Pooja Gupta
email: poojag@iiitd.ac.in

Angshul Majumdar
email: angshul@iiitd.ac.in

Emilie Chouzenoux
email: emilie.chouzenoux@centralesupelec.fr

Giovanni Chierchia
email: giovanni.chierchia@esiee.fr

SuperDeConFuse: A Supervised Deep Convolutional Transform based Fusion Framework for Financial Trading Systems

Keywords: information fusion, deep learning, convolution, transform learning, stock trading

This work proposes a supervised multi-channel time-series learning framework for financial stock trading. Although many deep learning models have recently been proposed in this domain, most of them treat the stock trading time-series data as 2-D image data, whereas its true nature is 1-D time-series data.

the stock trading systems are multi-channel data, many existing techniques treating them as 1-D time-series data are not suggestive of any technique to effectively fusion the information carried by the multiple channels. To contribute towards both of these shortcomings, we propose an end-to-end supervised learning framework inspired by the previously established (unsupervised) convolution transform learning framework. Our approach consists of processing the data channels through separate 1-D convolution layers, then fusing the outputs with a series of fully-connected layers, and finally applying a softmax classification layer. The peculiarity of our framework, that we call SuperDeConFuse (SDCF), is that we remove the nonlinear activation located between the multi-channel convolution layers and the fully-connected layers, as well as the one located between the latter and the output layer. We compensate for this removal by

Introduction

Financial time series forecasting, and particularly stock price forecasting, requires to determine the future value of a company's stock or any other form of a financial instrument traded on exchange as per the company. It plays a significant role in trading strategies to identify opportunities to buy and sell a stock and this process is known as stock trading. This future movement prediction of stock could capitulate the significant profit.

However, the problem of stock trading has been one of the most difficult problems for the researchers in finance data processing, and speculators. Struggles are mainly due to the uncertainties and noises of the samples. These samples are generated as a consequence of historical market behaviors. But their generation is also affected by other factors such as macroeconomy and investor feelings, hence it is not only dependent on historical information [START_REF] Sezer | Algorithmic financial trading with deep convolutional neural networks: Time series to image conversion approach[END_REF]. Two famous hypotheses emphasize how difficult it is to accurately predict a stock price.

First, the efficient market hypothesis introduced in [START_REF] Fama | Efficient capital markets: A review of theory and empirical work[END_REF] states that the current price of an asset always reflects all previous information available for it instantly.

Second, the random-walk hypothesis [START_REF] Malkiel | A random walk down wall street[END_REF] claims that stock price changes independently from its history. In other words, tomorrow's price will only depend on tomorrow's information regardless of today's price. Hence, automating the prediction of stock trends/movements is a very challenging task.

In past works, feature engineering played a key role in the prediction process. Features were extracted from the original stock data using technical analysis/indicators, which are in general used for analysing the stock market data.

Traditional statistical methods such as linear regression, autoregressive moving average (ARMA), and GARCH, were much beneficial for financial time series forecasting due to their interpretability. These statistical models were thus used on the extracted features, processed using technical indicators related to historical data for future value prediction [START_REF] Shynkevich | Forecasting price movements using technical indicators: investigating the impact of varying input window length[END_REF]. Previous works have also used the extracted features as input to machine learning models like Naive Bayes (NB), Logistic Regression (LR), Random Forest (RF), and k-nearest neighbors (kNN) [START_REF] Patel | Predicting stock and stock price index movement using trend deterministic data preparation and machine learning techniques[END_REF][START_REF] Sen | A robust predictive model for stock price forecasting[END_REF][START_REF] Ballings | Evaluating multiple classifiers for stock price direction prediction[END_REF].

In the last decade, deep learning based models/techniques have gained attention in multiple domains, and financial stock trading is one such domain.

Owing to the success of Convolutional Neural Networks (CNNs), there are previous studies that have used this model for the future prediction of the stock value. In [START_REF] Sezer | Algorithmic financial trading with deep convolutional neural networks: Time series to image conversion approach[END_REF], the features extracted using technical indicators for stock data are fed as a 2-D "image" matrix to the CNN, where each column represents shifted windows of the data. The work [START_REF] Long | Deep learning-based feature engineering for stock price movement prediction[END_REF] utilizes long short-term memory (LSTM), deemed most suitable for time series analysis as they were supposed to mimic memory, and CNN for the stock trading task. However, it is likely that the most natural and thus efficient way to process time-series is to consider its original form as 1-D data rather than a 2-D matrix. It is worth mentioning that, up to our knowledge, despite its multi-channel form, the problem of financial stock trading has been rarely treated as a fusion problem. We can only mention [START_REF] Yang | Deep convolutional neural networks on multichannel time series for human activity recognition[END_REF][START_REF] Yao | Deepsense: A unified deep learning framework for time-series mobile sensing data processing[END_REF] where a fusion framework is proposed, but only at the feature level rather than at the raw level.

In this work, motivated by the success of CNNs, we propose an end-toend supervised fusion framework for multi-channel time-series based financial trading systems that makes use of our recently introduced convolutional trans-form learning (CTL) approach [START_REF] Maggu | Convolutional transform learning[END_REF]. We call this framework -SuperDeConFuse (SDCF). 1 Our framework has the following contributions:

• It is an end-to-end framework that treats the multi-channel time-series stock data as univariate data corresponding to every channel, thus overcoming both the aforementioned issues present in the previous works solving the problem of stock trading.

• It promotes the learning of unique filters and hence a richer set of features, that was not guaranteed with CNNs, due to a "logarithm determinant" penalty applied to the transforms/filters.

• A non-negativity constraint on coefficients/features mitigates the dead neurons issue by removing the nonlinear activation of the fully-connected layers and the last convolution layer.

The remainder of this paper is organized as follows. Section 2 summarizes related works in the field of machine learning and deep learning that have been proposed for solving the stock trading problem/ stock market prediction.

Since our work focuses on a supervised multi-channel fusion framework, we will also review recent machine learning approaches for information fusion. Section 3 introduces the details of our proposed SuperDeConFuse (SDCF) approach, the mathematical tools involved and the training strategy that is retained. Section 4 discusses the considered dataset, data labeling, data preprocessing and the training methodology used. Section 5 provides the experimental results. Finally, Section 6 concludes this work.

Literature Review

Financial stock data analysis

In literature, different methodologies have been applied to the stock data for predicting future trading strategies (eg, buy and sell decisions). These include statistical methods, machine learning algorithms like Support Vector Machine (SVM) and Artificial Neural Networks (ANN), feature extraction approaches, deep learning models (eg, CNN, LSTM), that we briefly review in this section.

Statistical methods are probably the methods among others that are universally used for the prediction of financial stock trading strategies. In particular, many studies rely on the use of sequential statistical models, such as ARMA [START_REF] Kocak | Arma( p,q ) type high order fuzzy time series forecast method based on fuzzy logic relations[END_REF], ARCH [START_REF] Zumbach | Option pricing with realistic arch processes[END_REF], GARCH [START_REF] Lin | Modelling and forecasting the stock market volatility of sse composite index using garch models[END_REF] and [START_REF] Lk | Stock market's response to real output shocks in eastern european frontier markets: A varwal model[END_REF], Kalman filter [START_REF] Bisoi | A hybrid evolutionary dynamic neural network for stock market trend analysis and prediction using unscented kalman filter[END_REF].

Feature-based techniques are also considered as state-of-the-art. Technical indicators like Exponential moving average (EMA), Moving average convergence and divergence (MACD), Williams %R, etc. have been used in past studies to extract the features from the data. Text mining can also be used, to process financial analysis from newspapers [START_REF] Ming | Stock market prediction from wsj: Text mining via sparse matrix factorization[END_REF]. The features are then used as input to machine learning models, for example, SVM, ANN, kNN [START_REF] Shynkevich | Forecasting price movements using technical indicators: investigating the impact of varying input window length[END_REF]. Further studies have proposed hybrid machine learning models, based on the use of multiple types of base classifiers that operate on a common input and a meta classifier that learns from base classifiers' outputs to obtain a more precise stock return and risk predictions. Strategies such as Bagging, Boosting and AdaBoost, can be also applied to create diversity in classifier combinations [START_REF] Barak | Fusion of multiple diverse predictors in stock market[END_REF][START_REF] Weng | Predicting short-term stock prices using ensemble methods and online data sources[END_REF]. For example, a hybrid weighted SVM and weighted KNN model for predicting stock market indices is proposed in [START_REF] Chen | A feature weighted support vector machine and k-nearest neighbor algorithm for stock m[END_REF]. Similarly, a technique that combines Support Vector Regression (SVR), Random Forests and ANNs for predicting stock market index, is introduced in [START_REF] Patel | Predicting stock market index using fusion of machine learning techniques[END_REF]. Another study [START_REF] Ticknor | A bayesian regularized artificial neural network for stock market forecasting[END_REF] combines the statistical and probabilistic Bayesian Learning and the machine learning model ANN for the same. However, in all the aforementioned techniques, the relationship built between historical data and future value prediction may lack interpretation.

Hence, the performance of the methods are directly related to the quality of the features. Moreover, with machine learning techniques, overfitting is a major issue owing to their capability of non-linear mapping and fitting.

Deep learning based models have also been extensively used for solving stock forecasting problems. Recurrent Neural Networks (RNNs) are considered to be the most appropriate models for time-series analysis. LSTM is one such RNN which is regarded as the memory-mimicking model. Some studies use LSTM for the time-series stock forecasting [START_REF] Nelson | Stock market's price movement prediction with lstm neural networks[END_REF]. Another work uses LSTM on the technical indicators for the prediction [START_REF] Tingwei | Improving stock closing price prediction using recurrent neural network and technical indicators[END_REF]. However, despite the great performance obtained, the time complexity of training RNN via backpropagation has encouraged the users for searching for more tractable models and solutions. CNNs constitute another important deep learning model, apart from RNNs, which have been used profusely and have performed well in the stock time-series forecasting, especially 2-D CNNs. In [START_REF] Sezer | Algorithmic financial trading with deep convolutional neural networks: Time series to image conversion approach[END_REF], the said techniques have been used on stock prices for forecasting. A slightly different input is used in [START_REF] Tsantekidis | Forecasting stock prices from the limit order book using convolutional neural networks[END_REF], instead of using the standard variables (opening, closing, high, low and NAV), it uses high frequency data for forecasting major points of inflection in the financial market. In another work [START_REF] Gudelek | A deep learning based stock trading model with 2-d cnn trend detection[END_REF], a similar approach is used for modeling exchange traded fund (ETF). The 2-D CNN model performs similarly as LSTM or the standard multi-layer perceptron [START_REF] Persio | Artificial neural networks architectures for stock price prediction: Comparisons and applications[END_REF][START_REF] Hiransha | Nse stock market prediction using deep-learning models[END_REF] while being simpler to train. This apparent lack of performance improvement may be owing to the incorrect choice of CNN model, since these studies model an inherently 1D time series as an image.

Information Fusion

Many real world domains raise problems pertaining to the need for the fusion of information from multiple sources. Consider the problem of demand forecasting which requires estimating the power consumption at a future point given the available information until the current instant. At the building level forecasting, the inputs are usually power consumption, weather (temperature, humidity), and occupancy. This is a crucial problem in smart grids that ranges from planning electricity generation to preventing non-technical losses. Another area is biomedical signal analysis, for example, the problem of blood pressure estimation. The inputs are usually from two sources, namely the electrocardiogram (ECG) and pulsepleithismogram (PPG) [START_REF] Yoon | Non-constrained blood pressure monitoring using ecg and ppg for personal healthcare[END_REF], and the goal is to estimate the systolic and diastolic pressures. Transportation is also one such domain that needs the fusion of information from many sources to build intelligent trans-portation systems (ITS) [START_REF] Faouzi | Data fusion in intelligent transportation systems: Progress and challenges -a survey[END_REF][START_REF] Saadi | An efficient hierarchical model for multi-source information fusion[END_REF]. This is needed to improve passenger safety, reduced transportation time and fuel consumption, etc.

Image fusion is another area where the information from two or more images of an object has to be integrated into a single image that is more informative and appropriate for visual perception or computer analysis. It finds great application in medical imaging. One can mention for instance the fusion of MRI (Magnetic Resonance Imaging) and PET (Positron Emission Tomography) images using IHS (Intensity Hue Saturation) and RIM (Retina-Inspired Models) fusion methods to improve the functional and spatial information content of the PET images [START_REF] Daneshvar | Mri and pet image fusion by combining ihs and retina-inspired models[END_REF].

Deep learning has been widely used for analyzing multi-channel / multisensor signals. In such studies, all the sensors are stacked one after the other to form a matrix using 2-D CNN further to analyze these signals. For example, [START_REF] Yang | Deep convolutional neural networks on multichannel time series for human activity recognition[END_REF] uses the same explained model to analyze human activity recognition from multiple body sensors. Note that it must be distinguished from the studies mentioned before ( [START_REF] Sezer | Algorithmic financial trading with deep convolutional neural networks: Time series to image conversion approach[END_REF][START_REF] Tsantekidis | Forecasting stock prices from the limit order book using convolutional neural networks[END_REF][START_REF] Gudelek | A deep learning based stock trading model with 2-d cnn trend detection[END_REF][START_REF] Persio | Artificial neural networks architectures for stock price prediction: Comparisons and applications[END_REF][START_REF] Hiransha | Nse stock market prediction using deep-learning models[END_REF]), as the images in [START_REF] Yang | Deep convolutional neural networks on multichannel time series for human activity recognition[END_REF] are not formed from stacking windowed signals from the same signal one after the other, they are formed by stacking signals from different sensors. Note, however, that [START_REF] Yang | Deep convolutional neural networks on multichannel time series for human activity recognition[END_REF] does not account for any temporal modeling. This is rectified in [START_REF] Yao | Deepsense: A unified deep learning framework for time-series mobile sensing data processing[END_REF] where 2-D CNN is used on a time series window. The different windows are finally processed by GRU, thus explicitly incorporating time series modeling. In the aforesaid studies, there is however no explicit fusion framework. The information from raw signals is fused to form matrices and treated by 2-D convolutions. A true fusion framework was proposed in [START_REF] Zheng | Time series classification using multi-channels deep convolutional neural networks[END_REF]. Here, the fusion was happening at the feature level and not in the raw signal level as was in [START_REF] Yang | Deep convolutional neural networks on multichannel time series for human activity recognition[END_REF][START_REF] Yao | Deepsense: A unified deep learning framework for time-series mobile sensing data processing[END_REF].

Multi-modal data processing is another area that makes use of deep learning based fusion techniques. Although this problem is not multi-channel data processing per se, we will briefly review here some studies on this topic. In [START_REF] Ngiam | Multimodal deep learning[END_REF] a fusion scheme is proposed for audio-visual analysis, that uses a fusion scheme for deep belief network (DBN) and stacked autoencoder (SAE) for fusing the audio and video channels. Each of the said channels is processed separately and connected by a fully connected layer to produce fused features. These fused features are further processed for inference. The problem of video based action recognition is addressed in [START_REF] Feichtenhofer | Convolutional two-stream network fusion for video action recognition[END_REF]. It does not require audio data for the task; rather it proposes a fusion scheme for incorporating temporal information (processed by CNN) and spatial information (also processed by CNN). Experiments were carried out with different levels of early and late fusion. The fusion of multi-channel image datasets has also been investigated. In [START_REF] Eitel | Multimodal deep learning for robust rgb-d object recog-nition[END_REF], a fusion scheme is proposed for processing color and depth information (via 3-D and 2-D convolutions, respectively) with the objective of action recognition. In [START_REF] Chen | Deep fusion of remote sensing data for accurate classification[END_REF],

the authors consider fusing hyperspectral data (high spatial resolution) with Lidar (depth information), with the consequence of better classification results.

In [START_REF] Antropova | A deep feature fusion methodology for breast cancer diagnosis demonstrated on three imaging modality datasets[END_REF], it was shown that by fusing deeply learnt features (from CNN) with handcrafted features via a fully connected layer, can improve analysis tasks.

It is worthy to point out that the aforementioned time-series data based fusion studies do not process the time-series data as 1-D but as 2-D image/matrix. In the context of financial time-series, the state-of-the-art methods seem mostly based either on statistical and machine learning models or CNNs. For the former, the relationship built between historical data and future value prediction may lack interpretation; and hence, the performance of the methods is directly related to the quality of the features. While in the latter case of CNNs, there is no guarantee of unique filters learnt. In this work, we propose a novel framework that can tackle those issues.

Proposed Technique

This paper introduces a novel supervised framework for multi-channel data representation learning. A crucial element of the latter is our recently introduced CTL [START_REF] Maggu | Convolutional transform learning[END_REF]. For clarity, we first recall the important steps of the CTL technique.

Then, we propose an extension of this approach in order to handle a multilayer architecture. Finally, we present the overall SuperDeConFuse (SDCF) architecture.

Convolutional Transform Learning

As introduced in our seminal paper [START_REF] Maggu | Convolutional transform learning[END_REF], CTL learns some filters (t m ) 1≤m≤M operated on samples s (k) 1≤k≤K to generate the features (x

(k) m ) 1≤m≤M,1≤k≤K
. The inherent learning model is expressed by convolution operations (assuming suitable padding) defined as

(∀m ∈ {1, . . . , M } , ∀k ∈ {1, . . . , K}) t m * s (k) = x (k) m . (1)
A regularization is imposed on the filters to improve the representation ability and limit the overfitting issues, following from the original study on transform learning [START_REF] Ravishankar | Learning sparsifying transforms[END_REF]. Also, non-negativity constraint is imposed on the features, as it is commonly done in CNNs. The convolutional filters and the representation coefficients are learnt from the data during training. This is expressed as the following optimization problem: minimize

(tm)m,(x (k) m ) m,k 1 2 K k=1 M m=1 t m * s (k) -x (k) m 2 2 + ψ(x (k) m ) + µ M m=1 t m 2 2 -λ log det ([t 1 . . . t M ]), ( 2
)
where ψ is a suitable penalization function, and (µ, λ) are positive hyperparameters. It should be noted that the regularization term promotes unique filters to be learnt, something that is not easy to guarantee in CNNs. We can rewrite equivalently Equation (2) in matrix notation as2

F (T, X) = 1 2 T ⋆ S -X 2 F + Ψ(X) + µ T 2 F -λ log det (T ) , (3)
where 1) . . . t M * s (1) . . . . . . . . .

T = t 1 . . . t M , S = s (1) . . . s (K) ⊤ , X = x (k) 1 . . . x (k) M 1≤k≤K , T ⋆ S =      t 1 * s (
t 1 * s (K) . . . t M * s (K) ,      (4)
and Ψ amounts to applying the penalty term ψ column-wise on X and summing.

A local minimizer to (3) can be reached efficiently using the alternating proximal algorithm [START_REF] Attouch | Convergence of descent methods for semi-algebraic and tame problems: proximal algorithms,forward-backward splitting, and regularized gauss-seidel methods[END_REF][START_REF] Chouzenoux | A block coordinate variable metric forward-backward algorithm[END_REF][START_REF] Bolte | Proximal alternating linearized minimization for nonconvex and non-smooth problems[END_REF], which alternates between proximal updates on variables T and X. The proximity operator [START_REF] Combettes | Proximal splitting methods in signal processing[END_REF] at x ∈ H, with (H, • ) a Hilbert space, of a proper lower-semi-continuous convex function ϕ : H →] -∞, +∞] is defined as

prox ϕ (x) = arg min x∈H ϕ(x) + 1 2 x -x 2 . (5)
Then, the alternating proximal algorithm for CTL reads:

For n = 0, 1, ...     T [n+1] = prox γ1F (•,X [n] ) T [n] X [n+1] = prox γ2F (T [n+1] ,•) X [n] (6)
with initializations T [0] , X [0] of suitable dimensions, and γ 1 , γ 2 some positive constants. For more details on the derivations and the convergence guarantees, the readers can refer to [START_REF] Maggu | Convolutional transform learning[END_REF].

Deep Convolutional Transform Learning

Deep CTL consists in stacking multiple convolutional layers on top of each other to generate the features, as shown in Figure 1. Deep CTL depends on the key property that the solution X to the CTL problem, assuming fixed filters T , can be reformulated as the simple application of an element-wise activation function. That is:

argmin X F (T, X) = Φ(T ⋆ S), (7)
with Φ the proximity operator of Ψ [START_REF] Combettes | Deep neural network structures solving variational inequalities, Set-Valued and Variational Analysis[END_REF]. It is interesting to remark that, if Ψ is the indicator function of the positive orthant, then Φ identifies with the famous rectified linear unit (ReLU) activation function. Many other examples of mapping between proximity operators and activation functions are provided in [START_REF] Combettes | Deep neural network structures solving variational inequalities, Set-Valued and Variational Analysis[END_REF]. Consequently, we propose to compute deep features by stacking many such layers: where we set X 0 = S. Deep CTL consists of solving the problem minimize T1,...,TL,X

(∀ℓ ∈ {1, . . . , L -1}) X ℓ = Φ ℓ (T ℓ ⋆ X ℓ-1 ), (8)
F conv (T 1 , . . . , T L , X | S) (9)
with

F conv (T 1 , . . . , T L , X | S) = 1 2 T L ⋆ Φ L-1 (T L-1 ⋆ . . . Φ 1 (T 1 ⋆ S)) -X 2 F + Ψ(X) + L ℓ=1 (µ||T ℓ || 2 F -λ log det(T ℓ )). (10)
Deep CTL can thus be viewed as a natural and simple extension of the one-layer CTL formulation in (3).

Our Proposed Approach -SuperDeConFuse

We now present our novel approach, SuperDeConFuse (SDCF), which is a supervised fusion framework for multi-channel time-series stock data. This framework takes the channels of input data samples to separate branches of convolutional layers, leading to multiple sets of channel-wise features. The features obtained are thus decoupled. In order to couple (i.e., fuse) them, these are concatenated and passed to a fully-connected layer, which yields a set of unique coupled features via transform learning. These features are then fed to another linear fully-connected layer. This provides features that are finally inputted to the softmax layer that yields the probabilities for the classes. The complete architecture, called SuperDeConFuse (SDCF), is shown in Figure 2.

As the data considered is multi-channel, we learn a different set of convo-

lutional filters T (c) 1 , . . . , T (c)
L and features X (c) for each channel c ∈ {1, . . . , C}. Here P 1 × 1, . . . , P L × 1 represents the kernel size used in each layer ℓ ∈ {1, . . . , L}. Here, maxpooling is not performed after layer 4 due to the small window size/input sequence length.

We also learn the (not convolutional) linear transform T = ( T c ) 1≤c≤C to fuse the channel-wise features X = (X (c) ) 1≤c≤C , along with the corresponding fused features Z at the same time. The latter task is carried out by the cost function

F fusion ( T , Z, X) = 1 2 Z- C c=1 flat(X (c) ) T c 2 F +Ψ(Z)+ C c=1 µ T c 2 F -λ log det( T c ) (11)
where the operator "flat" transforms X (c) into a matrix where each row contains the "flattened" features of a sample.

Further, we learn the weight matrix θ of a multiclass classifier which takes the input features Z and yields the class probabilities. The cross-entropy (CE) loss associated with the final classification is given by

F CE (θ, Z | y) = K k=1 log V v=1 e z ⊤ k (θv-θy k ) , ( 12
)
where V is the number of classes, θ v is the v-th column of matrix θ, z ⊤ k is the k-th row of matrix Z, and y k ∈ {1, . . . , V } is the label of the k-th sample. This finally leads to the joint optimization problem defined as minimize

(T,X, T ,Z,θ) C c=1 F conv (T (c) 1 , . . . , T (c) L , X (c) |S (c) ) + F fusion ( T , Z, X) + F CE (θ, Z | y). J(T,X, T ,Z,θ) (13)
Conclusively, our formulation aims at jointly training the channel-wise convolutional filters T (c) l , the fusion coefficients T , and the multiclass classifier θ in an end-to-end fashion. We explicitly learn the features X and Z subject to the regularization Ψ, so as to avoid the problem of dead neurons. Moreover, the "log-det" regularization on both T (c) l and T breaks the symmetry and enforces the diversity in the learnt transforms, whereas the Frobenius regularization keeps the transform coefficients bounded.

Optimization algorithm

We propose to find a local minimizer to the nonconvex Problem (13) through the projected (sub)gradient descent, whose iterations read:

For n = 0, 1, ...              T [n+1] = T [n] -γ∇ T J(T [n]
, X [n] , T [n] , Z [n] , θ [n] )

X [n+1] = P + (X [n] -γ∇ X J(T [n]
, X [n] , T [n] , Z [n] , θ [n] ))

T [n+1] = T [n] -γ∇ T J(T [n] , X [n] , T [n] , Z [n] , θ [n] ) Z [n+1] = P + (Z [n] -γ∇ Z J(T [n] , X [n] , T [n] , Z [n] , θ [n] )) θ [n+1] = θ [n] -γ∇ θ J(T [n]
, X [n] , T [n] , Z [n] , θ [n] ) [START_REF] Lin | Modelling and forecasting the stock market volatility of sse composite index using garch models[END_REF] with P + = max{•, 0} (applied element-wise). In practice, we initialize it with some random matrices T

[0] , X [0] , T [0] , Z [0] , θ [0]
, we choice a suitable stepsize γ > 0, and we evaluate numerically the gradient step with the accelerated scheme initially introduced for the ADAM method in [START_REF] Kingma | A method for stochastic optimization[END_REF].

There are two remarkable advantages of the proposed optimization approach.

Firstly, we depend on automatic differentiation [START_REF] Paszke | Automatic differentiation in pytorch[END_REF] and stochastic gradient approximations to efficiently solve Problem [START_REF] Zumbach | Option pricing with realistic arch processes[END_REF]. Secondly, any sub-differentiable activation function Φ in [START_REF] Ballings | Evaluating multiple classifiers for stock price direction prediction[END_REF] can be plugged into our model, for instance SELU [START_REF] Klambauer | Self-normalizing neural networks[END_REF] or Leaky ReLU [START_REF] Mass | Rectifier nonlinearities improve neural network acoustic models[END_REF]. This flexibility will play a key role in the performance, as shown in the experimental section.

Computational Complexity of Proposed Framework -SuperDeConFuse(SDCF)

Table 1 summarizes the computational complexity of SuperDeconFuse(SDCF) architecture, both for training and test phases. We report the cost incurred for one input sample, either at an iteration of the training algorithm or at the testing phase. It is to be noted that the computational complexity of SDCF architecture is comparable to that of a standard CNN. The log-det regularization is the only addition that requires to compute the truncated singular value decomposition of T (c) ℓ and T c . However, as the size of these matrices is determined by the filter size, the number of filters, and the number of output features per sample, the training complexity is not worse than that of a CNN.

T (c) ℓ ∈ R P ℓ ×M ℓ 4. Frobenius norm on f.-c. layer O(I 2 C 2 ) flat(X (c) ) ∈ R K×I 5. log-det on conv. layers O(P 2 ℓ M ℓ C) Tc ∈ R I×O 6. log-det on f.-c. layer O(I 3 C 2 ) Z ∈ R K×O 7. output layer (classifier) O(V ) θ ∈ R O×V Test Step 1 + Step 2 + Step

Labeling

After curating the dataset for 15 stocks with values for the features -date, symbol, adjusted (adj.) close price, opening price, low price, high price, and net asset value, we have labeled the data. We will call the adj. close price as Close Price in the rest of the paper. In the labeling phase, we manually assign the labels to the daily close prices as Buy (0), Hold (1), Sell [START_REF] Fama | Efficient capital markets: A review of theory and empirical work[END_REF]. The labels are determined by performing a grid search on the list of holding percentages to identify the percentage change for which the stocks should be held to maximize the annualized returns for the company. Algorithm 1 gives the details of the labeling process.

if change > X[k] then 6: if CP [n + 1] > CP [n] then 7:
label == "Sell" AR.append(ar)

Training Details

We use the sliding walk forward validation technique which is used as the cross-validation technique in case of time-series data also shown in Figure 3.

As can be seen from Figure 3

Experimental Evaluation

We carry out experiments on the real world problem of stock trading. Stock trading is a classification problem, where the decision whether to buy or hold or sell a stock has to be taken at each time. The problem makes a decision that if the price of a stock at a later date is expected to increase, the stock must be bought; and if the stock price is expected to go down, the stock must be sold; and if there is no change in the price then it should be held, i.e., do nothing until the price increases. This is done in a way so as to maximize the annualized returns from the stock for the company's profit as mentioned in the labeling process.

We use the five raw inputs for both the tasks, namely open price, close price, high, low and net asset value (NAV). We chose to stay with the raw values.

However, one could compute technical indicators based on the raw inputs [START_REF] Sezer | Algorithmic financial trading with deep convolutional neural networks: Time series to image conversion approach[END_REF] but raw values allow here to keep up with the essence of the true nature of representation learning. Each of the five inputs is processed by a separate 1D processing pipeline. Each of these pipelines produces a flattened output (Figure 2). These flattened outputs are then concatenated and fed for fusion into the Transform Learning layer acting as the fully connected layer (Figure 2). Further, this is connected to another linear fully connected layer and finally, there is a softmax function. The softmax function gives the classification output which consists of the class probabilities for the three classes (BUY, HOLD and SELL).

We extend the architecture by adding CTL layers to 4 layers deep SDCF architectures. The details for all the four architectures are briefed in Table 2.

Maxpooling halves the input sequence length/window size/Time Steps with its every operation. Thus, after 3 layers, the size is reduced to the value that it cannot be employed after the 4 th CTL layer; and, hence, the architecture with 4 CTL layers of SDCF will not have maxpooling operation after layer 4. This is due to the small window size. Also, for making predictions on any day, the past 10 days will be analysed through the model which are labeled as Time Steps

shown in Figure 2. Additionally, to avoid the data leak, we do not predict the stock trading signal for the first 10 days of every test year. The predictions from every year totaling to 11 years will be saved and further, the metrics will be computed to analyse the performance of our model. We will compute two sets of metrics here, namely (i) classification metrics and (ii) financial metrics.

(i) Classification Metrics -This set of metrics includes class-wise F1 score, Precision and Recall to assess the performance from a classification point of view. We also calculate the weighted F1 Score, Precision and Recall to account for the class imbalance for every stock. Note that, in such case, the F1 score is not equivalent to the harmonic mean of Precision and Recall since it is weighted.

(ii) Financial Metrics -We also evaluate the performance of our framework and state-of-the-art from the financial point of view. We calculate, in specific, the Annualized Returns(AR) which is calculated using the predictions from all the models. The AR value will be calculated as mentioned in [START_REF] Sezer | Algorithmic financial trading with deep convolutional neural networks: Time series to image conversion approach[END_REF].

The starting capital will be Rs 10,00,00,000.0 and transaction charges will be Rs 10. We will use Indian currency to calculate the AR values since we have used all the Indian stocks. Note, however, that our metric is versatile and could be used to evaluate the model in any currency depending on the stocks analysed. We compare with three state-of-the-art time series based analysis models, out of which two techniques present the models proposed specifically for financial stock trading -CNN-TA [START_REF] Sezer | Algorithmic financial trading with deep convolutional neural networks: Time series to image conversion approach[END_REF] and MFNN [START_REF] Long | Deep learning-based feature engineering for stock price movement prediction[END_REF]; and the last technique presents a generic model for time-series based data -FCN(Fully Convolutional Network) [START_REF] Wang | Time series classification from scratch with deep neural networks: A strong baseline[END_REF]. The latter is used as it helps understand how generic the proposed model is if compared against both specific stock trading based and general time-series models. In all the techniques, processing pipelines are based on CNN. Other than CNN, MFNN [START_REF] Long | Deep learning-based feature engineering for stock price movement prediction[END_REF] is also based on the RNN type of network -LSTM. In [START_REF] Sezer | Algorithmic financial trading with deep convolutional neural networks: Time series to image conversion approach[END_REF],

                   layer1 : 1D Conv(1, 8, 3, 1, 1) 1 SELU + Maxpool(2, 2) 2 layer2 : 1D Conv(8, 16, 3, 1, 1) 1 Maxpool(2, 2) 2 layer3 : Fully Connected (TL) 3 layer4 : Fully Connected (Linear) Softmax SDCF 3L 5 ×                                  layer1 : 1D Conv(1, 4, 11, 1, 5) 1 SELU + Maxpool(2, 2) 2 layer2 : 1D Conv(4, 8, 7, 1, 3) 1 SELU + Maxpool(2, 2) 2 layer3 : 1D Conv(8, 16, 3, 1, 1) 1 Maxpool(2, 2) 2 layer4 : Fully Connected (TL) 3 layer5 : Fully Connected (Linear) Softmax 20 SDCF 4L 5 ×                                        layer1 : 1D Conv(1, 4,
the data is not used raw but processed as technical indicator values and passed as an image, hence uses 2D CNN whereas, in FCN [START_REF] Wang | Time series classification from scratch with deep neural networks: A strong baseline[END_REF], the data is processed via 2D CNN. The same hyperparameters for the benchmark techniques are used as given in the study except for FCN which is best tuned for our data. We have also compared our model to the simple CNN with the architecture same as that of our framework i.e. 3 convolutional layers deep architecture and used the same hyperparameters too except the kernel sizes of P 1 = 11, P 2 = 9 and P 3 = 7 for the convolutional layers ℓ = 1, 2 and 3 (padding size is P ℓ /2). The difference lies in the objective function of the convolutional learning in both the techniques i.e. our 3 layers deep SDCF and 3 layers deep simple 1D CNN. This is done to analyze the performance difference between the two supervised learning techniques. Additionally, we chose the architecture for CNN having 3 convolutional layers, since the results depleted after 3 convolutional layers for our framework and were best with 3 layers.

Classification Analysis

As mentioned previously, we first look at the Classification performance of our models. We test the framework for shallow -1 CTL layer and deeper versions -2, 3 and 4 CTL layers. The generated features from the fully connected layers are passed to the softmax and we get the probabilities for all the classes.

The one with the maximum probability is selected as the predicted label. The performance is calculated for every class. Specifically, metrics -F1 As we can observe, the performance for HOLD class decrease when increasing the number of layers for our model. However, we can also see that there is an increase in correct identification for BUY and SELL points despite the fact that BUY and SELL points appear extremely less in case of every stock as compared to HOLD points. The latter identification capacity is actually more crucial for the financial system as it directly influences the financial gains or loss. Moreover, the overall individual class performance indicate that the model captures all three classes i.e. BUY, HOLD and SELL well. This is also indicated in the confusion matrices, given for each of the shallow and deeper versions of our framework in Figure 4. With an increase in layers, the model starts to more correctly identify the BUY and SELL points. The HOLD signal has more false positives with shallow architecture (SDCF 1L) that decreases with the increase in layer number, which is important for the system in order to correctly classify other class points. Additionally, the overall performance with our model is better than the CNN. To better analyze the framework performance, we calculate the weighted F1 score, precision and recall metric values for all the stocks under consideration.

We calculate the weighted values to incorporate the class imbalance for every stock. The detailed and summary results are given in Table A.11 in Appendix A and Table 6. Again, the results comprise two sets of values marked in bold or red with the same err dif of 0.02. There are 6, 9, and 5 stocks with respect to the metrics F1 score, precision and recall for which the model performs greater than or equal to the performance given by the state-of-the-arts. Also, there are 6, 3 and 6 stocks for the metrics F1 score, precision and recall respectively for which the model gives the next best performance under 0.02 err dif. Although the BUY and SELL classes performance with the 4 CTL Layers deep architecture is better than the benchmarks compared against, but the overall performance from the average weighted metric is suggestive of the good performance with the 3 layers deep architecture classification wisely. This is also suggested from the financial results explained later.

Again analyzing explicitly for CNN, we have 4, 2 and 7 stocks with greater than or equal performance; and 3, 2 and 3 stocks under similar/next best performance for the metrics F1 score, precision and recall respectively. As can be referenced from the statistics presented here, our model is giving better results with greater than or equal and the next best/similar performances except for the number of stocks for recall metric are slightly more with CNN under greater than or equal to performance. However, the next best performance statistic for the recall metric is much better than CNN. Overall performance on an average is good with our proposed model as compared to the benchmarks and CNN which can be also referred from Table 6. For a deeper understanding of the aforementioned statistics, please refer to Table C.13 in Appendix C. With respect to CNN, there are only 2 stocks for which CNN performs better than any benchmarks and our proposed models, and 3 stocks for which it gives an equal performance. Thus, from the combined (greater than or equal to and next best / similar), average and the financial results, the CNN results are less performant than our model. This also indicates that the quality of predictions made with our model is better than CNN as the identified class labels give AR values quite close to the True AR values. This remains true for all the benchmarks. The statistics presented here can be deduced from Table C.13 in Appendix C for complete understanding. To further understand the better supervised learning for both regular CNN and our SDCF framework, we visualize channel-wise X c features for both the frameworks which are obtained after the last maxpool layer for the 3 convolutional layers deep framework. The following Figure 5 shows the visualizations of the features for one sample of the stock 'BSELINFRA.BO'.

Features generated by the proposed SDCF network.

Features generated by a standard CNN with a similar architecture. As can be seen from Figure 5, heatmap for each channel corresponding to the prices(Close, Open, High and Low) show no variation in the case of CNN as compared to the SDCF architecture. While it shows some variations for the features learnt corresponding to NAV, however, the features are still better learnt with SDCF. Also, darker the color in the heatmap, more it is indicative of the larger negative exponent values. In the case of CNN, hence, the values are very very small that are almost diminishing to zero. This also corroborates the fact that the filters learnt with our model are distinct due to the "log-det" term added which further gives different features with very less redundancy. Thus, the visualizations of these channel-wise features are also supportive of better supervised training with our framework than CNN.

Overall, our model performs better than benchmarks and CNN both classificationwise and financially, specifically, it gives the best performance with 3 CTL layers deep SDCF framework of all the 4 SDCF architectures. We also display the empirical convergence plots for a few stocks, namely INDRAMEDCO.BO and NATIONALUM.BO in Figure 6 for both shallow and deeper versions. We can see that the training loss decreases to a point of stability for each example considered.

Conclusion

In this work, we propose SDCF, a deep fusion end-to-end framework for the processing of stock trading data. Unlike other deep learning models, our framework is a fusion supervised framework. It relies on a novel deep version of our recently proposed CTL model. We have applied the proposed model for stock trading leading to very good performance. In particular, the classification results are better with the proposed SDCF model, than with the 1-D CNN approach. Also, the features X c visualized for each channel and each method indicate the better feature learning with SDCF. The results show that the proposed solution is superior to CNN and other state-of-the arts techniques in this problem.

We believe that the framework is generic enough to handle other multichannel fusion problems as well. In the future, we plan to extend the application to other fusion 1-D as well as 2-D multi-channel problems to test its generality.

For example, we plan to implement it in the biomedical domain to analyse PPG and ECG signals for the blood pressure estimation pertaining to the 1-D multichannel problem. In case of 2-D problems, we would like to do multi-spectral image classification using this technique. Currently, the shortcoming with our model is that it takes slightly more time than CNN, for its training. Thus, we will investigate on the reduction of the time complexity of our framework in order to make it more efficient from this viewpoint.

Acknowledgement

This work was supported by the CNRS-CEFIPRA project under grant NextGenBP PRC2017.

Figure 1 :

 1 Figure 1: Deep CTL architecture for L = 2 layers.

Figure 2 :

 2 Figure 2: SuperDeConfuse Architecture. The architecture is tested for L = 1, 2, 3, 4 layers.

7 .

 7 See above. D = input sample size -K = num. of samples -C = num. of channels -L = num. of layers P ℓ = filter size at layer ℓ -M ℓ = num. of filters at layer ℓ -D ℓ = output sample size at layer ℓ I = DLML is the num. of output features per sample and per channel at last convolution layer O = αIC (with α ∈]0, 1]) is the num. of output features per sample at the fully-connected layer V = num. of classes

Algorithm 1

 1 Labelling Method Input : CP -Array of Closing Prices, S -stock/symbol Parameter : X -array of K holding percentages, NUMDAYS -number of days for the current symbol or len(CP) Labels -2D array of size K x NUMDAYS Output : FinalLabels -Labelled Dataset for S 1: AR = [ ] //it is of size K 2: for k = 0, 1, 2, . . . , K -1 do 3: for n = 0, . . . , N U M DAY S -1 do 4: change = abs((CP [n + 1] -CP [n]/CP [n]) * 100) //where CP[n+1] is the next day closing price 5:

18: end for 19 :

 19 maxAr = Max(AR), maxIndex = index(Max(AR)) 20: HoldPercentage = X[maxIndex] 21: FinalLabels = Labels[maxIndex] 22: return FinalLabels 23: Repeat all steps till 22 for all the Stocks/Symbols in the dataset.

Figure 3 :

 3 Figure 3: Sliding walk-forward validation technique used for hyperparameters tuning

Figure 4 :

 4 Figure 4: Confusion matrices corresponding to the different number of CTL layers of the architecture: a) 1 layer of CTL (shallow version), b) 2 layers of CTL (deep version), c) 3 layers of CTL (deep version) and d) 4 layers of CTL (deep version) where 0 -BUY, 1 -HOLD, 2 -SELL signals.

X 5 NAVFigure 5 :

 55 Figure 5: Visualization of channel-wise features Xc for SDCF versus a standard CNN, for one sample of stock BSELINFRA.BO (with 16x1 as the shape of the features obtained and resized to 8x2 for better visualization)

Figure 6 :

 6 Figure 6: Evolution of the loss during training for few stock examples, of our proposed model with (a) CTL 1 layer, (b) CTL 2 layers, (c) CTL 3 layers and (d) CTL 4 layers.

Table 1 :

 1 Time complexity in training and test phases (for one input sample)

	Phase	Steps	Time	Dimension
			Complexity Description
	Training 1. Convolution layers	O(P

ℓ D ℓ M ℓ C) 2. Fully-connected (f.-c.) layer O(I 2 C 2 ) S (c) ∈ R K×D 3.

Frobenius norm on conv. layers O(P ℓ M ℓ C)

  , we use 10 years of data for training and the subsequent 1 year data for testing, i.e., the stock data from 1998-2007 is for training and the year 2008 for testing. Then we slide the training window by 1 year which implies that we next train it from 1999-2008 and test it on the following year 2009 data and this period is called as horizon. In general, we

train for 10 years, test it for the following year and then slide it by a 1 year horizon and again train and test and so on till the year 2018. Thus, 11 years of data from 2008 -2018 are used as test data. This way, we have 11 models and we select the set of hyperparameters that give the best results across all the 11 models. The set of hyperparameters that we tune includes µ, λ, kernel sizes, number of filters/kernels, learning rate, weight decay of the Adam optimizer, batch size, and number of epochs. Additionally, we randomly initialize the weights for each stock's training. This appears here as a very efficient technique to analyse the robustness of the architecture. In other words, we calculate the model performance every time a year's data becomes available for testing and we use previous 1 year test data for training. We standardize the training and the test data using Normalizer from Python library as prices and the NAV features/channels have a varied range of values.

Table 2 :

 2 Description of compared models

	Method	Architecture Description	Other Parameters
	SDCF 1L	5 × layer2 : Fully Connected (TL) 3   layer1 : 1D Conv(1, 16, 3, 1, 1) 1   Maxpool(2, 2) 2 

  13, 1, 6) 1

	SELU + Maxpool(2, 2) 2
	layer2 : 1D Conv(4, 8, 11, 1, 5) 1
	SELU + Maxpool(2, 2) 2
	layer3 : 1D Conv(8, 16, 9, 1, 4) 1
	SELU + Maxpool(2, 2) 2
	layer4 : 1D Conv(8, 16, 5, 1, 2) 1
	layer5 : Fully Connected (TL) 3
	layer6 : Fully Connected (Linear)
	Softmax

1

(in˙planes, out˙planes, kernel size, stride, padding)

2 (kernel size, stride) 3 TL -Transform Learning L -#CTL layers

Table 3 :

 3 Summary of BUY Class Classification Results for Stock Trading

	Method	Avg. BUY	Avg. BUY	Avg. BUY
		F1 Score	Precision	Recall
	SDCF 1L	0.0645	0.2182	0.0475
	SDCF 2L	0.0916	0.2356	0.0683
	SDCF 3L	0.1091	0.2205	0.0854
	SDCF 4L	0.1566	0.3242	0.1355
	CNN	0.0688	0.1179	0.0551
	FCN	0.0758	0.1446	0.0617
	CNN-TA	0.1205	0.1611	0.1263
	MFNN	0.0881	0.1672	0.2401

Table 4 :

 4 Summary of HOLD Class Classification Results for Stock Trading

	Method	Avg. HOLD	Avg. HOLD	Avg. HOLD
		F1 Score	Precision	Recall
	SDCF 1L	0.7983	0.7091	0.9446
	SDCF 2L	0.7912	0.7113	0.9164
	SDCF 3L	0.7813	0.7113	0.8842
	SDCF 4L	0.6684	0.5950	0.7960
	CNN	0.7909	0.7090	0.9239
	FCN	0.7825	0.7119	0.9051
	CNN-TA	0.7686	0.7142	0.8557
	MFNN	0.5161	0.6425	0.5718

Table 5 :

 5 Summary of SELL Class Classification Results for Stock Trading

	Method	Avg. SELL	Avg. SELL	Avg. SELL
		F1 Score	Precision	Recall
	SDCF 1L	0.0423	0.1778	0.0285
	SDCF 2L	0.0650	0.1752	0.0503
	SDCF 3L	0.0759	0.1574	0.0635
	SDCF 4L	0.1410	0.2139	0.1250
	CNN	0.0481	0.0946	0.0379
	FCN	0.0742	0.1658	0.0802
	CNN-TA	0.0679	0.1768	0.0487
	MFNN	0.0633	0.1034	0.1734
	The summary results for individual classes corresponding to every metric are
	given in Tables 3, 4, 5 above. The average metric values for which the model
	gives the best performance are average F1 score and precision for BUY class,
	average F1 score and recall for HOLD class, average F1 score and precision for

SELL class; where F1 score being important metric, as it is the harmonic mean of precision and recall, is the best with our model for all three classes.

Table 6 :

 6 Summary of Weighted Classification Results for Stock TradingIt is very important to analyse the performance from a financial perspective to understand the quality of predictions made by our model. For this, as explained earlier, we have calculated the AR values with the predictions generated by each of the techniques for every stock over 11 years. We also calculate the AR values with the True labels for every stock over the same period. Finally, we calculate the absolute difference/error between the AR values from Predictions and the AR values from True labels. We average the absolute difference values for all stocks yielding the so-called Mean Absolute Error. The detailed results are given in TableB.12. With our proposed model 5 stocks have the best performance whereas with CNN-TA there is 1 stock and 2 stocks under MFNN

	Method	Avg.	Avg.	Avg.
		F1 Score	Precision	Recall
	SDCF 1L	0.6169	0.6216	0.6941
	SDCF 2L	0.6229	0.6207	0.6867
	SDCF 3L	0.6250	0.6146	0.6784
	SDCF 4L	0.5345	0.5464	0.5890
	CNN	0.6182	0.5907	0.6898
	FCN	0.6090	0.6079	0.6725
	CNN-TA	0.6148	0.6161	0.6575
	MFNN	0.4162	0.5509	0.4676

and FCN. On the whole, the performance is good with our proposed model as also evident from the summary results in Table

7

where we have a mean of the absolute difference/error(MAE) between the True AR and Predicted AR. Also, there are 3 stocks for which the proposed model gives an equal performance as the other benchmark techniques. Here, this set of results is illustrating that, despite the higher capability of identifying the BUY and SELL points with 4 layers deep CTL, the AR values are better predicted with the 3 layers deep CTL framework.

Table 7 :

 7 Summary of Financial Results for Stock Trading

	Method	MAE AR
	SDCF 1L	22.5613
	SDCF 2L	20.7227
	SDCF 3L	20.5067
	SDCF 4L	22.8287
	CNN	21.1140
	FCN	23.7720
	CNN-TA	22.1380
	MFNN	22.3040

https://github.com/pooja290992/SuperDeConFuse.git

Note that T is not necessarily a square matrix. By an abuse of notation, we define the "log-det" of a rectangular matrix as the sum of logarithms of its singular values, taking infinity value as soon as one of those is non positive.

Appendix A. Class-Wise Classification Results for Stock Trading

This section displays all the tables with the Classification Metrics results, both class-wise and weighted, for stock trading.


R Avigo

O Wieland
email: oliver.wieland@mi.infn.it

A Bracco

F Camera

F Ameil

T Arici

A Ataç

D Barrientos

D Bazzacco

P Bednarczyk

G Benzoni

B Birkenbach

N Blasi

H C Boston

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S Brambilla

B Bruyneel

M Ciemała

E Clément

M L Cortés

F C L Crespi

D M Cullen

D Curien

F Didierjean

C Domingo-Pardo

G Duchêne

J Eberth

A Görgen

A Gadea

J Gerl

N Goel

P Golubev

V González

M Górska

A Gottardo

E Gregor

G Guastalla

T Habermann

L J Harkness-Brennan

A Jungclaus

M Kmiecik

I Kojouharov

W Korten

N Kurz

M Labiche

N Lalović

S Leoni

M Lettmann

A Maj

R Menegazzo

D Mengoni

E Merchan

B Million

A I Morales

D R Napoli

C Nociforo

J Nyberg

N Pietralla

S Pietri

Zs Podolyák

V Yu Ponomarev

A Pullia

B Quintana

G Rainovski

D Ralet

F Recchia

M Reese

P Regan

P Reiter

S Riboldi

D Rudolph

M.D Salsac

E Sanchis Aa

L G Sarmiento

H Schaffner

J Simpson

O Stezowski Ab

J J Valiente-Dobón

H J Wollersheim

Low-lying electric dipole γ-continuum for the unstable 62,64 Fe nuclei: strength evolution with neutron number

Keywords: 64 Fe, 62 Fe, nuclear structure, dipole excitation around neutron threshold

The γ-ray emission from the nuclei 62,64 Fe following Coulomb excitation at bombarding energy of 400-440 AMeV was measured with special focus on E1 transitions in the energy region 4-8 MeV. The unstable neutron-rich nuclei 62,64 Fe were produced at the FAIR-GSI laboratories and selected with the FRS spectrometer. The γ decay was detected with AGATA. From the measured γ-ray spectra the summed E1 strength is extracted and compared to microscopic quasi-particle phonon model calculations. The trend of the E1 strength with increasing neutron number is found to be fairly well reproduced with calculations that assume a rather complex structure of the 1 -states (three-phonon states) inducing a strong fragmentation of the E1 nuclear response below the neutron binding energy.

The interaction of the atomic nucleus with an electromagnetic field is a topic largely investigated both experimentally and theoretically. It provides useful and selective information on the nuclear structure. Among the possible excitation modes the electric dipole (E1) type is of particular interest. It is generally dominated by a strong, collective isovector nuclear vibration, the isovector giant dipole resonance (IVGDR) [1,[START_REF] Harakeh | Giant Resonances[END_REF]. However, the simple dipole oscillation of nucleons giving rise to the IVGDR mode is not the only focus of the investigation. Several studies concentrate on the details of the low-energy part of the electric dipole response, i.e. the so called pygmy resonance [START_REF] Bracco | [END_REF]. This is because, it provides relevant information, crucial for testing predictions related to the nuclear equation of state (see [4] and [5] and references therein), and additionally supplies inputs for calculations of the r-process nucleosynthesis [6] and astrophysics [7]. The r-process chain for fast neutron capture involves stable Fe isotopes as seed and goes through 62 Fe and 64 Fe towards the drip-line .But only little is known about their low-lying E1 strength, an important ingredient for this process. Model predictions suggest an increase of E1 strength with neutron number and a sizable difference between stable and unstable isotopes (see supplementary material for this manuscript). It is thus important to provide an experimental ground to test theory far from stability also in view of extending the calculations to the uncharted region of even more neutron rich nuclei. In connection with the present situation on the study of the pygmy resonance, it is important to stress that the main effort is concentrated in understanding the nature of the excitation mode. For this purpose, the investigations are moving in two directions: one is the study of selected nuclei with different probes, the so called multi-messenger approach, the other is to select one probe and investigate various isotopes of the same element, with the ambitious goal to reach neutron-rich nuclei far from stability. These concepts, together with many existing results on the pygmy resonance, are discussed in Refs. [1,[START_REF] Bracco | [END_REF]8,9,5]. Concerning the approach grounded on the use of different probes to populate low-lying 1 -states, the existing extensive work concerns mostly stable nuclei [10,11,12,13,14,15] for which the used reactions are the electromagnetic (γ, γ ) and nuclear inelastic scattering of p, α, C and O particles, at energies for which the interaction has a dominant isoscaler component. In all cases, for each given nucleus comparisons were made of the different excitation cross sections. The overar-ching picture is that up to approximately an excitation energy between 6-7 MeV the pygmy states have a large isoscalar component. Within the study of the evolution with neutron number of the dipole strength at energies below the neutron binding energy in stable nuclei, the cases of Sn and Xe were measured, as these elements have the longest chains of isotopes. The results for Sn and Xe isotopes show a marked increase of E1 strength and of its fragmentation with increasing neutron number (see [16,17,18] and references therein), which is found to be in rather good agreement with theory [19,20,21,22]. Several calculations exist suggesting that the isovector dipole response has a component shifted to low energy with an enhancement below the binding energy, becoming more and more pronounced by going further and further away from stability (see [5] and references therein). It is clear that the way to test this effect is to extend the investigations for the electromagnetic excitation to nuclei produced as radioactive beams. This can only be done by using virtual photons from the Coulomb excitation with a high Z target. Up to now there are only few measurements of the low-lying E1 strength in unstable nuclei. They concern the nuclei of 20,22 O, 26 Ne, 68,70 Ni and 130,132 Sn [1,15,23,24,25,26,27,28,29,30]. In 68 Ni and 130,132 Sn nuclei, low-lying E1 strength was found above the neutron separation energy at 9-11 MeV. In contrast, in the more recent work on 70 Ni [23], it was possible to infer the presence of strength above and also below the particle binding energy, covering the region 6-11 MeV. The present work concerns the nuclei 62,64 Fe. It was motivated by the search of E1 strength in nuclei far from stability below the neutron binding energy, an energy region which is particularly interesting for the modeling of the r-process where the E1 decay is competing with neutron capture. In addition, this choice was made to study the evolution of the strength over four isotopes since data for the two stable nuclei 54,56 Fe obtained with real photons are available. The experiment was performed in FAIR-GSI laboratories using 62,64 Fe beams at 400-440 AMeV impinging on a 1 g/cm 2 thick Au target. At this bombarding energy the cross section for Coulomb excitation is larger than 10 mb in the region of interest. The radioactive 62,64 Fe beams were produced by fragmentation of a primary 86 Kr beam delivered by the SIS-18 heavy ion synchrotron at 700 and 900 AMeV and focused on a natural 2.5 g/cm 2 thick Be target. The ions of interest were selected and separated in mass and charge with the in flight Fragment Separator (FRS) [31] and transported to the target area. The settings of the FRS were chosen to accept secondary fragments with a magnetic rigidity corresponding to a specific mass-over-charge ratio. This provided a beam cocktail containing 62,64 Fe in a large fraction. The different nuclei contained in the secondary beam were identified uniquely, according to their nuclear charge and mass number on an event-byevent basis. In particular, the incoming isotopes of interest were selected using reconstructed identification scatter plots, namely mass over ion charge (AoQ) versus atomic number (Z). The primary beam intensity of 86 Kr was between 8×10 8 and 8×10 9 pps, the count rate after the production target and the first stage of selection was between 1 to 5×10 5 pps and the total number of event-by-event traced and registered Fe isotopes up to the outgoing particle and beam identification at the end of the beamline was 1.4×10 5 for 64 Fe and 1.8×10 6 for 62 Fe. The corresponding identification plots are shown in the top part of Figure 1, panel a) for 62 Fe and panel b) for 64 Fe . The two beams of Fe isotopes, 62 Fe and 64 Fe, are clearly separated from the other isotopes. The outgoing ions, after interaction with the secondary Au target nuclei, were identified and selected using four pieces of information: energy loss, total energy, time of flight and deflection angle. All these quantities were measured with a special calorimetric detector system [32,33,34], called LYCCA (Lund-York-Cologne CAlorimeter) and placed behind the secondary target covering angles around zero degree and a system of fast scintillators. In the present case, particle multiplicity of one and very small deflection angles of 8±1 mrad, far below the grazing angle, were chosen in order to assure Coulomb excitation to be the main interaction mechanism. This corresponds to values of the impact parameter larger than 17 fm, at which the nuclear contribution of the dipole excitation cross section is strongly suppressed (see [23] and ref. therein). For the work presented in this paper, the γ rays were measured with AGATA (Advanced GAmma Tracking Array) [35] placed around the Au target. AGATA was installed in the PreSPEC (Pre SPECtroscopy) configuration [32,36,37,38,39,40] and consisted in 5 triple and 3 double clusters of segmented HPGe (High Purity Germanium) detectors mounted in forward angles between 15 to 60 degree (with an add back efficiency of 3% at 1 MeV determined with source calibration runs and validated with GEANT simulations). Energies up to 20 MeV were measured with the segmented HPGe detectors of AGATA which are equipped with digital electronics allowing pulse shape analysis (PSA) of the signals. This leads to a very precise information on the γ ray interaction position (σ ≈ 2 mm), which is crucial if one wants to minimize the Doppler broadening of the measured spectral lines. In addition, the γ-ray tracking allows to increase the peak-to-total ratio in the spectrum [41] and to lower the background. This is especially important at lower energies where a high atomic background is present. For what concerns high energy γ rays, the response of the AGATA was validated in ref. [42]. The γ rays detected with AGATA were measured in coincidence with incident and outgoing ions, whose position information was used to apply the Doppler correction. In this case γ rays were emitted from projectiles moving with a velocity up to 72% of the speed of light. The γ-ray spectra discussed below are all corrected for Doppler shift using the velocity and direction of the scattered beam particles with respect to the first interaction point of the γ ray, which is for energies higher than some hundreds of keV in general the one with the highest deposited energy. In addition to the pulse shape analysis, the add-back technique was used [42]. For the energies up to 1 MeV, the full tracking technique, suited for this energy region, was also exploited in order to better reconstruct the low-energy γ transitions. The γ rays from the first 2 + states in 62,64 Fe at energy of 0.746 and 0.877 MeV, respectively, are visible. The measured spectra in AGATA is shown up to 3 MeV in Fig. 1 c) and e). The inset in each panel displays the region around the first 2 + →0 + peak for 64 Fe the 600-800 keV and for 62 Fe the 800-1000 keV region. Since the energy and the E2 strength of the 2 + states of 62,64 Fe are well known (see [43] and ref. therein) their B(E2) values were used to calculate the population cross section via distorted wave Born approximation (DWBA) and relativistic coupled-channel (CC) approaches [44,45]. Using a Monte Carlo (GEANT4) code [46,47], the γray interaction distribution inside AGATA was simulated and analyzed with the same procedure employed in the experimental data analysis. An atomic X-ray background at low energies was also included. Peak structures before and after the 2 + →0 + peak of interest were included in the evaluation as background. They do not influence the result as Compton events are strongly suppressed by the very effective tracking. All uncertainties like beam straggling, spreading in the target, detector resolution, dead layers, time and space limitations and losses were included in the simulation made to deduce the measured counts for the first excited state. The obtained intensity of the simulated 2 + →0 + decays of the two 62,64 Fe isotopes was found to match the best reported value in the literature only after an additionally correction of 20% (similar to the findings in [48,49]). In this way, the known B(E2) values were used as a reference to extract new information on E1 transitions from higher lying states in the same nucleus. The γ-ray spectra measured at energy larger than 3 MeV are shown in panels d) and f) for the 64 Fe and 62 Fe nuclei, respectively. The binning was chosen to be consistent with the obtained statistics. This part of the spectra below the neutron threshold and the IVGDR, is the region of interest in which the search for E1 transitions, which could originate from 1 -states, is done. The detectors were calibrated in this region with 88 Y, PuC and AmBe(Ni) γ-ray sources emmitting γ rays of 1.836, 4.438, 6.13 and 9. MeV. The measured spectra shown in Fig. 1 d) and f) have a quasi-continuum fea-ture, with some peak structures in the region 4-6 MeV. Few counts are found up to 7.5 MeV and the statistics is not sufficient to detect γ decay events above the neutron separation energy, which is 7.4 MeV in 64 Fe and 8.0 MeV in 62 Fe.

To interpret the peak structures in terms of the detection features, Monte Carlo simulations were made to be compared with the spectral shape. This comparison is shown in the insets in Figure 1 d) and f) for calculated peaks with centroids at 4.4 and 5 MeV for 64 Fe and at 4.5 MeV for 62 Fe, all with a width of ≈200 keV. The simulation includes Doppler broadening, target energy broadening and straggling of the impinging beam and uncertainty in the direction of the emitting nucleus, plus the PSA uncertainty in the reconstruction of the interaction points of the γ rays. The yields of events outside the Coulomb excitation angular window and of time uncorrelated events (10 ns out of the prompt time peak) were measured simultaneously and properly normalized to the incoming beam statistics. These distributions are shown with the wide red shaded area and blue narrow shaded area, respectively. The detector response corrected contribution from statistical decay, assuming the standard Lorentzian as a tail of the IVGDR in the projectile, summed to cosmic background events [23,24] is shown with the green dashed line in the high energy spectra of Figure 1 d) and f). The comparison with the measured spectrum possibly shows a difference in size and shape indicating that the E1 strength is different in this case from the tail of a Lorentzian shape for the IVGDR systematics. We note that for the present configuration of AGATA the statistical decay from the target when corrected for the Doppler shift of the projectile, produces counts in the region below 2 MeV and thus can be very well decoupled from the projectile emission which is the focus of this experiment. At this bombarding energy, in the excitation region larger than 5 MeV the Coulomb excitation mechanism is dominated by exchange of virtual photons of E1 type. To verify this relevant point, the angular distributions of counts were measured for different energy regions of the γ-ray spectra. The difference in the angular distribution between E1 and E2 type due to the effect of the Lorentz boost is very small and far below the detection capabilities, but the ratio between them is detectable. For this reason, the angular distributions of γ transitions from the first 2 + states were used as reference. Figure 2 a) shows the ratio of γ-ray yields of 62,64 Fe with the ones of the corresponding 2 + -state decay, for energies at 3.5 MeV, 4.5 MeV and above 5 MeV. The data are compared with the corresponding predictions. In Figure 2 b) the data for 62 Fe are shown as a function of γ-ray energy and from there one sees that γ transitions above 5 MeV have E1 character. Therefore the continuum type spectrum above 5 MeV is the one for which the discussion in connection with theory is considered, for further analysis and interpretation. In particular, B(E1) strengths were obtained in this quasi-continuum region where the E1 character dominates. The measured spectra were converted directly into E1 strength considering the detector efficiency and a bin size of 160 keV. The counts were background subtracted. The background consisted in off prompt, no Coulomb excitation events, as shown in Figure 1, and in a continuum cosmic event background which was measured during and after the experiment for events with energy higher than 20 MeV. The total background contribution amounted to about 1 count per bin in the present configuration. The measured counts in the high energy continuum region were then corrected for the detection efficiency and response function. After subtracting the background, the counts were converted into cross section. The conversion into B(E1) strength used the method of the relativistic Coulomb excitation of Ref. [50,51,23] and assumed virtual photons of E1 type in the region above 6 MeV. In doing this the energy variation of the virtual photon yield was taken into account. We note that no valuable feeding to lower states was found in γ-γ coincidence spectra and no isomers were detected in the off prompt or in the beam like implantation spectra. The corresponding E1 strength values in the region above 6 MeV are shown in Figure 3 in the two top panels, for 64 Fe and 62 Fe. The error is approximately 35%. The integrated E1 strenght between 6 and 8 MeV is 0.26±0.10 and 0.10±0.03 e 2 fm 2 for 64 Fe and 62 Fe, respectively.

The measured spectra from which the E1 strength was obtained are dominated by a continuum structure due to the experimental conditions, and data are restricted to observable energy region up to ≈8 MeV. The main finding is an increase with neutron number of the total E1 strength and of its fragmentation, which shifts the strength towards lower energy. The experimental results are compared with the B(E1) values reported in the literature for 54 Fe and 56 Fe isotopes resulting from γ scattering experiments [52,53], they are displayed in the two bottom panels of Figure 3. One can see an increase of the E1 strength with the neutron excess and the shifting down of strength to lower energy in the case of the more neutron-rich isotopes of 62 Fe and 64 Fe. In addition, there is a noticeable increase of the strength in 64 Fe nucleus with respect to the one in 62 Fe, due to the addition of only two neutrons. A neutron effect is thus visible in the E1 strength below the neutron separation energy. In particular, the summed E1 strength is shown in panel a) of Figure 4. In 64 Fe, it is found to be approximately 2 to 4 times larger than that in 62 Fe. The spectra of the theoretical calculations (discussed below) are also shown. They have been binned in a way to be comparable to the experimental findings (the unbinned theoretical data can be found in the supplementary materials to this manuscript). ) and for the stable nuclei of 56 Fe and 54 Fe (panel c) and d)) [52,53].

To describe the obtained results calculations were made for the 62,64 Fe isotopes in the framework of the quasi-particle phonon model (QPM) [START_REF] Soloviev | Theory of Atomic Nuclei: Quasiparticles and Phonons[END_REF]. This model has been widely used from the late 90's [START_REF] Herzberg | [END_REF]56] to describe the extensive existing data of pygmy states from (γ, γ ) experiments. The QPM predictions are well suited to describe the E1 strength at low energy among the many available predictions of the dipole response (see, e.g. [START_REF] Bracco | [END_REF] for review). This is because the model considers complex configurations by going beyond the simple one-particle one-hole (1p-1h) states and including two-particle two-hole (2p-2h) or even three-particle three-hole (3p-3h) configurations. In particular, the QPM uses single-particle energies from a global Woods-Saxon parametrization obtained from a fit to experimental data over a wide mass range [57]. The simplest modes of nuclear excitation are treated as phonons (ph) which are obtained by solving quasiparticle RPA equations on 1p-1h basis. The excited levels are assumed to have more complex structure: their wave functions contain a mixture of 1ph and 2ph (and 3ph) components. The former plays a role of doorway states and carry the major part of the excitation E1 strength. The latter are the background configurations, their density increases exponentially with excitation energy and they participate in fragmentation of the strength of the doorway components. Following this approach the QPM calculations are found to give a rather realistic description and a rather good account of the strongly fragmented experimental strength seen in many stable neutron rich nuclei in different nuclear reactions [58]. Therefore, the description of the present data focuses on the comparison with the QPM predictions. The E1 strength distributions, calculated in the QPM, depends on the strength parameter of the isovector residual interaction and is shown and explained in Figure 4 in the supplementary material. [59]. In Figure 4 the QPM predictions for the summed E1 strength in 62 Fe and 64 Fe are shown in panel (b), while the corresponding experimental data are shown in panel a). One sees that the calculations follow well the experimental trend. In addition, the relative increase is also reasonably accounted. Indeed, the summed E1 strength in 64 Fe is found to be approximately 2 to 4 times larger than that in 62 Fe in the observed energy range, similarly to the experimental data. In order to emphasise this relative increase of the summed E1 strength as a function of energy the ratio of the strength in the two isotopes is shown both for the data and the calculations in panel (c). An additional prediction (dashed line) is also shown and this considers only up to 2ph configurations. This last calculation is far away from the experimental data and predicts a rather small variation of strength among the two isotopes. We notice that the lowest 1ph configurations in both 62,64 Fe isotopes has excitation energy of about 9 MeV and their B(E1) values are too small in comparison to data (see Fig. 1 in [59]). In the case of inclusion of 2ph only, a comparable behaviour to the observed E1 strength appears in this doorway picture only around 11 MeV and it is not much different between the two isotops. Only the coupling of the doorway states to more complex configurations pushes a part of the dipole strength to lower energies and dramatically improves the agreement with data. The coupling is stronger in a softer 64 Fe with more collective low-lying phonons, leading to the enhancement of the E1-strength below the threshold, in comparison to 62 Fe. Otherwise, at higher excitation energies, including the GDR the QPM predicts no essential differences in the E1-strength distribution between the two isotopes (see Fig. 1 in [59]). In the supplementary material to this manuscript the strength distribution in 1 phonon approximation is shown for 62,64 Fe isotopes in Figure 1 [59], for the 1+2 and 1+2+3 phonon approximation in stable 54,56 Fe and unstable 62,64 Fe isotopes in Figure 2 and 3 [59]. One can note that the low-lying strength for the stable isotopes is one to two orders of magnitude smaller than the exotic ones. The stable isotopes show an even more pronounced effect of increase of strength with neutron number below neutron threshold energy, which opened the question if this trend is still valid for unstable nuclei or damped. The 1 phonon approximation does not show any low-lying states, the two phonon approximation does show little difference in the strength sum between 62 Fe and 64 Fe. Only the three phonon approximation can reproduce the experimental findings in this manuscript. The finding of this work, that in the interval 6-7.5 MeV, the summed E1 strength is in 64 Fe a factor of 2 to 4 times larger than in 62 Fe is interesting and this trend is on the other hand in line with what found in stable neutron rich-nuclei [8]. However, the identification of E1 strength below binding energy in unstable nuclei is very relevant to test nuclear structure far from stability and how well the models, in this case the QPM, can also describe these nuclei precursors of stable nuclei in explosive nucleosynthesis processes. In summary, we have measured for the first time E1 transitions in the region above 5 MeV up to the neutron separation energy, for the two unstable nuclei 62 Fe and 64 Fe. These nuclei appear to have a continuum structure in spectra measured with the technique of virtual photon scattering, the only one which allows to access this excitation energy region and populate mostly 1 states. The comparison with QPM model predictions is made for the summed E1 strength as a function of energy. The trends and the relative increase going from the 62 Fe nucleus to the 64 Fe nucleus are reproduced rather well by the QPM predictions including complex configuration of 1 -states up to 3p3h states. In contrast, the inclusion of only 2p2h states cannot explain the data. The present findings shed more light on the understanding of the structure of the low-lying E1 strength, which is found in the energy region below the particle binding energy to be sizable and to increase with neutron number. Moreover, the fact that the main features for the two measured unstable isotopes are rather well reproduced by the QPM calculations including couplings up to the three particle level, gives a relevant test to this model, which so far was widely used for stable nuclei. However, the description of nuclei even more neutron rich might be more complex as more couplings might be required. The ambition for the future is to go further away from stability with this type of investigations which need also more statistics. With a more efficient AGATA detector, as it is planed to be in the future, and more intense fragmentation beams, this challenging program is expected to be feasible for example within the HISPEC project [60].

Supplementary materials (theoretical predictions): 54,56,62,64 Fe. The bottom part presents the accumulative strength, as a function of excitation energy. The calculation is performed in 1+2+3 phonon (left column) and 1+2 phonon (right column) approximation. The E1 strength under discussion in the manuscript is indicated by dashed ovals (see Fig. The E1 strength distributions, calculated in the QPM, depend on the strength parameter of the isovector residual interaction. This parameter is adjusted to the experimental value (or the value from systematics) of the GDR peak position. In Fig. 4 we present the accumulative E1 strength in 64 Fe calculated for three values of the strength parameter adjusted so that the energy centroid

< E > = i E i B i (E1) i B i (E1)
increases by 1 MeV. This corresponds to increase of the strength parameter by roughly 20%. The quantities E i and B i (E1) in this expression are the excitation energy and reduced transition probability of the individual 1 - i state (i= 1, 2, 3, . . .), respectively. Notice that while the E1 strength distribution in the GDR energy region substantially changes with the increase of the isovector strength (right part of Fig. 4), its low energy part is almost not affected (left part).

Figure 1 :

 1 Figure 1: Top panels a) and b) show the particle identification plots obtained when the fragment separator was set for the optimal transmission of the 62 Fe (left) and 64 Fe (right) secondary beams. The circles indicate the isotopes of interest. Panels c) to f) show the low energy part (left) and high energy part (right) of the measured γ spectra.The shaded areas in these figures display the spectral shapes obtained with gates corresponding to events outside the Coulomb excitation conditions (red area) or out of the prompt time of flight peak (blue area). The dotted lines (green) in panels d) and f) give the calculated statistical contribution to the spectra. The insets in the panels c) and e) display the known transitions from the first 2 + states, as obtained with full tracking together with the Monte Carlo simulations of the expected peak. In the insets of panels d) and f) the counts forming peak structures at around 4.5 MeV are compared with simulations taking into account the detection conditions. An increase of the bin size with energy has been chosen to take account for the efficiency loss in the detection of high energy γ rays.

Figure 2 :

 2 FIG3C

Figure 3 :

 3 FIG4C

Figure 4 :

 4 Figure 4: The summed E1 strength values, as a function of energy, for data and calculations. Panel a) shows the data for 64 Fe with solid line and for 62 Fe with dashed line. In panel b) the binned QPM calculations are shown: for the 3-phonon model (3ph) for 64 Fe (full line)and for 62 Fe (dashed line) and for the 2-phonon model (2ph) for 64 Fe (dotted line) and for 62 Fe (dash-dotted line). Panel c) shows the ratio of the summed strength for data and calculations for the two nuclei64 Fe and 62 Fe. The dashed curve corresponds to a calculation in which only 2p-2h states are included.

Figure 1 :

 1 Figure 1: B(E1) strength distribution over low-lying doorway (onephonon) 1 -states in 62,64 Fe.

Figure 2 :

 2 Figure 2: B(E1) strength distribution over 1 -states in54,56,62,64 Fe. The bottom part presents the accumulative strength, as a function of excitation energy. The calculation is performed in 1+2+3 phonon (left column) and 1+2 phonon (right column) approximation. The E1 strength under discussion in the manuscript is indicated by dashed ovals (see Fig.3for details.)

  Figure 2: B(E1) strength distribution over 1 -states in54,56,62,64 Fe. The bottom part presents the accumulative strength, as a function of excitation energy. The calculation is performed in 1+2+3 phonon (left column) and 1+2 phonon (right column) approximation. The E1 strength under discussion in the manuscript is indicated by dashed ovals (see Fig.3for details.)

Figure 3 :Figure 4 :

 34 Figure 3: B(E1) strength distribution over low-lying 1 -states in54,56,62,64 Fe, performed in 1+2+3 phonon approximation.

The authors would like to thank D. Peña Arteaga, X. Roca-Maza for fruitful discussions. The authors would especially thank the FRS team of GSI/FAIR for their work. This work was partially supported by INFN (Italy) and STFC (UK), the Swedish Research Council (Vetenskapsrådet, VR 2010-147, 2011-5253, and 2016-3969), BMBF NuSTAR-AGATA 05P18RDFN9 and BMBF NuSTAR 05P18RDFN1. The project was co-funded by the European Commission under the 7 th Framework Program for RTD(2007-2013) under the Capacities Program (Contract Number 262010, EN-SAR). This work was partially supported by the Ministry of Science, and Generalitat Valenciana, Spain, under the Grants SEV-2014-0398, FPA2017-84756-C4, PROMETEO/2019/005 and by the EU FEDER funds.