be valuable to the community. \n- Introduces many fashion-specific heuristic preprocesses, based on domain knowledge. \n\nWeakness\n- The description is often hard to follow possibly due to language issues. \n- The novelty of the prediction model looks weak. \n ", "was_revised": false, "final_justification": "", "rebuttal": "", "claims_and_evidence": "Yes", "methods_and_evaluation_criteria": "", "theoretical_claims": "", "experimental_designs_or_analyses": "", "relation_to_broader_scientific_literature": "", "essential_references_not_discussed": "", "other_strengths_and_weaknesses": "", "other_comments_or_suggestions": "", "questions_for_authors": "", "summary_of_paper": "", "summary_of_recommendation": "", "technical_quality": "", "clarity_of_presentation": "", "potential_impact": "", "robotics_focus": "", "quality_of_limitations": "", "questions_for_rebuttal": "", "reasons_to_accept": "", "reasons_to_reject": "", "questions_to_authors": "", "paper_topic_and_main_contributions": "", "excitement": null, "reproducibility": null, "ethical_concerns": "", "main_review": "", "summary_of_the_paper": "", "summary_of_the_review": "", "strength_and_weaknesses": "", "limitations_and_societal_impact": "", "review_text": "", "summary_of_contributions": "This paper makes two contributions. First, it provides a large fashion time series generated from a massive collection of images. Second, it proposes a forecasting model in the specific setting of fashion trend prediction. Each fashion trend is basically modeled independently as a univariate signal, but the model is designed to incorporate \"weak signals,\" which may result in some dependency between the univariate time series.  ", "claims_and_evidence_yn": "Yes", "claims_explanation": "", "audience_yn": "Yes", "audience_explanation": "", "requested_changes": "Improve the description of Section 2. \n- Clearly define the terms like trend and cloth type using examples. \n- Section 2 should be described in a top-down manner if the resulting dataset is claimed to be one of the main contributions. Describe first what the resulting data looks like, then explain how you generated it. \n- It is not clear if the authors are calming technical contributions in the preprocess approaches. If so, the authors need to validate the specific choices like (1). The figure looks like a cherry-picked example. \n     \n \nSection 3\n- The univariate model adopted is not clearly described. What is the model?  \n- The term \"error correction\" sounds misleading as there is not such an explicit algorithm to compute the error. Perhaps the second term of (3) should be called just the correction term. \n- I need methodological validation on (4). Apparently, the scaling of the input of RNN is critical to serving the RNN term as the coefficient of the correction term.  \n", "broader_impact_concerns": "Dataset is valuable and the paper makes a good contribution to the community. ", "additional_comments": "", "extra_scores": {}, "extra_text": {}}, {"review_id": "uBokTUmHX0", "reviewer": "Reviewer_hRzZ", "summary": "", "questions": "", "limitations": "", "rating": null, "confidence": null, "soundness": null, "presentation": null, "contribution": null, "strengths": "", "weaknesses": "", "quality": null, "clarity": null, "significance": null, "originality": null, "strengths_and_weaknesses": "S1. Creating and publishing a new fashion time series is significant because it promotes research in machine learning and data mining. The data is interesting in that it includes influencer data tagged by domain experts.\n\nS2. Although the proposed method is somewhat incremental, it attempts to design for the characteristics of the newly created data.\n\nS3. Experiments show that the proposed method performs better than the baseline in forecasting fashion trends.\n\nW1. The method of data set creation (especially human processing) is unclear. Discussion of the nature and reliability of the data set is also lacking.\n\nW2. There is room for improvement in the explanation of the proposed method.\n\nW3. Although the authors compared the proposed method with various baselines in the experiments, there is insufficient discussion of the advantages and disadvantages of the proposed method. The results need to be organized and discussed.", "was_revised": false, "final_justification": "", "rebuttal": "", "claims_and_evidence": "Yes", "methods_and_evaluation_criteria": "", "theoretical_claims": "", "experimental_designs_or_analyses": "", "relation_to_broader_scientific_literature": "", "essential_references_not_discussed": "", "other_strengths_and_weaknesses": "", "other_comments_or_suggestions": "", "questions_for_authors": "", "summary_of_paper": "", "summary_of_recommendation": "", "technical_quality": "", "clarity_of_presentation": "", "potential_impact": "", "robotics_focus": "", "quality_of_limitations": "", "questions_for_rebuttal": "", "reasons_to_accept": "", "reasons_to_reject": "", "questions_to_authors": "", "paper_topic_and_main_contributions": "", "excitement": null, "reproducibility": null, "ethical_concerns": "", "main_review": "", "summary_of_the_paper": "", "summary_of_the_review": "", "strength_and_weaknesses": "", "limitations_and_societal_impact": "", "review_text": "", "summary_of_contributions": "- **Constructing a new fashion time series.**\nThis paper presents large-scale time series data of fashion trends by processing photos posted on social networking services (SNS). The data is obtained from markets in five countries. A unique feature is the inclusion of posts by influencers tagged by domain experts.\n- **Neural network models for forecasting fashion time series.** The authors propose a method that combines parametric models that capture the local behavior of time series and RNN models to include external signals as input.\n- **Numerical experiments.** The effectiveness of the proposed method is evaluated using original and public data.", "claims_and_evidence_yn": "Yes", "claims_explanation": "", "audience_yn": "Yes", "audience_explanation": "", "requested_changes": "- The reliability of the data is important for the use of this data by third parties. Please explain in detail how the human processing was done. Also, to what extent can machine failures be included in image processing? Do humans correct them?\n\n- In publishing the data, it would be good to add a little more analysis of the characteristics of the data. For example, basic statistics for each market, statistics for the Weak signal $w_t^{f, i}$, etc.\n\n- It is written that $i$ represents the index of fashion trends, but it is unclear what exactly it refers to. Does it correspond to a product/item?\n\n- I didn't know what statistic $\\hat{y}^{c,g,m}$ was. Could you please explain it in detail?\n\n- It may be a good idea to modify the structure of Section 3. \n  - It is better to explain the hybrid model (4) at first. Then, elaborate on the motivation behind the formulation of (4). The output of the RNN appears to represent the weights in summing $f^n(\\cdot)$ and $\\bar{y}_T^n$. Is that correct? Also, when is one of $y^{pred}$ and $y^{corr}$ more dominant than the other?\n  - The cost function described in Section 4.1 should be moved to Section 3.\n\n- It would be good to add a discussion of the limitations of this study.\n  - Other external factors could be various, such as advertisements. This study focuses on the correlation between weak signals and trends, not causality. This should be clearly stated.\n  - As seen in Figure 2, the effect of the external factor is accompanied by a time delay. It would be interesting to analyze how such time delays vary by fashion category and market. It would also be interesting to incorporate this into machine learning models.\n\n- In Table 2, it would be good to clarify which methods can handle weak signals and which cannot. Then, please clarify by experiment that it is effective to construct a hybrid model by the weighted sum of two terms as in the proposed method (4).\n\n- Minor issues:\n  - The term \"causal inference\" appears in the abstract, but it is not discussed in this study and should be changed to another term.\n  - The proposed method, HERMES, does not specify what it stands for.\n  - Please cite Figure 1 in the text.\n  - Please change the \"t\" in \"At each time t\" in section 2.1 to italic.", "broader_impact_concerns": "N/A", "additional_comments": "", "extra_scores": {}, "extra_text": {}}, {"review_id": "pYmcUTdETp", "reviewer": "Reviewer_KPmd", "summary": "", "questions": "", "limitations": "", "rating": null, "confidence": null, "soundness": null, "presentation": null, "contribution": null, "strengths": "", "weaknesses": "", "quality": null, "clarity": null, "significance": null, "originality": null, "strengths_and_weaknesses": "Strengths:\n\n- The paper is clearly written and easy to follow \n\n- Contribution of a new time-series dataset in fashion domain\n\n- Competitive results on the M4 benchmark as well as on the proposed fashion dataset.\n\n- The work evaluates a number of models\n\n\nWeaknesses:\n\n- The proposed method only provides point-forecasts. It does not provide any uncertainty bounds around predictions.\n\n- Model optimization and evaluation only done for a single split across time (3 years train, 1 year validation, 1 test). \n\n- Results on smaller subsets of data do not show model performance with weak signals included as input. Trend classification performance also not reported for smaller datasets. \n", "was_revised": false, "final_justification": "", "rebuttal": "", "claims_and_evidence": "Yes", "methods_and_evaluation_criteria": "", "theoretical_claims": "", "experimental_designs_or_analyses": "", "relation_to_broader_scientific_literature": "", "essential_references_not_discussed": "", "other_strengths_and_weaknesses": "", "other_comments_or_suggestions": "", "questions_for_authors": "", "summary_of_paper": "", "summary_of_recommendation": "", "technical_quality": "", "clarity_of_presentation": "", "potential_impact": "", "robotics_focus": "", "quality_of_limitations": "", "questions_for_rebuttal": "", "reasons_to_accept": "", "reasons_to_reject": "", "questions_to_authors": "", "paper_topic_and_main_contributions": "", "excitement": null, "reproducibility": null, "ethical_concerns": "", "main_review": "", "summary_of_the_paper": "", "summary_of_the_review": "", "strength_and_weaknesses": "", "limitations_and_societal_impact": "", "review_text": "", "summary_of_contributions": "This work proposes a hybrid framework for time-series forecasting. Another contribution of the work is a new time-series dataset, which tracks aggregated social media posts reflecting different fashion trends in various clothing categories and regions over a period of five years. The dataset additionally tracks social media posts of influencers across trends, categories and regions, which is to be utilised as an auxiliary signal for forecasting common-user behavior/posts. The authors fine-tune pre-trained image classifiers to setup an automated image segmentation and categorization pipeline. The pipeline is used to detect and classify fashion articles in posted images into a predefined set of categories. Images within each category are analyzed by fashion experts to identify and track various trends over time.\n\nAt the core, the hybrid forecasting framework uses a statistical forecasting model, which is optimized individually for each time-series. A recurrent neural network (RNN) is trained across all time-series to correct prediction errors of the per time-series statistical models. The authors employ available implementations of a number of statistical models in their framework. \n\nBenchmarking results on the proposed dataset indicate that the hybrid combination of a statistical model and the error-corrector RNN outperforms other such combinations as well as individual instances of statistical and RNN models. They authors also compare variations of the RNN trained with and without the influencer activity signals to domonstrate that the best results are obtained when the signals are included as input. Experiments on smaller subsets of the fashion dataset are performed to demonstrate the applicability of the proposed model to small datasets. Finally the hybrid forecasting model and its ensemble version are shown to outperform other comparable approaches.\n", "claims_and_evidence_yn": "Yes", "claims_explanation": "", "audience_yn": "Yes", "audience_explanation": "", "requested_changes": "\nThe evaluation would be more complete if performance for model trained with weak signals is reported for smaller subsets of data. Also the classification performance should be reported for smaller datasets.\n\nThe authors should also consider reporting results for multiple temporal splits of data\n\nIt seems that there is no concept of trend birth and death in the proposed dataset. It would be nice if the authors can clarify if that is true and why is that the case ", "broader_impact_concerns": "The work releases a new dataset in the public domain, which is derived from social media activity of a large number of user across multiple regions. It would be in order if the authors clarify whether due to the data collection or the release of the fashion dataset, there will be any implications for individuals' privacy whose data were utilized for this work.", "additional_comments": "", "extra_scores": {}, "extra_text": {}}, {"review_id": "GqQD00cxKs", "reviewer": "Reviewer_ixuS", "summary": "", "questions": "", "limitations": "", "rating": null, "confidence": null, "soundness": null, "presentation": null, "contribution": null, "strengths": "", "weaknesses": "", "quality": null, "clarity": null, "significance": null, "originality": null, "strengths_and_weaknesses": "Strengths:\n1. The paper proposes a novel hybrid model that can leverage external weak signals to improve the forecasting accuracy and the detection of emerging and declining trends.\n2. The paper introduces a new fashion dataset that is large, diverse, and rich in information. The dataset can be useful for other researchers and practitioners in the fashion industry.\n3. The paper evaluates the performance of HERMES on two datasets and compares it with several statistical and deep learning benchmarks. The paper shows that HERMES outperforms the alternatives in terms of MASE and accuracy metrics.\n\nWeaknesses:\n1. The paper does not provide a clear motivation or literature review for the choice of per-time-series models.\n2. The paper could benefit from a more detailed discussion of the limitations of the proposed approach and potential avenues for future work.\n3. The paper could also benefit from a more thorough comparison with existing methods, including a discussion of their strengths and weaknesses in relation to the proposed approach.\n", "was_revised": false, "final_justification": "", "rebuttal": "", "claims_and_evidence": "Yes", "methods_and_evaluation_criteria": "", "theoretical_claims": "", "experimental_designs_or_analyses": "", "relation_to_broader_scientific_literature": "", "essential_references_not_discussed": "", "other_strengths_and_weaknesses": "", "other_comments_or_suggestions": "", "questions_for_authors": "", "summary_of_paper": "", "summary_of_recommendation": "", "technical_quality": "", "clarity_of_presentation": "", "potential_impact": "", "robotics_focus": "", "quality_of_limitations": "", "questions_for_rebuttal": "", "reasons_to_accept": "", "reasons_to_reject": "", "questions_to_authors": "", "paper_topic_and_main_contributions": "", "excitement": null, "reproducibility": null, "ethical_concerns": "", "main_review": "", "summary_of_the_paper": "", "summary_of_the_review": "", "strength_and_weaknesses": "", "limitations_and_societal_impact": "", "review_text": "", "summary_of_contributions": "This paper presents a new hybrid model for time series forecasting, called HERMES, that combines per-time-series parametric models with a global recurrent neural network (RNN) to correct the errors of the former. The paper also introduces a new fashion dataset that contains 10000 weekly fashion time series with external weak signals derived from social media influencers. The paper evaluates the performance of HERMES on the fashion dataset and the M4 weekly dataset, and compares it with several statistical and deep learning benchmarks. The paper shows that HERMES can leverage the external weak signals to improve the accuracy of forecasting and the detection of emerging and declining trends. The paper also demonstrates that HERMES is robust to different sizes of datasets and different choices of per-time-series models. The paper contributes to the literature on hybrid models for time series forecasting and provides a novel application to the fashion industry.", "claims_and_evidence_yn": "Yes", "claims_explanation": "", "audience_yn": "Yes", "audience_explanation": "", "requested_changes": "1. The paper should provide a clear motivation or literature review for the choice of per-time-series models. \n2. The paper should explain why exponential smoothing and TBATS are chosen as the predictors, and how they compare to other possible choices such as ARIMA or HMM. \n3. The paper should compare with more existing methods. \n4. The paper should explain how HERMES handles different types of time series, such as stationary, non-stationary, seasonal, or sporadic. \n", "broader_impact_concerns": "1. The paper should discuss the potential positive and negative impacts of the proposed model and dataset on the society and the environment.\n2. The paper should address the privacy and consent issues of using social media images and data to construct the fashion dataset and the weak signals. \n3. The paper should explain how the data is collected, processed, and anonymized, and what are the ethical and legal implications of this process.\n4. The paper should also acknowledge the potential risks or biases of using social media data to represent fashion trends, such as cultural diversity, inclusivity, or sustainability.\n", "additional_comments": "", "extra_scores": {}, "extra_text": {}}]

Author rebuttal:

Meta-review: This paper proposes a new fashion time-series data set and proposes a RNN-based time-series forecasting method. Specifically, it first offers an extensive fashion time series dataset generated from an extensive collection of images. Secondly, it introduces a forecasting model tailored to the realm of predicting fashion trends. While each fashion trend is treated as an independent univariate signal, the model is constructed to accommodate "weak signals," potentially introducing interdependencies among these individual time series.

Overall, the paper is well written. The issues raised by the reviewers have been adequately resolved. Therefore, I also recommend accepting it.

As a minor change, the size of the figures in Figure 12 is not consistent. Please make it consistent in the final version.
Decision: Accept (poster)