A new multilayer network construction via Tensor learning
AA new multilayer network construction viaTensor learning
Giuseppe Brandi ∗ , † , T. Di Matteo † , ‡ , § † Department of Mathematics, King’s College London, The Strand,London, WC2R 2LS, UK ‡ Department of Computer Science, University College London,Gower Street, London, WC1E 6BT, UK § Complexity Science Hub Vienna, Josefstaedter Strasse 39, A1080 Vienna, Austria
Abstract
Multilayer networks proved to be suitable in extracting and providingdependency information of different complex systems. The construction ofthese networks is difficult and is mostly done with a static approach, ne-glecting time delayed interdependences. Tensors are objects that naturallyrepresent multilayer networks and in this paper, we propose a new method-ology based on Tucker tensor autoregression in order to build a multilayernetwork directly from data. This methodology captures within and betweenconnections across layers and makes use of a filtering procedure to extractrelevant information and improve visualization. We show the applicationof this methodology to different stationary fractionally differenced financialdata. We argue that our result is useful to understand the dependenciesacross three different aspects of financial risk, namely market risk, liquidityrisk, and volatility risk. Indeed, we show how the resulting visualization isa useful tool for risk managers depicting dependency asymmetries betweendifferent risk factors and accounting for delayed cross dependencies. Theconstructed multilayer network shows a strong interconnection betweenthe volumes and prices layers across all the stocks considered while a lowernumber of interconnections between the uncertainty measures is identified.
Keywords:
Tensor regression, Multidimensional data, Multilayer networks, Frac-tional differentiation ∗ Corresponding author. Email: [email protected] a r X i v : . [ q -f i n . R M ] A p r ccepted in the ICCS 2020 Proceedings Network structures are present in different fields of research. Multilayer networksrepresent a widely used tool for representing financial interconnections, both inindustry and academia [1] and has been shown that the complex structure ofthe financial system plays a crucial role in the risk assessment [2, 3]. A complexnetwork is a collection of connected objects. These objects, such as stocks, banksor institutions, are called nodes and the connections between the nodes are callededges, which represent their dependency structure. Multilayer networks extendthe standard networks by assembling multiple networks ‘layers’ that are connectedto each other via interlayer edges [4] and can be naturally represented by tensors[5]. The interlayer edges form the dependency structure between different layersand in the context of this paper, across different risk factors. However, two issuesarise:1 The construction of such networks is usually based on correlation matri-ces (or other symmetric dependence measures) calculated on financial assetreturns. Unfortunately, such matrices being symmetric, hide possible asym-metries between stocks.2 Multilayer networks are usually constructed via contemporaneous intercon-nections, neglecting the possible delayed cause-effect relationship betweenand within layers.In this paper, we propose a method that relies on tensor autoregression whichavoids these two issues. In particular, we use the tensor learning approach estab-lish in [6] to estimate the tensor coefficients, which are the building blocks of themultilayer network of the intra and inter dependencies in the analyzed financialdata. In particular, we tackle three different aspects of financial risk, i.e. marketrisk, liquidity risk, and future volatility risk. These three risk factors are repre-sented by prices, volumes and two measures of expected future uncertainty, i.e.implied volatility at 10 days (IV10) and implied volatility at 30 days (IV30) ofeach stock. In order to have stationary data but retain the maximum amount ofmemory, we computed the fractional difference for each time series [7]. To im-prove visualization and to extract relevant information, the resulting multilayeris then filtered independently in each dimension with the recently proposed Polyafilter [8]. The analysis shows a strong interconnection between the volumes andprices layers across all the stocks considered while a lower number of interconnec-tion between the volatility at different maturity is identified. Furthermore, a clearfinancial connection between risk factors can be recognized from the multilayervisualization and can be a useful tool for risk assessment. The paper is structured2ccepted in the ICCS 2020 Proceedingsas follows. Section 2 is devoted to the tensor autoregression. Section 3 shows theempirical application while Section 4 concludes.
Tensor regression can be formulated in different ways: the tensor structure is onlyin the response or the regression variable or it can be on both. The literaturerelated to the first specification is ample [9, 10] whilst the fully tensor variateregression received attention only recently from the statistics and machine learn-ing communities employing different approaches [6, 11]. The tensor regression weare going to use is the Tucker tensor regression proposed in [6]. The model isformulated making use of the contracted product, the higher order counterpartof matrix product [6] and can be expressed as: Y = A + (cid:104) X , B (cid:105) ( I X ; I B ) + E (1)where X ∈ R N × I ×···× I N is the regressor tensor, Y ∈ R N × J ×···× J M is the responsetensor, E ∈ R N × J ×···× J M is the error tensor, A ∈ R × J ×···× J M is the intercepttensor while the slope coefficient tensor, which represents the multilayer networkwe are interested to learn, is B ∈ R I ×···× I N × J ×···× J M . Subscripts I X and J B arethe modes over winch the product is carried out. In the context of this paper, X is a lagged version of Y , hence B represents the multilinear interactions that thevariables in X generate in Y . These interactions are generally asymmetric andtake into account lagged dependencies being B the mediator between two separatein time tensor datasets. Therefore, B represents a perfect candidate to use forbuilding a multilayer network. However, the B coefficient is high dimensional.In order to resolve the issue, a Tucker structure is imposed on B such that it ispossible to recover the original B with smaller objects. One of the advantagesof the Tucker structure is, contrarily to other tensor decompositions such as thePARAFAC, that it can handle dimension asymmetric tensors since each factormatrix does not need to have the same number of components.
Tensor regression is prone to over-fitting when intra-mode collinearity is present.In this case, a shrinkage estimator is necessary for a stable solution. In fact,the presence of collinearity between the variables of the dataset degrades the If the imposed Tucker rank is lower than the dimension of the tensor dataset, we havedimensionality reduction. (cid:98) B = arg min T rk ( B ) ≤ R • (cid:107) Y − (cid:104) X , B (cid:105) ( I X ; I B ) (cid:107) F + λ (cid:107) B (cid:107) F (2)where λ > (cid:107)(cid:107) F is the squared Frobeniusnorm. The greater the λ the stronger is the shrinkage effect on the parameters.However, high values of λ increase the bias of the tensor coefficient B . Indeed, theshrinkage parameter is usually set via data driven procedures rather than inputby the user. The Tikhonov regularization can be computationally very expensivefor big data problem. To solve this issue, [13] proposed a decomposition of theTikhonov regularization. The learning of the model parameters is a nonlinearoptimization problem that can be solved by iterative algorithms such as the Al-ternating Least Squares (ALS) introduced by [14] for the Tucker decomposition.This methodology solves the optimization problem by dividing it into small leastsquares problems. Recently, [6] developed an ALS algorithm for the estimationof the tensor regression parameters with Tucker structure in both the penalizedand unpenalized settings. For the technical derivation refer to [6]. In this section, we show the results of the construction of the multilayer networkvia the tensor regression proposed in Eq. 1.
The dataset used in this paper is composed of stocks listed in the
Dow Jones (DJ). These stocks time series are recorded on a daily basis from 01/03/1994 upto 20/11/2019, i.e. 6712 trading days. We use 26 over the 30 listed stocks asthey are the ones for which the entire time series is available. For the purposeof our analysis, we use log-differenciated prices, volumes, implied volatility at 10days (IV10) and implied volatility at 30 days (IV30). In particular, we use thefractional difference algorithm of [7] to balance stationarity and residual memoryin the data. In fact, the original time series have the full amount of memorybut they are non-stationary while integer log-differentiated data are stationarybut have small residual memory due to the process of differentiation. In orderto preserve the maximum amount of memory in the data, we use the fractional4ccepted in the ICCS 2020 Proceedingsdifferentiation algorithm with different levels of fractional differentiation and thentest for stationarity using the Augmented Dickey-Fuller test [15]. We find that allthe data are stationary when the order of differentiation is α = 0 .
2. This meansthat only a small amount of memory is lost in the process of differentiation.
The tensor regression presented in Eq. 1 has some parameters to be set, i.e.the Tucker rank and the shrinkage parameter λ for the penalized estimation ofEq. 2 as discussed in [6]. Regarding the Tucker rank, we used the full rankspecification since we do not want to reduce the number of independent links. Infact, using a reduced rank would imply common factors to be mapped together,an undesirable feature for this application. Regarding the shrinkage parameter λ , we selected the value as follows. First, we split the data in a training setcomposed of 90% of the sample and in a test set with the remaining 10%. Wethen estimated the regression coefficients for different values of λ on the trainingset and then we computed the predicted R on the test set. We used a grid of λ = 0 , , , , , . and the predicted R is maximized at λ = 0 (no shrinkage). In this section, we show the results of the analysis carried out with the datapresented in Sec. 3.1. The multilayer network built via the estimated tensorautoregression coefficient B represents the interconnections between and withineach layer. In particular B i,j,k,l is the connection between stock i in layer j andstock k in layer l . It is important to notice that the estimated dependencies are ingeneral not symmetric, i.e. B i,j,k,l (cid:54) = B k,j,i,l . However, the multilayer network con-structed using B is fully connected. For this reason, a method for filtering thosenetworks is necessary. Different methodologies are available for filtering informa-tion from complex networks[8, 16]. In this paper, we use the Polya filter of [8] asit can handle directed weighted networks and it is both flexible and statisticallydriven. In fact, it employs a tuning parameter a that drives the strength of thefilter and returns the p-values for the null null hypotheses of random interactions.We filter every network independently (both intra and inter connections) using aparametrization such that 90% of the total links are removed. In order to assesthe dependency across the layers, we analyze two standard multilayer networkmeasures, i.e. inter-layer assortativity and edge overlapping. A standard way toquantify inter-layer assortativity is to calculate Pearson’s correlation coefficientover degree sequences of two layers and it represents a measure of association Using hard thresholding the results are qualitatively equivalent.
Figure 1:
Multilayer network assortativity matrix and edge overlapping matrix. Linearscale. Darker colour represents higher values.
Finally, we show in Figure 2 the filtered multilayer network constructed via thetensor coefficient B estimated via the tensor autoregression of Eq. 1. As it canbe possible to notice, the volumes layer has more interlayer connections ratherthan intralayer connections. Since each link represents the effect that one variablehas on itself and other variables in the future, this means that stocks’ liquidityrisk mostly influences future prices and expected uncertainty. The two volatilitynetworks have a relatively small number of interlayer connections despite beingassortative. This could be due to the fact that volatility risk tends to increaseor decrease through a specific maturity rather than across maturities. It is alsopossible to notice that more central stocks, depicted as bigger nodes in Figure2, have more connections but that this feature does not directly translate in ahigher strength (depicted as darker colour of the nodes). This is a feature alreadyemphasized in [3] for financial networks.6ccepted in the ICCS 2020 Proceedings Figure 2:
Estimated multilayer network. Node colours: loglog scale; darker colouris associated to higher strength of the node. Node size: loglog scale; darker colour isassociated to higher k-coreness score. Edge colour: uniform.
From a financial point of view, such graphical representation put together threedifferent aspects of financial risk: market risk, liquidity risk (in terms of vol-umes exchanged) and forward looking uncertainty measures, which account forexpected volatility risk. In fact, the stocks in the volumes layer are not stronglyinterconnected but produce a huge amount of risk propagation through prices andvolatility. Understanding the dynamics of such multilayer network representationwould be a useful tool for risk managers in order to understand risk balances andpropose risk mitigation techniques.
In this paper, we proposed a methodology to build a multilayer network via theestimated coefficient of the Tucker tensor autoregression of [6]. This methodology,in combination with a filtering technique, has proven able to reproduce intercon-nections between different financial risk factors. These interconnections can beeasily mapped to real financial mechanisms and can be a useful tool for moni-toring risk as the topology within and between layers can be strongly affectedin distressed periods. In order to preserve the maximum memory information in7ccepted in the ICCS 2020 Proceedingsthe data but requiring stationarity, we made use of fractional differentiation andfound out that the variables analyzed are stationary with differentiation of order α = 0 .
2. The model can be extended to a dynamic framework in order to analyzethe dependency structures under different market conditions.
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