MD a ; Anna Taieb

MD b ; Florence Jeune

MD, PhD Said Lebbah

MD Matthieu Schmidt

MD Romain Deransy

; Vaillant

MD Charles-Edouard Luyt

MD, PhD Christophe Trésallet
email: christophe.tresallet@aphp.fr

MD Alain Combes

MD Nicolas Bréchot

Emergency abdominal surgery outcomes of critically Ill patients on ExtraCorporeal Membrane Oxygenation: A case-matched study with a propensity score analysis

Keywords: digestive surgery extracorporeal membrane oxygenation, venoarterial, venovenous, noncardiac surgery, bleeding, hemostasis

Background: Extracorporeal membrane oxygenation (ECMO) induces coagulation disorders increasing the risk of bleeding during invasive procedures. We aimed to describe the outcomes of critically ill ECMO patients undergoing emergency abdominal surgery compared to nonECMO patients.

Study Design: Retrospective case-matched single-center study with propensity score analysis in a tertiary ICU. All patients who underwent abdominal surgery were included.

Results:

From 2006 to 2014, 77 patients admitted in our ICU underwent emergency abdominal surgery, 35 were on ECMO. Surgery indications were comparable for ECMO and nonECMO patients: mostly intestinal ischemia (42%) and cholecystectomy (25%). Postoperative bleeding was significantly more frequent in ECMO group vs. nonECMO: 77% vs 40% transfused, with medians of 13 (6-22) vs. 3 (0-5) packed red blood cell; 9 (3-17) vs 0 (0-4) fresh-frozen plasma and 12 (3-22) vs 0 (0-8) platelet units (p<0.001 for all items). Reintervention for haemorrhage was required in 20% vs 2% respectively, p = 0.02. At multivariable analysis, ECMO was strongly associated with bleeding (OR, 5.6 [95% CI,]; p = 0.001). ICU mortality was higher for ECMO-treated patients (69% vs. 33%; p = 0.003) but perioperative mortality remained comparable between groups (11% vs. 12%, NS). More frequent and severe bleeding was confirmed for highly comparable ECMO and nonECMO patients (21 each), propensity score-matched to be ECMO-implanted.

Conclusions: Abdominal surgery procedures on ECMO-treated patients are associated with a higher risk of haemorrhage compared to nonECMO ICU patients. Further studies are needed to optimize ECMO-patient management during such interventions.

ABBREVIATIONS

ARDS, acute respiratory distress syndrome ECMO, extracorporeal membrane oxygenation PRBCs, packed red blood cell units SAPS, Simplified Acute Physiology Score SOFA, Sepsis-Related Organ-Failure Assessment VA-ECMO, venoarterial-extracorporeal membrane oxygenation VV-ECMO, venovenous-extracorporeal membrane oxygenation

INTRODUCTION

Extracorporeal membrane oxygenation (ECMO) is increasingly used as rescue therapy during major cardiac failure or acute respiratory distress syndrome (ARDS), (1-3) but remains associated with major complications. ECMO-treated patients are at high-risk of haemorrhage during invasive procedures, due to major coagulation disorders induced by the patient's critical state and the device itself. [START_REF] Zangrillo | Extracorporeal membrane oxygenation (ECMO) in patients with H1N1 influenza infection: a systematic review and meta-analysis including 8 studies and 266 patients receiving ECMO[END_REF][START_REF] Marasco | Thoracotomy in adults while on ECMO is associated with uncontrollable bleeding[END_REF][START_REF] Aubron | Factors associated with outcomes of patients on extracorporeal membrane oxygenation support: a 5-year cohort study[END_REF] In this context, emergency surgery, when needed, is perceived as extremely detrimental. The outcomes of ECMO-assisted patients undergoing noncardiac surgery have only been poorly described to date. More specifically, no study focused on abdominal surgery in this setting. This may be of particular interest in order to improve peroperative management of these patients.

The aim of the present case-matched study, using a propensity score analysis, was to assess the outcomes of patients on ECMO who required emergency abdominal surgery, and to compare the postoperative outcomes with those of nonECMO patients matched for initial severity.

METHODS

Patients and Data Collection

All our intensive care unit (ICU) patients who underwent emergency abdominal surgery between July 2006 and August 2014 were eligible for the study. Patients were screened from the operating room registry of the surgery department. Data were extracted from the computerized ICU charts. In accordance with our hospital's Institutional Review Board (Committee for the Protection of Human Subjects) ethical standards, informed consent for demographic, physiological and hospital-outcome data analyses was not required because this observational study did not modify existing diagnostic or therapeutic strategies.

ECMO Implantation and Management

All patients underwent Doppler echocardiography to evaluate hemodynamic status before ECMO implantation. Venoarterial-ECMO (VA-ECMO) was indicated for refractory cardiogenic shock, defined as evidence of tissue hypoxia (e.g., extensive skin mottling or elevated blood lactate) associated with low cardiac index (<2.2 L/min/m 2 ), despite infusion of high catecholamine dose and adequate fluid loading. [START_REF] Combes | Outcomes and long-term quality-of-life of patients supported by extracorporeal membrane oxygenation for refractory cardiogenic shock[END_REF] Venovenous-ECMO (VV-ECMO) was indicated for ARDS refractory to conventional treatment. [START_REF] Brodie | Extracorporeal membrane oxygenation for ARDS in adults[END_REF] Management of ECMO-implanted patients in our ICU was described previously. [START_REF] Bréchot | Venoarterial extracorporeal membrane oxygenation support for refractory cardiovascular dysfunction during severe bacterial septic shock[END_REF][START_REF]Extracorporeal Membrane Oxygenation for Severe Acute Respiratory Distress Syndrome (EOLIA trial)[END_REF] All VA-ECMO devices were surgically inserted with femoral-femoral 23F to 29F-15F to 18F

cannulation. An additional 7F catheter was systematically inserted into the femoral artery to prevent leg ischemia. Pump speed was adjusted to obtain blood flow of 3.5-4.5 L/min. Patients were assessed daily for possible ECMO weaning using clinical and echocardiographic criteria, as previously described. [START_REF] Aissaoui | Predictors of successful extracorporeal membrane oxygenation (ECMO) weaning after assistance for refractory cardiogenic shock[END_REF] VV-ECMO was inserted percutaneously with femoral-jugular 25F to 29F-18F to 21F cannulation. Weaning was achieved as previously described. (9) Experienced perfusionists checked the circuit daily.

A 5000-IU heparin bolus was injected at ECMO initiation and intravenous (IV) unfractionated heparin was given to maintain the activated partial thromboplastin time (aPTT) ratio at 1.5-2 times normal. The heparin dose was adapted at least once daily. Heparin was stopped when bleeding occurred or the platelet count decreased significantly (< 50 G/L). [START_REF] Lamarche | Thromboembolic events in patients on extracorporeal membrane oxygenation without anticoagulation[END_REF] The membrane oxygenator was changed prophylactically when the platelet count decreased significantly (< 50 G/L), blood oxygenation declined sharply or significant hemolysis appeared. [START_REF] Lubnow | D-dimers as an early marker for oxygenator exchange in extracorporeal membrane oxygenation[END_REF]

Abdominal Surgery

All procedures were performed by trained abdominal surgeons. Cholecystectomy used retrograde or anterograde (when pediculitis was too severe) techniques.

For small bowel or colon ischemia, the length of intestinal resection depended on the extension of ischemic lesions. The mesentery or mesocolon was ligated with a thermo-fusion instrument (Ligasure ® , Covidien, Minneapolis, Minnesota, USA) and intestinal stoma were created with 3.0 Vicryl ® . Bowel anastomoses when performed, used a 65 or 80 GIA ® stapler (AutoSuture Company, United States Surgical Corporation, Norwalk, CT).

Outcome Definitions

The following definitions were applied:

Preoperative period: the 24 hours preceding surgery; immediate postoperative period: the 24 hours postsurgery; and late postoperative period: the time between the 24 hours postsurgery and ICU discharge or death; perioperative bleeding: the need of transfusion of at least three units of packed red blood cell units (PRBCs) per procedure and within the 72-hours postsurgery.

Statistical Analyses

Continuous variables, expressed as medians (1 st -3 rd interquartile range [IQR]), were compared with Mann-Whitney U-tests. Categorical variables, expressed as n (%), were compared with Fisher's exact test. For multivariable analyses of factors associated with perioperative bleeding, all those associated with the dependent variable in univariable analyses with p < 0.2 were included using a backward-stepwise logistic-regression model. For practical purposes, continuous variables were transformed into categorical variables, with the median as the threshold (except for the Sepsis-Related Organ-Failure Assessment (SOFA) score which was trichotomized). All potential explanatory variables included in the multivariable analyses were subjected to a correlation matrix for analysis of collinearity and variables associated among one another were not included in the model. Thereafter, multiple backward-stepwise logisticregression analyses eliminated variables with an exit threshold set at p > 0.10.

For sensitivity analyses, baseline characteristics of ECMO and nonECMO patients were matched by their propensity scores to receive ECMO [START_REF] Agostino | Propensity score methods for bias reduction in the comparison of a treatment to a non-randomized control group[END_REF]. Baseline covariates (age, sex, body mass index, sepsis, underlying diagnosis, Simplified Acute Physiology Score (SAPS) II and SOFA score) were used to build an initial logistic-regression model predicting ECMO implantation. After calculating each patient's propensity score to be ECMO-implanted, ECMO recipients were matched (1:1) to nonECMO ICU patients for the analysis, without replacement, based on the propensity score log odds ("logit"). Using the estimated logits, a randomly selected ECMO patient was matched to the nonECMO patient with the closest estimated logit value with a maximum 20% difference between the two logits. Investigators were blinded to other patient data during the matching procedure.

Analyses were computed with SPSS v20 statistics package (IBM Corporation, Armonk, NY), Prism 4.0c software (GraphPad Software, La Jolla, CA), SAS v9.3 (SAS Institute Inc, Cary, NC) and R software (R foundation for statistical computing, Vienna, Austria), with p < 0.05 defining significance.

RESULTS

Study Population

Among 1166 ECMO-implanted patients during the study period, 35 (3%) required abdominal surgery (Fig. 1). The two main indications were digestive ischemia and acute cholecystitis in 15 and 8 patients respectively. The other indications are reported in Supplementary Table 1. Patients' demographic characteristics are given in Table 1. The cohort consisted predominantly of VA-ECMO patients implanted for cardiogenic shock, who required secondary surgery for abdominal complications. Only 20% were admitted primarily for gastrointestinal disease. Patients were young and very severely ill, as reflected by the median SOFA score of 16 and the median SAPS II of 71 at admission. Forty-two nonECMO patients underwent abdominal surgery during the study period. They had the same underlying diagnoses and indications for surgery profiles as the ECMO-treated patients, but were older and slightly less severely ill.

Perioperative Management

Patients' preoperative status is described in Table 2. Both groups had comparable hemodynamic impairments. Notably, the ECMO group had thrombocytopenia and lower fibrinogenemia than nonECMO patients. Surgical characteristics are reported in Table 2.

Surgery was predominantly cholecystectomy, digestive resection and stoma creation via laparotomy. A significantly higher percentage of ECMO patients (66%) experienced bleeding during surgery requiring transfusion than nonECMO patients (48%). Perioperative bleeding (defined as transfusion of > 3 PRBCs during the first 72-hours postsurgery) was significantly more frequent for ECMO-treated patients (74% vs 33%, p < 0.001).

Outcomes

ECMO patients' postoperative periods were marked by significantly more bleeding than nonECMO patients (Table 3), with 77% vs 40% receiving PRBC, fresh-frozen plasma or platelet units (p < 0.001 for all items). Reintervention for haemorrhage was significantly more frequent in the ECMO group (20%) than in the nonECMO group (2%; p = 0.02). Importantly, bleedingcomplication rates were comparable for VA-and VV-ECMO-treated patients (Supplementary

Table 2).

Perioperative mortality (<72-hours postsurgery) was comparable between groups (Table 3):

four patient in ECMO-treated group (three died of refractory multiorgan failure after cardiogenic shock with digestive ischemia, and 1 of hemorrhagic shock following abdominal necrosectomy for acute pancreatitis) and three in the nonECMO group (all with episodes of refractory multiorgan failure following cardiogenic shock with digestive ischemia, and two with secondary ventilator-associated pneumonia). Five (14%) ECMO-group patients' deaths were directly triggered by abdominal surgery complications (one with postoperative peritonitis, and four with hemorrhagic shock following bleeding at the surgical site) vs. one (2%) nonECMO patient (postoperative peritonitis) (p = 0.08). Finally, 24 (69%) ECMO-treated patients died in-ICU vs.

14 (33%) nonECMO patients (p = 0.003). ECMO-group deaths were distributed as follows: 10 of refractory multiorgan failure, five of hemorrhagic shock, four of ventilator-associated pneumonia, three without myocardial recovery and limitation of active therapeutics, one without pulmonary recovery and one of peritonitis. In the nonECMO group, refractory multiorgan failure accounted for seven in-ICU deaths, ventilator-associated pneumonia for two, absence of neurological recovery for two, peritonitis for one and hemorrhagic shock for two. ICU mortality reached 80% for ECMO patients treated surgically for digestive ischemia and 50% of patients undergoing cholecystectomy.

Factors Associated with Postoperative Bleeding

At univariable analysis, age, body mass index, SOFA score and ECMO implantation were associated with perioperative bleeding, whereas laparotomy was associated with a protective effect (Supplementary Table 3). At multivariable analysis ECMO was independently associated with a 5.6-fold increased risk of bleeding, while preoperative SAPS II >70 increased it by 2.9 (Fig. 2).

For sensitivity analyses, 21 ECMO patients could be matched to 21 nonECMO patients based on their propensity scores to be ECMO-implanted. Characteristics of matched and unmatched patients are given in Supplementary Table 4. Propensity matching yielded two groups of highly comparable patients in terms of underlying diagnosis, demographic characteristics, type of abdominal procedure and preoperative disease severity (Tables 1 and2).

A markedly higher rate of perioperative bleeding was confirmed for ECMO patients compared to matched nonECMO controls, and the notably higher amounts of blood products received by the former.

DISCUSSION

In this study, we assessed the outcomes of patients on ECMO who required emergency abdominal surgery. We found that, although such invasive procedures were feasible, they were associated with a significantly increased risk of bleeding compared to nonECMO ICU patients.

To date, outcomes of abdominal surgery under ECMO have not been reported. Up to 30% of ECMO-treated patients experience bleeding complications, questioning the feasibility of these invasive interventions.(4,6,7) Among 563 ECMO patients who underwent 149 miscellaneous noncardiac surgical procedures, ECMO was associated with a 28% bleeding-complication rate, without bleeding episodes impacting in-ICU mortality. ( 14) A greater need for blood products was also demonstrated in two populations of patients receiving mechanical circulatory support, including 11 and 14 ECMO-treated patients, who had undergone miscellaneous noncardiac surgical interventions. [START_REF] Taghavi | Noncardiac surgery in patients on mechanical circulatory support[END_REF][START_REF] Chestovich | Surgical procedures for patients receiving mechanical cardiac support[END_REF]. In another study, thoracotomy on ECMO led to unusual and uncontrollable massive hemorrhages. [START_REF] Marasco | Thoracotomy in adults while on ECMO is associated with uncontrollable bleeding[END_REF] In our cohort, 34% of ECMO-treated patients experienced postoperative bleeding, requiring reintervention in 20% of them and consumption of large amounts of blood products. However, perioperative mortality remained comparable to that of nonECMO patients. Patients on ECMO frequently develop complex coagulation disorders. Contact with the foreign circuit surfaces activates platelets and the coagulation cascade [START_REF] Lubnow | D-dimers as an early marker for oxygenator exchange in extracorporeal membrane oxygenation[END_REF][START_REF] Cheung | The mechanisms of platelet dysfunction during extracorporeal membrane oxygenation in critically ill neonates[END_REF][START_REF] Straub | Selective inhibition of the platelet phosphoinositide 3-kinase p110beta as promising new strategy for platelet protection during extracorporeal circulation[END_REF][START_REF] Malfertheiner | Hemostatic Changes During Extracorporeal Membrane Oxygenation: A Prospective Randomized Clinical Trial Comparing Three Different Extracorporeal Membrane Oxygenation Systems[END_REF], responsible for a hypercoagulation state. In turn, clots forming on the membrane trigger intense fibrinolysis, which, in addition to platelet consumption, is responsible for a secondary hypocoagulation state. [START_REF] Malfertheiner | Hemostatic Changes During Extracorporeal Membrane Oxygenation: A Prospective Randomized Clinical Trial Comparing Three Different Extracorporeal Membrane Oxygenation Systems[END_REF][START_REF] Dornia | D-dimers Are a Predictor of Clot Volume Inside Membrane Oxygenators During Extracorporeal Membrane Oxygenation[END_REF] An acquired von Willebrand syndrome was also described under ECMO. [START_REF] Heilmann | Acquired von Willebrand syndrome in patients with extracorporeal life support (ECLS)[END_REF] This pathophysiology opens an important area of research to limit the risk of bleeding during invasive procedures in this particular setting.

Intensivists need to evaluate more rigorous anticipation of those coagulation disorders by systematically replacing the ECMO circuit before surgery or providing greater quantities of blood products, particularly fibrinogen and platelets. Antifibrinolytic therapy with tranexamic acid achieved markedly lower bleeding rates in a large, multicenter, randomized trial during coronary-artery surgery under cardiopulmonary bypass, and could also be of interest during abdominal surgery.( 22) Finally, better characterization of the patient's coagulation state using thromboelastometry and platelet aggregometry might also help optimize those patients' coagulation states in the future. [START_REF] Nair | Prospective observational study of hemostatic alterations during adult extracorporeal membrane oxygenation (ECMO) using point-of-care thromboelastometry and platelet aggregometry[END_REF] In our cohort, ECMO patients had lower presurgery platelet counts and fibrinogenemia than nonECMO controls, consistent with more pronounced fibrinolysis. Whether or not a more aggressive protocol could manage coagulation disorders during invasive procedures might be a promising avenue of research.

For surgeons, coagulation disorders considerably modify the benefit/risk ratio of invasive surgery, with important potential implications. Bowel resection of a not viable segment might be prioritized over bowel preservation during ischemia, to avoid, as much as possible, the need for a second-look. Systematic cholecystectomy should be reevaluated in this specific setting and should be avoided if it is not essential. Lastly, noninvasive approaches, like medical treatment of cholecystitis, should be prefered. Taken together, such complex interventions further emphasize the need to manage ECMO-treated patients in centers highly experienced in its medical and surgical aspects.

In our cohort, although ECMO patients' in-ICU mortality was very high, perioperative mortality and direct surgical complication-triggered mortality remained low and comparable to that of the nonECMO group. It must be kept in mind that our patients predominantly had medical conditions with secondary abdominal complications. Although mortality related reasons can be a matter of debate for this cohort, we think that the underlying disease and extreme severity of ECMO patients accounted, in large part, for the observed mortality. Notably, the cohort contained a high percentage of patients with refractory cardiogenic shock complicated by digestive ischemia, a pathology also associated with high mortality of nonECMO patients.

Finally, our study has several limitations. Although it is the first homogeneous cohort of abdominal surgery procedures under ECMO, the sample size was small. Secondly, our cohort comprised selected medical ICU patients, limiting the external validity of our results. Lastly, our matching procedure used a propensity score to be ECMO-implanted to match ECMO and nonECMO patients. Although this procedure yielded highly comparable groups, in terms of demographics and preoperative characteristics, and the bleeding-event results for the matched cohort were consistent with multivariable analyses of the whole cohort, it cannot be excluded that factors associated with the patients' underlying conditions necessitating ECMO implantation may have accounted for the bleeding rate observed in matched ECMO patients.

We finally observed that the the results of our study, undertaken to assess the outcomes of ECMO-treated patients who required emergency abdominal surgery, revealed that such invasive procedures were feasible, but associated with a markedly increased risk of haemorrhage. Further studies are needed to improve the management of such surgical interventions for patients on ECMO.

Supplemental Digital Content

Supplementary Tables 1-4 Results are expressed as number (%) or median (1 st -3 rd IQR). Significant differences are in bold type.

a Defined as having received ≥3 PRBC units within the first 72-h postoperative.

b Defined as Dobutamine dose + 100 x (Epinephrine dose + Norepinephrine dose), all in µg/kg/min. aPTT = activated partial thromboplastin time; SAPS = simplified Acute physiology score; SOFA = Sepsis-related Organ Failure Assessment Score.

Figure 1 .

 1 Figure 1. Study flow chart. ICU = intensive care unit; ECMO = extracorporeal membrane

Figure 2 .

 2 Figure 2. Multivariable analysis results of factors independently associated with perioperative

a Defined as Dobutamine dose + 100 x (Epinephrine dose + Norepinephrine dose), all in µg/kg/min.

b Curative anticoagulation was defined as an unfractionated heparin dose >12000 IU per day.


Matthieu Roy

Stéphane Paquelet

Matthieu Crussière

Degrees of Freedom of Ray-Based Models for mm-Wave Wideband MIMO-OFDM

Keywords: ray-based model, MIMO, Saleh-Valenzuela

In this paper we analyze the number of degrees of freedom needed to accurately capture and model wideband MIMO-OFDM channels. Two modeling approaches are considered, either examining each coefficient of the conventional inputoutput MIMO channel matrix or investigating the components constituting the physical ray-based propagation between the antenna arrays. As our analysis accounts for wideband channels, a time domain decomposition of MIMO channels into tapped delay lines is performed in each case. The efficiency of each approach is then evaluated in terms of number of taps to determine the most appropriate representation between either the conventional antenna-based one or the propagation-based one.

This analysis is performed analytically using the Saleh-Valenzuela model which is recognized for its consistency for wideband as well as spatial modeling. We exploit the second-order statistics of the Channel Impulse Response to derive the taps weights.

I. INTRODUCTION

Over the last decades, the envisioned size and complexity of Multiple-Input-Multiple-Output (MIMO) systems increased tremendously. From Single-Input-Single-Output (SISO) to small-scale MIMO systems, each MIMO link was modeled by an equivalent SISO (correlated or independent) link. As a consequence MIMO signal processing techniques were usually designed for SISO then mapped to MIMO channels [START_REF] Stuber | Broadband MIMO-OFDM wireless communications[END_REF]. However this approach becomes increasingly more complex as the antenna arrays get wider.

To tackle this issue, new scalable signal processing techniques for large-scale MIMO have emerged and constitute an highly active research topic today [START_REF] Magoarou | Parametric channel estimation for massive MIMO[END_REF], [START_REF] Khlifi | Hybrid LS-LMMSE Channel Estimation Technique for LTE Downlink Systems[END_REF]. Those advanced techniques take advantage of the clustered structure of the MIMO channel in the angular domain. Such feature is of particular importance in the millimeter wave band [START_REF] Magoarou | Parametric channel estimation for massive MIMO[END_REF].

From a channel model point of view, most of the current stochastic modeling techniques are based on the framework of the well-known Saleh-Valenzuela channel model [START_REF] Saleh | A statistical model for indoor multipath propagation[END_REF]. Originally introduced for SISO communications, it represents the propagation channel as a sum of a few clusters of decreasing average power with delay. The angular dimension of the propagation has been added later on for MIMO modeling in what is called the Extended Saleh-Valenzuela channel model [START_REF] Spencer | Modeling the statistical time and angle of arrival characteristics of an indoor multipath channel[END_REF].

From its introduction, the Saleh-Valenzuela model has been the focus of many studies in terms of verification through measurement campaigns and of characterization and calibration for various environments and frequencies [START_REF] Saleh | A statistical model for indoor multipath propagation[END_REF], [START_REF] Spencer | Modeling the statistical time and angle of arrival characteristics of an indoor multipath channel[END_REF], [START_REF] Gustafson | On mm-Wave Multipath Clustering and Channel Modeling[END_REF], [START_REF] Yong | G3c channel modeling sub-commitee final report[END_REF]. Despite its physical consistency, this model is considered as too complex for analytical computations and has essentially been used for simulation purposes. Only a few articles provide analytical results. Among which are the papers [START_REF] Derpich | Second-Order Spectral Statistics for the Power Gain of Wideband Wireless Channels[END_REF], [START_REF]On the second order power spectral statistics of wideband indoor microwave channels[END_REF], [START_REF] Meijerink | On the Physical Interpretation of the SalehValenzuela Model and the Definition of Its Power Delay Profiles[END_REF].

Contributions. In this paper we conduct an analytical analysis of the number of time domain coefficient (taps) required to model a wideband Saleh-Valenzuela MIMO-OFDM channel.

More precisely, we propose to compare two channel representations. For the first approach, each wideband link from a transmitting antenna to a receiving antenna is considered separately. This conventional approach is efficient for small-scale MIMO systems but scales poorly when increasing the antenna counts. As for the second approach, the clustered structure of the channel in the angular domain is exploited and we rather examine the frequency dependent cluster gains. This approach is expected to become more efficient in terms of number of taps for wide arrays as it no longer depends on the number of antennas.

We compare the two approaches using the Saleh-Valenzuela calibration measurements done in the 60 GHz band by Lund University [START_REF] Gustafson | On mm-Wave Multipath Clustering and Channel Modeling[END_REF] and the IEEE 802.15.3c [START_REF] Yong | G3c channel modeling sub-commitee final report[END_REF]. We show that only a few taps are sufficient to capture the channel behavior. The residual error is characterized herein by the model parameters through its first and second order statistics.

Notations. Upper case and lower case bold symbols are used for complex matrices and vectors of arbitrary size. .,. denote the inner product between two vectors of C N . u stands for a three-dimensional (3D) vector. a • u denote the inner product between two 3D vectors. z * denotes the conjugate of z. [H] p,q is the element of matrix H at row p and column q. H F stands for the Frobenius norm. H H and H T denotes the conjugate transpose and the transpose matrices. E{.} and Var {.} denote the expectation and variance. denotes the cardinal of a set. Throughout this paper, the indexes i, j, k, n denote iterators on receive antennas, transmit antennas, OFDM sub-carriers and time domain coefficients, respectively.

II. MIMO CHANNEL MODEL

Let us consider the ray-based model widely used in the literature [START_REF] Magoarou | Parametric channel estimation for massive MIMO[END_REF], [START_REF] Spencer | Modeling the statistical time and angle of arrival characteristics of an indoor multipath channel[END_REF], [START_REF] Liu | Angle of arrival extended S-V model for the 60 GHz wireless indoor channel[END_REF]. Following such approach, the MIMO channel is decomposed into several multipath components called clusters. Each cluster is defined by a frequency-dependent complex gain, a direction of departure and a direction of arrival.

A. Multipath components

The frequency gain of each cluster is described by a sum of P intra-cluster rays

c q (f ) = P p=1 β p,q e -2πjf τp,q (1)
where τ p,q is the arrival time of ray p in cluster q. β p,q is the complex gain of ray p of cluster q. The MIMO channel between the transmitter and the receiver is then given by the sum over all the clusters

H(f ) = N t N r Q q=1 c q (f )e -2πjf τq
e r ( u rx,q )e t ( u tx,q ) H

(2)

where Q denote the number of clusters, τ q the arrival time of cluster q relatively to the first cluster. u tx,p,q ( u rx,p,q ) are 3D unit norm vectors denoting the direction of departure (arrival) of cluster q, also abbreviated DoD (DoA).

Considering the plane wave assumption, the functions e t ( u) and e r ( u) are called the steering vectors of the antenna array defined by

e t ( u) = 1 √ N t e 2jπ a tx,1 • u λ ,•••,e 2jπ a tx,N t • u λ T ( 3
)
where the antenna positions in the array are defined by the 3D vectors a tx,j at the transmitter ( a rx,i at the receiver respectively).

The Saleh-Valenzuela model relies on such ray-based model and specifies the distributions of the random variables τ q , τ p,q and β p,q as detailed hereafter and illustrated in Fig. 1.

B. Times of arrival

The times of arrival of clusters follow a Poisson process of parameter Λ, called the cluster arrival rate. The first cluster arrives at τ 1 = 0. The other times of arrival are then defined by the probability density function p(τ q |τ q-1 ) = Λe -Λ(τq-τq-1) , τ q > τ q-1 .

The times of arrival of intra-cluster rays also follow a Poisson process of parameter λ, called the ray arrival rate. The first ray arrives at τ 1,q = τ q . Then the other times of arrival are defined by the following probability density function p(τ p,q |τ p-1,q ) = λe -λ(τp,q-τp-1,q) , τ p,q > τ p-1,q .

C. Complex gains

Each ray is also characterized by a complex gain β p,q that follows a centered complex gaussian distribution. In this paper we normalize the average power of the very first ray to E |β 1,1 | 2 = 1. The variances of the coefficients follow a double exponential decay function parametrized by Γ the cluster decay parameter and γ the ray decay parameter E |β p,q | 2 = e -τq/Γ e -τp,q/γ .

III. SECOND ORDER STATISTICS

We first establish the expressions for the second order statistics of the channel, namely the frequency autocorrelation and the Power Spectral Density (PSD) as defined in [START_REF] Hajek | Random Processes for Engineers[END_REF], for both a single cluster and the full Saleh-Valenzuela channel. In this context the PSD matches the Power Delay Profile (PDP). Those expressions have already been derived in the context of UWB channel modeling in [START_REF] Meijerink | On the Physical Interpretation of the SalehValenzuela Model and the Definition of Its Power Delay Profiles[END_REF]. However our proofs (which are given in the appendices) neither require advanced point process theory or use Campbell's theorem.

A. Single Cluster Analysis

In this section we derive the expression of the autocorrelation and PDP of a single cluster of the Saleh-Valenzuela channel model. The complex gain of a cluster is given by c q (f ) = P p=1 β p,q e -2πjf τp,q .

(4)

Due to the exponentially decaying intra-cluster powers with increasing delay, only a finite number of dominant rays are distinguishable. P is chosen high enough to embrace the whole multi-path power while keeping computational complexity low for simulation purposes. For analytical computation convenience, we will consider that P → ∞. The details of the derivations are given in appendix A-A. We obtain the formula

R cq (∆f ) = E |β 1,q | 2 1+ λγ 1-2πj∆f γ . ( 5
)
It is noticeable that this expression consists of two terms, the former being constant and the latter vanishing as the frequency difference increases. This feature is also observed in [START_REF] Derpich | Second-Order Spectral Statistics for the Power Gain of Wideband Wireless Channels[END_REF]. We obtain the PDP by applying the Fourier Transform on the autocorrelation,

S cq (τ ) = E |β 1,q | 2 δ(τ )+λe -τ /γ u(τ ) (6)
where u(τ ) is the Heaviside step function. The average power of the cluster is

P cq = E |c q | 2 = E |β 1,q | 2 (1+λγ).
Using a recurrence we compute

E |β 1,q | 2 E |β 1,q | 2 = E |β 1,q-1 | 2 1 1+ 1 ΛΓ = 1 1+ 1 ΛΓ q-1 . (7)

B. Saleh-Valenzuela Analysis

In this section we derive the expression of the autocorrelation and PDP of the full Saleh-Valenzuela. The complex frequency gain of the channel is given by

H(f ) = Q q=1 P p=1 β p,q e -2πjf (τp,q+τq) . ( 8
)
As previously done, we will consider that P → ∞ and Q → ∞. The detail of the calculus of the autocorrelation function is given in appendix A-B. In particular we find that the random process H is WSS. This yields the expression

R H (∆f ) = 1+ ΛΓ 1-2πj∆f Γ 1+ λγ 1-2πj∆f γ . (9)
This expression can be expressed as the product of the autocorrelations of two single-exponentially decaying channels. Then we apply the Fourier Transform to the autocorrelation function to get the PDP

S H (τ ) =δ(τ )+Λe -τ /Γ u(τ )+λe -τ /γ u(τ ) +Λλ Γγ Γ-γ e -τ /γ -e -τ /Γ u(τ )
where u(τ ) is the Heaviside step function.

The average power of the Saleh-Valenzuela channel is then given by

P H = E |H| 2 = (1+λγ)(1+ΛΓ).

IV. MODELING SISO CHANNELS

In this section we consider a SISO-OFDM setup that operates on a bandwidth B = [-B/2,B/2], divided into N f subcarriers spaced by ∆B = B/N f . The frequency-domain channel gain at subcarrier k is denoted H[k]. The model complexity is measured in terms of the number of time domain coefficients. To do so we perform an IFFT on the frequency domain channel gain. The N f taps are given by

h n = 1 N f N f -1 k=0 H[k]e 2πjnk/N f ,n ∈ 0,N f -1 . ( 10
)
Although the sum (10) can be very large, the channel power is in practice concentrated on a small set of time domain coefficients N ( N N f ) so most of them are not significant and can be discarded. As a consequence we will only keep a small set of coefficients N

H N [k] = n∈N h n e -2πjnk/N f .
This complexity reduction introduces an error that is quantified herein by the Mean Square Error (MSE) criterion. This operation is typically performed in OFDM channel estimation for noise reduction. The resulting systematic error is then called the bias. Using the orthogonality between the complex exponential functions, the MSE induced by this simplification is given by

E HN -H 2 = N f n ∈N E |h n | 2 = N f n ∈N Var(h n ). (11)
The variances can be computed by solving the integral

Var(h n ) = ∞ -∞ S H (τ )D 2 N f π n-τ B N f dτ (12)
where D N (x) denotes the normalized Dirichlet kernel This expression of the variance provides a simple way to compute the error induced by a particular truncation of the time domain coefficients. Variance formula ( 12) is illustrated on Fig. 2. The leftmost bin denoted n = -1 is a replicate of the last bin n = 5 as the Dirichlet kernel is a periodic function.

D N (x) = sin(N x) N sin(x) .

V. MODELING MIMO CHANNELS

We consider two modeling approaches. First we analyze each frequency-dependent MIMO channel matrix coefficient. Each one of those coefficients represents the equivalent SISO channel from an emitting antenna to a receiving antenna. We called this method antenna based modeling.

On one other hand, we consider the channel as a sum of several multipath components also called clusters. We can take advantage of this feature and analyze the channel through those frequency-dependent clusters. We called this method propagation based modeling. Both approaches are illustrated by Fig. 3 and Fig. 4.

In this section, H is a 3-dimensional tensor whose elements are denoted H i,j [k]. The approximated channel matrix is denoted H N . The error is evaluated with the MSE criterion,

E H N -H 2 = E    i,j,k |H N i,j [k]-H i,j [k]| 2    . (13)

A. Antenna based modeling

The channel gain from antenna j to antenna i on subcarrier k is given by H i,j [k]. Each one of the N r × N t channels can be decomposed into time domain coefficients as in [START_REF] Meijerink | On the Physical Interpretation of the SalehValenzuela Model and the Definition of Its Power Delay Profiles[END_REF]. The resulting h i,j,n are obviously correlated over the different antennas. We can then apply the methodology of IV. Considering that only the subset N of coefficient is not discarded on each SISO-equivalent link, the MSE is given by

H 2,1 [k] T x 1 T x 2 Rx 1 Rx 2 n h2,1,n
E HN -H 2 = N r N t N f n ∈N Var(h n ).
One can notice that the results doesn't depend on the correlations between the time domain coefficients.

B. Propagation based modeling

The gain of each cluster at sub-carrier k can be obtained by sampling of c q [k] (1). Then we follow the same methodology as in IV. This cluster gain can be represented in the time domain

h q,n = 1 N f N f -1 k=0 c q [k]e 2πjnk/N f . ( 14
)
Cluster q can then be represented by a set of N f time domain coefficients h q,n , most of those being not significant and can be discarded. For each cluster q we keep a set of coefficients N q . The MSE introduced by this truncation is given by

E HN -H 2 = N r N t Q q=1 E c q,Nq -c q 2
where the right hand side is given by the uniform independent phases of the cluster gains. The terms of the sum are given by

E cq,N -c q 2 = N f k ∈Nq E |h q,n | 2 = N f n ∈Nq
Var(h q,n ).

(15) It is expressed as a sum of variances, whose expression is : Based on the expression of S cq in (16), equation ( 15) can be further simplified using ( 7)

Var(h q,n ) = ∞ -∞ S cq (τ )D 2 N f π n-τ B N f dτ. ( 16
)
E cq,N -c q 2 = N f 1 1+ 1 ΛΓ q-1 n ∈Nq Var(h 1,n ).
Once the variances are evaluated, the error induced by the choices of N q can be easily calculated.

VI. RESULTS

Parameters Λ, Γ, λ, γ characterizing the Saleh-Valenzuela have been measured in a wide variety of situations in both GHz as well as millimeter waves frequency bands. We used the 60 GHz parameters from Lund University [START_REF] Gustafson | On mm-Wave Multipath Clustering and Channel Modeling[END_REF] and the IEEE 802.15.3c office and library setups [START_REF] Yong | G3c channel modeling sub-commitee final report[END_REF] which are summarized in Table I. In the previous sections we saw that the total channel power depends on Λ, Γ, λ, γ. To come by with comparable results we will consider the relative error

E HN -H 2 E{ H 2 } = E HN -H 2 N r N t N f (1+ΛΓ)(1+λγ)
.

We performed numerical evaluations of the relative error for a typical LTE frame structure (also valid for 5G NR frames) of 10 M Hz total bandwidth (In fact the occupied bandwidth is B = 9 M Hz due to edge effects), an inter-carrier spacing of ∆f = 15 kHz and N f = 600 sub-carriers. We evaluate the error for both antenna based modeling and propagation based modeling. The MSE of the channel can then be derived using [START_REF] Liu | Angle of arrival extended S-V model for the 60 GHz wireless indoor channel[END_REF] and (15). The results are given in Fig. 5. The antenna based approach (dashed lines) yields a smaller MSE for small arrays while the propagation based approach (red solid line) is preferable for large arrays.

Those MSE formulas can be inverted using iterative techniques to assess the number of taps required to achieve a desired MSE. This yields Fig. 6 where the boundary between both regimes is even further evidenced. The antenna based approach is as expected more efficient for small number of antennas whereas the propagation based approach is more efficient for large-scale MIMO. The transition from the former to the later approaches happens surprisingly fast at N r ×N t > 10 at most. This configuration is easily attained if we consider multiple receive antennas.

We neglected the cluster angular spread. Indeed for mm-waves setups where N r × N t < 10 this hypothesis is reasonable. Wider arrays could be considered by splitting the clusters into multiple subclusters depending on N r × N t . The flat curves of Fig. 6 would be slightly increasing and the previous observations and conclusions wouldn't change. VII. CONCLUSION In this paper the number of time domain coefficients required to model a MIMO-OFDM Saleh-Valenzuela channel for a given error has been analyzed. This evaluation is based on the secondorder channel statistics which enabled the comparison between two representation methods. The matrix coefficient (resp. multipath) based approach is suitable for small-scale (resp. large-scale) MIMO. We performed extensive numerical evaluations using calibrated Saleh-Valenzuela models available in the literature and found out that the transition between those two representations arises at surprisingly small number of antennas. To highlight this property, we made the assumption that array angular resolution is limited resulting in unresolvable intra-clusters DoDs and DoAs.

In further studies this assumption could be removed by splitting clusters into multiple sub-clusters with parametrization scaled accordingly.

The channel model bias, induced by the discarded taps and derived in this study, can be further exploited for channel estimation error evaluation.

APPENDIX A STATISTICAL AUTOCORRELATION FUNCTIONS A. Cluster channel gain

In (4) the phases of the complex gains β p,q are all independent resulting in E β p1,q β * p2,q = δ p1,p2 E {|β p1,q β p2,q |}. The autocorrelation is then given by

R cq (f,f ) = ∞ p=1 E |β p,q | 2 e -2πj(f -f )τp,q .
The random process is Wide Sense Stationary (WSS). We set ∆f = f -f . Then using conditional expectation on τ p,q and the exponential decay E |β p,q | 2 τ p,q = E |β 1,q | 2 e -τp,q/γ we get

R cq (∆f ) =E |β 1,q | 2 ∞ p=1 E τp,q e (2πj∆f -1/γ)τp,q . (17)
Then using a recurrence over p we can prove that E τp,q e (2πj∆f -1/γ)τp,q = λ p-1 1 ( 1 γ +λ-2πj∆f ) p-1 . Equation ( 17) is a geometric series whose ratio r = 1 1 λγ +1-2πj∆f /λ always verify |r| < 1 so that the series converges to

R cc (∆f ) = E |β 1,q | 2 1+ λγ 1-2πjγ∆f .
B. Saleh-Valenzuela channel gain Still using the independence of the phases of the complex gains in (8), we get a simplified expression. We find that this channel is also WSS. We obtain the expression

R H (∆f ) = ∞ q=1 ∞ p=1 E |β p,q | 2 e 2πj∆f (τp,q+τq) . (18)
Using a conditional expectation on the τ p,q and the τ q and using the double exponential decay formula of the Saleh-Valenzuela channel E |β p,q | 2 τ p,q = e -τq/Γ e -τp,q/γ , we get an expression similar to (17)

R H (∆f ) = ∞ q=1 ∞ p=1 E e -(1/Γ-2πj∆f
)τq e -(1/γ-2πj∆f )τp,q .

Because the random variables τ q and τ p,q are independent, we can split the expectation. Using the result of (A-A) on both parts, the equation can be rewritten as the product of two geometric series yielding the result [START_REF]On the second order power spectral statistics of wideband indoor microwave channels[END_REF].

APPENDIX B VARIANCES OF TIME DOMAIN COEFFICIENTS

The variance of the time domain coefficient h n is

E{h n h * n } = 1 N 2 f N f -1 k1=0 N f -1 k2=0 E{H[k 1 ]H[k 2 ] * }e -2πj(k2-k1)n/N f .
Where we can recognize the autocorrelation

E{H[k 1 ]H[k 2 ] * } = R H,H [k 2 -k 1 ]
. We use the autocorrelation expression from (18) with ∆f = (k 2 -k 1 )∆B

E{H[k 1 ]H[k 2 ] * } = ∞ q=1 ∞ p=1 E |β p,q | 2
e 2πj(k2-k1)∆B(τp,q+τq) .

We insert this expression into the variance formula and get

E{h n h * n } = 1 N 2 f ∞ q=1 ∞ p=1 E |β p,q | 2 N f -1 k1=0 N f -1 k2=0
e 2πj(k2-k1)∆B(τp,q+τq-n/(N f ∆B))

where the sums equals D 2 N f (π∆B(τ p,q +τ q -n/B)) resulting in

E{h n h * n } = ∞ q=1 ∞ p=1 E |β p,q | 2 D 2
N f (π∆B(τ p,q +τ q -n/B)) .

We introduce the continuous function defined for x ∈ R

f (x) = ∞ q=1 ∞ p=1 E |β p,q | 2 D 2
N f (π∆B(τ p,q +τ q -x/B)) .

After a first Inverse Fourier Transform and substituting y = Bτ p,q +Bτ q -x ∞ q=1 ∞ p=1 E |β p,q | 2 e 2πjν(τp,q+τq)

∞ -∞ D 2
N f (πy/N f )e -2πjνy/B dy.

The first line matches the autocorrelation function of the channel R(ν). The second part is the Fourier Transform of the squared Dirichlet kernel. We retrieve the function f using a FT:

f (x) = ∞ -∞ S H (τ )D 2 N f (π(τ B -x)/N f )dτ.
Applying this formula for x = n, we get the expression.

Fig. 1 .

 1 Fig. 1. A realization of a Saleh-Valenzuela channel.

1 Fig. 2 .

 12 Fig. 2. Relation between the PDP of a channel and the variance of the time domain taps. Each tap is obtained by the convolution of the corresponding Dirichlet kernel with the PSD of the channel.

Fig. 3 .Fig. 4 .

 34 Fig. 3. Antenna based modeling.

Fig. 5 .

 5 Fig.5. MSE as a function of the number of taps. The antenna based approach is represented by the dashed lines and the propagation based approach is represented by the red solid line.

Fig. 6 .

 6 Fig.6. Number of taps required to achieve a desired MSE. After Nr ×Nt > 10, the propagation based approach is always more efficient. The boundary between both regimes for any MSE is given by the red solid curve.


Maarja Kruuse
email: maarja@to.ee

Elmo Tempel
email: elmo.tempel@to.ee

Rain Kipper
email: rain.kipper@to.ee

Radu Stoica
email: radu-stefan.stoica@univ-lorraine.fr

The bivariate J-function to analyse positive association between galaxies and galaxy filaments

Keywords: Methods, statistical -galaxies, statistics -large-scale structure of Universe

The compelling and intrinsic network of galaxies builds up a complex structure for the Universe. In this work positive association between long bridging structures named galaxy filaments and a photometric redshift galaxy dataset is under investigation. Possible positive association is studied by the use of a bivariate J-function.

Introduction

Large scale galaxy surveys show the intrinsic structure of the Universe: bridging vast filamentary patterns, galaxy clusters, sheets, superclusters and immense regions almost devoid of galaxies named voids [START_REF] Jõeveer | [END_REF]). These maps are dominated by galaxy filaments, which connect the structures into a web [START_REF] Pimbblet | [END_REF]). A mathematical framework named the Bisous model (Stoica et al. (2010)) has been applied on observed large scale galaxy datasets in the paper Tempel et al. (2014). In the latter paper they estimated the filamentary pattern from Sloan Digital Sky Survey (York et al. ( 2010)) (SDSS) spectroscopic galaxy data (these galaxies have precise distances in redshift 1 space).

In this study we analyse whether these galaxy filaments (Tempel et al. (2014)) are positively associated with the Sloan Digital Sky Survey photometric redshift galaxies dataset (these galaxies have rather uncertain distance estimations in redshift space) compiled in [START_REF] Beck | [END_REF]). The galaxies can be viewed as points in 3-dimensional Euclidean space [START_REF] Kruuse | Statistics of the galaxy distribution[END_REF]). The Cartesian coordinates of these galaxies are point locations in 3-dimensional Euclidean space and other properties can be considered as marks. Thus the galaxies situated in a survey region can be seen as a realization of a marked point process in a compact set.

1 Characteristic that estimates the measure of distance in cosmology. Figure 1: Visualization of the two datasets in spherical sky-coordinates: kernel smoothed 2-dimensional density from the distribution of photometric galaxies on the viewed region of the sphere (grey background density) and the filamentary spines (orange spines). All the objects locate in the distance range of 200 -400 Mpc and the darker grey denotes a higher density of photometric galaxies.

1

Data

In this section the analysed photometric redshift galaxies dataset and the filamentary pattern catalogue are characterized. Limitations will be set on all of the observed objects dependent on their distance to the observer. Figure 1 depicts the two studied datasets in spherical sky coordinates η and λ. From visual inspection one can notice that the higher density of filamentary spines seems to follow the higher density of photometric redshift galaxies. Further on this possible clustering effect is analysed with the use of the bivariate J-function.

Filamentary network

The filamentary spines catalogue holds information about 46 403 objects. They form a random set of objects Y fil in the 3-dimensional Euclidean space. The mapping of a point in R 3 to the spherical coordinates latitude η and longitude λ is done using

η = 2 arctan y x 2 + y 2 -x , λ = arcsin z (x 2 + y 2 + z 2 ) . (1)
In paper (Kruuse et al. submitted) the mean distance used to characterise spine Y distance to the observer O is defined as:

d(O, Y ) = 1 l(Y ) Y d(O, y)dy, (2)
where l(Y ) is the associated spine length and d(O, y) is the distance from the observer O to the point y of the spine Y . The integral in ( 2) is computed along the considered spine. These mean distances will be used to limit the filamentary spines into the distance range of 200 -400 Mpc. Filamentary spines, which satisfy this condition, are further analysed.

The dataset Y fil will be of 20 367 filamentary spines. The chosen distance range carries the bulk of the filamentary spines.

Photometric galaxies

The rather uncertain redshift z photo estimations (δz photo ≤ 0.05) in the catalogue (Beck (2005)) are used to limit the galaxies into the distance range of 200 -400 Mpc. Which leaves us with the dataset of 236 850 photometric galaxies. Their locations are defined on the sphere by their radian latitude η and longitude λ. They are viewed as a configuration of points X photo in a sphere S 2 .

3 The bivariate J-function

The datasets studied are fit for being analysed with spatial statistics tools [START_REF] Baddeley | Spatial point patterns: methodology and applications with R[END_REF]; [START_REF] Kruuse | Statistics of the galaxy distribution[END_REF]). One can consider the photometric galaxies set like a point processes realization and the filamentary set as a realization of a random set. The bivariate J-function can be used to study possible positive association between these objects.

Thorough mathematical description of the theory of point processes and probabilistic models therein are written in van Lieshout (2000), [START_REF] Møller | Statistical inference and simulation for spatial point processes[END_REF], [START_REF] Illian | Statistical analysis and modelling of spatial point patterns[END_REF], [START_REF] Chiu | Stochastic Geometry and its Applications[END_REF] and [START_REF] Baddeley | Spatial point patterns: methodology and applications with R[END_REF]. Here we will present the border corrected estimations for the empty space function, nearest-neighbour function and the bivariate J-function as they are described in van Lieshout & [START_REF] Van Lieshout | [END_REF], [START_REF] Foxall | [END_REF] and [START_REF] Baddeley | Spatial point patterns: methodology and applications with R[END_REF]. All the described summary statistics have exact formulas for the homogeneous Poisson point process, which represents a completely random pattern. A rigorous representation of the Poisson point process is written [START_REF] Baddeley | Spatial point patterns: methodology and applications with R[END_REF].

Empty space function

First of all the shortest distances from a point w ∈ W to a subset A ⊂ W is noted by the following form d(w, A) = inf a∈A w -a .

The border corrected estimator for the empty space function from any arbitrary point w i to the random set of Y is of the following form

F Y (r) = i 1{d(w i , W c ) ≥ r}1{d(w i , Y ) ≤ r} i 1{d(w i , W c ) ≥ r} (3)
with W c the border of W and {w i , i = 1, 2, . . .} a finite family of arbitrary points in W . The estimation F is a cumulative distribution function of smallest distances from arbitrary points to the closest object in the random set.

Nearest-neighbour function

The estimator of the nearest neighbour distribution from any point in X to the random set Y is of the following form

G X,Y (r) = i 1{d(x i , W c ) ≥ r}1{d(x i , Y ) ≤ r} i 1{d(x i , W c ) ≥ r} , (4)
where {x i , i = 1, . . .} is the observed finite point configuration of X. The nearestneighbour function is a cumulative distribution function of smallest distances from a point of the point process X to the nearest object in the random set Y . The estimation of G measures association between the objects of type X and type Y . If the point process X and the random set of objects Y are independent, then G X,Y ≡ F X .

Bivariate J-function

The bivariate J function is obtained by the nearest-neighbour function G X,Y (r) and empty space function F Y (r) in the following form: The J-function measures association between X and Y [START_REF] Foxall | [END_REF]. If X and Y are independent, then J X,Y = 1. Values of J X,Y close to 1 suggest independence, values higher than 1 suggest negative association and values lower than 1 suggest positive association.

J X,Y (r) = 1 -G X,Y (r) 1 -F Y (r) . (5)
4 Applying the bivariate J-function

In this section we will present the results obtained to analyse possible positive association between the photometric galaxies X photo and filamentary spines Y f il , which locate at the distance range of 200 -400 Mpc from the observer. The border corrected empty space function F Y fil , border corrected nearest-neighbour function G X photo ,Y fil (r) and the bivariate J-function J X photo ,Y fil are presented.

Figure 2 left-hand panel represents the results for the border corrected estimations of the G and F statistics. The values for the empty space function are below of the values of the nearest neighbour function for small values of distance r. To investigate whether positive association between the photometric galaxies and filamentary spines exists the bivariate J-function is presented in the Fig. 2 right-hand panel.

Figure 2 right-hand panel draws the result for the bivariate J-function (eq. ( 5)). The decreasing J X photo ,Y fil values below the theoretical reference case (which would indicate independence between observed sets) represent positive association between the photometric galaxies and all filamentary spines. Indicating that the photometric galaxies carry information about the filamentary network, which has been detected from the spatial distribution of spectroscopic galaxies.

Conclusion

The bivariate J-function is well fit to study possible positive association between complex filamentary structures and galaxies, which can be described as a realisation of a point process. The statistical signal of positive association indicates that the photometric galaxies hold information about the location of the filamentary network. The result obtained from the bivariate J-function encourages to use the photometric galaxies. They would increase the number density of galaxies per volume of space, which will contribute to a more detailed analysis of the cosmic web. A thorough analysis with the bivariate J-function to study eventual correlation between galaxies and galaxy filaments is done in Kruuse et al. (submitted to A & A).

Figure 2 :

 2 Figure 2: On the left panel: the results of the empty space function F Y fil (red dotted line) and nearest-neighbour function G X photo ,Y fil (r) (black continuous line). On the right panel: the results of the bivariate J-function J X photo ,Y fil with the 0.95-confidence interval. The theoretical case of independence is noted by the dashed dark line.


John Barzman

A paper at Session 12 « Politics : Party and Otherwise