Outlier detection for patient monitoring and alerting We develop and evaluate a data-driven approach for detecting unusual (anomalous) patient-management decisions using past patient cases stored in electronic health records (EHRs). Our hypothesis is that a patient-management decision that is unusual with respect to past patient care may be due to an error and that it is worthwhile to generate an alert if such a decision is encountered. We evaluate this hypothesis using data obtained from EHRs of 4486 post-cardiac surgical patients and a subset of 222 alerts generated from the data. We base the evaluation on the opinions of a panel of experts. The results of the study support our hypothesis that the outlier-based alerting can lead to promising true alert rates. We observed true alert rates that ranged from 25% to 66% for a variety of patient-management actions, with 66% corresponding to the strongest outliers. Despite numerous improvements in health-care practice, the occurrence of medical errors remains a persistent and serious problem [1, 2] . The well-known Institute of Medicine's report To Err Is Human -Building a Safer Health System estimated that between 44,000 and 98,000 Americans die each year as a result of medical errors [1] . The number of patients suffering morbidities due to such errors is estimated to be much higher. A 1999 study [3] estimates that the total national cost of injuries per year due to medical errors to be at least 17 billion dollars and that preventable injuries during hospital care affect 2% of hospital patients. More recent studies support that the actual rate of medical errors may be even higher than the above estimates [4] . A 2010 report [5] estimates that 13.5% of hospitalized Medicare beneficiaries experienced adverse events during their hospitalizations and that 44% of these events were preventable, yielding a medical error rate of about 6% (0.135 Â 0.44). A study of hospitals in North Carolina estimated that the medical error rate was between about 6% (external reviewers) and 16% (internal reviewers) [6] .. The urgency and the scope of the medical errors problem have prompted the development of solutions to aid clinicians in eliminating such mistakes. Current computer tools for monitoring patients are primarily knowledge-based; the ability to monitor depends on the knowledge represented in the computer and extracted a priori from clinical experts. Unfortunately, these systems are time consuming to build and their clinical coverage is quite limited. This paper presents a new data-driven monitoring and alerting framework that relies on stored clinical information of past patient cases and on statistical methods for the identification of clinical outliers (anomalies). It provides an expanded description of the methods and results originally reported in [7] .. Our conjecture is that the detection of anomalies corresponding to unusual patient-management actions will help to identify medical errors. This outlier-based (or anomaly-based) monitoring and alerting approach can complement the use of knowledge-based alerting systems, thereby improving overall clinical coverage of alerting. In clinical sub-areas where knowledge-based alerting is not yet available, this new approach can serve as a standalone system.. Typical outlier detection methods identify unusual data instances that deviate from the majority of examples in the dataset [8, 9] . In our approach the objective is different: we want to identify outliers in a given patient's care, where individual patient-management actions depend strongly on the condition of the patient. We build upon conditional outlier detection methods [7, 10] to identify outliers in such settings. Our approach aims to identify patient-management actions for a given patient that are highly unusual with respect to past patients with condition(s) that are similar to the ones that the given patient suffers from. Once a deviation is detected, it may be used to generate a patient-specific alert for consideration by the clinician(s) who are caring for the patient.. Ideally we would like to have all statistical outliers correspond to medical errors. Hence their detection would lead to useful clinical alerts. However, in reality a patient-management action that is unusual from a statistical point of view does not always correspond to a helpful clinical alert. Nevertheless, our belief (and hypothesis) is that unusual patient-management actions that can be found by outlier detection methods will identify errors often enough to be worth alerting on. We report here our investigation of the relationship between conditional outliers and clinically useful alerts. We conducted experiments on data obtained from electronic health records (EHRs) of 4486 post-cardiac surgical patients and by having a subset of the 222 alerts generated by our approach evaluated by a panel of 15 experts in critical care medicine. Our results show that statistical outlier detection is positively correlated with clinically meaningful alerts, which provides support for our hypothesis that outlier-based alerting can be clinically useful. We proposed a new data-driven approach for identifying outliers in clinical care and for medical-error alerting. According to expert judgment, this approach is capable of identifying clinically valid outliers at reasonably high true positive alert rates across a broad range of clinical conditions and patient-management actions. The methodology has the advantage that it can be built directly from data in EHRs and does not depend on the construction of alerting models by experts. Hence it offers a complementary system to rule-based alerting systems for detecting potential medical errors.. A limitation of our evaluation study is that it was performed offline on retrospective patient cases outside of the regular clinical workflow and hence did not account for issues such as possible workflow interrupt and alert fatigue. Another, limiting factor of the study is the number of patient cases and alerts the expert reviewers could assess, which prevented us from studying many different aspects of the outlier-based alerting system. First, in order to study alerts across a broad range of actions, we limited the alert instances for each action to be among the top 125 anomalies and the top 20 alerts for that action. Studying a broader range of anomalies and alerts for each individual action could provide additional insights on how to better control action-specific alerts and their alert rates. Second, the skewed selection of alerts towards higher alert scores and towards patients with a larger number alerts (to make the review process more efficient) could also influence the obtained results. For example, patients with multiple alerts may be more likely to be hospitalized for longer periods of time and consequently may represent more complicated cases. We plan to address these limitations in future studies of the approach.. Finally, we believe many improvements of this new approach are possible. First, our current outlier models are built for each patient management action individually and do not consider multivariate relations among them; hence they may fail to identify multivariate outliers. For example, an outlier identified in the current medication order may ignore other alternative medications that could be used to substitute for it, and hence the outlier model may incorrectly generate an omission alert if the medication was substituted by an alternative. Second, our current method relies on the time segmentation of 24 h that was used both for segmenting the patient record into patient instances and for predicting actions. This segmentation resolution may lead to reduced ability to reliably distinguish and predict orders of medications with shorter time-horizon effects, such as epinephrine that is given in acute care life-support settings. Third, while our current predictive features were very good for many actions, they still failed to yield good predictive models for others. The investigation of new, more suitable feature sets that characterize complex time-series data may lead to further improvements and better coverage of patient management actions with such models. The experimental results reported above support that the proposed outlier-based methodology can generate clinically useful alerts with true alert rates ranging from 0.25 for weak alerts corresponding to weak outliers to 0.66 for stronger alerts.. Clinical alerting systems in the medical informatics literature are typically evaluated in terms of alert override rates that compare accepted and dismissed alerts [23, 24, 34, 35] . The override rates may be influenced by multiple factors, such as the frequency (or the number) of the alerts and the quality of alerts [35] [36] [37] . In general, high frequency and low quality alerts can lead to alert fatigue and subsequently to high override rates [23, 24, [34] [35] [36] . In such a case, it is possible that overrides may include both unimportant and important (or useful) alerts. The override alert rates for a variety of drug safety systems reported in the literature were in the 0.49-0.96 range [23, 24, [34] [35] [36] .. The analysis of override alert rates is typically associated with online deployed systems. Our evaluation study was conducted offline using retrospective data and hence it did not account for all aspects of the deployed alerting systems, such as alert fatigue, interruption of the workflow, and other factors. However, assuming that override rates of 0.49-0.96 in the drug safety literature closely approximate false alert rates, then the true alert rates reported in our study, which range from 0.25 (for weaker) to 0.66 (for stronger alerts), compare favorably to these numbers. Similarly, our results compare favorably to false alert rates of 0.86-0.99 reported for clinical monitoring systems in [38] .. In addition, we showed that true alert rates are positively correlated with alert scores. This suggests the adjustment (control) of the alerting system toward desired true alert rates may be possible. Current approaches for controlling the false alert (or override rates) in the deployed systems include alert prioritization or alert tiering [36, 39] . Other approaches include alert override prediction [40] . Our approach is a new direction for controlling alert system performance that uses empirically derived measures, rather than using annotated alert data.. In general, incorrect alerts identified during this study were due to information present only in text report data, interactions among medications, correlations among laboratory test orders that are part of the same panel, the use of less common alternatives to monitor or treat a patient, or the inability of the system to follow temporal patient management patterns (such as transition to warfarin from heparin in patients who have had valve replacement surgery). We believe it is likely that further improvements of the statistical models and the representation of additional information in the EHRs will lead to even higher true alert rates. As a method of improving patient care, hospitals with EHRs often employ patient monitoring and alerting systems. These systems rapidly analyze patient data streams in order to identify events or conditions that may require the attention of clinical personnel. The notifications regarding the presence of these (typically adverse) events can come in the form of reminders or alerts. One important class of alerts aims to identify potential patient management errors. Examples include alerts that are designed to identify omission of an important medication, omission of an important laboratory test, or a prescription for a medication that is not consistent with a patient's condition and care.. Current computer systems that detect errors and alert on their occurrence are typically knowledge-based. Clinical knowledge acquired from domain experts is codified and represented (typically) as rules, which are then applied to patient data to detect adverse conditions or events [11, 12] . Rule-based alerting systems have been developed for a range of medication decision support, such as drug-allergy checking, automated dosing guidelines, identifying drug-drug interactions, and detecting potential adverse drug reactions [13] [14] [15] [16] . Such systems have also been developed for other clinical tasks, including monitoring of treatment protocols for infectious diseases and detection of deviations from such protocols [17] , detection of growth disorders [18, 19] , and detection of clinically important events in the management of chronic conditions, such as diabetes [20] and congestive heart failure [21] .. Rule-based systems have several advantages. They are based on clinical knowledge, and thus, are likely to be clinically useful. The rules are easy to automate and can be readily applied to patient data that are available in electronic form. However, rule-based systems suffer from several disadvantages. The creation of rules requires input from human experts, which can be tedious and time-consuming. Rules typically have limited coverage of the large space of potential adverse events, particularly more complex adverse events. Rulebased alerting systems are rigid and difficult to tune to achieve clinically acceptable performance in the environments in which they are deployed. It is not uncommon for alert rules to be retired (turned off) due to unacceptably high rates of false alerts [22] . Even when such rules remain active, the alerts generated by them may be ignored due to high false alert fatigue [23, 24] . This paper describes the development and evaluation of a new type of alerting approach, one that is based on identifying clinical care outliers. This approach utilizes conditional outlier-based methods [7, 10] to identify patient-management actions that are highly unusual with respect to past patients with same or similar conditions. Outliers thus represent actions that are unusual and may indicate patient-management errors. The advantages of the approach are that (1) it does not require expert input to develop a detection system, (2) clinically relevant outliers are derived empirically using a large set of prior patient cases and can be continually updated to reflect usual practice patterns, and (3) alert coverage can be broad and deep. Hence, this new approach has significant potential for wide, positive impact on clinical care. Our outlier-based alerting approach consists of two stages: a model-building stage and a model-application stage (see Fig. 1 ). In the model-building stage, cases from the EHR repository are used to learn outlier models that summarize when (under what patient conditions) certain patient-management actions are typically made. This stage builds multiple outlier models to cover different patient-management actions, such as medication orders and laboratory test orders. For example, the heparin model captures patient subpopulations for which heparin is typically prescribed, subpopulations for which it is not, and subpopulations for which it is typically discontinued. In the model-application stage the outlier models are applied to new patient data to identify those actions that are unusual and deviate from the prevalent pattern of care, as represented in the outlier models. Details of the two stages are described in the following sections.

O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O 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 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 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 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 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 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 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 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


A New Method of Matrix Spectral Factorization A new algorithm of matrix spectral factorization is proposed which can be applied to compute an approximate spectral factor of any positive definite matrix function which satisfies the Paley-Wiener condition. 1 S PECTRAL factorization plays a prominent role in a wide range of fields in Communications, System Theory, Control Engineering and so on. In the scalar case arising for single input and single output systems, the factorization problem is relatively easy and several classical algorithms exist to tackle it (see the survey paper [1] ) together with the reliable information on their software implementations [2] . There are also some recent claims as to their improvement [3] . Matrix spectral factorization which arises for multidimensional systems is essentially more difficult (see Section II, where the mathematical reasons of this fact are explained). Since Wiener's original efforts [4] to create a sound computational method of such factorization, tens of different algorithms have appeared in the literature (see the survey papers [5] , [1] and the references therein), but none of them is thought to have an essential superiority over all others (see [5, p. 1077 ]; [6, p. 206] ). Besides, most of these algorithms impose extra restrictions on matrix spectral densities (e.g., to be real or rational or nonsingular on the boundary), while the Paley-Wiener necessary and sufficient condition [see (2) ] will do for the existence of spectral factorization (see Section II).. In the present paper, a new method of matrix spectral factorization developed by the first two authors for 2 2 matrices in [7] is extended to matrices of any dimension. We do not impose additional restrictions on a matrix spectral density except the Paley-Wiener condition. To describe our method of matrix spectral factorization in a few words, we can say that it carries out an approximate spectral factorization of left-upper submatrices step-by-step,. . It is shown that in this process the decisive role is played by unitary matrix functions of certain structure (see Theorem 1), which eliminates many technical difficulties connected with computation. The explicit construction of such matrices given in the proof of Theorem 1 is an essential component of the algorithm. Recently, a close relationship of these unitary matrix functions with compactly supported wavelets has been discovered, which makes it possible to construct compact wavelets in a fast and reliable way and to completely parameterize them [8] .. Preliminary numerical simulations confirm the potential of the proposed algorithm (see Section VI).. The paper is organized as follows. In the next section, an exact mathematical formulation of the problem is given. In Section III, the notation used throughout the paper is introduced. In Section IV, we provide a theoretical background of the proposed method. In Section V, the computational procedures of the new matrix spectral factorization algorithm are described, and some illustrating examples of numerical simulations are presented in Section VI. A new algorithm of matrix spectral factorization is developed, which factorizes any matrix spectral density that admits spectral factorization. The advantage of the algorithm is illustrated by the examples of numerical simulations.

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


Strongly Noetherian rings and constructive ideal theory We give a new constructive definition for Noetherian rings. It has a very concrete statement and is nevertheless strong enough to prove constructively the termination of algorithms involving "trees of ideals". The efficiency of such algorithms (at least for providing clear and intuitive constructive proofs) is illustrated in a section about Lasker-Noether rings: we give constructive proofs for the existence of the minimal primes over an ideal, of its radical, of its primary decomposition, in a wide class of polynomial rings. In the year 1890, David Hilbert published the following result (cf. Hilbert, 1890 . There exists n ∈ N such that for all i ∈ N, the polynomial f i can be written as f i = a 1 · f 1 + · · · + a n · f n where a 1 , . . . , a n are homogeneous polynomials. * Tel.: . 1 Most of this paper was written in theÉquipe de Mathématiques de Besançon, UMR CNRS 6623, Université de Franche-Comté, Besançon, France. Note that the article quoted above, as well as Hermann's and Gordan's articles quoted below, can be downloaded on the Göttinger Digitalisierungs-Zentrum, web address http://www.gdz.sub.uni-goettingen.de/.. This highly non-constructive result was difficult to accept at the time. The reaction of Gordan (authentic or not) "Das ist nicht Mathematik, das ist Theologie!" is often quoted. Indeed, Hilbert used this result in the context of invariant theory to prove the existence of a finite basis of a system of invariants, without actually providing such a basis, which was both deceiving and shocking. Later, Hilbert gave a constructive proof for this last result.. Nowadays, "Hilbert's basis theorem" refers to the following equivalent statement:. Theorem B. Let F be a field; every ideal of the ring R = F[X 1 , . . . , X d ] admits a finite basis.. In this paper, we adopt a constructive point of view in the sense of Bishop (1967) , which is in our opinion the prolongation of previous works of Kronecker or Hermann (Hermann, 1926) (an English translation is available in Hermann, 1998) .. From this point of view, this result does not hold: it is definitively impossible to give an algorithmic way to extract a finite basis from an explicitly given infinite enumeration of polynomials.. More precisely, for a given Turing machine T , define elements f i of a (non-trivial) field F as follows: f 0 = 0, f i = 1 if T stops at step i , else f i = 0. One can compute as many of the f i as wanted, so this enumeration is explicitly given. Now, to give a basis of the ideal of F generated by f 0 , f 1 , . . . means to solve the halting problem for T , which is impossible.

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A distributed framework for trimmed Kernel k-Means clustering This document is made available in accordance with publisher policies. Please cite only the published version using the reference above. Data clustering is an unsupervised learning task that has found many applications in various scientific fields. The goal is to find subgroups of closely related data samples (clusters) in a set of unlabeled data. Kernel k-Means is a state of the art clustering algorithm. However, in contrast to clustering algorithms that can work using only a limited percentage of the data at a time, Kernel k-Means is a global clustering algorithm. It requires the computation of the kernel matrix, which takes O(n 2 d) time and O(n 2 ) space in memory. As datasets grow larger, the application of Kernel k-Means becomes infeasible on a single computer, a fact that strongly suggests a distributed approach. In this paper, we present such an approach to the Kernel k-Means clustering algorithm, in order to make its application to a large number of samples feasible and, thus, achieve high performance clustering results on very big datasets. Our distributed approach follows the MapReduce programming model and consists of 3 stages, the kernel matrix computation, a novel matrix trimming method and the Kernel k-Means clustering algorithm. The objective of data clustering is to divide a given group of unlabeled data samples in subgroups (clusters), so that data samples belonging to the same cluster are similar to each other and dissimilar to data samples belonging to any other clusters. Clustering has found many applications in different scientific fields. Despite the fact that there has been an extremely rich bibliography on this subject for years now [1] , it is still an active research field.. One of the earliest clustering methods is the k-Means algorithm [2] . It is a basic textbook approach. Yet it is still popular, despite its age. It involves an iterative process, in which each data sample is assigned to the closest of the k cluster centers and then each cluster center is updated to the mean of all data samples assigned to this cluster.. The initial cluster assignment can be random. The process continues, until there are no changes, or until a maximum number of iterations has been reached. The main drawback of this approach is the fact that the surfaces separating the clusters can only be hyperplanes. Thus, if the clusters are not linearly separable, the standard k-Means algorithm will not be able to give very good results.. In order to overcome this limitation, the classical algorithm has been extended into the Kernel k-Means [3] . The basic idea behind kernel approaches is to project the data into a higher, or even infinite dimensional space. It is possible for a linear separator in that space to have a non-linear projection back in the original space, thus solving the non-linear separability issue. The kernel trick [4] allows us to circumvent the actual projection to the higher dimensional space. The trick involves using a kernel function to implicitly calculate the dot products of vectors in the kernel space using the feature space vectors. Let a i , i = 1, . . . , n be the data sample set to be clustered and x i ∈ R d , i = 1, . . . , n their d-dimensional feature vectors. If φ(x i ), φ(x i ) are the projections of the feature vectors x i and x j on the kernel space, then κ(x i , x j ) = φ(x i ) T φ(x j ) is a kernel function. Different kernel functions correspond to different projections. Finally, Euclidean distances in the kernel space can be measured using dot products. Kernel k-Means provides a popular starting point for many state of the art clustering schemes [5, 6, 7, 8] . A recent survey on kernel clustering methods can be found in [9] , while [10] presents a comparative study which supports the superiority of kernel clustering methods, over more conventional clustering approaches.. A convenient way to have quick, repeated access to the dot products without calculating the kernel function every time, is to calculate the function once for every possible combination of two data samples. The results can be stored in a n × n matrix K called the kernel matrix, where K ij = κ(x i , x j ). This means that the i-th row of the kernel matrix contains the kernel function entry for every possible sample combination that 2 includes x i . Interestingly, it has been proven that Kernel k-Means, Spectral Clustering and Normalized Graph Cuts are closely related tasks [11] . The kernel matrix can, therefore, be viewed as the weighted adjacency matrix of a full graph, whose nodes are the data samples a i and whose edge weights are the kernel function values. Obviously, when there are n data samples, the size of the kernel matrix is n × n and, therefore, grows quadratically with respect to n.. Additionally, there have been approaches attempting to take advantage of the local area information around each sample, in order to improve performance, make the kernel matrix sparse, or determine the number of clusters. It is possible to use only a small number of entries in each row of the kernel matrix, instead of the entire matrix [12] .. This can be accomplished by either working on the k-Nearest Neighbor graph, where each sample is only connected to its k closest samples [13, 14] , or only using information from samples that are sufficiently close to each other [15, 16] , which is referred to as -ball, -neighborhood or r-graph clustering. Furthermore, it is also possible to introduce additional weights to the connections between samples, based on estimating the local scaling parameter, i.e., by taking into account how densely or sparsely populated the area around each sample is [17, 18] . The Hartigan Dip Test for unimodality [19] is used in [20] , in order to determine whether a cluster should be further divided into subclusters. The Dip Test is applied for each sample, referred to as a viewer in [20] , on the relative distances between itself and other samples of the cluster. If there are viewers for which the unimodality test fails, then the cluster is further split.. An arising research trend is the so-called Big Data research. With the recent advances in technology, digital data is being generated, stored and broadcast at unprecedented rates. Digital cameras, including those in cell phones, are widely available and people all over the world are taking pictures or shooting video clips. The bandwidth of Internet Service Providers has also improved to the point where broadband connections are very common. The Web itself has been growing at ever increasing rates. The connection graphs of various social networks easily number nodes in the millions. As of late 2013, the Web itself has over 2 billion indexed pages. Bid data clustering is a challenging problem.. It has been observed that fuzzy c-means and other fuzzy clustering algorithms face 3 problems, when dealing with high dimensionality data, or large data sets [21] . Methods that involve eigenanalysis, such as the normalized graph cut approach [22] and even recent state of the art approaches [5] are also problematic. Kernel or similarity based clustering methods also have scalability issues, with respect to the required matrix calculation. Additionally, it is also possible that, even for manageable kernel or similarity matrix sizes, the data dimensionality can be in the order of several millions.. The solution to this problem usually involves sub-sampling the features of each data sample, using e.g., Random Projections [23] or Conditional Random Sampling [24] approaches. In this paper, however, we will assume that the dimensionality of the data samples is small, compared to the number of data samples. This is the case, e.g., of moderately sized image clustering and image clustering after feature extraction [25] .. Since the kernel matrix size is of O(n 2 ), while time complexity is still an issue, it is the memory requirements that make it impossible to use Kernel k-Means to cluster datasets of this magnitude on average PCs, or even single high-end single machine systems. One way to work around this problem is provided by the Approximate Kernel k-Means algorithm [25] . Instead of using the full kernel matrix, only a user defined set of rows are calculated and used to measure distances and perform the clustering task.. Distributed computing can provide the means to handle problems on very large datasets that would otherwise be almost impossible to solve [26] . It provides virtually limitless memory and processing power. Provided that a task can be split into many independent subtasks, then it can theoretically be performed in a reasonable amount of time, regardless of the data size, given enough processing units. A distributed approach that can work with any serial clustering algorithm entails using the serial algorithm on data subsets, then merging the clusters [27] . Distributed versions of other clustering algorithms related to Kernel k-Means, like classic k-Means [28] and k-Medians [29] have already been discussed. However, to the best of our knowledge, a distributed approach to Kernel k-Means has not been proposed yet.. In this paper, we propose a distributed implementation of the Kernel k-Means clustering algorithm. We follow the MapReduce programming model [30] , which is a high level framework for distributed processing on a computing cluster. The implementation uses Apache Spark [31] , a cluster computing framework, which is similar to and compatible with Hadoop [32] . The computing cluster can include a wide variety of hardware from high-end, multiprocessor computers with large amounts of RAM, to average modern PCs. The focus of the proposed implementation is to avoid requiring the storage of n 2 kernel matrix entries into the distributed memory at the same time, if possible. In order to achieve this goal, we employ a novel kernel matrix trimming algorithm, which enables us to significantly reduce the number of non-zero entries in the kernel matrix. This allows us to use memory-saving adjacency lists, instead of the full matrix, while also increasing clustering performance. The proposed distributed clustering scheme is divided into three major parts: kernel matrix computation, kernel matrix trimming algorithm and, finally, Kernel k-Means itself. The application used to perform the experimental evaluation of the proposed approach is image clustering.. The paper is organized as follows. Section 2 provides a brief introduction to the Kernel k-Means algorithm. Section 3 details the novel kernel matrix trimming algorithm. Section 4 describes the distributed computing approach to all the relevant algorithms. Section 5 presents the experiments carried out to evaluate the performance of the proposed method and study the scalability of its distributed implementation. Section 6 concludes the paper. In this paper, we have proposed a novel kernel matrix trimming algorithm that both improves the performance of Kernel k-Means, while significantly reducing the kernel matrix size. Through a voting scheme, we are able to estimate the cardinality of the cluster that each individual data sample belongs to. This provides a threshold, using which we can separate within cluster kernel matrix entries, i.e., entries connecting the data sample to other samples of its cluster, from between cluster entries, i.e., entries connecting the data sample to samples of other clusters. During the justification of our motivation, we also provided some insight into how cluster density in the kernel space can affect the assignment of data samples to particular cluster types, depending on their density.. Experimental results strongly indicate that our approach consistently provides an improvement over baseline Kernel k-Means, which is not possible by the static kernel matrix trimming, despite the fact that is does not use ground truth knowledge. Our approach is even almost equivalent in clustering accuracy to a state of the art face clustering approach, which though takes advantage of video tracking trajectory-derived constraints.. Additionally, we have provided a distributed implementation of all three steps of the proposed Trimmed Kernel k-Means framework. The distributed implementation is designed to minimize memory usage, thus ensuring that either the entire problem can fit inside the distributed memory, or that the spilling of data to the disk is also minimized. The running times that were recoded while deploying our implementation to clusters of different numbers of cores reasonably follow the rectangular hyperbola curve with respect to the number of cores. In this section, we will present the results of the conducted experiments, in order to evaluate the clustering performance and speed of the proposed distributed clustering framework. Initially, we used the MATLAB implementation of the methods involved.. However, for the really big datasets, we used the distributed implementation described above. The first set of experiments concerns the direct comparative evaluation of our kernel trimming algorithm against baseline and Approximate Kernel k-Means [25] , in terms of both clustering performance and kernel matrix size reduction. The second set of experiments concerns a comparison with a state of the art facial image clustering approach. The third set of experiments was conducted, in order to study the run time speed up of our method, vs the number of available computing cores.

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 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 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 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 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 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 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 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 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 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 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 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 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 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 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 O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O 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 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 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 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


Distinguishing the rates of gene activation from phenotypic variations Background: Stochastic genetic switching driven by intrinsic noise is an important process in gene expression. When the rates of gene activation/inactivation are relatively slow, fast, or medium compared with the synthesis/degradation rates of mRNAs and proteins, the variability of protein and mRNA levels may exhibit very different dynamical patterns. It is desirable to provide a systematic approach to identify their key dynamical features in different regimes, aiming at distinguishing which regime a considered gene regulatory network is in from their phenotypic variations. Results: We studied a gene expression model with positive feedbacks when genetic switching rates vary over a wide range. With the goal of providing a method to distinguish the regime of the switching rates, we first focus on understanding the essential dynamics of gene expression system in different cases. In the regime of slow switching rates, we found that the effective dynamics can be reduced to independent evolutions on two separate layers corresponding to gene activation and inactivation states, and the transitions between two layers are rare events, after which the system goes mainly along deterministic ODE trajectories on a particular layer to reach new steady states. The energy landscape in this regime can be well approximated by using Gaussian mixture model. In the regime of intermediate switching rates, we analyzed the mean switching time to investigate the stability of the system in different parameter ranges. We also discussed the case of fast switching rates from the viewpoint of transition state theory. Based on the obtained results, we made a proposal to distinguish these three regimes in a simulation experiment. We identified the intermediate regime from the fact that the strength of cellular memory is lower than the other two cases, and the fast and slow regimes can be distinguished by their different perturbation-response behavior with respect to the switching rates perturbations.We proposed a simulation experiment to distinguish the slow, intermediate and fast regimes, which is the main point of our paper. In order to achieve this goal, we systematically studied the essential dynamics of gene expression system when the switching rates are in different regimes. Our theoretical understanding provides new insights on the gene expression experiments. In this paper, we proposed a methodology to distinguish the regime of genetic switching rates from phenotypic variations in a simulation experiment. To accomplish this task, we put forward different reduction and investigation methods to reveal the essential dynamics of the genetic switching system when the switching rates are relatively slow, fast, or medium compared with the degradation rates of proteins. Although in the fast and slow regimes, the observed steady state distributions of the mRNAs and proteins are similar, we showed that the underlying mechanisms are qualitatively different. Based on our results, we provided a simulation experiment to distinguish these three regimes. The intermediate regime can be identified from the fact that the strength of cellular memory is lower than the other two cases, and the fast and slow regimes can be distinguished by their different perturbation-response behavior with respect to the switching rates perturbations. We also discussed the robustness of our results with respect to different choices of parameters and feedback function forms. This study provides insights to the experimental analysis of the gene expression system. It will be interesting to investigate similar problems in different experimental setups in the future. Understanding the dynamical features and the energy landscapes of the gene regulatory network is an interesting and challenging topic in quantitative biology. Some approaches focused on the regime of fast switching rates have been developed in recent years, whereas the slow regime which many eukaryotic cells correspond to attracts less attention. In this paper, besides the theoretical analysis of the essential dynamics in the slow, intermediate and fast regimes, we provide a method to distinguish these three regimes. The result is meaningful and enlightening for the design of corresponding biological experiments.. With regard to a previous work about stochastic dynamics of single regulating genes [19] , we provide a systematic study to understand the genetic switching dynamics when the switching rates are in different regimes. Although the global quasi-potential energy landscape and other properties of the system can be obtained from brutal force through numerical simulations, the theoretical reductions can reveal the intrinsic mechanisms. We analyze the model theoretically in the slow and fast regimes and successfully reconstruct the global energy landscape which reflects the inherent pattern of system behavior. But we obtain little theoretical results in the intermediate regime, since the nonexistence of time-scale separation makes this problem more challenging. Recent new ideas proposed in this field may be helpful in the study of this regime [36, 37] .. A related but different concept with multi-layer landscapes for the DNA fast switching case was mentioned in a recent literature [36] . However, we are mainly concerned with the two-layer case when the system is in slow regime. Besides understanding the system as decoupled layers, we further study the properties about transition paths and average switching time, and we construct the global potential energy landscape through simple Gaussian mixture and modified Gaussian mixture approximations. These pictures are quite different from the case of fast regime.. The MST is a key characterization of gene regulatory networks and has been studied analytically with respect to different genetic switching rates in [38] . Compared with the detailed theoretical analysis for a typical onedimensional self-activating switch model, we can not give fully theoretical results in the fast regime, i.e. the strictly adiabatic regimes considered in [38] , since our model involves both mRNA and protein explicitly. However, we present efficient numerical methods to make the quantitative estimates. The results for the extremely slow regime are also consistent with those obtained for non-adiabatic regime (i.e. Eq. 7). We also suggested the landscape as an intuitive tool to understand the essential dynamics of our model. Overall, the shape of the MST curve we obtained as a function of κ is qualitatively similar as in [38] as well.. The proposed quasi-potential energy function S(x) in the fast regime has close connection with the potential function considered in previous studies. In Wang et al's approach [14, 39] , they proposed to define the energy landscape as U(x) = −ln P ss (x), where P ss (x) is the steady state distribution. We have the connection that S(x) = lim V →∞ V −1 U(x) [15, 40] since U(x) itself depends on the system size V. We also note that the proposed quasi-potential S(x) satisfies the same Hamilton-Jacobi equation as the potential defined in [41, 42] in the diffusive approximations. But in addition, we provided a variational minimization approach to explicitly construct the S(x). Nonetheless, this construction is only valid for the fast regime and it is quite different from the picture we showed in the slow regime.. Although our four-state genetic switching model is based on chromatin remodeling, this is no longer expressed in the final simplified two-state model. In fact, some prokaryotic cells also have similar gene regulatory networks like Fig. 1b [17] . Our study mainly focuses on the two-state model, which can be suitable for a wide variety of cellular systems rather than the motivating chromatin remodeling process. Our simulation experiment to distinguish the slow, intermediate and fast regimes is an important point of the paper. So far, we are not able to perform real experiments to make the verification. The real experimental systems may be also more complicated than the model considered here. But our simulation experiment is logically true and self-consistent, and there have been related experiments performing well in Th2 cells [16] . Finding suitable system and validating our proposal will be an interesting task in the future.. Our study has potential application in single-cell experiments [43] [44] [45] . In the single-cell gene expression analysis, the protein level of each cell at different time points can be obtained through tracking and recording the behavior of each cell by fluorescence microscopy. With these data, one can obtain the switching time as the time difference between two consecutive states of a cell. Thus the MST can be roughly estimated. Similar as the procedure in flow cytometry experiments, the information about MST can be utilized to identify the intermediate regime, and the fast and slow regimes can be distinguished by perturbing κ. However, these points remain to be investigated in detail in the follow-up researches. Stochasticity is an inherent property of living cells. Stochastic perturbation comes not only from the noisy environment that cells live in, but also from all processes going on inside cells [1] . Especially when the copy number of molecules like mRNA or protein gets smaller, stochastic fluctuations can have significant effect on the *Correspondence: tieli@pku.edu.cn 1 LMAM and School of Mathematical Sciences, Peking University, No. 5. Yiheyuan Road, 100871 Beijing, China Full list of author information is available at the end of the article behavior of a cell. There have been many studies focused on discovering the effect of noise in living systems [2, 3] , and some of the results show that cells do not only evolve to resist noise perturbation better, but also form massive mechanisms which are the benefits of stochasticity [4, 5] . Genetic switching is a typical example of noisedriven cellular behavior which can be performed spontaneously under particular conditions in living cells, such as the switch of lac operon in Escherichia coli [6] , the formation of biofilm [7] , and the entrance into competence state [8] in Bacillus subtilis, etc.. Several recent studies discovered strong connections between noise and gene regulation mechanisms through measuring variability in protein and mRNA levels [9, 10] . Both experimental and theoretical results suggested that the switching rates of gene between its active and inactive states have significant effects on the noise of gene expression [11, 12] and are highly responsible to different gene expression dynamics. Previous theoretical studies mainly focused on the regime with fast switching rates [13] [14] [15] , however, recently some unique phenomena are observed especially when switching rates are slow, either due to the chromatin remodeling precess [16] or the transportation of transcription factor into nucleus [17] . Thus, it is important and desirable to provide an approach which can effectively identify the key dynamical features of gene expression system when switching rates are slow, and to distinguish the regime of gene regulatory networks from different phenotypic variations.. In this paper, we aim at providing a method to distinguish the rates of genetic switches from phenotypic variations. First, we systematically investigated the dynamical and stochastic properties of different genetic regulatory models in a comparative way when biochemical rates for gene activation/inactivation, transcription, translation, and mRNA/protein degradation vary over a wide range. We focused on a two-state gene-expression model with positive feedback loop, which is simplified from a four-state gene-expression network taking account of chromatin opening/closing step and regulatory protein binding/unbinding step [16, 18] . In the regime when the rates of gene activation/inactivation are relatively small compared with the synthesis/degradation rates of mRNAs and proteins, we found that the effective dynamics can be reduced to independent evolutions on two separate layers. We reconstructed the energy landscape of the system through Gaussian mixture approximations, and it is qualitatively different from the previously known picture in [15, 19] . In the regime with relatively fast switching rates, we briefly discussed about the system based on our previous study with large deviation theory [15] . In the regime with intermediate switching rates, we utilized the mean switching time (MST) to investigate the stability of the system in different parameter ranges. In the following paragraphs, we will call these three regimes as slow, fast, and intermediate regimes for convenience. Finally, based on our theoretical understanding of the essential dynamics of the slow, intermediate and fast regimes, we designed a simulation experiment to distinguish the three regimes. We identified the intermediate regime from the fact that cellular memory is much weaker in this regime than the other two regimes. And we distinguished the slow and fast regimes through their different response behavior with respect to switching rates perturbations. Our study gives a global characterization of the genetic switching dynamics in various cells and has potential guidance in real experiments. We consider a four-state gene-expression model with positive feedback loop ( Fig. 1a ). Proteins are self-activators in this system, since they can bind to DNA and activate the transcription step. Chromatin opening becomes relatively easy when DNA is bounded by a monomer regulatory protein. Open chromatin structure is permissible for transcription whereas closed chromatin is not. The rate of mRNA production reaches its maximum on the bounded and open state of DNA. Additionally, proteins and mRNAs are subject to degradation, which is not shown in Fig. 1a . Assuming that binding and release of regulatory proteins are in fast equilibrium [20, 21] , this four-state model can be simplified into a two-state model ( Fig. 1b ) through the quasi-steady-state approximation (QSSA) (see Additional file 1 for derivation details) [22, 23] . We will say that the gene is open (closed) when the chromatin is open (closed), and the system switches between open and closed gene states through gene activation/inactivation steps.. In the two-state model, there are six reactions occurring among four chemical species. All of the reactions involved in this system are summarized in Table 1 . Here m and n are the numbers of protein molecules and mRNA molecules, respectively. The jump rate into the open state and the transcription rate are controlled by the relative occupancy of protein at DNA binding site (i.e. O P ), whereas the jump rate into the closed state is not affected by the number of proteins. O P has the form O P = n/(n + K), where K is a constant by reduction and represents the strength of positive feedback. It is actually the conditional probability of state (iv) in Fig. 1a given that the chromatin is open. The rates k G O P + k G0 and k R O P + k R0 characterize the gene activation and mRNA synthesis, which are dependent of the number of proteins. The constants d G , k P , d R and d P correspond to the rates of gene inactivation, protein synthesis, mRNA and protein degradation. In fact, if we assume that dimer proteins (or other multimer-type of proteins) regulate the gene instead of the monomer protein, O P will have the form O P = n k /(n k + K k ). The value of k will not influence the essential dynamics of the gene expression model, on condition that we choose proper range of parameters with respect to different k. Under some reasonable assumptions, most of the derivations and results in the next section are still correct (see Additional file 1 for details).. Due to the intrinsic noise, the dynamics of the system can be described by the chemical master equation (CME) [1, [24] [25] [26] [27] as . where P α (m, n) stands for the probability that there are m mRNA molecules and n protein molecules in the system when the gene is open (α = 1) or closed (α = 0). One possibility to study the behavior of the system is to numerically solve Eqs. 1-(2) by truncating the domain and setting the boundary conditions P α (m, n) = 0 when m M, n N (M, N is large enough, α = 0, 1). But simply obtaining this solution does not mean we understand the essential mechanism of the dynamics, and the computational effort may be huge when the mean number of molecules is quite large in this system. Thus, we aim to find other ways to learn about the dynamics.. We define a key parameter κ to compare two characteristic timescales of the problem: the timescale d −1 P on which the gene state changes and the timescale d −1 G on which the number of protein and mRNA molecules changes, where d −1 P is the lifetime of a single protein and d −1 G is the lifetime of the open gene state. The ratio κ = d G /d P can take any positive value, which will lead to notable differences in the dynamics. Small ratios (κ 0.01) describe long timescales of the change in gene states, compared with the timescale on which mRNA and protein numbers change. In this regime, there is enough time for the system to reach a steady state when gene is either open or closed. Transition from one steady state to another always occurs immediately after the genetic switches. Large values of the ratio (κ 1) indicate that the gene states change very rapidly, whereas the synthesis and degradation of protein and mRNA molecules are relatively slow. As κ goes to infinity, the system exhibits the deterministic features [13, 15] . When κ is neither large nor small enough, genetic switches, transcription and translation will take place on the same timescale. The dynamics is complicate in this regime, and analytical results are hard to reach. So we resort to numerical simulations and extract useful information. In all of the three regimes, metastable states may occur with proper choice of parameters. Indeed, we will utilize a set of biologically relevant parameters in later computations (see Additional file 1 for details). For convenience, we will call the steady state with relatively large number of proteins as on state, whereas the steady state with relatively small number of proteins is off state.

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


GSM: gateway selection mechanism for strengthening inter-cluster coordination in cognitive radio ad hoc networks In the open access spectrum paradigm, the dynamics of the available spectrum heterogeneity due to the geographical location of various cognitive radio/secondary users (SUs) and primary users (PUs) prevent the assignment of a single common control channel to every SU. Thus, depending on channel availability, SUs are forced to form clusters to solve the problem of control channel assignment. However, the sudden appearance of a PU may lead to loss of intra-cluster or inter-cluster connectivity among SUs. This problem can be resolved by improving the cluster formation process and selecting suitable gateway nodes for inter-cluster coordination. In this paper, our goal is to improve inter-cluster connectivity by selecting more appropriate gateway nodes for inter-cluster coordination. We therefore propose an inter-cluster gateway node selection mechanism that strengthens inter-cluster coordination. Our scheme shows significant improvement in terms of inter-cluster connectivity compared with related work. Cognitive radio (CR) networks have gained much attention in the research community in the last decade and have emerged as a promising approach to solving the issues of spectrum scarcity [1] . CR is the key technology underlying the dynamic spectrum access (DSA) paradigm. In DSA, CR networks are overlaid over the licensed networks, also known as primary networks, for opportunistic spectrum access [2] . Consequently, in a network that utilizes DSA, users are classified as primary if they are licensed to operate in a particular frequency band; and secondary if otherwise. Secondary users (SUs) can operate in the vacant portion of the licensed band only if they do not interfere with the primary users (PUs) [3] . When a cognitive radio ad hoc network (CRAHN) is set up, SUs need to cooperate by exchanging control information to achieve reliability in seeking vacant frequency bands. As a result, the establishment of a control channel is essential for exchanging control information [1] [2] [3] . A comprehensive survey of control channels in CR networks is given in [4] .. When setting up a CRAHN, a control channel is essential in the carrying out of functions such as spectrum sensing, sharing, decision, and mobility as all these require extensive control information exchange [1] . Several techniques have been proposed in the literature pertaining to the design of control channels for CRAHNs. Most of the proposed techniques can be divided into two categories: (1) dedicated common control channels and (2) group/cluster-based control channels. Using a dedicated common control channel simplifies the process of coordination among SUs. However, reliance on a dedicated common control channel results in several limitations. Firstly, there is a single point of failure; for example, a dedicated common control channel can result in loss of connectivity among SUs in cases of saturation or jamming. Secondly, in the case of CRAHNs, the dynamically changing availability of the spectrum in the temporal and spatial domain makes it difficult to maintain a dedicated common control channel that is globally available.. Two major challenges arise in the assignment of a dedicated control channel for CRAHNs, namely, PU activity and spectrum heterogeneity. With regard to PU activity, in a CRAHN, a PU has higher priority than an SU. If an SU is operating on a particular frequency band and a PU comes back to that frequency band, the SU has to vacate that channel in favor of the PU and migrate to some other unused frequency band. Thus, in environments where PU activity is not known, it is impossible to guarantee that the particular frequency band of the designated control channel will always remain available in the presence of PU activity. With regard to spectrum heterogeneity, every SU observes a different set of available channels depending on its geographical location, each of which is a subset of the set of all the licensed channels. Due to this spectrum heterogeneity, very rarely can a channel commonly available to all SUs at a particular instance be found. To account for PU dynamics and spectrum heterogeneity issues, several researchers have proposed cluster-based approaches for control channel design of CRAHNs [5] [6] [7] [8] . In cluster-based control channel schemes for CRAHNs, a neighbor discovery (rendezvous) process is performed to set up an initial network topology. Based on the information exchanged during the rendezvous process, the nodes are partitioned into clusters depending on highly correlating spectrum opportunities. A comprehensive survey on rendezvous process for cognitive radio is presented [9] . After clustering, one or more channels that are common to nodes within each cluster are designated as the control channels.. In [9] , Zhao et al. proposed a distributed cluster-based approach for control channel design in CRAHNs. This scheme partitions the nodes into clusters, with the largest possible number of nodes in each cluster. This method reduces the number of distinct frequencies used for control channels thereby incurring less cluster management overhead. However, the scheme requires frequent re-clustering in response to PU activity. Another similar approach for clustering CR networks is proposed in [10] .. To avoid frequent re-clustering in response to PU activity and spectrum heterogeneity, the concept of backup control channels has been introduced. Bahl et al. [11] proposed a control channel scheme for centralized CR networks in which one main channel and one backup channel are designated as control channels. The main and backup channels are then interchanged according to the spectrum dynamics incurred due to PU activity. In [12] , Liu et al. extended this concept of backup control channel in CRAHNs by proposing a cluster-based control channel allocation scheme called spectrum opportunity-based clustering (SOC). Instead of making the cluster with the largest number of nodes as proposed in [13] or with just one backup channel as proposed in [11] , they proposed that each cluster be formed in such a way that the product of the number of common channels among cluster members and the number of cluster members in a cluster are maximized. In other words, clusters are formed in such a way that a balance is created between the cluster size and the number of common channels.. By maximizing the number of common channels among the nodes within a cluster, intra-cluster connectivity is strengthened and frequent re-clustering averted. However, the overall connectivity of a cluster-based CRAHN includes intra-cluster as well as inter-cluster connectivity. Thus, in this study, our focus is on inter-cluster connectivity, and we carry forward and enhance the concept introduced by Liu et al. [12] to improve inter-cluster connectivity. The inter-cluster connectivity is strengthened by designating more channels as control information between adjacent clusters.. Generally, a clustered ad hoc network consists of three kinds of nodes: cluster head nodes, gateway nodes, and ordinary nodes. Cluster head nodes are vested with the responsibility of managing the cluster. For example, in the case of a cluster-based control channel design for a CRAHN, the cluster head node is responsible for allocating a set of control channels out of the set of channels available to every member of that cluster. The coordination between two adjacent clusters is conducted through the gateway nodes. A node of a cluster can act as a gateway only if it has a neighboring node as well as a channel in common with the designated control channel of its neighboring cluster. All nodes other than the cluster head node and gateway node are ordinary nodes. Both gateway nodes and ordinary nodes are also known as cluster members and are managed by the cluster head node.. As mentioned earlier, in a cluster-based CRAHN, the gateway node is responsible for coordination between adjacent clusters. It addition, it is well known that inter-cluster connectivity becomes stronger the more channels there are between adjacent clusters in the case of CRAHNs. There can be nodes in a cluster that have one or more channels in common with the set of channels common among nodes in the adjacent cluster. Any of these nodes can be a gateway node candidate. However, the node with the maximum number of channels in common with the control channels in the adjacent cluster is the most suitable option because selecting such a node as a gateway strengthens the coordination with the adjacent clusters. Therefore, in this paper, we propose a mechanism for selecting gateway nodes and analyze its impact on the connectivity of CRAHNs.. The remainder of this paper is organized as follows: section 2 presents the system model used for our simulated environment. Section 3 discusses our proposed framework and describes the algorithm proposed for the selection of the gateway node for inter-cluster coordination in CRAHNs. Section 4 shows the impact of variations in SU and PU node densities on the proposed algorithm by presenting simulation results. Finally, section 5 concludes the paper. In this study, we examined the control channel establishment problem in CRAHNs. More specifically, we looked at the effect of spectrum heterogeneity and PU dynamic force in the formation of a group-based control channel assignment for CRAHNs. To avoid frequent reclustering, SOC nodes in such a way that more backup channels are available in each cluster. Whenever a channel gets occupied by a PU, instead of re-clustering, control information exchange migrates from the occupied channel to one of the other backup channels. We extended this concept from the intra-cluster level to the inter-cluster level. We surmised that inter-cluster coordination can be strengthened if more channels are between the adjacent clusters and proposed a technique for gateway node selection that ensures that more channels are available for control channel establishment between adjacent clusters. Our proposed approach provides significant improvement over SOC in terms of the average number of inter-cluster control channels.

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


Diffusion Tensor Imaging of the Spinal Cord Diffusion tensor imaging (DTI) provides a measure of the directional diffusion of water molecules in tissues. The measurement of DTI indexes within the spinal cord provides a quantitative assessment of neural damage in various spinal cord pathologies. DTI studies in animal models of spinal cord injury indicate that DTI is a reliable imaging technique with important histological and functional correlates. These studies demonstrate that DTI is a noninvasive marker of microstructural change within the spinal cord. In human studies, spinal cord DTI shows definite changes in subjects with acute and chronic spinal cord injury, as well as cervical spondylotic myelopathy. Interestingly, changes in DTI indexes are visualized in regions of the cord, which appear normal on conventional magnetic resonance imaging and are remote from the site of cord compression. Spinal cord DTI provides data that can help us understand underlying microstructural changes within the cord and assist in prognostication and planning of therapies. In this article, we review the use of DTI to investigate spinal cord pathology in animals and humans and describe advances in this technique that establish DTI as a promising biomarker for spinal cord disorders. and subsequent improvements in this technique have led to the development of DTI as a tool to delineate white matter tracts in the brain.DTI of the spinal cord in humans was initially inadequate because of the small area of the cord, susceptibility artifacts, and cardiac and respiratory motion artifacts.3,4 Improvements in scanning protocols have allowed usable diffusion images of the spinal cord. Spinal cord DTI, initially performed in animals, is now used to evaluate spinal cord disorders in humans. Investigators have shown that DTI is able to detect cord damage in regions of the cord that appear normal on T2-weighted images. 5, 6 Spinal cord DTI therefore represents an important advancement in the field of neuroimaging, and its use is being expanded both for prognostication and for guiding therapy.In this article, we review the literature on spinal cord DTI in both animal models and humans. We provide a summary for the clinical use of spinal cord DTI in a few neurosurgical conditions. We hope that by providing a review of the current status of spinal cord DTI, we may be able to better direct future efforts in this field.Diffusion MR imaging (MRI) provides a measure of the displacement of water molecules in tissues. Displaced water molecules produce an attenuated signal during diffusion MR scanning. By its nature, the axonal architecture in the white matter of the central nervous system promotes diffusion of water molecules in a direction predominantly parallel, rather than perpendicular, to axon fibers. 2, 7, 8 Diffusion perpendicular to the fibers seems to be limited by cell membranes DTI provides unique insight into the pathophysiology and microstructural alterations associated with spinal cord disorders. Although initial studies in rat models have primed this modality for human research, more data are required on the accuracy and reliability of DTI indexes in defining cord pathology. DTI of the spinal cord shows promise in certain neurosurgical conditions such as traumatic SCI, cervical spondylotic myelopathy, and spinal cord tumors. However, scanning protocols and image processing need to be refined and standardized. Once these challenges are overcome, we can expect the use of DTI in mainstream clinical practice both to prognosticate and to monitor patients with spinal cord disease.

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Field Inhomogeneity Correction Based on Gridding Reconstruction for Magnetic Resonance Imaging Abstract-Spatial variations of the main field give rise to artifacts in magnetic resonance images if disregarded in reconstruction. With non-Cartesian k-space sampling, they often lead to unacceptable blurring. Data from such acquisitions are usually reconstructed with gridding methods and optionally restored with various correction methods. Both types of methods essentially face the same basic problem of adequately approximating an exponential function to enable an efficient processing with fast Fourier transforms. Nevertheless, they have commonly addressed it differently so far. In the present work, a unified approach is pursued. The principle behind gridding methods is first generalized to nonequispaced sampling in both domains and then applied to field inhomogeneity correction. Three new algorithms, which are compatible with a direct conjugate phase and an iterative algebraic reconstruction, are derived in this way from a straightforward embedding of the data into a higher dimensional space. Their evaluation in simulations and phantom experiments with spiral k-space sampling shows that one of them promises to provide a favorable compromise between fidelity and complexity compared with existing algorithms. Moreover, it allows a simple choice of key parameters involved in approximating an exponential function and a balance between the accuracy of reconstruction and correction. The efficient reconstruction of non-Cartesian acquisitions faces the problem of a nonequispaced sampling in the spatial frequency domain. It is adequately solved by both the adjoint NFFT and gridding reconstruction, which rely on the same approximation of an exponential function (A-4). Field inhomogeneity, however, introduces an exponential function with irregular sampling in both domains. We showed that (A-4) can be generalized to this case, leading to the similar approximation (B-1). In this way, the field inhomogeneity correction and the reconstruction of non-Cartesian acquisitions may be founded on the same basic approximation. Based on this concept, we derived three new algorithms in the present work.. Among the gridding-based algorithms, the 3-D NNFFT approach shows the highest error. We attribute this to the unnecessary interpolation for all spatial dimensions. As already pointed out in [6] , gridding remains an approximation even if the samples are equispaced. The usual discretization of images on a Cartesian grid should, therefore, be exploited. The inferior accuracy of the 2-D 1-D NFFT-F approach compared to the 3-D The results in Fig. 3 underline the advantage of using a local linear combination in conjunction with a weighting in the transformed domain instead of a local linear combination only, since the accuracy of the gridding-based approach is substantially superior to that of the Hanning interpolation from a certain minimum number of segments on. It then also exceeds the accuracy of the Man interpolation, which may be considered as a nonoptimal, global linear combination. Moreover, for a calculable, slightly higher number of segments, the 2-D NFFTs limit the achievable overall accurancy, thus equalizing it for the gridding-based approach and the least squares interpolation. The results in Table III highlight the relevance of the amount of data to be regridded for the complexity. As in gridding reconstruction, the use of a local neighborhood in the interpolation appears to be advantageous. However, the primary benefit is in this case the reduced effort for the 2-D NFFTs rather than for the linear combination. Previously, it has been proposed to eliminate most of the regridding from both the direct CPR [12] and the iterative algebraic reconstruction [15] . While a detailed comparison remains to be done, the apparent advantage is often offset by either a restriction on the supported k-space trajectories or the requirement of higher oversampling factors to avoid excessive backfolding, as demonstrated for a related problem in parallel imaging [16] .. The required number of segments also has a considerable influence on the complexity. For the gridding-based approach, the application of the well-known concept of oversampling permits a simple choice of an adequate number of segments. Moreover, the unified treatment of reconstruction and correction enables a balance of the accuracy of both. Thus, the accuracy of one is not increased beyond the limit set by the other. The same choice may, in principle, be used for the least squares interpolation. Fig. 3 suggests, however, that the error flattens out with fewer segments compared to the gridding-based approach. While this observation may be exploited to further reduce the complexity especially for the direct CPR in this particular case, it is not necessarily generalizable.. Finally, the effort involved in the initialization affects the complexity. It is mainly determined by the calculation of the weights for the linear combination from a given field map. Those approaches which require little such effort are clearly preferable for small number of iterations. The Hanning and the Man interpolation, but also the gridding-based approaches are among these, since they solely need to evaluate a given function. By contrast, the considerably higher effort for the least squares interpolation usually pays off for large number of iterations only.. The framework we chose for field inhomogeneity correction assumes a piecewise constant field map. While this is usually a sufficiently accurate model, a more precise one is obtained by also taking intravoxel gradients into account. These lead to a distortion of the actual k-space trajectory, which varies with the spatial position. For a more detailed description of the problem and a potential solution, the reader is referred to [17] . The framework we proposed for reconstruction includes a weighting in the linear system of equations to be solved. This weighting entails in theory a signal-to-noise (SNR) penalty [5] . In our experience, however, the loss in SNR was insignificant and was far outweighed by the acceleration of convergence it provided. APPENDIX A In this appendix, the NFFT and its adjoint, i.e., the nonuniform FFT of type 1 and type 2 [9] , are outlined for the one dimensional case, and their close relation to gridding reconstruction is highlighted. For a more detailed description, the reader is referred to [18] . where has been replaced by . The latter is the one periodization of a truncation of defined by. The support of is determined by , the so-called kernel size. Typically, is chosen such that . The truncation thus reduces the complexity of the evaluation of the right-hand side of (A-4) considerably.. The NFFT, i.e., the nonuniform FFT of type 2, evaluates the trigonometric polynomial , where is a diagonal matrix with entries , an oversampled Fourier matrix, which includes the factor , and a sparse matrix with entries . The adjoint NFFT, i.e., the nonuniform FFT of type 1, evaluates the sum for given nonequispaced samples at given equispaced positions . It may be considered as a multiplication of with . As pointed out in [7] , [18] , gridding reconstruction is simply a fast algorithm for the application of to a vector of nonequispaced samples. Including a sampling density compensation, it performs the following [6] 1) a weighting of the data, i.e., a multiplication with a diagonal matrix , 2) a convolution with a window function and a resampling to an oversampled Cartesian grid, i.e., a multiplication with , 3) an inverse FFT, i.e., a multiplication with , 4) a deapodization, i.e., a multiplication with . Obviously, the adjoint NFFT and gridding reconstruction rely on the same approximation and involve very similar processing. Moreover, the Kaiser-Bessel window, or its Fourier transform, is a particularly good choice for in both cases [18] , [19] . More sophisticated approaches based on scaling vectors [20] , a minimization of the Frobenius norm of certain error matrices [21] , or a min-max interpolation [22] did not prove significantly superior.

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