Natural climate reconstruction in the Norwegian stave churches through time series processing with variational autoencoders

1. Introduction

Studying the microclimate of heritage buildings has always been fundamental for assessing the impact that temperature, relative humidity, mixing ratio and other atmospheric variables have on historic materials conservation. In particular, the variations of these atmospheric variables affect organic hygroscopic materials and may trigger deterioration mechanisms, potentially catastrophic (Camuffo, 2019). For this reason, it is mandatory to identify the microclimate conditions that are desirable and suitable for the studied heritage and historical buildings, in order to slow down their decay and deterioration. In this framework, being able to gather as much information as possible on the ambient physical conditions surrounding a given building or building asset may be of extreme help. Generally, the microclimatic situation is depicted by recovering and/or collecting time-series of temperature and relative humidity of the selected environment; however, this often implies to carry out extensive and time-consuming on-the-field monitoring campaigns (Frasca et al., 2017; Olstad et al., 2020; Varas-Muriel et al., 2014). To avoid too long and expensive monitoring campaigns, exploiting tools of Building Simulation (Kramer et al., 2013; de Wit, 2006) could be of help, for instance. These kinds of tools allow to compute and forecast the indoor environmental conditions by simulating the building behavior, subjected to the external climate. In addition, the modern approaches of ML have been recently gaining more and more interest due to their exceptional capability of handling huge datasets, recognizing anomalies and, in some cases, filtering them out (Wen and Keyes, 2019). Among the several ML techniques, the VAEs (Mehta et al., 2019) are appealing as they are able to reconstruct realistic time-series and to reliably fill data gaps. This capability may be groundbreaking in the field of Heritage Science that is often based on studies that either need too many and too expensive experimental observations and, sometimes, rely on corrupted data.

In general, time-series forecasting and reconstruction is a task common to many fields, such as IoT (Han et al., 2021), solar physics (Arslan and Sekertekin, 2019), building energy minimization (Liguori et al., 2021) and cultural heritage (Bertolin et al., 2015). In literature, three main different approaches are proposed: missing data are reconstructed through parametric models based on a priori knowledge about the system (American Society of Heating Refrigerating and Air-Conditioning Engineers, 2011; Ciulla et al., 2010; Giaconia and Orioli, 2000), through data-driven approaches (such as deep learning) (Arslan and Sekertekin, 2019; Liguori et al., 2021), or through a mixture of the two (Han et al., 2021).

The first method (Ciulla et al., 2010; Giaconia and Orioli, 2000) is consolidated in the field, but it does not generalize to problems with non-negligible stochasticity (as, for example, the cases in which the impact of human actions is evident and not trivial to include in a deterministic model). Therefore, fitting a polynomial model with such spoiled data would not give good results (Liguori et al., 2021). Even so, the application of pure deep learning could not be advisable as, in case of lack of data, any effective learning may be undermined.

The third approach (Han et al., 2021), through an initial pre-processing and augmentation of the data, makes it possible to build a sufficiently large dataset to feed a machine learning algorithm able to model the system, including its stochastic parts. In this way, good use is made of both clean and spoiled data, which still carry meaningful information and should not thus be simply discarded. Even though the pre-processing phase makes this approach less generalizable to other kind of tasks, it seems to be the only way to tackle a problem with high stochasticity and low data availability. However, in literature the focus is on the forecast of a single measurement, given its history (Arslan and Sekertekin, 2019; Han et al., 2021).

The current work is based on reconstructing the whole natural climate time-series of a wooden heritage building located in Norway: the Ringebu Stave Church. Based on the literature framework, the existing methods seem to be not directly applicable to the current case study as they mainly focus on one-step-ahead predictions (Arslan and Sekertekin, 2019; Han et al., 2021) or short intervals reconstruction (less than 20 h) (Liguori et al., 2021). In the case of this work, instead, the measurements cover one-year period and data gaps that last some days, up to a week, are present as well. To the authors' knowledge, the task of reconstructing such a wide time interval is not common. For this reason, a different approach from the one recently tested by (Liguori et al., 2021) has been implemented: the choice is to exploit the powerful representation and regularization capabilities of Variational Auto Encoders (VAEs) (Mehta et al., 2019). This has been carried out by analyzing microclimatic data collected in Ringebu Stave Church, a Norwegian medieval wooden building. The study case is presented in Section 2; then, the time series reconstruction has been carried out in Section 3, by using a simple transfer function (Section 3.1) and the VAEs (Section 3.2). The main results are discussed in Section 4 and the main conclusions are highlighted in Section 5.

2. Materials

The data used in this work were collected during a monitoring campaign conducted within the framework of the symbol – Sustainable Management of Heritage Building in a long-term perspective project in the Ringebu Church, a stave church located in the region of Innlandet, Norway. Stave churches are medieval Christian churches entirely made of wood, built from the 11th century onward. In particular, Ringebu Church dates back to the 13th century, coming to the present days with only a few modifications. Stave churches are a great heritage for Norwegian culture, and among the over 1,000 churches that were built in the middle age, only 28 still survive. The oldest ones collapsed because the wood was dug directly into the soil, a fact that led to a fast deterioration. In the following centuries new techniques were developed, and the wood was put above a stone base, allowing the remaining stave churches to last in time. However, the risk of losing these masterpieces in recent years is getting stronger and stronger. This is due to the fact that in order to use the churches for celebration, heating is used to create a comfortable internal climate for the churchgoers. This significantly threatens the wood composing the churches architecture which arrived till nowadays through many decades of history and through adaptation to the natural climate of Norway. The work of reconstruction of the natural climate of the church has been performed by using time-series of temperature and relative humidity collected in Ringebu Church using three data loggers installed during a monitoring campaign that started in 2019. Two of these sensors (denoted as DL1 and DL2, with a reading acquisition time of 5 min) are located inside the church, at different positions, whereas the third one (DL3 with a reading acquisition time of 15 min) is placed outside the building, providing external climate measurements. To quantify the similarity between the DL1 and DL2 indoor temperature series, the normalized root mean square error (NRMSE) was used

As easily noticeable from Figure 1, during the considered year there were sporadic peaks of the internal temperature that lay beyond the natural (external) variability, particularly during the cold season. These events are classified as anomalous, as they introduce perturbations in the microclimatic time-series, and they are caused by the artificial heating that is activated in view of celebrations. In particular, there was a week in which the heating was kept on without interruptions, causing the huge plateau that can be spotted in the inset of Figure 1a. Besides, additional data (metadata) carrying information about the use of artificial heating are available as well, allowing to precisely identify anomalous (i.e., artificial) events.

3. Methods

Clearly, using a machine learning-based strategy for reconstructing the natural climate of the Church by having at disposal a unique and flawed time series might be tricky and may lead to unreliable results, if a proper training phase is not carried out. A possible solution could be based on the assumption that the internal temperature of the building is easily obtained from the external one (Giaconia and Orioli, 2000) by means of a deterministic transfer function plus a sub-leading noise contribution. In other words, the building's indoor temperature is inevitably dependent on the outdoor one and it is influenced by the presence of the building envelope acting as a buffer between the outdoor and indoor environments. Indeed, high-resolution historical external temperature and relative humidity time series are provided by the weather forecast station E6 Frya, which is located 5 km away from the Ringebu Church. In particular, it was possible to gather data collected from the weather forecast station in the period September 2014–April 2020, although with some missing data.

At this point, the temperature time-series recorded by the station was compared to the one collected by DL3. An NRMSE of 0.21 was obtained, confirming that the two series are indeed very similar to one another. Hence, the idea is to collect the available data from the station and to apply a properly calibrated transfer function that returns a synthetic internal climate of the church, as it would have been without any human intervention. Eventually, the weather station’s time series, along with the reconstructed one, can be used as a simulated DL3 and DL1 series in order to train the VAE. From now on, this dataset will be referred to as “station dataset”.

The same procedure was implemented for the relative humidity as well, because adding this information can be reasonably beneficial for the VAE, in order to recognize possible connections between temperature and relative humidity time series.

4. Results and discussion

After 50 epochs (i.e., the times that the learning algorithm will work through the entire training dataset) of training, a stabilization of the training error was noticed, thus the training phase was stopped in order to move to the performance testing phase.

The architecture VAE 1 had a TE score of 0.28, whereas VAE 2 scored −0.32. As discussed above, the interpretation is that the thermal ellipse reproduced by VAE 1 is broader than the original one, whereas VAE 2 tends to underestimate the thermal inertia of the building by generating a narrow thermal ellipse. Figure 6 shows the thermal ellipses of VAE 1 and the two stages of VAE 2 compared with the fitted original one.

As for the single sample processing, shown in Figure 7, the daily periodicity is present in the reconstruction (red curve) of each feature (internal humidity, Figure 7a, internal temperature, Figure 7b, external humidity, Figure 7c, and external temperature, Figure 7d) and the signals (blue dashed curve) of both humidity and temperature are filtered, resulting in more regular curves that still have the local unique variations typical of a natural climate. In Figure 7 it can be noticed how the reconstructed signals tend to be more regularized than the original ones, avoiding abrupt deviations from the average pattern. This is because, ideally, the reconstruction has to be interpreted as the average of many realizations of a stochastic process.

Moving to the whole year reconstruction, it can be subdivided into two sections: cold seasons (autumn and winter) and warm seasons (spring and summer).

The months of cold seasons are affected by the use of artificial heating, which corrupts the original signal. By feeding those samples to the trained VAE, as shown in Figure 8 (Goodfellow et al., 2016), it recognizes that the heating peaks are anomalous and thus filters them, retrieving a time series that shows a strong and clean daily periodicity, resembling the one of a natural climate.

Instead, warm seasons are less affected by artificial heating, but still some effects due to the presence of visitors within the church are present. As shown in Figure 8, when feeding the VAE with these kinds of time series, the natural behavior is respected.

The auto encoder is also able to reconstruct the behavior of the temperature when some data are missing (or removed because too corrupted): up to a week of observations can be artificially restored by feeding the VAE with the values corresponding to a linear interpolation of the extremes of the missing interval, as can be seen in Figure 9.

Longer missing intervals should be tackled with a different approach being the climate variations substantial and a simple linear interpolation not sufficient for a proper reconstruction.

Finally, a simple assessment of the latent space may be carried out by plotting the scatterplots of the training samples encodings in Figure 10. The most important aspect is the presence of seasonal clusters, as it is an index of the correct encoding of the yearly periodicity.

An important remark has to be done concerning the training procedure. Typically, the machine learning way to proceed is to update the weights of the model based on the training samples' reconstruction and to monitor overfitting over a test set. However, the setup used in the current work is quite different. Indeed, there is just a 4–5 years series, i.e., repeated measurements of each time window. This is a very limiting factor, because it means no significant pieces of the data may be extracted for comparing the reconstruction of a training sample with the corresponding sample of a test set. Another option could be to compare the reconstruction of a sample with a nearby one used as a test sample, under the assumption that they are similar. In practice, however, this last assumption is quite arguable, let alone the fact that this would prevent the use of a conspicuous part of the available data. Moreover, the goal is to start from the original, spoiled, time series and to generate another series similar to the previous one but without any anomalous peak. The goodness of the reconstruction can be therefore assessed by the capability of the VAE to either generate a resembling time series and to cut off the anomalies. These arguments drove the decision to use the entire dataset of the station as a training set, and to look at the reconstruction obtained by feeding the VAE with the original Ringebu series. Then, relying on the score defined in Section 3.2 and also on a qualitative assessment, the results have been evaluated.

Moreover, the approach followed using the station dataset for the training phase is the best possible to the authors' knowledge, but a note of caution has to be taken since it is just a plausible reconstruction of the possible series that would have been expected in case many years of records had been available.

With respect to other classical VAE architectures, the one presented in this work outperforms the others mainly thanks to two key aspects. First, the introduction of the LSTM layer, which allows the encoder to detect important features of the time series and to keep memory of them for a long period of time. In addition, it contains a larger number of trainable parameters and makes the training a little slower, but its contribution to the final results is fundamental. The second beneficial aspect is the implementation of convolutional layers with kernel sizes equal to the daily and half-day periodicities, which were evident in the dataset.

Finally, the results have shown that the VAE is able to reconstruct the time series even in the absence of some fractions of input data.

5. Conclusions

In this work an attempt to implement an unsupervised machine learning method for the problem of reconstructing the internal climate time series of a heritage building, a stave church in this case, has been proposed. As previously underlined, there are some criticalities within this approach, which are due to the poor amount of available data.

Nevertheless, it has been successfully managed to reconstruct, for the vast majority of the year period, a realistic time series of the natural internal climate of the church, providing a method that is able to reliably fill missing data of many hours' time span, up to some days.

There is surely room for improvements and further studies. For example, the current work has been focused on denoising the original time series; for this kind of task, possibly even a simpler auto encoder architecture could have worked as well (although a VAE is typically more versatile). In other words, the generating properties of the VAE have not been exploited thoroughly, because the latent space of such a complex dataset is very difficult to explore, and it is hard even to define what is hoped to be thrown out from it. However, future perspectives could deal with:

1. Finding a way to understand the geometry of the latent space, to generate completely new samples;

2. Studying and implementing conditional variational auto encoders, which allow generating a complete time series from the input of just some control parameters (in this case the sine and cosine of day and year, for instance). Using them, it could be even possible to generate new data by just specifying the needed time window, provided that the architecture is able to learn the seasonality of one sample.

This way, optimized tools would be available and could be used in several different fields. For example, it can be used to forecast financial time series and business metrics such as stock market price, sales. It could be useful for filling gaps in data from obsolete weather stations and, in general, it could be implemented in any signal processing analysis and control system. In conclusion, these approaches would be of extreme usefulness in studying and predicting the indoor microclimate of historical buildings; as a matter of fact, they would lead towards smarter choices for the management of the indoor microclimate, in the framework of preservation and conservation of the historical and cultural heritage, by limiting, when possible, time and money consuming on-the-field monitoring campaigns.