The complexity of finding arc-disjoint branching flows The concept of arc-disjoint flows in networks was recently introduced in [2]. This is a very general framework within which many well-known and important problems can be formulated. In particular, the existence of arc-disjoint branching flows, that is, flows which send one unit of flow from a given source s to all other vertices, generalizes the concept of arc-disjoint out-branchings (spanning out-trees) in a digraph. A pair of out-branchings B + s,1 , B + s,2 from a root s in a digraph D = (V, A) on n vertices corresponds to arc-disjoint branching flows x1, x2 (the arcs carrying flow in xi are those used in B + s,i , i = 1, 2) in the network that we obtain from D by giving all arcs capacity n − 1. It is then a natural question to ask how much we can lower the capacities on the arcs and still have, say, two arc-disjoint branching flows from the given root s. We prove that for every fixed integer k ≥ 2 it is• an NP-complete problem to decide whether a network N = (V, A, u) where uij = k for every arc ij has two arc-disjoint branching flows rooted at s.• a polynomial problem to decide whether a network N = (V, A, u) on n vertices and uij = n − k for every arc ij has two arc-disjoint branching flows rooted at s.The algorithm for the later result generalizes the polynomial algorithm, due to Lovász, for deciding whether a given input digraph has two arc-disjoint out-branchings rooted at a given vertex. Finally we prove that under the so-called Exponential Time Hypothesis (ETH), for every > 0 and for every k(n) with (log (n)) 1+ ≤ k(n) ≤ n 2 (and for every large i we have k(n) = i for some n) there is no polynomial algorithm for deciding whether a given digraph contains two arc-disjoint branching flows from the same root so that no arc carries flow larger than n − k(n). Notation follows [3] . We denote the vertex set and arc set of a digraph D by V (D) and A(D), respectively and write D = (V, A) where V = V (D) and A = A(D). Unless otherwise specified, the numbers n and m will always be used to denote the number of vertices, repectively arcs in the digraph in question. The digraphs may have parallel arcs but no loops. Paths and cycles are always directed unless otherwise specified. We will use the notation [k] for the set of integers {1, 2, . . . , k}.. An (s, t)-path in a digraph D is a directed path from the vertex s to the vertex t. The underlying graph of a digraph D, denoted U G(D), is obtained from D by suppressing the orientation of each arc. A digraph D is connected if U G(D) is a connected graph. When xy is an arc of D we say that x dominates y. For a digraph D = (V, A) the out-degree, d + D (x) (resp. the in-degree, d − D (x)) of a vertex x ∈ V is the number of arcs of the kind xy (resp. yx) in A, where we count parallel arcs. When X ⊆ V we shall also write d + X (v) to denote the number of arcs vx with x ∈ X. An out-tree rooted at s, also called an s-out-tree, is a tree containing the vertex s in U G(D) such that every vertex v different from s has exactly one arc entering in the tree. Equivalently, s has a unique directed path to every other vertex of the tree using only arcs of the tree. An s-outbranching is a spanning s-out-tree. We use the notation T + s , B + s to denote an s-out-tree, respectively an s-out-branching.. Branchings in digraphs are important from a practical point of view and appear in many applications and hence it is relevant to consider quality measures for branchings. This has been done in a number of papers, see e.g. [5, 8, 9, 13] . An s-out-branching B + s is k-safe if no matter which out-neighbour v of s in B + s we consider, the s-out-tree T + s that we will obtain after deleting all the vertices of the out-tree T + v rooted at v in B + s contains at least k vertices (s and at least k − 1 other vertices). In applications (where an s-out-branching is used to route information or similar from the root s) it is desirable to use an out-branching which is k-safe for a high value of k, because this means that no matter which arc we cut, s can still reach k other vertices using only arcs from the remaining s-out-tree.. In terms of protection against arc-faults, branchings are not a very good way of sending information from one source to all other vertices: If we insist that the vertex sets of the routes that we use to send the information from s to all other vertices may only intersect in a prefix of each route (this is equivalent to saying that the union of the routes is an s-out-branching), then the set of routes may be very vulnerable to arc-deletions. As an example consider the digraph H consisting of vertices s = v 1 , v 2 , . . . , v 2p+1 and arcs {sv 2 , sv 3 , v 2 v 3 } ∪ {v 3 v i |i ≥ 4}. This digraph contains no 3-safe s-outbranching. On the other hand, if instead we use each of the arcs sv 2 , sv 3 on p of the paths from s to V − s, then deletion of one of the arcs sv 2 , sv 3 disconnects s from only half of the other vertices. We may then ask what is the best way to route the information from s to all other vertices, while preserving a high degree of protection against arc-faults. This leads to the study of branching flows. Before we can formally define these, we need to recall a bit of flow theory.. A network N = (V, A, u) is a digraph D = (V, A) equipped with a non-negative capacity function u : A→R 0 on its arcs. A flow in N is any non-negative function x : A→R 0 which satisfies that x ij ≤ u ij for every ij ∈ A, where x ij , u ij denote, respectively, the flow value on ij and the capacity of ij. The balance-vector of a flow x is the function b x on V which to each vertex i ∈ V associates the value b x (i) = ij∈A x ij − pi∈A x pi .. If N = (V, A, u, b), that is, there is also a balance-vector specified for N , then a flow. Two flows x, y in a network N are arc-disjoint if x ij · y ij = 0 for every arc ij of N .. An important result in flow theory is the following which states that it is possible to decide in polynomial time whether or not there exists a feasible flow for a given network N = (V, A, u, b) (see e.g. [3, Section 4.8]). Theorem 1.1 For a given network N = (V, A, u, b) with arc-capacities given by u and vertex-balances prescribed by b one can check, using one max-flow calculation, whether there exists a feasible flow x in N .. A path flow along the path P (resp. cycle flow along the cycle C) in a network N is a flow x which has x ij = k for every arc on P (resp. C) for some positive value k and x ij = 0 for all arcs not on P (resp. C). An s-branching flow in a network N is a flow x in N with balance vector b x (v) = −1 for v = s and b x (s) = n − 1, where n denotes the number of vertices in N .. The following folklore result (see e.g. [1, Section 3.5] or [3, Section 4.3.1]) is very useful when working with flows.

O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O B O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O B I O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O B I O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O


System modeling and identification in indicator dilution method for assessment of Ejection Fraction and Pulmonary Blood Volume Clinically relevant cardiovascular parameters, such as pulmonary blood volume (PBV) and ejection fraction (EF), can be assessed through indicator dilution techniques. Among these techniques, which are typically invasive due to the need for central catheterization, contrast ultrasonography provides a new emerging minimally invasive option. PBV and EF are then measured by a dilution system identification algorithm after detection of multiple dilution curves by an ultrasound scanner. In this paper, dilution systems are represented by parametric models. Since the measured indicator dilution curves (IDCs) are corrupted by measurement artifacts and outliers, the use of conventional least square error (LSE) estimator for estimating system parameters is not optimal. Different estimators are therefore proposed for estimating the system parameters. Comparison of these estimators with the LSE estimator in assessing EF and PBV is performed on simulated, in vitro and patient data. The results show that the proposed total least absolute deviation estimator (TLAD) outperforms other estimators. The measured IDCs are highly corrupted by noise, which affect the estimation of EF and PBV. Therefore, a two stage denoising method capable of removing outliers is also proposed for removing noise in IDCs. Indication dilution measurements are invasive in nature, which involve central catheterization of the patient [1] . After the injection of an indicator bolus, the indicator concentration is measured at a specific region of the circulatory system over a period of time to generate a time series, referred to as indicator dilution curve (IDC) [2] . The measured IDC is then analyzed for assessment of several cardiovascular parameters. Such parameters include cardiac output (CO), ventricular ejection fraction (EF) [3] and pulmonary blood volume (PBV), which are valuable indicators of cardiac preload.. The injection of an ultrasound contrast agent (UCA) bolus allows an IDC echographic measurement, resulting in a minimally invasive technique [4, 5] . UCAs are micro-bubbles (of diameter from 1 to 10 m) encapsulated in a shell of biocompatible material. When subjected to an ultrasonic beam, UCAs oscillate and backscatter a large part of the acoustic energy [6, 7] . As a result, UCAs are easily detected by an ultrasound transducer. Their passage in a specific region-of-interest (ROI) over time can be detected by B-mode ultrasound imaging to derive an UCA IDC. Several IDCs can be measured in different ROIs and directly interpolated and interpreted by specific models for a minimally invasive quantification of PBV and CO [8, 9] . An UCA bolus is injected into a peripheral (arm) vein and its movement through the cardiac chamber is tracked by a transthoracic ultrasound transducer. This will measure the acoustic intensity backscattered by the diluted UCA. A four-chamber view allows one to measure an IDC from each cardiac chamber. Fig. 1 shows an example of four-chamber view with three ROIs overlapped on the right ventricle (RV), left atrium (LA), and left ventricle (LV), respectively. In fact, estimation of EF and PBV requires identification of the dilution system between two sites of the circulatory system: LA and LV for EF, RV and LA for PBV [9, 10] .. To identify the dilution system using the measured acoustic intensity, a linear relationship between UCA concentration and detected acoustic intensity is necessary. It is shown in [9, 11] that at low UCA concentration and low mechanical index (MI) of the ultrasound scanner, the relationship between UCA concentration and the detected acoustic intensity is linear. The use of specific contrast detection modes [12] , such as power modulation, allows reducing UCA concentrations while increasing the image signal-tonoise ratio (SNR). Yet the SNR in IDC measurements is corrupted by several noise sources, comprising bad UCA mixing as well as measurement artifacts due to pressure variations and blood acceleration in the cardiac chambers [9] . For IDC data acquisition, the adopted ultrasound imaging mode was power modulation at 1.9 MHz with low mechanical index (MI = 0.1). The MI determines the mechanical interaction between ultrasound waves and UCA; therefore, a low MI helps in limiting the bubble disruption [13, 14] .. In this paper, the dilution systems are assumed to be linear and, therefore, the impulse response is sufficient to represent the entire dilution system. The estimated impulse response of the dilution system between the two detection sites can then be interpolated by specific models for the assessment of PBV and EF [10, 15] . Once the IDCs are measured using the ultrasound scanner, linear dilution system can be identified through deconvolution techniques. In discrete domain, convolution can be viewed as a system of linear equations, y = Ah, where y, A and h are the output signal, circulant convolution matrix (composed of the input signal) and the dilution system impulse response, respectively. Then the estimation of the impulse response, h, is an inverse problem, whose least square error (LSE) solution isĥ = (A T A) −1 A T y. Although this approach is straight forward, the matrix A T A will be nearly singular due to low SNR of the measured IDCs, and hence its inversion will be difficult. Singular value decomposition (SVD) can be used to find approximate inverse of A T A, but this will result in poor estimates of EF and PBV. An adaptive Wiener deconvolution filter is adopted to estimate the impulse response of the dilution system [10, 15] . However, the Wiener deconvolution filter requires model interpolation of the estimated impulse response to calculate EF and PBV. This increases the variance of the estimated EF and PBV. Also at low SNR, Wiener filter acts as a low-pass filter [16] . As a result, the model interpolation of the impulse response is inaccurate. Besides, the model interpolation will be inaccurate when the frequency component of the input IDC is lower than those representing the transfer function of the dilution system. Since the estimation of EF and PBV depends on the quality of model interpolation of the estimated impulse response, this may result in erroneous assessment of EF and PBV.. To overcome the problems associated with the Wiener filter, a new parametric deconvolution method was proposed [16] . This method employs specific parametric models that represent the dilution system. As a result, the system identification reduces to finding a few system parameters rather than all the frequency components of the dilution system transfer function. In the parametric deconvolution method, system parameters are estimated without model interpolation of the impulse response; therefore, the estimated system parameters show less variance. This improves the measurement of EF and PBV. In this method, the system parameters are identified by minimizing the squared error between the measured and the estimated output IDC of the dilution system represented by the parametric model. In [16] , a low-pass finite impulse response (FIR) filter is used for denoising the measured IDCs and the system parameters are estimated using an LSE estimator. It is observed that the measured IDCs contain outliers and nonlinearities. LSE is not a robust estimator in the presence of outliers and also the FIR filter is not capable of removing outliers. As a result, the estimated EF and PBV will not be robust and may result in poor estimates.. In order to overcome the problem with LSE estimator, we propose many robust estimators which can take care of outliers present in the IDCs. In this paper, the low-pass FIR filter proposed in [16] for removal of the IDC outliers is replaced by a two stage denoising method. The remainder of the paper is organized as follows: In Section 2, the modeling of the dilution system is described. In Section 3, different estimators used for system identification are proposed. The denoising methods used and the LA IDC attenuation compensation are discussed in Section 4. A comparison of the proposed estimators using a set of measurements in patients is presented in Section 5. Conclusions and future course of work are discussed in Section 6. Estimation of EF and PBV using indicator dilution methods is studied, where we have used a parametric system identification technique to calculate EF and PBV. Two system models are proposed for the LV dilution system and one for the transpulmonary dilution system. Analysis of the two models (Model-1 and Model-2) proposed for EF estimation suggests that Model-1 approximates the LV dilution system better than Model-2.. Different estimators are proposed for estimating the LV and transpulmonary dilution system parameters and performance analysis of these estimators in assessing EF and PBV is done. In EF estimation, analysis of the results show that the LAD, HFR, MHFR and IRLS estimators perform better than the conventional LSE estimator. Since all the measured IDCs are corrupted by noise, TLS and TLAD are more suitable for system identification. Results given in Tables 3-7 show that the TLAD method, which accounts for measurement errors in both the input and the output IDCs, together with its robustness to outliers, is best suited for dilution system identification. In PBV estimation, only two estimators, LSE and LAD, are compared. In vitro results show that the LAD performs slightly better than LSE, but the difference is not significant.. The two stage denoising method, namely, TV-Rloess, proposed to denoise IDCs, gives significant improvement over single stage denoising methods, namely, FIR filtering and TV denoising in estimating EF and PBV. The proposed dilution systems and the error functions for estimating EF and PBV requires a more extensive validation. EF estimates were compared with the bi-plane method, which is not a gold standard and has to be compared with magnetic resonance imaging method [31] . PBV estimates should be compared with transpolmunary dye or thermodilution measurements.. The values of regularization parameter, and local window length, l used in TV and Rloess denoising methods will effect the estimation of EF and PBV. In this paper, the values of and l are found empirically and the same values are used for denoising all measured IDCs. Future work can also include a method to find the optimal values and l adaptively for each IDC. Finding the values of and l based on the IDC will prevent over or under smoothing of the IDCs and will therefore result in better EF and PBV estimation.

O O O O O B I I O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O B I I O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O


Enabling network inference methods to handle missing data and outliers Background: The inference of complex networks from data is a challenging problem in biological sciences, as well as in a wide range of disciplines such as chemistry, technology, economics, or sociology. The quantity and quality of the data greatly affect the results. While many methodologies have been developed for this task, they seldom take into account issues such as missing data or outlier detection and correction, which need to be properly addressed before network inference. Results: Here we present an approach to (i) handle missing data and (ii) detect and correct outliers based on multivariate projection to latent structures. The method, called trimmed scores regression (TSR), enables network inference methods to analyse incomplete datasets by imputing the missing values coherently with the latent data structure. Furthermore, it substitutes the faulty values in a dataset by proper estimations. We provide an implementation of this approach, and show how it can be integrated with any network inference method as a preliminary data curation step. This functionality is demonstrated with a state of the art network inference method based on mutual information distance and entropy reduction, MIDER.The methodology presented here enables network inference methods to analyse a large number of incomplete and faulty datasets that could not be reliably analysed so far. Our comparative studies show the superiority of TSR over other missing data approaches used by practitioners. Furthermore, the method allows for outlier detection and correction. This work has presented an enhancement of network inference methods consisting of two preprocessing modules for handling incomplete and faulty datasets. The first one is capable of imputing values for lost measurements coherently with the latent structure of data. The second module detects univariate and multivariate outliers and replaces the faulty measurements with new values coherently with the available data.. A comparison of different methodologies for handling missing data has led to two main conclusions. First, traditional approaches used by practitioners, like complete case analysis (CC), linear interpolation (LI), nearest neighbor algorithm (NN) and unconditional mean imputation (MI), have problems when the datasets have high percentages of missing values and when the complexity of the network increases. Second, since trimmed scores regression (TSR) performs significantly better than the previous methods in BM1-BM4, including the iterative algorithm (IA), it represents the best approach to deal with missing values in network inference.. Likewise, the module for detecting and correcting outliers proposed here also applies TSR for replacing faulty observations. The good performance of this approach has been shown by means of simulations with five benchmark problems. The results of the network inference in four benchmarks with simulated multivariate outliers are statistically better when the new module is used as a preprocessing step; results obtained with univariate outliers in the fourth benchmark are not statistically significant. Due to the long computation time required to analyse the fifth benchmark, a single simulation is performed in this large-scale network. Therefore, no statistical differences are computed in this case study. However, the results of TSR in the comparative study and the outlier detection and correction are coherent with the results of BM4.. By extending MIDER with these new functionalities we have combined two different approaches for data analysis: information-theoretic and variance-based. The joint use of both methodologies increases significantly the number of datasets that can be used for network inference.. Crucially, both MIDER and the new preprocessing modules are general-purpose methods, which may be applied to networks of any kind -not only biological, but also from other areas of science -without requiring prior knowledge from the user. Furthermore, the missing data and outlier detection and correction modules can be used as a preprocessing step for other network inference methods. The problem of inferring complex networks is frequently addressed in several research areas [1, 2] such as social sciences (networks between academics, labor relations), engineering (electric power grid), chemistry (chemical reaction networks) or biology (ecological or cellular networks). In the context of cellular networks the individuals (network nodes) are biochemical entities such as genes, proteins, or metabolites. Many different approaches have been applied to date in this area, none of which can be singled out as the best one for all problems. Comparisons often find large discrepancies between the predictions of *Correspondence: abfolfor@upv.es 1 Departamento de Estadística e Investigación Operativa Aplicadas y Calidad, Universitat Politècnica de València, Camino de Vera s/n, 46022 Valencia, Spain Full list of author information is available at the end of the article different algorithms, making network inference an open problem in bioinformatics research [3] [4] [5] [6] [7] .. Network reconstruction usually begins with collecting data from each individual that participates in the network. Then, based on the relationships detected among them, links between the individuals are established. When no prior knowledge is introduced, the reconstruction of complex networks depends solely on the available data. In many cases, the data collection represents a challenge itself when experimental measurements are involved. Network inference methods rely on estimating quantities such as correlation or mutual information, whose calculation requires simultaneous measurements of several variables. When the data collection in a time point fails for a particular variable, resulting in an unmeasured value, the scientist has to decide whether to discard the information regarding the entire experiment at this time point or to impute an appropriate value. This problem is also faced in process monitoring, for example when sensors fail, when a particular measurement is outside the range of the instrument, or when a failure is produced between the control system and the instrumentation [8, 9] .. The problem of missing data has been widely studied using projection to latent structure methods such as principal component analysis (PCA) [10] . Several methods have been developed using PCA in their cores, like the iterative algorithm (IA) [11] , the NIPALS algorithm [12] , and regression-based methods like known data regression (KDR) and trimmed scores regression (TSR) [13] , of which the latter has been shown to be one of the best performers [14] . These methods exploit the existing correlations between variables to impute the missing values coherently with the latent structure of the dataset. Other methods commonly applied to impute missing data are based on clustering algorithms, like the k-nearest neighbor algorithm and its variants [15] .. Projection to latent structure methods, especially PCA, can also be used for outlier detection [16] . Multivariate statistical process control allows the detection of abnormal situations during process monitoring [17] . Latent structure-based methods allow monitoring a large number of process variables, reducing the dimensionality of the original space to a few latent variables and statistics that are easy to control using charts. These methods are able to detect not only univariate outliers (i.e. values outside the usual range of a particular variable) but also changes in the latent structure (i.e. correlation structure) of the original variables, which classical univariate control charts usually miss [18] . Latent structure-based monitoring methods can be easily extrapolated to network inference. In this context, the outliers would be abnormal behaviors for certain individuals of the original dataset, or odd measurements for concentrations in time points during an experiment.. In this work we provide two new functional modules for curating the data that will be used as input to network inference procedures. The first module is devoted to handle missing data. If the toolbox detects that the raw dataset has missing values, the TSR method is applied in order to fill the holes in the dataset coherently with the latent structure of data. The second module detects extreme outliers in the raw dataset, and if there exist, they are first replaced by missing values and then recalculated using TSR. TSR was originally proposed in [14] for PCA model building with missing data, however, the application of TSR to correct outliers is new. The way both preprocessing modules act can be visualized in Fig. 1 . To evaluate the performance of TSR a comparison with other missing data methods commonly used by practitioners is carried out using several network inference benchmark problems. Likewise, several univariate and multivariate outliers are included in the datasets in order to check the ability of the outlier detection module.. Additionally, to illustrate how a network inference method can be augmented with these functionalities, we use them here in combination with a state of the art technique called MIDER (Mutual Information Distance and Entropy Reduction) [19] . MIDER is a general purpose method that exploits information-theoretic concepts such as entropy and mutual information [20, 21] . Informationtheoretic methods have strengths such as good scalability and the ability of detecting nonlinear relationships, and are widely used for the reverse-engineering of biological networks [22] , with examples such as CLR [23] , ARACNE [24] , MRNET [25] , MI3 [26] , or TD-ARACNE [27] . In [19] the performance of MIDER was benchmarked against these methods, obtaining competitive results. We emphasize that MIDER is used for demonstration purposes, but any other network inference method -such as those mentioned before -could be used as well. The data curation modules presented here are of general purpose, and work on the data independently of the reverse engineering procedure.. To facilitate their use, a Matlab/Octave implementation of the two data curation modules is included in the Additional files section (Additional files 1 and 2). Additionally, they have been included in a new version of the MIDER toolbox, MIDERv2 (http://gingproc.iim.csic. es/~gingproc/mider.html and https://sites.google.com/ site/midertoolbox/).. Network inference studies that take into account the missing data imputation problem have been more common in the social sciences than in the biological sciences; however, some examples of the latter type can also be found. Thus, the works [28] [29] [30] [31] [32] report network inference results obtained with datasets with missing values. Wu et al. [28] presented a network inference method with an interpolation controller, providing three selections of data interpolation approaches. In [29, 30] missing data is handled with a weighted k-nearest neighbor method. Hurley et al. [32] illustrated the use of a suite of GRN analysis tools imputing missing data using the LSImpute missing value estimation method. It should be noted that the aforementioned methods [28] [29] [30] 32] are specific for GRN inference with gene expression data, and the approaches chosen to handle missing data are not justified nor compared to other alternatives. In [31] a network inference tool designed for GRNs, NetGenerator, is extended in several ways in order to predict pathogen-host interactions (PHIs). Remarkably, NetGenerator is applicable to datasets with missing values, although it requires complete data for the last time point, which means that missing data imputation procedures may still be needed in some cases. There are also studies that have addressed the Fig. 1 Missing data and outlier detection and correction modules. Initially, if there are missing values in the raw dataset, trimmed scores regression (TSR) method is used to impute the missing data. Then, if the dataset has extreme outliers, first a missing value is generated for the faulty observation, and secondly the observation is reconstructed using again TSR issue of missing data in biological applications, but outside the context of network inference, e.g. [33] .. The present contribution differs from the aforementioned works in that it (i) presents a general method for handling missing data and outliers, (ii) compares its performance with that of other common approaches, (iii) provides an implementation of the methodology, and (iv) combines it with a freely available general-purpose network inference method. In this respect, there may be more resemblances with a recently published paper [34] , which presents a framework for network inference in Cytoscape and includes the possibility of using three built-in missing data "apps": row average, zero imputation, and Bayesian principal component analysis. The present manuscript is complementary to [34] since (i) the missing data imputation methods are different, and (ii) here, outlier detection and correction is also taken into account. This section begins with a description of the methodology for handling missing data, presenting the different methods used in the comparative study. Afterwards, the procedure for the detection and correction of outliers is defined. Then the main features of the MIDER network inference method are summarized, including an introduction of the concepts of entropy and mutual information, which are common to other information-theoretic methods. Finally, the case studies used for testing the methodology are described.

O B I O O O B I O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O B I O B I O O O O O O O O O B I O O O O O O O O O B I O O O O O O O O O B I I I O O O B I I O O B I O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O B I O O O O O O O O O O O O O O O O O B I O O O B I O O O O O O O O O B I O O O O O O O O O O O O O O O O O O O O O O O O O O O O O B I O O O O O O O O O B I O O O O O O O O O B I O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O B I O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O B I I O O O O O O O O O O O O O O O O O O O O O O O O O O B I O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O B I O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O B I O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O B I O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O B I O B I O O O O O O O O O O O O B I O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O B I O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O B I O O O O O O O O O B I O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O B I O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O B I I O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O B I O O O O O O O O O O O O O O O O O O O O O O O O O O O B I O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O B I O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O B I O O O O O O O O B I O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O B I O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O B I O O O O O O O O O O B I O O O O O O O O O O O O O O O O O O O O O B I O O O O O O B I O O O O O O O O O O O O O O O O O O O O O O B I O O O O O O O O O O O O O O O O O O O O B I O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O B I O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O B I O O O O O O B I O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O B I O O O O O O O O O O O O O B I O O O O O O O O O O O O O O O B I O O O O O O O O O O O O O O O O O O O O O O O B I O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O B I O O O O O O O O O O O B I O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O B I O O O O O O O O O O O O O O O O O O O B I O B I O O O O O O O O O O O O O B I I O O O O O O O B I O O O O O O O O O O O O O O O O O O O O O O O O O O O O B I O O O O O O O O B I O O O O O O O O O O O O O O O O O O O O O B I O O O O O O O O O O O O O O O O O O O O O O O O O O O O B I O O O O O O O O O O O O O O O O O O O O O B I O O O O O O O O O O B I O O O O O O O O O O O O O O O O O O O O B I O O O O O O O B I O O O O O O O O O O O O O O O O O O B I O O O O O O O O O O O O O O O O O O O O O O O O O O O O B I O O O O O O O O O O O O B I O O O O O O O O O O O O O O O O O O


Optimal throughput-delay scaling in wireless networks - part I: the fluid model Abstract-Gupta and Kumar (2000) introduced a random model to study throughput scaling in a wireless network with static nodes, and showed that the throughput per source-destination pair is 2 1 log . Grossglauser and Tse (2001) showed that when nodes are mobile it is possible to have a constant throughput scaling per source-destination pair. In most applications, delay is also a key metric of network performance. It is expected that high throughput is achieved at the cost of high delay and that one can be improved at the cost of the other. The focus of this paper is on studying this tradeoff for wireless networks in a general framework. Optimal throughput-delay scaling laws for static and mobile wireless networks are established. For static networks, it is shown that the optimal throughput-delay tradeoff is given by ( ) = 2( ( )), where ( ) and ( ) are the throughput and delay scaling, respectively. For mobile networks, a simple proof of the throughput scaling of 2(1) for the Grossglauser-Tse scheme is given and the associated delay scaling is shown to be 2( log ). The optimal throughput-delay tradeoff for mobile networks is also established. To capture physical movement in the real world, a random-walk (RW) model for node mobility is assumed. It is shown that for throughput of 1 log , which can also be achieved in static networks, the throughput-delay tradeoff is the same as in static networks, i.e., ( ) = 2( ( )).Surprisingly,foralmostanythroughput of a higher order, the delay is shown to be 2( log ), which is the delay for throughput of 2(1). Our result, thus, suggests that the use of mobility to increase throughput, even slightly, in real-world networks would necessitate an abrupt and very large increase in delay. A wireless network consists of a collection of nodes, each capable of transmitting to or receiving from other nodes. When a node transmits to another node, it creates interference for other nodes in its vicinity. When several nodes transmit simultaneously, a receiver can successfully receive the data sent by the desired transmitter only if the interference from the other nodes is sufficiently small. An important characteristic of wireless networks is that the topology of the nodes may not be known. For example, it may be a sensor network formed by a random configuration of nodes with wireless communication capability. The wireless nodes could also be mobile, in which case the topology could be continuously changing.. As the complexity of wireless networks increases, there is a need to develop better understanding of the fundamental tradeoffs that govern their behavior. How much does interference limit throughput? How much does cooperation between the users help combat such interference? How does mobility affect network performance? Attempting to answer such questions by studying instances of wireless networks is not likely to lead to answers applicable to most of them.. In their seminal paper [11] , Gupta and Kumar introduced a random network model for studying throughput scaling in a static wireless network, i.e., when the nodes do not move. They defined a random network to consist of nodes distributed independently and uniformly on a unit disk. Each node has a randomly chosen destination node and can transmit at bits per second provided that the interference is sufficiently small. Thus, each node can be simultaneously a source, S, a destination, D, and a relay for other source-destination (S-D) pairs. They showed that for almost all realizations of the random network, throughput scales as per S-D pair. 1 Their result also showed that cooperation among users is essential to combating the adverse effects of interference. In [13] , throughput scaling in static networks is studied and results similar to that of [11] are obtained using a different and simpler approach. In a related line of work (see, e.g., [20] , [14] , [12] ), the information-theoretic notion of network capacity is studied.. In [10] , Grossglauser and Tse showed that by allowing the nodes to move, throughput scaling changes dramatically. Indeed, if node motion is independent across nodes and has a uniform stationary distribution, a constant throughput scaling per S-D pair is feasible. Later, Diggavi, Grossglauser, and Tse [5] showed that a constant throughput per S-D pair is feasible even with a more restricted mobility model.. In most networking applications, delay is also a key performance metric along with throughput. Further, throughput that can be obtained from a network at the cost of increase in delay may not be useful. In this context, the understanding of throughput-delay tradeoff is key to achieving the quality of service required by the application. Indeed, a more useful description of the network capability would be in terms of its delay-constrained throughput. Most previous work has focused mainly on achieving the highest throughput with no consideration for delay or the tradeoff between throughput and delay. In [2] , a random network model with both mobile and static nodes is considered. The authors propose a routing algorithm that is almost optimal in terms of throughput and study its delay. In [18] , throughput-delay tradeoff in a mobile network is studied using an independent and identically distributed (i.i.d.) mobility model. In this model, each node is equally likely to be in any part of the network at each time instant.. In [7] , we studied throughput and delay scaling for static and mobile networks and established optimal throughput-delay tradeoffs. This paper provides a more complete treatment of our earlier results reported in [7] as well as some extensions. Further, we provide results on throughput-delay tradeoff for mobile networks under a RW model instead of the less realistic "hierarchical" Brownian motion model assumed in [7] . 2 In addition to establishing the optimal throughput-delay tradeoff in wireless networks, the schemes we developed to achieve the optimal tradeoff should help enhance our understanding of the fundamental interplay between such essential features of wireless networks as uncontrolled placement of nodes, common medium of communication leading to interference, and node mobility.. From previous throughput results in [11] and [10] , one may make the following inferences about the tradeoff between throughput and delay: i) In a static random network a small transmission range is necessary to limit interference and hence to obtain a high throughput. This results in multihopping, and consequently leads to large delay. ii) On the other hand, mobility allows nodes to approach each other closely. This not only allows the use of smaller transmission ranges, but more crucially, it allows the use of a single-relay node, which boosts throughput to . However, the delay is now dictated by the time it takes the relay to carry the packet to the neighborhood of the destination, which can be very large. The preceding observations point out three important features that influence the throughput and delay of any communication scheme in networks: i) the number of hops, ii) the transmission range, and iii) the node mobility and velocity. We exploit these three features, to different degrees, to obtain the throughput-delay tradeoff in an optimal way.. In Section II, we provide needed definitions and summarize our main results. In Section III, we establish the throughput-delay tradeoff for static wireless networks. In Section IV, we consider a mobile network model, present a scheme that achieves constant throughput, and determine its the corresponding delay. In Section V, we established the tradeoff for mobile networks. Optimal throughput scaling with the number of nodes in static and mobile wireless networks was established in [11] and [10] using a random network framework. Delay scaling, however, was not as well addressed in earlier work. This paper established the optimal tradeoff between throughput and average packet delay in static and mobile wireless networks using a similar random network framework to that in [11] and [10] .. For static networks, we showed that the optimal throughputdelay tradeoff is given by . For mobile networks, we presented a scheme similar to the Grossglauser-Tse scheme in [10] and showed that the delay corresponding to throughput scales as when the nodes move according to independent random walks. Further, we described and analyzed a scheme that achieves the optimal throughput-delay tradeoff for mobile networks by varying the number of hops, the transmission range, and the degree to which node mobility is used. For the low-throughput range achieved in static networks, we found that the tradeoff for mobile networks is the same as that for static networks. For higher throughputs, there is almost no tradeoff between throughput and delay-the same maximum delay is incurred regardless of the throughput. Our results utilized a unified framework for static and mobile networks and our random network model resulted in simpler proofs of previous throughput results. The framework and proofs should provide a better understanding of the essential characteristics of large wireless networks.. Several questions remain to be tackled for a better theoretical understanding of the data communication aspect of random wireless networks. With the exception of Scheme 2, the paper assumed a fluid model in which the packet size is allowed to be arbitrarily small. A priori, it is not clear what the tradeoff will be with the additional constraint that packet size remains constant. In the second part of this paper [9] , we show that for static networks, the tradeoff remains unchanged even with packets of constant size. We believe that the same should hold for mobile networks, but such a result remains to be established.. The Protocol model used in this paper and the Physical model have been used by many researchers. These models, however, do not take transmission energy into account. Modeling energy consumption and determining the tradeoffs between throughput, delay, and energy is an important issue with initial steps already taken by several researchers (e.g., [3] , [8] , [6] ).. In this paper, we assumed an RW model for node mobility. One expects that the scaling results would be the same for a larger class of Markovian motion models. Determining the class of such motion models would be another future challenge.

O O O O B I O O O O O O O O O O O O O O O O B I O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O B I O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O B I O O O O O O B I O O O O B I O O O O O O O O O O O B I O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O B I O O O O O O O O O O O O O O O O O O O O O O B I O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O B I O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O B I O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O B I O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O B I O O O O O O O O O O O O O O O O O O O O O O O O O O O O B I O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O B I O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O B I O O O O O O O O O O O O O O O O O O B I O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O B I O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O B I O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O B I O O O O O O O O O O O O O O O O O O O O O O O O O O O B I O B I O O O O O O O O O O O O O O O O O O O O O O O B I O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O B I O O O O O O O O O O O O O O O O O O O O O O B I O O B I O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O B I O O O O O O O O O B I O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O B I O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O B I O O O O O O O O O O O B I O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O B I O O O O O O O O O O O O O O O O O O B I O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O


Online and Adaptive Pseudoinverse Solutions for ELM Weights The ELM method has become widely used for classification and regressions problems as a result of its accuracy, simplicity and ease of use. The solution of the hidden layer weights by means of a matrix pseudoinverse operation is a significant contributor to the utility of the method; however, the conventional calculation of the pseudoinverse by means of a singular value decomposition (SVD) is not always practical for large data sets or for online updates to the solution. In this paper we discuss incremental methods for solving the pseudoinverse which are suitable for ELM. We show that careful choice of methods allows us to optimize for accuracy, ease of computation, or adaptability of the solution. Since its invention by Huang and colleagues [1] the Extreme Learning Machine (ELM) has become widely used as a method of regression or classification for large and complex data sets. The feature that perhaps most distinguishes the standard ELM from other methods is the analytical solution of the hidden layer weights by means of a matrix pseudoinverse operation. This method allows the rapid and accurate determination of the weights without an incremental or approximate learning process, and makes a significant contribution to the ease of use of ELM. However, there are two ways in which the standard method creates difficulties for users, which are related to the conventional computation of the matrix pseudoinverse by means of a singular value decomposition (SVD) or QR decomposition. The first problem is that these methods are intrinsically batch methods, in which it is assumed that all the input data and all the output target values are available to the user, so that one single computation can be performed; and thereafter the computed hidden layer weights will be fixed and unchanging. The second problem is that as the data sets get large, these methods for computing the pseudoinverse become computationally intractable, as they require both a very large working memory and very large matrix multiplications.. A third issue, which is not as critical for most users, is that an ELM with weights computed using a batch method is not adaptive. If the underlying data-generating process is stationary in a statistical sense, this is not a problem, but in most real-world problems there will be changes in the generative process as time goes by, and the optimal solution will therefore change. It might be helpful to have a weights update rule that gave greater value to the more recent data available, and gradually reduced the contribution of older data.. In this paper we approach these problems by focusing on incremental solutions to the pseudoinverse of a matrix, extending the work on online sequential ELM (OS-ELM) presented in [2] . The purpose of the paper is threefold:. 1. To describe the application of Greville's method in ELM so as to produce a weights matrix that is identical to the weights matrix produced by SVD. 2. To outline simplifications to Greville's method, developed by the authors, which reduce the computation required in the incremental algorithm with negligible loss of accuracy. 3. To introduce an adaptive version of Greville's method which allows for nonstationary data. In this paper we have described the application of Greville's method to Extreme Learning Machines to obtain an online pseudoinverse update method (OPIUM) that produces a weights matrix identical to that produced by SVD in batch mode. When the condition of equation (10) holds, the derived update method is equivalent to the OS-ELM as derived in [2] , when updating the weight matrix every single input-output training pair. Our derivation via Greville's method also provides an update method for the case that equation (10) does not hold. However, in practical implementations, it will be difficult, if not impossible, to test the condition exactly. Therefore, we provide an alternative initialization method after which condition (10) no longer needs to be tested.. We have presented a simplification of OPIUM, OPIUM light, that is very fast, but not as precise. Finally we presented an algorithm designed to cope with a nonstationary mapping between input and output vectors. The inclusion of a free parameter, , allows the user to control the speed of adaptation in this version. For the static mapping case, the MNIST database, with 60,000 training patterns and 10,000 test patterns, was classified by an ELM using a 728 neuron input layer, a 7280 neuron hidden layer, and a 10 neuron output layer. A simulation using OPIUM was performed in Python and took approximately 20 hours to process all the training patterns once and test all test patterns, while consuming up to 1.8GB of RAM. The classification error, without any pre-processing, such as de-skewing of the digits, was 2.75% [8] . Using OPIUM light the simulation consumed only 1GB of memory and the same full MNIST database was processed in under 7 minutes on the same computer, resulting in a speed up by a factor 175. The classification error increased to 3.7%, so depending on the application, the increase in speed may well outweigh the reduction in precision. In our experiments, g was arbitrarily set to 1, but simulations with g equal to 100, 4, 0.4, and 0.01 produced very similar results, indicating that the choice of g is not critical. For OPIUM, = 0.001 was used so that this initialization only had a small effect on the final weights matrix. The code for these simulations is available from the authors' website [9] . To investigate performance and execution time dependence on the number of training samples, we repeated the experiment with only 10,000 training patterns, randomly drawn from the full set and again tested the full 10,000 test patterns. The OPIUM simulation took just under four hours and produced an error rate of 10.3%, while OPIUM light only took just under two minutes for an error rate of 6.7%. Note that the memory consumption for both online algorithms is independent of the number of training patterns.. In order to test the performance for a nonstationary mapping, we simulated OPIUM, OPIUM light, and the dynamic version of OPIUM on a task where the mapping between input and output changes halfway through the task. A network with ten input neurons, one hundred hidden neurons and one output neuron was trained to predict the current sample (at time t) of a sine wave with frequency f and unit amplitude from ten past input samples of the target time series, taken at t-10dt, t-20dt, …, t-100dt, where dt is the sample interval. The weights from the input neurons to the hidden neurons were drawn from a uniform distribution between -0.5 and 0.5, and the hidden neuron activation was calculated as the hyperbolic tangent of the sum of its weighted inputs. In our simulation, dt = 1ms, and f = 10Hz for the first 500ms, and f = 20Hz for the second 500ms. For OPIUM light, g = 1, and for dynamic OPIUM, = 1.003. We should point out that in these examples, there are two abrupt transitions of the mapping between inputs and outputs, of which only the second, the change in frequency, is shown. The first change stems from the initialization with (23) & (24), which maps individual activations of hidden layer neurons ( !×! ) to the zero matrix !×! . In the above example we have chosen to exaggerate this transition for illustrative purposes by choosing = 10, whereas normally would be chosen to be much smaller than that. From Figure 1 and the results in Table 1 we can see that the standard OPIUM algorithm copes poorly with these changes (Figure 1a) , and that the dynamic OPIUM performs very well (Figure 1c ). OPIUM light, while not designed for the case where the mapping changes, performs rather well.. We repeated the experiment with an excerpt from a recorded electrocardiogram (ECG) signal to demonstrate the performance with a less predictable signal. In this case, we chose a much smaller = 0.01, so that the initialization only has a very small effect on the learned weight matrix. Furthermore, we simply used the ten most recent samples of the target time series as the input values for the network to predict the current sample of the target series. All other settings, including the values of the random weights, were the same as in the previous experiment. The results are shown in Figure 2 . It can be seen that, with an appropriate initialization, OPIUM performs much better, but that the dynamic version of OPIUM still results in a further improvement.

O O O B O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O B O O O O O O O O O O O O O O O O O B O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O B O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O B I I O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O B O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O B O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O B O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O B O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O B O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O


A Virtual Element Method for elastic and inelastic problems on polytope meshes We present a Virtual Element Method (VEM) for possibly nonlinear elastic and inelastic problems, mainly focusing on a small deformation regime. The numerical scheme is based on a low-order approximation of the displacement field, as well as a suitable treatment of the displacement gradient. The proposed method allows for general polygonal and polyhedral meshes, it is efficient in terms of number of applications of the constitutive law, and it can make use of any standard black-box constitutive law algorithm. Some theoretical results have been developed for the elastic case. Several numerical results within the 2D setting are presented, and a brief discussion on the extension to large deformation problems is included. The Virtual Element Method (VEM), introduced in [2] , is a recent generalization of the Finite Element Method which is characterized by the capability of dealing with very general polygonal/polyhedral meshes and the possibility to easily implement highly regular discrete spaces [11, 6] . Indeed, by avoiding the explicit construction of the local basis functions, the VEM can easily handle general polygons/polyhedrons without complex integrations on the element (see [4] for details on the coding aspects of the method). The interest in numerical methods that can make use of general polytopal meshes has recently undergone a significant growth in the mathematical and engineering literature. Among the large number of papers, we cite as a minimal sample [10, 5, 28, 19, 12, 30, 31, 17, 18, 16] . Indeed, polytopal meshes can be very useful for a wide range of reasons, including meshing of the domain (such as cracks) and data (such as inclusions) features, automatic use of hanging nodes, use of moving meshes, adaptivity.. In the framework of Structural Mechanics, recent applications of Polygonal Finite Element Methods, which is a different technology employing direct integration of complex non-polynomial functions, have shed light on some very interesting advantages of using general polygons to mesh the computational domain. This include, for instance, the greater robustness to mesh distortion [13] , a reduced mesh sensitivity of solutions in topology optimization [12, 20] , better handling of contact problems [7] and crack propagation [24] . Unfortunately, Polygonal Finite Elements suffer from some serious drawbacks, such as the strong difficulties in the three dimensional case (polyhedrons) and in the use of non convex elements. On the contrary, the VEM is free from the above-mentioned troubles, and thus it represents a very promising approach for Computational Structural Mechanics problems.. Aim of the present paper is to initiate the investigation on the VEM when applied to non-linear elastic and inelastic problems in small deformations. More precisely, we mainly focus on the following cases: 1) non-linear elastic constitutive laws in a small deformation regime which, however, pertain to stable materials; 2) inelastic constitutive laws in a small deformation regime as they arise, for instance, in classical plasticity problems. We remark that we are not going to consider here situations with internal constraints, such as incompressibility, which require additional peculiar numerical treatment. Virtual elements for the linear elasticity problem where introduced in [3, 21] . The scheme in the present paper is one of the very first developements of the VEM technology for nonlinear problems, and it is structured in such a way that a general non linear constitutive law can be automatically included. Indeed, on every element of the mesh the constitutive law needs only to be applied once (similarly to what happens in one-point Gauss quadrature scheme) and the constitutive law algorithm can be independently embedded as a self-standing black-box, as in common engineering FEM schemes. Therefore, in addition to the advantage of handling general polygons/polyhedra, the present method is computationally efficient, in the sense that the constitutive law needs to be applied only once per element at every iteration step. The risk of ensuing hourglass modes is avoided by using an evolution of the standard VEM stabilization procedure used in linear problems. However, we highlight that the proposed method is described for general d-dimensional problems (d = 2, 3), but the performed numerical experiments are confined to the two dimensional setting.. A brief outline of the paper is as follows. In Section 2 we describe the continuous problems we are interested in. In particular, we distinguish between the elastic, possibly non-linear, case (Section 2.1), and the general inelastic case (Section 2.2). Section 3 deals with the VEM discretization. After having introduced the approximation spaces and the necessary projection operators (Section 3.1), we detail the discrete problems for the elastic case in Section 3.2, and for the inelastic case in Section 3.3. In Section 4, combining ideas and techniques from [14] and [2] , we provide some theoretical results concerning the convergence of the proposed scheme in the elastic situation. We remark that our analysis is confined to cases where the non-linear costitutive law fulfills suitable continuity and stability properties, as stated at the beginning of the Section. Section 5 presents several numerical examples which asses the actual behaviour of the proposed scheme. In Sections 5.1 and 5.2 we consider non-linear elastic cases, while in Section 5.3 a von Mises plasticity problem with hardening is detailed. Furthermore, an initial brief discussion about a possible extension to large deformation problem is included (Section 5.4). Finally, we draw some conclusion in Section 6.. Throughout the paper, we will make use of standard notations regarding Sobolev spaces, norms and seminorms, see [8] , for instance. In addition, C will denote a constant independent of the meshsize, not necessarily the same at each occurrence. Finally, given two real quantities a and b, we will write a b to mean that there exists C such that a ≤ Cb. We have presented a Virtual Element Method to deal with fairly general non-linear elastic and inelastic problems. Our scheme is based on a low-order approximation of the displacement field, together with a suitable treatment of the numerical displacement Table 6 : Computed displacements using triangular (T1,...,T4), square (Q1,...,Q4), and hexagonal Voronoi (V1,...,V4) meshes. gradient. The proposed method allows for general polygonal/polyhedral meshes, is efficient in terms of number of applications of the constitutive law, and can make use of any standard black-box constitutive law algorithm. We have presented several numerical tests assessing the computational performance of the proposed methodology. However, we remark that this study is intended as a first step towards the design of efficient Virtual Element Methods for non-linear Computational Mechanics problems. Many possible extensions and improvements could be of interest. For instance, large deformation problems require a much deeper investigations, and other inelastic cases such as perfect plasticity or damage could be considered.

O B I I O O O O O O O O O O O B I I O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O B I I O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O B O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O B I I O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O


Noninvasive Three-Dimensional Cardiac Activation Imaging From Body Surface Potential Maps: A Computational and Experimental Study on a Rabbit Model were also consistent with our computer simulations conducted in well-controlled and realistic conditions. The present study suggest that 3-DCAI can noninvasively capture some important features of ventricular excitation (e.g., the activation origin and the activation sequence), and has the potential of becoming a useful imaging tool aiding cardiovascular research and clinical diagnosis of cardiac diseases. The present study is aimed at quantitatively evaluating the biophysical model-based 3-DCAI algorithm, detailed in [17] , through both in vivo animal experiments and computer simulation studies. The comparison between the noninvasively imaged activation sequence based on 3-DCAI and the measured activation sequence based on the intracardiac mapping suggests that: 1) 3-DCAI is capable of reconstructing the overall pattern of the 3-D ventricular excitation profile induced from a single pacing site, 2) the localization error of 3-DCAI for localizing the origin of ventricular activation was mm, and 3) 3-DCAI has relatively lower spatial resolution than the intracardiac mapping.. The fact that both the intramural bipolar recording and the equivalent current density are proportional to the spatial gradient of the TMP points to the common physical principle shared by 3-DCAI [17] and the intracardiac mapping [18] , [19] in terms of estimation and visualization of the activation sequence throughout the entire myocardium. This commonality also justifies the use of the 3-D intracardiac mapping as an ideal way to evaluate the activation sequences imaged by using 3-DCAI.. The intracardiac mapping technique based upon intramural bipolar recording has been employed to investigate ventricular tachycardia (VT) [18] , [19] . These successful applications have confirmed the efficacy of using the TMP spatial gradient to detect the activation time for both normal and abnormal myocardial cells. Note that the TMP spatial gradient is essentially proportional to the imaging contrast (i.e., the equivalent current density) of the present 3-DCAI algorithm. Therefore, 3-DCAI can be thought of as a noninvasive alternative technique for the intracardiac mapping. More importantly, it also suggests that 3-DCAI is in principle applicable to image the intrinsic cardiac activation under pathological conditions when the intracardiac mapping is applicable.. However, the need to solve a highly ill-posed linear inverse problem in order to estimate the ECD from body surface measurements poses a great challenge to 3-DCAI. Solving such an inverse problem is essentially a task of seeking a spatial filter (or inverse operator) that projects the recorded data in the sensor space to the solution in the source space. It is often experienced in the ECG (as well as EEG/MEG) inverse problem that applying such a spatial filter cannot lead to a source reconstruction in perfect agreement with the true source distribution [26] . Instead, the source estimates may appear as a spatially blurred image, from which one may still resolve the true sources if they are distributed in a discrete or focal manner [10] . But when the current sources are continuously distributed over a wavefront during the paced ventricular activation, the imaged ECD distribution at any specific time point is highly distinct from a transmural surface as one may expect for the instantaneous excitation wavefront.. In 3-DCAI the activation time is determined exclusively from the time course of local ECD estimates. We found that the ill-posed nature caused much less distortion to the normalized temporal waveform of ECD estimates than it does to the instantaneous ECD spatial distribution. At a noisy condition, the ECD spatial distribution was found to be hardly interpretable, whereas the estimated ECD waveform at each individual myocardial site still demonstrated a clear peak at the instant deemed as the activation time. Regarding this finding, our speculated explanation is two fold. First, the activation time is a magnitude-independent temporal feature of local ECD estimates; therefore it is less sensitive to the spatial distortion of the inverse solution. It is true that the torso volume conductor behaves as a spatial filter and the estimation of the equivalent current density is essentially the deconvolution of this spatial filter. However, even though both forward and inverse procedures operate upon the spatial domain, the bias and blurring of the inverse solution in space also necessarily affects the time course of the source estimate. This effect may be well accounted for by a low-pass filter. Importantly, this effect does not significantly bias the peak instant which marks the activation time, since the "peak instant" of the source time course is a "low-frequency" feature. But this effect may cause the ambiguity and potential bias of other competitive methods based on determining the activation time from the maximum positive or minimum negative derivative, which is a "high-frequency" feature, of the estimated TMPs, or the estimated epicardial or endocardial potentials. In this sense, it represents another technical merit and robustness of 3-DCAI. Second, the cardiac excitation process is typically continuous and smooth in nature and so is the point spread (or crosstalk) function of the 3-D ECG inverse solution. In other words, the source estimates at any specific location are primarily affected by the source activities at neighboring sites with similar activation times.. In future studies, we will attempt to provide more rigorous theoretical insights with regards to these interpretations. The ill-posed problem of the ECD estimation is still the primary source of the activation imaging bias of 3-DCAI [17] . Because of the spatial blurring and bias of the inverse solution, the estimated ECD time course was also smooth over time, whereas a delta function was observed in the intramural bipolar recording at the same site. This distinction accounted for the reduced specificity of activation time estimation. As a consequence, different myocardial locations within an extended region, particularly those that are activated around the middle of the QRS interval, often had the identical activation time, although they were actually activated progressively. In addition, it also caused the total activation time (TAT) obtained from the imaged activation sequence to be shorter than the TAT determined from the intracardiac data.. In the present study, we have only demonstrated experimentally the feasibility of 3-DCAI in imaging ventricular activation induced by single-site pacing. Such single-site pacing provides a well-controlled experimental setting for evaluating the performance of 3-DCAI in imaging ventricular activity with focal activation, such as ectopic beats or some automatic VT. The present results suggest the potential use of 3-DCAI in guiding interventional procedures (e.g., catheter ablation) in treatment of automatic VT of focal nature [19] . As we described in Section II, the imaging principle of 3-DCAI is not limited to single-site pacing-induced activities. 3-DCAI is in principle applicable to imaging complex cardiac activation. Further in vivo experimentation needs to be conducted to validate 3-DCAI in imaging more complex cardiac activation such as reentry VT or other arrhythmias.. In the present study, computer simulations were also conducted to provide additional assessment, which helps in the interpretation of our experimental results. We have obtained an averaged LE of 4.56 mm in singe-site pacing during computer simulation, which is smaller than the averaged LE of 5.49 mm on a human model [17] . However, the LE does not linearly scale to the geometry size of heart-torso model. The primary source for these errors was caused by the intrinsic ill-posedness of the inverse problem. For example, the imaged area of early activation was a region rather than a single point. The shape of the time course of the estimated ECD was much smoother than the shape of "delta function" that simulated the "true" time course of ECD. Also, in the present computer simulations, we used 53 body surface electrodes to simulate the real in vivo experimental setup, while in the previous simulation study on a human model [17] , there were 200 body surface electrodes. Such imaging parameters may influence our imaging results.. The averaged LE of 5.47 mm in the in vivo experiments is larger than our computer simulation results of 4.56 mm. Various reasons related to the in vivo experimental procedures may account for the slightly larger error, including the localization errors of body-surface electrodes, the localization errors of plunge-needle electrodes, and the volume conductor modeling errors [16] , [27] . These errors are mainly determined by experimental procedures, and do not necessarily scale to the geometry size of the subject in a linear manner. Considering the area of catheter ablation typically ranges from 5 to 8 mm [28] , the present results represent reasonable accuracy for assessing the imaging ability of the 3-DCAI technique.. Compared with our previous in vivo study of a cellular automaton heart model-based approach [16] , the 3-DCAI approach obtained similar results in terms of LE and RE, but lower resolution, considering that TAT obtained from imaged activation sequence was shorter than the TAT obtained from measured activation sequence. The cellular automaton heart model-based approach utilizes the a priori electrophysiological knowledge as a constraint and thus is capable of producing higher resolution imaging results. In comparison, 3-DCAI does not need to build a sophisticated heart model. It only uses the biophysical relationships governing the cardiac electrical activity, and has a notable feature of minimal dependence on the physiological knowledge of cardiac electrical properties. Therefore, 3-DCAI is potentially applicable to imaging various cardiac activities including sinus rhythm and arrhythmias.. In the present study, we have used a well-established 3-D intracardiac mapping procedure to test the feasibility of the 3-DCAI algorithm in imaging activation sequence under an in vivo situation. Compared with other studies reported in literature [4] , [8] , [29] , [30] for heart surface imaging approaches, the 3-D intracardiac mapping represents a desirable validation protocol as the 3-DCAI results are distributed within the 3-D myocardium.. Last but not least, 3-DCAI provides the functional data throughout the entire myocardium instead of only over the epicardium and/or endocardium, which is distinct from existing heart surface imaging techniques [4] , [7] , [8] . The intramural cardiac function is of interest when the excitation profile propagates transmurally from the endocardium to the epicardium. To account for a more accurate volume conductor modeling, it may be preferable to use the finite element method (FEM) rather than the BEM [31] , given the recent development of diffusion tensor imaging (DTI) which holds a potential to obtain the information of anisotropic conductivity [32] .. In summary, the present study suggests the feasibility of the biophysical source model-based 3-DCAI in imaging the 3-D ventricular activation sequence, as assessed by the invasive 3-D intracardiac mapping. The present results obtained from both computer simulations and in vivo experimental studies indicate a reasonable performance of the 3-DCAI algorithm in reconstructing the overall activation sequence and localizing the site of origin of activation. The present experimental finding is encouraging and suggests that the biophysical model-based 3-DCAI technique may have potential to become a useful noninvasive imaging means for mapping cardiac electrical activity noninvasively throughout the 3-D myocardium. In Section II, we reveal the physical correlation between the ECD and the intramural bipolar potential. This correlation suggests that the ECD estimates should ideally be proportional to the recorded intramural bipolar potentials. Therefore, we compared the time course of the ECD estimates at a given myocardial site with the intramural bipolar electrogram recorded at the same site. In Fig. 3(b) , both the intramural electrograms and the local ECD estimates at two representative locations are displayed as functions of time. Note that both time courses were normalized before comparison since the activation time was magnitude-independent. It was found that the bipolar electrogram always appeared as a delta function, which indicated a transient TMP spatial gradient as a consequence of the steep upstroke of the TMP undergoing depolarization. Although it was anticipated that the real ECD waveform should be similar to the intramural electrogram, the estimated ECD by solving the linear inverse problem evolved much slower than the intramural bipolar potential. This is in line with our findings obtained in previous computer simulation studies in a human heart-torso model [17] . Nevertheless, the activation times, determined as the instant with peak amplitude in the intracardiac bipolar electrogram [18] , [19] , [25] and as the instant with maximum ECD estimate [17] , respectively, aligned well around the same time point for both examples shown here. Fig. 4 shows examples of comparison between the measured and imaged activation sequences for four paced beats recorded from all four rabbits (one pacing beat per rabbit). For each beat, the activation time distributions on six representative axial slices were displayed along with the 3-D realistic heart geometry. We found that the imaged activation sequences (left column) were qualitatively consistent with the measured ones (right column). The origin of activation was localized to be around the true pacing site, whereas the imaged activation origin tended to be slightly closer to the epicardium or the base. This tendency probably represents a localization bias of the current 3-DCAI algorithm. In the imaged activation sequences, the regions in the middle activation (shown in light blue and yellow) are also found to be larger than in the measured activation map, particularly for Fig. 4(B) . The imaged excitation process appeared to be more discrete instead of being continuous and smooth as observed in the intracardiac measurements, suggesting the limited spatial resolution of the current 3-DCAI algorithm. The quantitative comparison returned a rms of 6.78 ms, a RE of 0.21, and a LE of 3.67 mm for rabbit 1, a rms of 7.02 ms, a RE of 0.26, and a LE of 7.48 mm for rabbit 2, a rms of 7.97 ms, a RE of 0.21 and a LE of 5.20 mm for rabbit 3, and a rms of 7.90 ms, a RE of 0.27 and a LE of 5.51 mm for rabbit 4, respectively. On average, rms, RE, and LE were ms, , and mm, respectively, for this pilot study. Fig. 5 shows examples of comparison between the imaged activation sequence and simulated "true" activation sequence during computer simulation. As shown in Fig. 5(a) , the imaged pattern coincided well with the simulated activation sequence, with . The LV pacing site was localized with a 3.86-mm LE. Similar imaging performance was observed when the activation was induced by pacing at the middle lateral wall of the right ventricle, as shown in Fig. 5(b) , with and mm. Fig. 5(c) shows an example of simultaneous dual-site pacing. The imaged activation sequence is consistent with the simulated activation sequence, with which shows that 3-DCAI can reconstruct the activation sequence throughout the ventricles for dual-site pacing at two separate locations. Quantitative comparison between imaged activation sequence and simulated "true" activation sequence under both single-site pacing and dual-site pacing with respect to the different location of the origin of the ventricular excitation are summarized in Tables I and II . On average, RE was and LE was mm for single-site pacing. For dual-site pacing, averaged RE was .

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