Original: RECOWNs: Probabilistic Circuits for Trustworthy Time Series Forecasting
Revised: RECOWNs: Probabilistic Circuits for Trustworthy Time Series Forecasting

Original: Abstract
Revised: Abstract

Original: Time series forecasting is a relevant task that is performed in several real-world scenarios such as product sales analysis and prediction of energy demand.
Revised: Time series forecasting is a relevant task that is performed in several real-world scenarios such as product sales analysis and prediction of energy demand.

Original: Given their accuracy performance, currently,
Revised: Given their accuracy performance, currently,

Original: Recurrent Neural Networks (RNNs) are the models of choice for this task.
Revised: Recurrent Neural Networks (RNNs) are the models of choice for this task.

Original: Despite their success in time series forecasting, less attention has been paid to make the RNNs trustworthy.
Revised: Despite their success in time series forecasting, less attention has been paid to make the RNNs trustworthy.

Original: For example,
Revised: For example,

Original: RNNs can not naturally provide an uncertainty measure to their predictions.
Revised: RNNs can not naturally provide an uncertainty measure to their predictions.

Original: This could be extremely useful in practice in several cases e.g. to detect when a prediction might be completely wrong due to an unusual pattern in the time series.
Revised: This could be extremely useful in practice in several cases e.g. to detect when a prediction might be completely wrong due to an unusual pattern in the time series.

Original: Whittle Sum-Product Networks (WSPNs), prominent deep tractable probabilistic circuits (PCs) for time series, can assist an RNN with providing meaningful probabilities as uncertainty measure.
Revised: Whittle Sum-Product Networks (WSPNs), prominent deep tractable probabilistic circuits (PCs) for time series, can assist an RNN with providing meaningful probabilities as uncertainty measure.

Original: With this aim, we propose RECOWN, a novel architecture that employs RNNs and a discriminant variant of WSPNs called Conditional WSPNs (CWSPNs).
Revised: With this aim, we propose RECOWN, a novel architecture that employs RNNs and a discriminant variant of WSPNs called Conditional WSPNs (CWSPNs).

Original: We also formulate a Log-Likelihood Ratio Score as better estimation of uncertainty that is tailored to time series and Whittle likelihoods.
Revised: We also formulate a Log-Likelihood Ratio Score as better estimation of uncertainty that is tailored to time series and Whittle likelihoods.

Original: In our experiments, we show that RECOWNs are accurate and trustworthy time series predictors, able to “know when they do not know”.
Revised: In our experiments, we show that RECOWNs are accurate and trustworthy time series predictors, able to “know when they do not know”.

Original:  INTRODUCTION
Revised:  INTRODUCTION

Original: Time series forecasting is the task to predict the future course ( Y ) of a time series given its past X , also known as context.
Revised: Time series forecasting is the task to predict the future course ( Y ) of a time series given its past X , also known as context.

Original: Currently, Recurrent Neural Networks [Rumelhart et al., 1985] are models of choice when it comes to time series forecasting.
Revised: Currently, Recurrent Neural Networks [Rumelhart et al., 1985] are models of choice when it comes to time series forecasting.

Original: Recent advancements in RNNs research fostered their adoption in practice surpassing established models as e.g. ARIMA (Autoregressive Integrated the RECOWN architecture.
Revised: Recent advancements in RNNs research fostered their adoption in practice surpassing established models, e.g. ARIMA (Autoregressive Integrated Moving Av-

Original: The context X is transformed using STFT with a window size of T w , a) determining the weights of the CWSPN via the Neural
Revised: The context X is transformed using STFT with a window size of T w , a) determining the weights of the CWSPN via the Neural

Original: Network (NN) and b) serving as input to the (complex)-
Revised: Network (NN) and b) serving as input to the (complex)-

Original: Gated Recurrent Units (GRU) [Chung et al., 2014] of the RNN, resulting in the prediction of the Fourier coefficients D ′1 , D ′ 2 , D ′ 3 .
Revised: Gated Recurrent Units (GRU) [Chung et al., 2014] of the RNN, resulting in the prediction of the Fourier coefficients D ′ 2 , D ′ 3 .

Original: Those are then provided to the CWSPN, which computes the conditional Whittle log-likelihood ℓ ( D ′1 , D ′ 2 , D ′ 3 | D 0 , D 1, D 2 ) (see Section 2.3).
Revised: Those are then provided to the CWSPN, which computes the conditional Whittle log-likelihood ℓ ( D ′ 2 , D ′ 3 | D 0 , D 1 ) (see Section 2.3).

Original: Moving Average)
Revised: erage)

Original: [Siami-Namini et al., 2018] in several scenarios [Fei and Yeung, 2015, Li et al., 2019, Li and Cao, 2018].
Revised: [Siami-Namini et al., 2018], in several scenarios [Fei and Yeung, 2015, Li et al., 2019, Li and Cao, 2018].

Original: However, in complex real-world applications, time series are highly subject to several influence factors which are often hard to capture.
Revised: However, in complex real-world applications, time series are highly subject to several influence factors which are often hard to capture.

Original: For example, in the case of grocery demand, the demand for ice cream or BBQ-related products is expected to be higher as long as the weather is warmer than usual.
Revised: For example, in the case of grocery demand, the demand for ice cream or BBQ-related products is expected to be higher as long as the weather is warmer than usual.

Original: Furthermore, exceptional events like a pandemic can significantly influence demands as well.
Revised: Furthermore, exceptional events like a pandemic can significantly influence demands as well.

Original: In [Figure]
Revised: In such cases, the prediction of a model will likely be less accurate compared to usual circumstances. To properly detect such cases, a mea-

Original: [Figure]
Revised: [Figure]

Original: P o w e r ( zsc o r e nor m a li z ed )
Revised: no

Original: [Figure]
Revised: [Figure]

Original: Ground Truth Prediction
Revised: ed

Original: [Figure]
Revised: [Figure]

Original: −
Revised: o l d

Original: [Figure]
Revised: It e m s ( z

Original: [Figure]
Revised: [Figure]

Original: S o l d It e m s ( zsc o r e no r m a li zed )
Revised: o r e no r m a li z

Original: [Figure]
Revised: [Figure]

Original: processes.

Original: such cases, the prediction of a model will likely be less accurate compared to usual circumstances.

Original: To properly detect such cases, a measure of uncertainty about the prediction is valuable [Guo et al., 2017, Laptev et al., 2017, Gal and Ghahramani, 2016].
Revised: sure of uncertainty about the prediction is valuable [Guo et al., 2017, Laptev et al., 2017, Gal and Ghahramani, 2016].

Original: It would make the predictions trustworthy and it would better support the users in decision-making processes.
Revised: It would make the predictions trustworthy and it would better support the users in decision-making processes.

Original: A relevant approach to achieve this is provided by Gaussian Processes (GPs)
Revised: A relevant approach to achieve this is provided by Gaussian Processes (GPs)

Original: [Rasmussen and Williams, 2006].
Revised: [Rasmussen and Williams, 2006].

Original: However, GPs are comparably computationally expensive and therefore not suited for large datasets [Seeger, 2004].
Revised: However, GPs are comparably computationally expensive and therefore not suited for large datasets [Seeger, 2004].

Original: Several methods have been proposed to scale GPs on large datasets, e.g. by modeling the mixture of subspaces using GP experts with Sum-Product-Networks (SPNs) [Trapp et al., 2020, Bruinsma et al., 2020, Yu et al., 2021b].
Revised: Several methods have been proposed to scale GPs on large datasets, e.g. by modeling the mixture of subspaces using GP experts with Sum-Product Networks (SPNs) [Trapp et al., 2020, Bruinsma et al., 2020, Yu et al., 2021b].

Original: Still, for time series forecasting, RNN architectures [Alpay et al., 2016, Wolter et al., 2020, Koutnik et al., 2014] have shown to be superior in terms of prediction accuracy. 
Revised: Anyhow, for time series forecasting, RNN architectures [Alpay et al., 2016, Wolter et al., 2020, Koutnik et al., 2014] have shown to be superior in terms of prediction accuracy. Rasul et al.

Revised: [2021] already examined this direction, by employing autoregressive denoising diffusion models in the time domain to equip RNN-predictions with a measure of uncertainty on the level of time steps.

Revised: Still, they do not provide an uncertainty measure on the level of sequences, which enables users to quickly detect potentially problematic forecasts.

Revised: Furthermore, modeling in the spectral domain is beneficial for prediction performance [Wolter et al., 2020].

Original: Therefore, we introduce REcurrent COnditional Whittle
Revised: Therefore, we introduce REcurrent COnditional Whittle

Original: Networks (RECOWNs), a deep architecture that makes use of Short Time Fourier Transform (STFT) and integrates a
Revised: Networks (RECOWNs), a deep architecture that makes use of Short Time Fourier Transform (STFT) and integrates a

Original: Spectral RNN [Wolter et al., 2020] and a conditional  Whittle SPN(CWSPN) [Yu et al., 2021a]  i.e. a discriminative probabilistic circuits tailored for time series.
Revised: Spectral RNN [Wolter et al., 2020] together with a conditional variant of a Whittle SPN [Yu et al., 2021a] (CWSPN) i.e. a discriminative probabilistic circuits tailored for time series.

Original: The overall architecture is depicted in Figure 1.
Revised: The overall architecture is depicted in Figure 1.

Original: In this way, RECOWN can keep the good forecasting performance of RNNs while, thanks to the CWSPN, providing meaningful probabilities as a measure of uncertainty of the predictions.
Revised: In this way, RECOWN can keep the good forecasting performance of RNNs while, thanks to the CWSPN, providing meaningful probabilities as a measure of uncertainty of the predictions.

Original: A showcase of its potential can be found in Figure 2, showing how the CWSPN correctly predicts when the forecast of the Spectral RNN increasingly diverges from the ground truth.
Revised: A showcase of its potential can be found in Figure 2, showing how the CWSPN correctly predicts when the forecast of the Spectral RNN increasingly diverges from the ground truth.

Original: RECOWNs can be trained end-to-end by gradient descent and have modular nature, in fact, one can employ any differentiable density estimator instead of a CWSPN.
Revised: RECOWNs can be trained end-to-end by gradient descent and have modular nature, in fact, one can employ any differentiable density estimator instead of a CWSPN.

Original: However, in our experiments, CWSPNs resulted the best choice compared to another state-of-the-art density estimators, namely MAF [Papamakarios et al., 2017], especially when the model capacity is reduced.
Revised: However, in our experiments, CWSPNs resulted the best choice compared to another state-of-the-art density estimator, MAF [Papamakarios et al., 2017], especially when the model capacity is reduced.

Original: Furthermore, CWSPNs are also more flexible given that they can answer to a wider range of exact probabilistic queries in a tractable way.
Revised: Furthermore, CWSPNs are also more flexible given that they can answer to a wider range of exact probabilistic queries in a tractable way.

Original: Similar to Yu et al.
Revised: Similar to Yu et al.

Original: [2021a], RECOWNs model time series in the spectral domain, but employ a different kind of
Revised: [2021a], RECOWNs model time se- ries in the spectral domain, but employ a different kind of

Original: Fourier Transform – the STFT – to account for changes in the frequency domain as the signal changes over time.
Revised: Fourier Transform – the STFT – to account for changes in the frequency domain as the signal changes over time.

Original: Whilein Yu et al.
Revised: While Yu et al.

Original: [2021a] the authors focused on modeling the joint distribution of the multivariate time series, in this work we shift our focus on predictions, providing them with probabilities as a useful measure of uncertainty.
Revised: [2021a] focus on modeling the joint distribution of the multivariate time series, in this work we shift our focus on predictions, providing them with probabilities as a useful measure of uncertainty.

Original: As successfully done in Yu et al.
Revised: As successfully done in Yu et al.

Original: [2021a] and other previous methods [Tank et al., 2015], modeling  in the spectral domain enables us to make use of theso-called Whittle Approximation Assumption [Whittle, 1953] which facilitates the modeling of Fourier coefficients of a time series.
Revised: [2021a] and other previous methods [Tank et al., 2015], modeling time series in the spectral domain enables us to make use of the Whittle Approximation Assumption [Whittle, 1953] which facilitates the modeling of Fourier coefficients of a time series.

Original: Additionally, since RECOWNs model the time series in the spectral domain, we propose Log-Likelihood Ratio Score (LLRS) which enables to compute the confidence intervals  back in the time domain.
Revised: Additionally, since RECOWNs model the time series in the spectral domain, we propose Log-Likelihood Ratio Score (LLRS) which enables to compute the confidence intervals of the predictions back in the time domain.

Original: Experiments show that, compared to stateof-the-art, RECOWNs are more accurate and trustworthy.
Revised: Experiments show that, compared to state-of-the-art models, RECOWNs are more accurate and trustworthy.

Original: Our contributions are the following:
Revised: Our contributions are the following:

Original: • We introduce RECOWN, the first deep tractable model for time series forecasting that is accurate and that provides an uncertainty measure for its predictions, making them more trustworthy.
Revised: • We introduce RECOWN, the first deep tractable model for time series forecasting that is accurate and that provides an uncertainty measure for its predictions on sequence level as well as in the time domain, making them more trustworthy.

Original: • We introduce a data sample weighting strategy based on the Mean Squared Error.
Revised: • We introduce a data sample weighting strategy based on the Mean Squared Error.

Original: • We formulate the (Whittle) Log-Likelihood Ratio Score, tailored to better estimate the uncertainty of time series predictions based on the Whittle likelihood.
Revised: • We formulate the (Whittle) Log-Likelihood Ratio Score, tailored to better estimate the uncertainty of time series predictions based on the Whittle likelihood.

Original: The paper is structured as follows: We start by introducing
Revised: The paper is structured as follows: We start by introducing

Original: STFT, CWSPNs as well as Spectral RNNs, i.e. the main components of RECOWNs in Section 2.
Revised: STFT, CWSPNs as well as Spectral RNNs, i.e. the main components of RECOWNs in Section 2.

Original: Then, in Section 3, we describe the experimental setting and analyze the results showing that RECOWNs areboth accurate and trustworthy and that the LLRS is a valid tool to assess the uncertainty and the quality of the predictions.
Revised: Then, in Section 3, we describe the experimental setting and analyze the results, showing that RECOWNs are accurate as well as trustworthy and that the LLRS is a valid tool to assess the uncertainty and the quality of the predictions.

Original: We conclude in Sectionwhere we also point out future directions.
Revised: We conclude in Sectionwhere we also point out future directions.

Original:  RECOWN: RECURRENT CONDITIONAL WHITTLE NETWORKS 2.1 SHORT TIME FOURIER TRANSFORM
Revised:  RECOWN: RECURRENT CONDITIONAL WHITTLE NETWORKS 2.1 SHORT TIME FOURIER TRANSFORM

Original: Using discrete Fourier Transformation (DFT), a signal can be mapped from the time to the spectral domain using a decomposition into linear combinations of sinus functions.
Revised: With Discrete Fourier Transformation (DFT), a time series can be mapped from the time to the spectral domain with a decomposition into linear combinations of sinus functions.

Original: For a multivariate time series X = x 1 ,..., x T with x t ∈ R p and length T , we can define the discrete Fourier coefficients d k ∈ C p at frequency λ k = 2 π kT , k = 0 , ··· , T − 1, using the Fourier transformation F as follows [Tank et al., 2015]:
Revised: For a multivariate time series X = x 1 , ..., x T with x t ∈ R p and length T , we can define the discrete Fourier coefficients d k ∈ C p at frequency λ k = 2 πkT , k = 0 , · · · , T − 1 , using the Fourier transformation F as follows [Tank et al., 2015]:

Original: [Equation]
Revised: [Equation]

Original: Moreover, given Fourier coefficients D = ( d 0 ,..., d T − 1 ) , we can apply the inverse DFT to regain a time-domain representation:
Revised: Moreover, given Fourier coefficients D = ( d 0 , ..., d T − 1 ) , we can apply the inverse DFT to project the frequencies back to the time domain:

Original: [Equation]
Revised: [Equation]

Original: However, based on the standard definition, infinite support of theinput signal is required in order to apply DFT.
Revised: However, to account for potential changes in the frequency domain as the signal changes over time, we apply STFT.

Original: To overcome this limitation, we apply STFT: Instead of approximating the whole signal at once, the input is divided into overlapping segments and each segment is then approximated separately. 
Revised: The input is divided into overlapping segments and each segment is then approximated separately. STFT is introduced by

Original: Mathematically speaking, considering segments of length T w , extracted every S time steps, STFT is introduced by Griffin and Lim [1984]:
Revised: Griffin and Lim [1984] by considering segments of length T w , extracted every S time steps:

Original: [Equation]
Revised: [Equation]

Original: with w being a window function, m denoting a respective shift and the remaining defined analogues to Equation 1.
Revised: with w being a window function, m denoting the corresponding shift and the remaining defined analogously to Equation 1.

Original: This results in n s = TS windows in total.
Revised: This results in n s = T − T w S +3 windows in total when applying a padding of size S at the start and end of sequences.

Original: As with the regular Fourier transform, STFT can also be inverted:
Revised: As with the regular Fourier transform, STFT can also be inverted:

Original: [Equation]
Revised: [Equation]

Original: Due to Hermitian symmetry, fora real-valued time series, the negative Fourier coefficients are redundant and we only need to model T = (cid:4) T w 2 (cid:5) + 1 Fourier coefficients for a window size of T w .
Revised: Due to Hermitian symmetry, for real-valued time series, the negative Fourier coefficients are redundant. Therefore, we only need to model T = (cid:4) T w 2 (cid:5) + 1 Fourier coefficients for a window size of T w .

Original: Furthermore, we apply a low-pass filter, to filter out noise and further reduce the number of parameters in the model.
Revised: Furthermore, we apply a low-pass filter to filter out noise, which further reduces the number of parameters in the model.

Original: Details on the extent of filtering are described in the Experimental Evaluation section.
Revised: Details on the extent of filtering are described in Section 3.

Original: The window function can be selected by hand, but it may also be learned by an optimizer as part of a machine learning model, as e.g. described by Wolter et al.
Revised: The window function can be determined either by hand or learned by an optimizer [Wolter et al., 2020].

Original: Analogues to their approach, we apply a truncated Gaussian window:
Revised: We apply a truncated Gaussian window at position n :

Original: [Equation]
Revised: [Equation]

Original: The standard deviation σ is learned by the optimizer:
Revised: The standard deviation σ is learned by the optimizer.

Original: By increasing σ , the window approaches a more rectangular shape, while it gets narrower with decreasing σ .
Revised: By increasing σ , the window approaches a more rectangular shape, while it gets narrower when decreasing σ .

Original: 2.2 WHITTLE LIKELIHOOD
Revised: 2.2 WHITTLE LIKELIHOOD

Original: The Whittle likelihood modelsthe multivariate time series in the spectral domain.
Revised: The Whittle likelihood models multivariate time series in the spectral domain.

Original: With X , x t and Tgiven as  in Section 2.1, x t is Gaussian stationary for t ∈ Z , if:
Revised: Given X , x t and T as defined in Section 2.1, x t is Gaussian stationary for t ∈ Z , if:

Original: [Equation]
Revised: [Equation]

Original: With X 1: N = { X 1 ,..., X N } being N independent realizations of X , d n , k ∈ C p denotes the discrete Fourier coefficient of the n th sequence at frequency λ k :
Revised: Given X 1: N = {X 1 , . . . , X N } being N independent realizations of X and d n,k ∈ C p represent the discrete Fourier coefficient of the n -th sequence at frequency λ k :

Original: [Equation]
Revised: [Equation]

Original: Based on the Whittle approximation assumption [Whittle, 1953], the Fourier coefficients are independent complex normal RVs with mean zero:
Revised: Based on the Whittle approximation assumption [Whittle, 1953], the Fourier coefficients are independent complex normal RVs with mean zero:

Original: [Equation]
Revised: [Equation]

Original: with S k ∈ C
Revised: with S k ∈ C

Original: p × p being the spectral density matrix .
Revised: p × p being the spectral density matrix .

Original: For a stationary time series, it is defined as:
Revised: For a stationary time series, it is defined as:

Original: [Equation]
Revised: [Equation]

Revised: where the infinite sum may be approximated.

Original: Finally, the Whittle likelihood of the X 1: N is given by: 
Revised: Finally, the Whittle likelihood of the X 1: N is given by: p ( X 1: N | S

Original: [Equation]
Revised: [Equation]

Original: The Whittle approximation holds asymptotically with large
Revised: The Whittle approximation holds asymptotically with large

Original: T and has been used  inthe Bayesian context.
Revised: T and has been used also in Bayesian context [Tank et al., 2015].

Original: We will make use of it and place a Complex Conditional SPN over the frequencies, resulting in CWSPNs, modeling the conditional Whittle Log-Likelihood (CWLL).
Revised: We will make use of it and place a Complex Conditional SPN over the frequencies, resulting in CWSPNs, modeling the conditional Whittle Log-Likelihood (CWLL).

Original: 2.3 COMPLEX CONDITIONAL SPN
Revised: 2.3 COMPLEX CONDITIONAL SPN

Original: Originally introduced by Poon and Domingos [2011], SPNs have gained recent popularity, showing also promising results on univariate time series  [Melibari et al., 2016].
Revised: Sum-Product Networks [Poon and Domingos, 2011] are deep probabilistic circuits with tractable and exact inference. Among several applications, they have been successfully employed also for univariate time series modeling [Melibari et al., 2016].

Original: Recent approaches build upon randomly generated structures, RATSPNs, introduced by Peharz et al.
Revised: Recent tensor- SPN implementations such as RAT-SPNs [Peharz et al., 2020b] build upon randomly generated structures based on the notion of region graphs [Dennis and Ventura, 2012].

Original: [2020b], enabling scalability as well as training in an end-to-end fashion combined with NNs
Revised: They are more scalable and enable training in an end-to-end fashion which allows to be optimized jointly with NNs.

Original: In order to provide a measure of how good a prediction ( Y ) is with respect to a context ( X ), we aim for modeling the conditional likelihood P ( Y | X ) .
Revised: In order to provide a measure of how good a prediction ( Y ) is with respect to a context ( X ), we aim for modeling the conditional likelihood P ( Y | X ) .

Original: Although any SPNs modeling the joint  could be employed for this task (since the conditional can be derived from the joint using P ( Y | X ) =
Revised: Although any SPN modeling the joint distribution could be employed for this task (since the conditional can be derived from the joint using P ( Y | X ) =

Original: P ( X , Y ) P ( X ) ), the 
Revised: P ( X,Y ) P ( X ) ), the Conditial SPN (CSPN)

Original: Conditial SPN (CSPN) is a  natural choice. 
Revised: [Shao et al., 2020] is a more natural choice. CSPN parameters are not learned directly, instead, a seperate general function approximator g – in our case a neural network – is employed to provide parameters θ to the SPN based on the input x, i.e. g ( x ) = θ , while only y is provided as input to the SPN on the leaf layer.

Original: Originally proposed by Shao et al.
Revised: Analogously to Shao et al.

Original: [2020], the CSPN parameters are not learned directly. Instead, a neural network is employed to provide parameters to the SPN based on the input X, while only Y is provided as input to the SPN on the leaf layer.
Revised: [2020], we structure g as two separate, fullyconnected networks g L and g S with ReLU activations, learning the leaf and sum node parameters θ = ( θ L , θ S ) respectively, i.e. g ( x ) = ( g L ( x ) , g S ( x )) .

Original: Training this architecture – using gradient descent – results in modeling P ( Y  | X  ) .
Revised: Training this architecture – using gradient descent – results in modeling P ( Y = y | X = x ) : = P ( Y =

Original:  The structure of CSPN israndomly generated  based on RAT-SPNs [Peharz et al., 2020b].
Revised: y ; θ ) . The structure of CSPN is generated at random, based on RAT-SPNs [Peharz et al., 2020b].

Revised: In this way, we can take advantage of the benefits provided by a tensorized SPN implementation.

Original: We alter this approach to account for the complex values of the Fourier coefficients analogues to the Complex-Valued
Revised: We alter this approach to account for the complex values of the Fourier coefficients similarly to the Complex-Valued

Original: SPNs (CoSPNs) [Yu et al., 2021a]. Input to the leaves of the
Revised: SPNs (CoSPNs) [Yu et al., 2021a], resulting in Complex

Original:  CoCSPN will be the Fourier coefficients of Y at frequency k and shift m , i.e. d mk = STFT ( Y ) mk .
Revised: Conditional SPNs (CoCSPNs). The input for the leaves of the CoCSPN are the Fourier coefficients of Y at frequency k and shift m , i.e. d mk = STFT ( Y ) mk .

Original: Based on the Whittle assumption, we know that the Fourier coefficients d mk are normal distributed.
Revised: Based on the Whittle assumption, we know that the Fourier coefficients d mk are normal distributed.

Original: Therefore, also their real and imaginary parts will be jointly normal distributed.
Revised: Therefore, also their real and imaginary parts are jointly normal distributed.

Original: To account for  the correlations between the two parts, they will be jointly modeled by a single pairwise Gaussian leaf node, parameterized by a vector of means µ d mk ∈ R 2 and a covariance matrix Σ d mk ∈ R 2 × 2 .
Revised: To account for [Figure] the correlations between the two parts, they are jointly modeled by a single pairwise Gaussian leaf node, parameterized by a vector of means µ d mk ∈ R 2 and a covariance matrix Σ d mk ∈ R 2 × 2 .

Original: In total, the CoCSPNtherefore encodes the conditionaldensity
Revised: Thus, CoCSPN encodes the conditional

Original: [Equation]
Revised: [Equation]

Original: (12) resulting in the CWLL 
Revised: where d mk = [Re( d mk ) , Im( d mk )] . Based on this and Equation 11, we define the CWLL as:

Original: [Equation]
Revised: [Equation]

Revised: which models the probability of the predicted STFTwindows given the STFT-windows of the context,

Revised: i.e. the

Revised: Fourier coefficients of STFT ( X ) are provided to the neural network g .

Original: The structural constraints of completeness and decomposability for SPNs still hold, see Yu et al. [2021a]. [Figure]
Revised: The structural constraints of completeness and decomposability for SPNs still hold, as in [Yu et al., 2021a].

Original: 2.4 SPECTRAL RNN
Revised: 2.4 SPECTRAL RNN

Original: We employ Spectral RNN
Revised: We employ Spectral RNN

Original: [Wolter et al., 2020] for univariate time series forecasting.
Revised: [Wolter et al., 2020], designed for univariate time series forecasting, to provide predictions in RECOWN.

Original: Compared to a standard RNN, the recurrent steps are performed over the windows retrieved from STFT.
Revised: Compared to a standard RNN, the recurrent steps are performed over the windows retrieved from STFT.

Original: Therefore, for a window with width T w and a step size S , it only has to perform n s instead of T time steps for an input of length T .
Revised: Therefore, for a window with width T w and a step size S , it only has to perform n s instead of T time steps for an input of length T .

Original: Mathematically, the SRNN is defined as follows:
Revised: The SRNN is defined as follows:

Original: [Equation]
Revised: [Equation]

Original: with τ = [ 0 , n s ] enumerating the total number of segments n s .
Revised: with τ = [0 , n s ] enumerating the total number of segments n s .

Original: Since X τ ∈ C T × 1 is a complex signal, the RNN cell either needs to operate in the complex space or needs to provide projections I : C T (cid:55)→ R n i , O : R n o (cid:55)→ C T for  in- and  outputs respectively.
Revised: Since X τ ∈ C T × 1 is a complex signal, the RNN cell either needs to operate in the complex space or needs to provide projections I : C T (cid:55)→ R n i , O : R n o (cid:55)→ C T for n i -dimensional in- and n o -dimensional outputs respectively.

Original: Since complex units have not shown superior in our experiments (similar to findings in Wolter et al. [2020]), we employ standard Gated Recurrent Units (GRU)
Revised: Since complex units have not shown superior in preliminary experiments (similar to [Wolter et al., 2020]), we employ standard Gated Recurrent Units (GRU)

Original: [Chung et al., 2014].
Revised: [Chung et al., 2014].

Original: As projections, we employ concatenation and splitting respectively, i.e. I ( X τ ) =
Revised: As projections, we employ concatenation and splitting respectively, i.e. I ( X τ ) =

Original: ( Re ( X τ ) , Im ( X τ )) 
Revised: ( Re ( X τ ) , Im ( X τ )) ,

Original: , O ( h τ ) =
Revised: O ( h [Equation] =

Original: h 1 ,..., T τ + h T + 1 ,..., 2 T τ i and n i = n o = 2 T .
Revised: h 1 ,..., T τ + [Equation] [Equation] · i and n i = n o = 2 T .

Original: Therefore, the weights have the form of h τ ∈ R n h × 1 , W c ∈ R n h × n h , V c ∈ R n h × 2 T , b c ∈ R n h × 1 and
Revised: Thus, [Equation]

Original:  W pc ∈ R n h × 2 T , where n h is the size of the hidden state and T the reduced amount of frequencies in the STFT.
Revised: and W pc ∈ R n h × 2 T , where n h is the size of the hidden state and T the reduced amount of frequencies in the STFT.

Original: Further details on SRNN may be found in Wolter et al.
Revised: Additional details on SRNN are in [Wolter et al., 2020].

Original: 2.5 TRAINING RECOWNS
Revised: 2.5 TRAINING RECOWNS

Original: Training SRNN and CWSPN can be achieved jointly in a co-ordinate descent fashion, as shown in Figure 3, resulting in RECOWN.The overall network structure of RECOWN is shown in Figure 1.
Revised: The architecture of RECOWN is shown in Figure 1. SRNN and CWSPN can be trained end-to-end in a co-ordinate descent fashion, as shown in Figure 3, resulting in RECOWN.

Original: In each optimization step, the weights of the SRNN are updated first by minimizing the Mean Squared Error (MSE-loss), i.e. MSE ( Y GT , Y Pred ) .
Revised: In each optimization step, the weights of the SRNN are updated first by minimizing the Mean Squared Error (MSE-loss), i.e. MSE ( Y GT , Y Pred ) .

Original: Afterwards, the SRNN weights are fixed and the CWSPN is to be optimized w.r.t.
Revised: Afterwards, the SRNN weights are fixed and the CWSPN is to be optimized w.r.t.

Original: minimizing an MSE-weighted negative Log-Likelihood (LL) loss on M samples of data:
Revised: minimizing an MSE-weighted negative LogLikelihood (LL) loss on M samples of data:

Original: [Equation]
Revised: [Equation]

Original: where SE denotes the Squared Error.
Revised: where SE denotes the Squared Error.

Original: Therefore, the CWSPN learns a skewed conditional distribution of predictions given context.
Revised: Therefore, the CWSPN learns a skewed conditional distribution of predictions given the context: Predictions of the SRNN are directly passed to the SPN-part of CWSPN, while the Fourier coefficients of the context are passed to the NN g of CWSPN.

Original: By using the inverse weighting by the SE, we simulate that less accurate predictions (high SE) occur less frequently in the distribution, while better predictions (low SE) are simulated to appear more often in the distribution.
Revised: By using the inverse weighting with the SE, we assume that less accurate predictions (high SE) occur less frequently in the distribution, while better predictions (low SE) are assumed to appear more often in the distribution.

Revised: We opt for the square of the scaling factor to account for the exponentiallyshaped curve of the MSE as it can be seen in Figure 1.

Original: This helps the CWSPN to learn a more usable likelihood, as we will show in section 3.
Revised: This helps the CWSPN to learn a more meaningful likelihood, as we will show in Section 3.

Original: Furthermore, this allows for any gradient-descent-based density estimatorto be placed in the RECOWN structure.
Revised: One can employ any gradientdescent-based density estimator in RECOWN.

Original: Our experiments, however, as we will see in the next section, show CWSPNs  to be especially suited to be used in the RECWON structure.
Revised: However, as shown in the next Section, CWSPNs turned to be more suitable for RECOWN. [Figure] [Figure]

Original:  EXPERIMENTAL EVALUATION
Revised:  EXPERIMENTAL EVALUATION

Original: To show the benefits of using RECOWNs, we investigate the following research questions:
Revised: To show the benefits of using RECOWNs, we investigate the following research questions:

Original: (Q1) Quality of Predictions Can RECOWNs differentiate “bad” from “good” predictions on a sequence level?
Revised: (Q1) Quality of Predictions Can RECOWNs distinguish between “bad” and “good” predictions on a sequence level?

Original: (Q2) Uncertainty of Predictions Using the proposed LLRS, can RECOWNs estimate the uncertainty of predictions in the time domain?
Revised: (Q2) Uncertainty of Predictions Using the LogLikelihood Ratio Score (proposed in Section 3.3), can RECOWNs estimate the uncertainty of predictions in the time domain?

Original: 3.1 DATASETS
Revised: 3.1 DATASETS

Original: For our experiments  we evaluate the performance on two differentdatasets. We apply z-score normalization to normalize the data for all of our experiments.
Revised: For our experiments 1 we evaluate the performance on two different z-score normalized datasets.

Original: Firstly, the problem of forecasting Power consumption, using the 15 minute-frequency data from the European Net- [Figure] P r ed i c t i on [Figure]
Revised: The first dataset is Power consumption, which uses theminute-frequency data from the European Network of Trans- mission System Operators for Electricity 2 .

Original: SE

Original: [Figure]

Original: ond i t i

Original: [Figure]

Original: h

Original: i tt l

Original: [Figure]

Original: LL

Original: [Figure]

Original: [Figure]

Original: P r ed i c t i on M

Original: SE

Original: [Figure]

Original: C ond i t i ona l W h

Original: i tt l e

Original: LL

Original: [Figure]

Original: [Figure]

Original: r

Original: [Figure]

Original: i c t i [Figure] work of Transmission System Operators for Electricity 1 . Given 14 days of context, the network needs to predict the power load from noon to midnight  the following day (i.e. 1. days).
Revised: Given 14 days of context, the network has to predict the power load from noon to midnight of the following day (i.e. 1.5 days).

Original: As window size, we choose 96, which corresponds to a full day given the 15-minute sampling rate.
Revised: As window size, we choose 96, which corresponds to a full day given the 15-minute sampling rate.

Original: Secondly, the task of forecasting the Retail demand, using data from a retail location of a big german retailer, spanning  2 years and  roughly 4000 different products with a daily sampling rate.
Revised: The second dataset regards the task of forecasting the Retail demand, using data from a retail location of a big German retailer 3 , spanning over 2 years and including roughly 4k different products with a daily sampling rate.

Original: Here, the task is to predict three weeks of demand given half a year of context.
Revised: Here, the task is to predict three weeks of demand given half a year of context.

Original: Since no sales data of Sundays is present, we filter them out, making a window size of 12 a reasonable choice,  spanning 2 weeks of data.
Revised: Since no sales data of Sundays is present, we filter them out, making a window size of 12 a reasonable choice, i.e. spanning 2 weeks.

Original: We deliberately use a smaller window size hereas compared to the Power dataset, in order to verify that the approach works for different window sizes.
Revised: We deliberately use a smaller window size here compared to the Power dataset in order to verify that the approach works also with different window sizes.

Original: Examples for sequences of both datasets can be found in Figure 4, together with exemplary predictions of the SRNN. Regarding the low-pass filter of STFT, we apply it with a factor of four to the Power and with a factor of two to the Retail data. 
Revised: Regarding the low-pass filter of STFT, we apply it with a factor of 4 to the Power and with a factor of 2 to the Retail data. Examples of sequences of both datasets can be found in Figure 4, together with the predictions of the SRNN.

Original: 3.2 CLASSIFICATION ACCURACY
Revised: 3.2 CLASSIFICATION ACCURACY

Original: As the target of our approach is to use the likelihood of the CWSPN as an indicator for the quality of predictions, we will focus our evaluation on whether this correlation is true for a significant part of our test sequences or not.
Revised: Since the target of our approach is to use the likelihood of the CWSPN as an indicator for the quality of predictions, we will focus our evaluation on whether this correlation is true for a significant part of our test sequences or not.

Original: To do so, wewill first have a look at the correlation using plots, while accessing it afterwards mathematically using a correlation error.
Revised: To do so, we first inspect the correlation visually through plots, and revise it more formally later by using a correlation error.

Original: 3.2.1 Correlation Plots
Revised: 3.2.1 Correlation Plots

Original: To visually access the quality of the CWLL as an indicator for the prediction quality, we first calculate the CWLL and
Revised: To visually inspect the quality of the CWLL as an indicator for the prediction quality, we first calculate the CWLL and

Original: MSE for all test sequences of each dataset and plot those sorted ascending  by MSE.
Revised: MSE for all test sequences of each dataset and plot those in ascending order by MSE.

Original: As shown in Figure 5, a clear  correlation between CWLL and MSE exists: The higher the MSE, the lower the conditional Whittle likelihood.
Revised: As shown in Figure 5, a clear [Table] correlation between CWLL and MSE exists: The higher the MSE, the lower the CWLL.

Original: The same applies for the magnitude, i.e. the exponential-shaped curve of the MSE is reflected in the CWLL.
Revised: The same happens for the magnitude, i [Table] e. the exponentially-shaped curve of the MSE is reflected in the CWLL.

Original: Based on this observation, we  assume that the CWLL can be a good indicator for the prediction quality, and may therefore fulfill the research question (Q1).
Revised: Based on these observations, we can assume that the CWLL is a good indicator for the prediction quality, and it fulfills (Q1) .

Original: A hypothesis, we want to mathematically analyze further ins section 3.2.2.
Revised: We show this assumption holds more formally in Section 3.2.2.

Original: As in the application case, the ground truth is not (yet) known and the MSE is therefore missing, one could use the 
Revised: As in real world scenarios, where usually the ground truth is not (yet) known and the MSE is therefore missing, one could use the CWLL to sort sequences from potential high

Original: CWLL to sort sequences from high risk (aka low CWLL) to  low risk (aka high CWLL) andfurther analyze  those high-risk frequencies.
Revised: MSE (low CWLL) to probably low MSE (high CWLL) and analyze further those potential high MSE frequencies.

Original: This use case is depicted in Figure 6, showing again a good quality correlation between a rising CWLL and a lower MSE of prediction.
Revised: Such use case is depicted in Figure 6. It shows once again a strong correlation between an increasing CWLL with a lower MSE of predictions.

Original: To access it mathematically, we provide the following application example:
Revised: To have a quantitative perspective, we provide the following example of a potential use-case:

Original: Selecting the 5% sequences with the lowest CWLL 
Revised: Selecting the 5% sequences with the lowest CWLL from CWSPN on

Original: [Table]

Original: from CWSPN on Power , we find that 75% of all sequences with the worst 5% of MSEs are included.
Revised: Power , we find that 75% of all sequences with the worst 5% of MSEs are included.

Original: Looking at the 10% sequences with the lowest CWLL covers 98.5% of all sequences with the worst 5% of MSEs, displaying the effectiveness of RECOWN.
Revised: Looking at the 10% sequences with the lowest CWLL, we find 98.5% of all sequences with the worst 5% of MSEs. This highlights how RECOWNs are able to distinguish between good and bad predictions.

Original: 3.2.2 Correlation Error (CE)
Revised: 3.2.2 Correlation Error (CE)

Original: In order to provide a correlation error for each test sequence n , we calculate a relative prediction error( S nPred ) and a likelihood score ( S n ℓ ) respectively:
Revised: In order to provide a correlation error for each test sequence n , we calculate a relative prediction error S nPred = (cid:115) [Equation] and a likelihood score ( S nℓ ) respectively:

Original: [Equation]
Revised: [Equation]

Original: The square root is taken to  accountfor the exponential shape of the conditional CWLL, as can be seen in Figure 5.
Revised: The square root is taken to take into account the exponential shape of the conditional CWLL, as can be seen in Figure 5.

Original: Note that the calculation of S n ℓ corresponds to the likelihood ratios introduced in section 3.3.

Original: Given that the MSE reflects the gold standard on where a sequence should be placed on the range from “bad” to “good”, we define the Correlation Error (CE) for the CWLL as the quadratic distance of the scores:
Revised: Given that the MSE reflects the gold standard on where a sequence should be placed on the range from “bad” to “good”, we define the Correlation Error (CE) for the CWLL as the quadratic distance of the scores:

Original: [Equation]
Revised: [Equation]

Original:  Since S nPred , S n ℓ ∈ [ 0 , 1 ] by definition, we have CE n ∈ [ 0 , 1 ] .
Revised: [Equation] Since S nPred , S nℓ ∈ [0 , 1] by definition, we have CE n ∈ [0 , 1] .

Original: In order toenable a better assessment of this new proposed error, we provide a random baseline, which draws likelihood scores randomly from a uniform distribution, i.e. S n ℓ random ∼ U ( 0 , 1 ) .
Revised: In order to better assess this novel score, we provide a random baseline, which draws likelihood scores randomly from a uniform distribution, i.e. S nℓ random ∼ U (0 , 1) .

Original: For the evaluations of the correlation error, we comparethe CWSPN with Masked Autoregressive Flow (MAF)
Revised: To evaluate the correlation error, we compare CWSPN with Masked Autoregressive Flow (MAF)

Original: [Papamakarios et al., 2017], a state-of-the-art neural density estimator.
Revised: [Papamakarios et al., 2017], a state-of-the-art neural density estimator.

Original: MAF is integrated into the joint training just as the CWSPN, therefore also follows the same training objective as given in Equation 18.
Revised: MAF is integrated into the joint RECOWN architecture like CWSPN, therefore, it follows the same training objective as given in Equation 18.

Original:  For each system, we report scores on the Fourier coefficients as well as on the original time series (for CWSPN, modeling the time series in the time domain results in astandard CSPN).
Revised: We refer to this architecture as RECOWN-MAF . For each model, we report scores from the spectral domain as well as on the original time series (for CWSPN, modeling the time series in the time domain results in a CSPN).

Original: Furthermore, we evaluate three different model sizes, Small,Medium, and
Revised: Furthermore, we evaluate three different model sizes, Small,

Original:  Large .
Revised: Medium, and Large .

Original: Their respective amount of  parameters is given in Table 1.
Revised: The results and the number of trainable parameters are given in Table 1.

Original: In general,we see that modeling in the spectral domain (i.e. using STFT) is mostly superior to operating in the time domain, seeming to improve parameter efficiency.
Revised: In general, modeling in the spectral domain (by using STFT) is beneficial than operating in the time domain, while improving also parameter efficiency.

Original: This is especially true for MAF.
Revised: This is more prominent for MAF.

Original: Furthermore,while MAF achieves the best scores on bigger model sizes, CWSPN is especially goodat modeling the CWLL with a smaller model size.
Revised: Furthermore, MAF achieves the best scores on bigger models, in comparison, CWSPN is particularly good with reduced model capacity.

Original: In general, the  error differences between the models are quite small, making especially CWSPN and MAF in all cases a good choice, enabling us tosuccessfully answerthe research question (Q1)  – RECOWN can differentiate “bad” from “good” predictions on a sequence level.
Revised: Overall, the correlation error obtained with the different architectures is relatively low, also on Retail which is a more difficult dataset. Moreover, it is much better than the random baseline. This allows us to answer (Q1) affirmatively. – RECOWN can distinguish between “bad” from “good” predictions on a sequence level.

Original: Note that SPN-architectures offer more inference options – like computing the MPE, all marginals in linear time, and similar – than MAF.
Revised: However, it is important to remark that SPN architec- tures can naturally answer to a wider range of queries than MAF.

Original: Additionally,  our experiments showed thatthe CWSPN is more stable w.r.t. changes in the hyperparameters compared to MAF.
Revised: Additionally, during our experiments we observed that CWSPN is also less sensitive to hyperparameter tuning.

Original: Finally, the predictions on the Retail dataset seem to be a bit harder to access overall compared to the Power data.

Original: 3.3 PROVIDING UNCERTAINTY TO PREDICTIONS
Revised: 3.3 PROVIDING UNCERTAINTY TO PREDICTIONS

Original: As an additional benefit, the conditional Whittle Likelihood can also be used to access the uncertainty for a prediction in the time domain, allowing to take insight into how it changes at different steps in time.
Revised: The conditional Whittle Likelihood can also be used to estimate the uncertainty for a prediction in the time domain. This allows to take insights on how it changes over time.

Original: To allow for this, we use the notion of likelihood ratios, leveraging the window function w  to project the likelihood to the time domain at time step n :
Revised: With this aim, we use the notion of likelihood ratios, leveraging the window function w , to project the likelihood to the time domain at time step n :

Original: [Equation]
Revised: [Equation]

Revised: Here, ℓ ( Y pred ( n ) | X ) denotes the conditional likelihood of prediction window n given the context.

Revised: Every other window in the prediction is marginalized.

Revised: Note that the calculation of λ LR has high similarities to the likelihood-score

Revised: S nℓ introduced in Section 3.2.2.

Original: Furthermore, we define the maximum likelihood ratio occurring in the training data:
Revised: Furthermore, we define the maximum likelihood ratio occurring in the training data:

Original: [Equation]
Revised: [Equation]

Original: Using λ LRMax as normalization for λ LR , we can estimate the uncertainty of the prediction via the Log-LikelihoodRatio
Revised: Using λ LRmax as normalization for λ LR , we can estimate the uncertainty of the prediction via the Log-Likelihood

Original:  Score (LLRS):
Revised: Ratio Score (LLRS):

Original: [Equation]
Revised: [Equation]

Original: As with the correlation error, we take the square root to account for the exponential shape of the CWLL, as shown in Figure 5.
Revised: As with the correlation error, we take the square root to account for the exponential shape of the CWLL (see e.g. Figure 5).

Original: Using this technique, a likelihood equally worse as the worst training  likelihood (i.e. ℓ ( Y pred | X ) 
Revised: In this way, a likelihood equally worse as the worst training sample likelihood (i.e. ℓ ( Y pred | X ) =

Original: = min Y , X ∈ D Train ℓ ( Y | X ) ) results inan LLRS of one: LLRS = 1.
Revised: min Y,X ∈ D Train ℓ ( Y | X ) ) results in LLRS = 1 .

Original: Bigger likelihoods (i.e. ℓ ( Y pred | X ) > min Y , X ∈ D Train ℓ ( Y | X ) ) result in scoressmaller one ( LLRS < 1), smaller likelihoods (i.e. ℓ ( Y pred | X ) < min Y , X ∈ D Train ℓ ( Y | X ) ) inlarger scores ( LLRS > 1).
Revised: Larger likelihoods (i.e. ℓ ( Y pred | X ) > min Y,X ∈ D Train ℓ ( Y | X ) ) result in scores LLRS < 1 , smaller likelihoods (i.e. ℓ ( Y pred | X ) < min Y,X ∈ D Train ℓ ( Y | X ) ) in LLRS > 1 .

Original: With z-score normalization, LLRS can be applied to any magnitude of the original data.
Revised: Thanks to z-score normalization, LLRS can be applied without concerns about the magnitude of the original data.

Original: To show its potential, we ran a long-time forecastingexperiment on  the Power dataset over 40 days, performed by
Revised: To evaluate LLRS potential, we run long time range forecasting on both datasets. For the Power dataset, we predict 40 days as long-range prediction, performed by RECOWN trained only for 5 days prediction (“standard prediction”). For the Retail dataset, we predict 16 weeks, performed by

Original: RECOWN trained  for 5 days of prediction.
Revised: RECOWN trained only for 8 weeks prediction.

Original: We choose this setting, since we noticed an increasingly worsening prediction by the SRNN in the long run and wanted to make sure, that this unwanted behaviour is captured by the uncertainty estimated with CWLL.
Revised: Our assumption is that the predictions made by the SRNN are less accurate over time and our aim is to make sure that this undesirable behaviour is captured by the uncertainty estimated with LLRS.

Original: As it can be seen in Figure 2, the  uncertainty grows together with an increasingly divergent prediction.
Revised: As it can be seen in Figure 2, the more the prediction diverges from the ground truth over time, the more the uncertainty grows.

Original: This shows, that the uncertainty  can be a good warning sign for users to detect such cases, even though the ground truth may not be available yet.
Revised: This shows that the uncertainty estimated by LLRS can be a good indicator for users to detect such cases, even when the ground truth might be not available.

Original: Therefore, (Q2) can be answered affirmatively: RECOWNs can estimate the uncertainly of predictions in the time domain.
Revised: Therefore, (Q2) can be answered positively: RECOWNs can estimate the uncertainty of predictions in the time domain and this can support users in decision-making.

Original:  CONCLUSION
Revised:  CONCLUSION

Original: In this paper, we proposed RECOWNs, a novel architecture that jointly trains an SRNN and a probabilistic circuit in order to provide the SRNN with a measure of uncertainty of its predictions.
Revised: In this paper, we proposed RECOWNs, a novel architecture that jointly trains an SRNN and a probabilistic circuit in order to provide the SRNN with a measure of uncertainty of its predictions, on sequence level as well as in the time domain.

Original: This yields a trustworthy model for time series forecasting being able to inform the users when it is not confident of its predictions.
Revised: This yields a trustworthy model for time series forecasting being able to inform the users when it is not confident of its predictions.

Original: We leveraged the conditional likelihood of in- and output together with the Whittle approximation to introduce CWSPNs, which can provide a likelihoodmeasure of the prediction given the context in the spectral domain.
Revised: We leveraged the conditional likelihood of context and predictions together with the Whittle approximation to introduce CWSPNs, which can provide a likelihood of the prediction given the context in the spectral domain.

Original: Furthermore, we introduced LLRS, an effective score to evaluate the uncertaintyinterval for any time point of a prediction.
Revised: Furthermore, we introduced LLRS, an effective score to evaluate the uncertainty for any time point of a prediction.

Original: Our experiments on real-world datasets show that RECOWNs are both accurate and trustworthy.
Revised: Our experiments on real-world datasets show that RECOWNs are both accurate and trustworthy.

Original: In this context, a probabilistic circuit tailored for time series based on the Whittle approximation showed to be superior to MAF, a state-of-the-art density estimator.
Revised: In this context, a probabilistic circuit tailored for time series based on the Whittle approximation showed to be superior to MAF, a state-of-the-art density estimator.

Original: We hope our results will inspire further research on trustworthy models for time series forecasting.
Revised: We hope our results will inspire further research on trustworthy models for time series forecasting.

Original: Future work may extend our contributions in several ways.
Revised: Future work may extend our contributions in several ways.

Original: Since RECOWN can be trained in an end-to-end fashion, we envision more sophisticated strategies for joint training.
Revised: Since RECOWN can be trained in an end-to-end fashion, we envision more sophisticated strategies for joint training.

Original: For example, gradients from the CWSPN could be used to improve the predictions of the neural predictor.
Revised: For example, gradients from the CWSPN could be used to improve the predictions of an SRNN.

Original: Moreover, it could be explored whether RECOWNs are robust against adversarial attacks.
Revised: Moreover, it could be explored whether RECOWNs are robust against adversarial attacks.

Original: Besides, tractableand general inference of CWSPNs could be exploited to gain knowledge about factors of influence for the prediction, e.g., similarly to what done in Yu et al.
Revised: Besides, tractable general inference of CWSPNs could be exploited to gain knowledge about factors of influence for the prediction.

Original: [2021a], by constructing a Bayesian Network using the CWSPN.

Original: In the same direction, modeling the joint distribution of the coefficients instead of the conditional, thus, being able to compute any marginal, could open new opportunities.
Revised: In the same direction, modeling the joint distribution of the coefficients instead of the conditional, thus, being able to compute any marginal, could open new opportunities.

Original: For this scope, employing Einsum Networks [Peharz et al., 2020a], fast and scalable probabilistic circuits, could be an effective solution. 
Revised: For this scope, employing Einsum Networks [Peharz et al., 2020a], fast and scalable probabilistic circuits, could be an effective solution. Acknowledgements

Revised: This work was supported by the Federal Ministry of Education and Research (BMBF; project “MADESI”, FKZ 01IS18043B, and Competence Center for AI and Labour; “kompAKI”, FKZ 02L19C150), the German Science Foundation (DFG, German Research Foundation; GRK 1994/ “AIPHES”), the Hessian Ministry of Higher Education, Research, Science and the Arts (HMWK; projects “The Third

Revised: Wave of AI” and “The Adaptive Mind”), and the Hessian research priority programme LOEWE within the project “WhiteBox”.

Revised: The authors thank German Management Consulting GmbH for supporting this work.