text stringlengths 128 2.05k |
|---|
and low [MATH] about equal to lung compliance. The parameter [MATH] characterized the approximate dynamic chest wall compliance, which was calculated as the slope [MATH] . Parameter values for [MATH] [MATH] [MATH] , and [MATH] were estimated from previously published studies |
. The viscoelastic parameters [MATH] and [MATH] were manually tuned to obtain idealized tidal volume and end-expiratory lung volume rather than the magnitude of the hysteresis. |
FRC is the volume at the resting position of the respiratory system i.e. where [MATH] . The naturally high compliance of the healthy full-term and especially preterm infant (with even steeper [MATH] ) lowers [MATH] and decreases FRC to about 20% of vital capacity (VC), compared to at about 35-40% of VC in the adult |
. A nominal value for FRC for a given set of static compliance curves is obtained by first computing volumes using a vector of physiological pressures [-20…40] cm H O. Lung and chest recoil pressure vectors are then added in the [MATH] direction to obtain [MATH] , and the index where [MATH] is used to determine FRC usi... |
Table gives values and formulas / sources for parameters that remain unchanged between simulations. These values as well as the FRC, respiratory pressure amplitude, chest wall compliance, and upper airway resistance parameters in Table that vary between simulation conditions were manually tuned to best obtain the repor... |
Computational procedures All simulations proceeded with an initial respiration rate of 60 breaths/min ( [MATH] ), initial minute ventilation [MATH] , and initial tidal volume [MATH] ml, with the expectation that tidal volume changes with changes in dynamic lung compliance. The motivation for ths choice was twofold: One... |
Simulated conditions were chosen to demonstrate the model dynamics with high and low chest wall compliance, under two interventions and two states of permanent alveolar closure. An infant often exhibits compensatory mechanisms such as laryngeal braking (grunting) and increased activity of diaphragm and intercostal musc... |
to increase end-expiratory pressure in order to keep EELV above the volume at which alveolar units start to collapse during expiration. Laryngeal braking is simulated with a 10-fold increase in expiratory upper airway resistance [MATH] . CPAP is simulated with an increase of [MATH] from 0 to 5 triggered at [MATH] chara... |
The system of differential equations ( ), together with the constitutive relations ( 13 ), were solved using MATLAB R2016b (MathWorks, Natick, MA) with the differential equations solver ode15s . Initial conditions were set at physiological values as given in Table . The equations were solved for each new breath using t... |
Results Parameterized static compliance curves for [MATH] and [MATH] are shown in Fig for high [MATH] (left) and low [MATH] (right). The hysteretic tidal breathing loops are superimposed on the curve [MATH] for normal [MATH] in black and increased [MATH] in grey. Hysteresis is caused in the model by the viscoelastic pa... |
Fig shows the impact of high vs low [MATH] and normal vs. high [MATH] on the five states [MATH] , and [MATH] . Increased [MATH] increases [MATH] almost threefold, but [MATH] has very little impact. However, decreased [MATH] increases [MATH] significantly, effectively raising it higher on the lung PV curve. Increasing [... |
in which esophageal (pleural) pressure, airflow, and tidal volume were approximately -2 to -6 cm H O, -30 to 30 ml/s, and 8 ml, respectively. |
Table presents the 14 simulations and their time to failure, defined for this study as 90% volume loss. Dynamics were comparable between simulations with the major difference being the timing, therefore only representative or significant results are presented in figures. Our model consistently indicates a faster loss o... |
The breath-to-breath change in EELV and [MATH] under high and low [MATH] conditions with no interventions are given in Fig (Simulations 1 and 3). The high [MATH] simulation reaches accelerated loss of volume and eventually failure at 0.3 hours, much more quickly than the low [MATH] at 2.5 hours. This depicts a possible... |
Fig shows changes in dynamic lung compliance and tidal volume with high and low [MATH] without changes in [MATH] , then adding CPAP to the high [MATH] condition at three different levels (c.f. Table , simulations 1,3,11,13-14). CPAP was simulated by an increase in mouth pressure [MATH] to 5 cm H O when [MATH] , which h... |
. Increasing [MATH] moves the resulting PV loop higher up on the lung compliance curve but does not change the nature of the curve, thus eventually the influence of high [MATH] on dynamics induces the same lung volume loss without other mitigating actions. |
Discussion In summary, we have developed a lumped-parameter respiratory mechanics model tuned with parameters specific to the extremely preterm infant weighing 1 kg. The model includes a novel representation of derecruitment based on alveolar pressure and volume expansion compensating for collapsed alveoli. Model simul... |
and a later one by Pandit et al gave the best insight into the respiratory dynamics of an extremely preterm infant, making these the standard against which our results were qualitatively validated. We therefore claim that this effort is a “proof of concept” that will be further explored in future investigations using p... |
As mentioned briefly in Nonlinear compliance constitutive relations , recruitment/decrecruitment may have a time component , in that the time it takes for an airway or alveolus to open may be a function of how far away its pressure is from its critical opening pressure. Earlier studies have developed models that incorp... |
. These previous studies considered recruitment resulting from one or two hyperinflations but not long-term derecruitment. In our model breath-to-breath derecruitment is manifested as the change of the lung compliance curve during normal spontaneous breathing as described in Progressive volume loss , and the hysteresis... |
). While such a modification may affect the overall trends in observed states such as EELV, the differential impact between high and low [MATH] would be expected to remain. |
The noninvasive ventilatory intervention CPAP shifts the tidal volume loop to a higher position on the lung compliance curve, operating with a higher EELV and end-expiratory lung elastic recoil. Our model suggests that the timing of administration of CPAP and the permanent closure or injury state of alveoli may impact ... |
Prolonged shallow breathing has been associated with increased surface tension and decreased surface area that further hinders breathing |
. We safely assume in our model that derecruitment is a continuous process that will eventually induce loss of lung volume if left uncompensated |
. In a healthy lung in the absence of fatigue, permanent alveolar collapse (due to injury or disease), and/or high chest wall compliance, this process is on a much longer time scale than the natural compensation mechanisms that compensate for and recoup volume loss (such as grunting in the infant). One such mechanism i... |
by re-opening air spaces that collapse naturally under tidal breathing via increased pressure and surfactant activation and possibly affect neurorespiratory control. Sighing occurs more frequently and at relatively larger magnitude in the infant vs adults |
. A natural extension of our model would be incorporating the restorative actions of sighing and testing the hypothesis that spontaneous deep breaths mitigate or reverse volume loss. |
Several features of the physiology of preterm infants are not currently addressed in this model but should be considered in future model enhancements for further investigations. Preterm infants commonly exhibit diaphragm weakness and dysfunction and paradoxical breathing. While a sinusoidal waveform is used in the clin... |
). Components that differentiate between abdominal and rib cage movements (see e.g. ) may model the paradoxical chest movement. Another limitation of this model is the absence of any feedback mechanisms compensating for loss of volume. More sophisticated models of central pattern generators have been developed in conju... |
that could potentially be incorporated with ours. A chemoreflex model, see for example , may also augment our model. Despite these limitation, we expect that the timing of dynamics of individual simulations may change with model enhancements but that time to failure would still be extended under low chest wall complian... |
Conclusion Respiratory mechanics models have been investigated for several years and many formulations exist; the challenge to be appreciated is the customization to the preterm infant with significantly different physiological features than adults and even term infants. Hence future model modifications must always kee... |
Acknowledgments This research was supported in part by the Atlantic Pediatric Device Consortium via FDA grant 5P50FD004193-07 (H. Rozycki, L. Ellwein, M. Brandes) and the VCU College of Humanities and Sciences Faculty Research Council (L. Ellwein, L. Linkous). |
Appendix We analyze the inherent stability of the model under constant non-oscillatory muscle pressure by examining the eigenvalues of the Jacobian at the nominal parameter set and varying parameters by multiples of 2 and 10. To obtain steady-states, [MATH] is set at a constant called [MATH] and the system of ODE’s is ... |
[EQUATION] It follows that [MATH] and [MATH] . Given the relations [MATH] [MATH] , and [MATH] , we have 3 equations with 6 unknowns [MATH] ). Three additional equations come from incorporating loop summations such that [MATH] [MATH] , and [MATH] . Noting that [MATH] is defined implicitly by Eq. (10) and using the compl... |
[EQUATION] we obtain two equations [EQUATION] that are solved numerically for two unknowns [MATH] for each modified parameter set using an iterative algorithm. [MATH] From these steady-state values we can calculate the remaining state variables. |
In order to linearize the system and determine asympototic behavior we rewrite the system in terms of the state variables [MATH] using previously described relationships: |
[EQUATION] where the quantity [MATH] The Jacobian [EQUATION] was found using symbolic computation then used numerically to calculate eigenvalues. All parameter variations gave stable solutions except [MATH] , and [MATH] . However, varying these parameters by 2 and 10 is not actually physiological and the instability co... |
# Source: arxiv 1805.06682 # Title: Analyzing order flows in limit order books with ratios of Cox-type intensities # Sections: all # Downloaded: 2026-03-03T04:48:02.180668+00:00 |
Analyzing order flows in limit order books with ratios of Cox-type intensities Abstract We introduce a Cox-type model for relative intensities of orders flows in a limit order book. The model assumes that all intensities share a common baseline intensity, which may for example represent the global market activity. Para... |
Keywords : order book models; point processes; Cox processes; Hawkes processes; ratio models; trading signals; imbalance; spread. |
Introduction The limit order book is the central structure that aggregates all orders submitted by market participants to buy or sell a given asset on a financial market. Buy offers form the bid side, sell offers form the ask side. Simplified representations of possible interactions with a limit order book usually cons... |
The rise of electronic markets, the accelerating rate of trading on financial markets, the development of optimal trading strategies by brokers, etc., have pushed for better investigation and modeling of limit order books. Chakraborti et al. , ( 2011 Gould et al. , ( 2013 and Abergel et al. , ( 2016 provide some overvi... |
, ( 2012 limits tests on their estimation procedure to two-hour periods to avoid seasonality effects; Lallouache & Challet, ( 2016 uses time-dependent piecewise-linear baseline intensity (with nodes spaced every few hours). However, even at these scales, global market activity varies and may limit the reliability of es... |
In this work we continue previous investigations on the influences of financial variables (state of the order book or other trading signals) on the processes of order submissions. We assume that orders submissions are modeled by point processes with Cox-like intensities depending on given covariates. However, we do not... |
In Section we describe the general model of ratios of intensities. In Section we show that the quasi-likelihood estimators of the model are consistent and asymptotically normal. Section discusses the method with respect to information criteria and penalization. Finally, Section illustrates the benefits of the model wit... |
Model description Let [MATH] and [MATH] for some strictly positive integers [MATH] and [MATH] . Let [MATH] be some counting processes. We assume that the intensities [MATH] of these counting processes share a common baseline intensity [MATH] [MATH] is neither observable nor specified as a function of observables and pa... |
[EQUATION] where [MATH] is the [MATH] -th observable covariate process and [MATH] a parameter vector. We are not interested in the value of the coefficient [MATH] , modeling the specific response of the counting process [MATH] to the covariate [MATH] , but rather in the relative responses of the intensities compared to... |
[EQUATION] be the relative response of process [MATH] compared to process [MATH] with respect to covariate [MATH] [MATH] Obviously, [MATH] since the process [MATH] is taken as an arbitrary reference, and [MATH] , i.e. the differences between absolute and relative responses are equal. Instead of the standard intensities... |
[EQUATION] where [MATH] denotes the new parameter vector. Notation is justified by the following computation: [EQUATION] As hinted in the introduction, this framework is convenient to model a limit order book since daily movements are known to exhibit both intraday seasonality and sudden bursts of activities, sometimes... |
One should also note that by construction [MATH] . This gives another important feature of the model, which is that the intensities ratios [MATH] are directly interpretable in terms of probability. Let us assume that the counting process [MATH] [MATH] , are counting the number of events of (mutually exclusive) type [MA... |
Likelihood analysis In this section, we show that the quasi-maximum likelihood estimator and the quasi-Bayesian estimator of the ratio model of Equation ( ) are consistent and asymptotically normal. Two close formulations are provided. In the first one, stationarity of covariates is assumed to obtain consistency and as... |
Let us introduce some notations used in the following analysis. For a tensor [MATH] , we write [EQUATION] for [MATH] ,…, [MATH] . Brackets [MATH] stand for a multilinear mapping. We denote by [MATH] the [MATH] times tensor product of [MATH] Let [MATH] , where [MATH] . We will write [MATH] for [MATH] for notational simp... |
3.1 Case of stationary covariates Let [MATH] . To estimate [MATH] based on the observations on [MATH] , we consider the quasi-log likelihood (log partial likelihood) |
[EQUATION] Obviously, the model is continuously extended to the closure [MATH] , and [MATH] is extended to there as a continuous function. A quasi-maximum likelihood estimator (QMLE) is a measurable mapping [MATH] satisfying |
[EQUATION] for all [MATH] The quasi-Bayesian estimator (QBE) with respect to a prior density [MATH] is defined by [EQUATION] The QBE takes values in [MATH] , the convex hull of [MATH] We assume [MATH] is continuous and satisfies [MATH] These estimators are called together quasi-likelihood estimators. We now investigate... |
Let [MATH] In this first step, in order to simplify the statements, we assume that the process [MATH] is stationary. Denote by [MATH] the true value of [MATH] and assume that [MATH] Denote by [MATH] the [MATH] -field |
[MATH] for [MATH] Let [EQUATION] for [MATH] We consider the following conditions. [A1] The process [MATH] is stationary, [MATH] and |
[MATH] for all [MATH] [A2] The function [MATH] is rapidly decreasing, that is, [MATH] for every [MATH] Let [EQUATION] for [MATH] and |
[MATH] Let [EQUATION] for [MATH] and [MATH] Denote by [MATH] the variance matrix of the [MATH] -dimensional multinomial distribution |
[MATH] with [MATH] [MATH] and [MATH] is [MATH] ’s [MATH] -element. Let [MATH] Define a symmetric tensor [MATH] by [EQUATION] for [MATH] where |
[MATH] The matrix [MATH] is nonnegative definite. We assume [A3] [MATH] The non-degeneracy of [MATH] is a local condition. However, the global identifiability condition follows from this condition in the present model, as seen later. |
Let [MATH] denote the true value of [MATH] Denote by [MATH] the space of continuous functions on [MATH] of at most polynomial growth. We write |
[MATH] and [MATH] Denote by [MATH] [MATH] -dimensional standard Gaussian vector. The quasi-likelihood analysis ensures convergence of moments as well as asymptotic normality of the quasi-likelihood estimators. |
Theorem 3.1 Suppose that [MATH] [MATH] and [MATH] are satisfied. Then [EQUATION] for any [MATH] and [MATH] Proof is given in the appendix. |
Example 1 Let [MATH] be a Hawkes process with exponential kernel with parameters [MATH] . For the sake of illustrating Theorem 3.1 , we consider a version of model ( ) with [MATH] [MATH] , and in which the common baseline intensity [MATH] is the intensity of the process [MATH] . We thus have: |
[EQUATION] where [MATH] is a Markov chain with states [MATH] and infinitesimal generator [MATH] Then we obviously have [MATH] , i.e. [MATH] is the single parameter to be fit. In this specific case, one may explicitly compute the asymptotic variance of Equation ( 12 ): |
[EQUATION] We run 1000 simulations of the model with Hawkes parameters [MATH] , covariate parameter [MATH] , and horizon [MATH] . True values of the parameters are [MATH] [MATH] so that [MATH] . For each simulation we estimate [MATH] . The empirical density of the estimated values of [MATH] is plotted on Figure in dots... |
Example 2 This paper does not focus on the baseline intensity. However, in the case of a parametric model where the baseline intensity is fully specified, then the ratio approach is particularly helpful as it reduces the dimension of the space of the parameters, and can thus help improving numerical estimations by choo... |
[EQUATION] Maximum likelihood estimators of this model are directly computable when the process [MATH] is observable. This requires an optimization on 10 parameters. As an alternative estimation procedure, we can estimate the ratio model (4 parameters [MATH] [MATH] [MATH] ) and then estimate the full model likelihood w... |
“Full” denotes the standard maximum likelihood estimation. “Combined” denotes the mixed method involving a ratio estimation as a first step. Optimizations are carried out with a Nelder-Mead algorithm (Python scipy implementation) with random starting points (using a standard Gaussian distribution). With these parameter... |
3.2 Repeated measurements We complete the previous analysis with a formulation of our estimation result that may be convenient in finance and in other fields of applications. We consider a sequence of observations of intraday data. The observations may be non-ergodic each day. We shall consider intervals [MATH] [MATH] ... |
[MATH] have no common jumps. The stationarity of each process [MATH] is not assumed here. We are interested in estimation of the parameter [MATH] defined earlier by Equation ( ). The estimation will be based on the random field [MATH] re-defined by |
[EQUATION] Then the QMLE and the QBE are defined by Equations ( ) and ( ), respectively, but for [MATH] given by Equation ( 17 ). |
Let [MATH] . Let [MATH] and let [MATH] . The [MATH] -mixing coefficient for [MATH] is defined by [EQUATION] Let us then define the new conditions. |
[C1] The sequence [MATH] is identically distributed. Moreover, [MATH] and [MATH] for all [MATH] [C2] For every [MATH] [MATH] Define the symmetric tensor [MATH] by |
[EQUATION] for [MATH] . The matrix [MATH] is nonnegative definite. More strongly we assume [C3] [MATH] In a way similar to Theorem 3.1 , it is possible to prove the following theorem. |
Theorem 3.2 Suppose that [MATH] [MATH] and [MATH] are satisfied. Then [EQUATION] as [MATH] for any [MATH] and [MATH] The proof is omitted. |
Information criteria and penalization Since the ratio model flexibly incorporates various covariates processes and since it may have a large number of parameters, we need information criteria for model selection and other regularization methods for sparse estimation. Though the inference of the ratio model is based on ... |
Let [MATH] be a sequence of positive numbers such that [MATH] and [MATH] as [MATH] . Let [MATH] . We consider a sub-model [MATH] of [MATH] such that |
[EQUATION] Let [EQUATION] where [MATH] (depending on [MATH] ) is denoting the QMLE or QBE in the sub-model [MATH] and [MATH] is the dimension of [MATH] . More precisely, the QMLE [MATH] is defined as an estimator that satisfies |
[EQUATION] and the QBE [MATH] is defined by [EQUATION] for a continuous prior density [MATH] on [MATH] satisfying [MATH] . We denote by [MATH] the minimum model that includes [MATH] , in other words, [MATH] if and only if [MATH] . The QMLE and QBE for [MATH] restricted to the sub-model [MATH] |
are generically denoted by [MATH] In particular, [EQUATION] Proposition 4.1 Suppose that [MATH] [MATH] and [MATH] are satisfied. Then |
(i) If [MATH] , then [EQUATION] (ii) If [MATH] and [MATH] , then [EQUATION] A proof of Proposition 4.1 is given in Appendix for selfcontainedness. According to Proposition 4.1 , we should select a model [MATH] that attains [MATH] . Then the selection consistency holds as |
[EQUATION] Based on the quasi-likelihood function [MATH] , the criterion [MATH] gives the quasi-consistent AIC (QCAIC) when [MATH] , and the quasi-BIC (QBIC) when [MATH] among many other possible choices of [MATH] . A Hannan and Quinn type of quasi-information criterion (QHQ) is the case where [MATH] for [MATH] . The q... |
Recently penalized quasi-likelihood analysis for sparse estimation has been developed. We consider a penalty function [MATH] such that [MATH] [MATH] [MATH] is non-decreasing on [MATH] [MATH] is differentiable except for [MATH] , and [MATH] for some [MATH] . This class of penalty functions includes the ones of LASSO ( T... |
[EQUATION] and the penalized QMLE [MATH] (depending on [MATH] ) is associated by [EQUATION] In the case [MATH] , the polynomial type large deviation inequality is inherited from [MATH] to [MATH] , as a result, we obtain asymptotic properties of [MATH] as well as [MATH] -boundedness of the error. Asymptotic distribution... |
Another approach is toward the adaptive LASSO ( Zou, ( 2006 ) and the least square approximation method ( Wang & Leng, ( 2007 ). De Gregorio & Iacus, ( 2012 took this approach to sparse estimation of ergodic diffusions. Since we can assume strong mode of convergence for the initial estimator constructed within the QLA ... |
Related to regularization methods for point processes, Fan & Li, ( 2002 extended the nonconcave penalized likelihood approach to the Cox proportional hazards model. Yue & Loh, ( 2015 treated variable selection in spatial point processes. Hansen et al. , ( 2015 derived probabilistic inequalities for multivariate point p... |
Empirical results - Dependencies analysis This section describes the high-frequency financial data (Section 5.1 ) and several empirical studies conducted with it. A detailed example of financial analysis allowed by the ratio model is provided in Section 5.2 , where the determination of the sign of the next trade in a l... |
5.1 Data We use Reuters TRTH tick by tick data for 36 stocks traded on the Paris stock Exchange on most of the year 2015. The list of the stocks is given in Appendix . For each stock and each trading day, the orders flow is reconstructed using the method described in Muni Toke, ( 2016 , and we keep all trades and quote... |
All models are numerically estimated using by quasi-likelihood maximization. For practical purposes, we provide practical explicit expressions for the log partial likelihood. Using Equation ( ), one may write Equation ( ) with respect to [MATH] as: |
[EQUATION] Equation ( 17 ) is similarly written. In the example case [MATH] with three counting processes, the quasi-log likelihood given at Equation ( 31 ) is written: |
[EQUATION] An important feature of the model may be underlined here. Many mathematical models of limit order books, and models in high-frequency finance in general, rely on simple point processes, i.e. processes for which one may not observe two simultaneous events. Confronting such modeling to empirical data thus ofte... |
5.2 Trades signs in a limit order book It is well known that imbalance is good proxy to the sign of the next trade. Empirical observations can be found in e.g. Lipton et al. , ( 2013 Lehalle & Mounjid, ( 2017 , among others, is an attempt to incorporate this signal in trading strategies. Let us define the imbalance in ... |
[EQUATION] where [MATH] and [MATH] are the number of shares available at time [MATH] at the best quote on the ask side and bid side respectively. An imbalance close to [MATH] indicates a very small available volume on the ask side and consequently a probable upward price move, while an imbalance close to [MATH] increas... |
[EQUATION] where [MATH] is a potentially random baseline intensity. Parameters [MATH] [MATH] can be straightforwardly estimated by maximization of the quasi-likelihood. As mentioned above, a very interesting feature of our intensity model is that the intensities ratios [MATH] and [MATH] are then directly interpretable ... |
The model is fitted monthly for all stocks, from January 2015 to November 2015, which, excluding gaps in the database, gives 390 fits of the model. On Figure , we plot for two samples the empirical probability that for a given imbalance, the next trade occurs on the ask side, and the numerical estimation of the theoret... |
The simple ratio model with one covariate provides a very good fit of the probability of the next trade. But we can obviously investigate further possible factors influencing the sign of the next trade. It has been shown that times series of signs of trades ( [MATH] for an ask trade at time [MATH] [MATH] for a buy trad... |
[EQUATION] Figure plots the updated results for two samples. For representativity, we again select the two samples representing the [MATH] and [MATH] quantile measured in mean [MATH] error. All 390 fits are available upon request. |
It is interesting to observe the strong effect of the sign of the last trade on the imbalance predicting power. While a close to zero imbalance unconditionally indicate a close to [MATH] probability of an ask trade (see Figure ), this dramatically changes when the sign of the last trade comes into play. We now observe ... |
Subsets and Splits
No community queries yet
The top public SQL queries from the community will appear here once available.