True Moments Research Articles

Polynomial approximation of discounted moments

AbstractWe introduce an approximation strategy for the discounted moments of a stochastic process that can approximate the true moments for a large class of problems. These moments appear in pricing formulas of financial products such as bonds and credit derivatives. The approximation relies on a high-order power series expansion of the infinitesimal generator and draws parallels with the theory of polynomial processes. We demonstrate applications to bond pricing and credit derivatives. In the special cases that allow an analytical solution, the approximation error decreases to around 10 to 100 times machine precision for higher orders. When no analytical solution exists, we numerically compare the approximation with existing numerical techniques.

Bianca Torre Is Afraid of Everything by Justine Pucella Winans

Reviewed by: Bianca Torre Is Afraid of Everything by Justine Pucella Winans Natalie Berglind Winans, Justine Pucella Bianca Torre Is Afraid of Everything. Clarion, 2023 [384p] Trade ed. ISBN 9780358721642 $19.99 E-book ed. ISBN 9780358721659 $10.99 Reviewed from digital galleys Ad Gr. 7-10 Sixteen-year-old lesbian Bianca Torre has a laundry list of the many things they fear, from public speaking to sheep to beautiful people. Their closest friend, Anderson, is a fellow anime aficionado, but otherwise, they interact with the world through their telescope, allegedly birding (but usually people watching). When they spy a shut-in neighbor get murdered in his window, they understandably can't keep quiet—but someone wants them to, as evidenced by bloody threats left across their door. Now, they feel compelled to investigate a murder that the police ruled as a suicide, but along with trying out new pronouns themself and acting on their crush on a fellow birder, they've got their hands full. While Bianca's narration is sardonic and witty, they don't stand out from other YA protagonists with similar personalities, and the story is bogged down by frequent recitations of their long list of fears and anime references that don't add to the text. The mystery itself is interesting, but the author doesn't lean into the ridiculousness of a money-swindling birding group set on murder enough to truly be campy, and the book is never serious enough to create tension. The true golden moments come from Bianca's down-to-earth experiences—their journey through their acceptance of the nonbinary label, acknowledging Anderson as their best friend through their shared hobby, and navigating their crush are much more compelling elements than the thriller ones. Copyright © 2023 The Board of Trustees of the University of Illinois

Multivariate cumulants in flow analyses: The next generation

We reconcile for the first time the strict mathematical formalism of multivariate cumulants with the use of cumulants in anisotropic flow analyses in high-energy nuclear collisions. This reconciliation yields to the next generation of estimators to be used in flow analyses. We review all fundamental properties of multivariate cumulants and use them as a foundation to establish two simple necessary conditions to determine whether some multivariate random variable is a multivariate cumulant in the basis they are expressed in. We argue that properties of cumulants are preserved only for the stochastic variables on which the cumulant expansion has been performed directly, and if there are no underlying symmetries due to which some terms in the cumulant expansion are identically zero. We illustrate one possibility of new multivariate cumulants of azimuthal angles by defining them event-by-event and by keeping all nonisotropic terms in the cumulant expansion. Furthermore, we introduce new cumulants of flow amplitudes named asymmetric cumulants, which generalize recently introduced symmetric cumulants for the case when flow amplitudes are raised to different powers. Finally, we present the new concept of cumulants of symmetry plane correlations and provide the first realization for the lowest orders. The new estimators can be used directly to constrain the multivariate probability density function of flow fluctuations, since its functional form can be reconstructed only from its true moments or cumulants. The new definition for cumulants of azimuthal angles enables separation of nonflow and flow contributions and offers first insights into how the combinatorial background contributes for small multiplicities to flow measurements with correlation techniques. All the presented results are supported by Monte Carlo studies using state-of-the-art models.

Simulating the Cox–Ingersoll–Ross and Heston processes: matching the first four moments

We implement 15 simulation schemes for the Cox–Ingersoll–Ross (CIR) square root process and 10 schemes for Heston’s stochastic volatility model to generate draws that we investigate for the quality of their mean, variance, skewness and kurtosis estimates. Simulations of continuous-time processes require discretization, and we therefore investigate the quality of currently known simulation techniques from both an accuracy perspective and a timing perspective. We show that no method fits all situations, and we advise the use of different simulation techniques. We also provide an extension to Andersen’s quadratic exponential method to generate returns with skewness or kurtosis closer to their theoretical values in certain settings. A simulation experiment focusing on the estimation of return skewness and kurtosis demonstrates the relevance of using the correct simulation technique and reveals the limitations on the convergence of those estimates to the true moments when volatility is generated by a CIR process for which the Feller condition is not satisfied and the sample is not of relatively large size.

Pileup corrections on higher-order cumulants

We propose a method to remove the contributions of pileup events from higher-order cumulants and moments of event-by-event particle distributions. Assuming that the pileup events are given by the superposition of two independent single-collision events, we show that the true moments in each multiplicity bin can be obtained recursively from lower multiplicity events. In the correction procedure the necessary information are only the probabilities of pileup events. Other terms are extracted from the experimental data. We demonstrate that the true cumulants can be reconstructed successfully by this method in simple models. Systematics on trigger inefficiencies and correction parameters are discussed.

Intermittent Clutter Suppression Method Based on Adaptive Harmonic Wavelet Transform for L-Band Radar Wind Profiler

Boundary layer radar (L-band) wind profilers frequently encounter a significant problem arising from the contamination of intermittent clutter, produced by seasonal and nocturnal migrating birds, which often yields erroneous wind velocity and boundary layer information. Classical harmonic wavelet transforms (HWTs) are inadequate in removing the transient clutter contamination under certain conditions, particularly when the clutter is significant. We implemented an adaptive complex harmonic discrete wavelet transform with an advanced statistical method to overcome the shortcomings of the classical wavelet method. This algorithm effectively eliminates the bird contamination even where the classical method fails. Finally, a multiple peak-picking (MPP) algorithm was added to select true atmospheric signals and estimate accurate moments. The obtained wind velocity measurements were compared with those derived using the conventional method and validated with global positioning system radiosonde winds. The comparison shows that the proposed method is more effective than the conventional one.

Bulk flow and shear in the local Universe: 2MTF and cosmicflows-3

The low-order kinematic moments of galaxies, namely bulk flow and shear, enables us to test whether theoretical models can accurately describe the evolution of the mass density field in the nearby Universe. We use the so-called etaMLE maximum likelihood estimator in logdistance space to measure thesemoments from a combined sample of the 2MASS Tully-Fisher (2MTF) survey and the COSMICFLOWS-3 (CF3) compilation. Galaxies common between 2MTF and CF3 demonstrate a small zero-point difference of -0.016+-0.002 dex.We test the etaMLE on 16 mock 2MTF survey catalogues in order to explore how well the etaMLE recovers the true moments, and the effect of sample anisotropy. On the scale size of 37 Mpc/h, we find that the bulk flow of the local Universe is 259 +- 15 km/h in the direction is (l,b)=(300+-4, 23+-3) (Galactic coordinates). The average shear amplitude is 1.7+-0.4 h km/s/Mpc. We use a variable window function to explore the bulk and shear moments as a function of depth. In all cases, the measurements are consistent with the predictions of the L cold dark matter (LCDM) model.

Removing unwanted background phase with a reference phantom for applications in susceptibility quantification

PurposeA method of removing the background phase with a reference phantom but without overcorrecting the induced phase from objects of interest is proposed. Several factors during the imaging procedure and post-processing are investigated for their accuracies. MethodsA method using a reference phantom to remove eddy currents as well as using the least squares fit to quantify susceptibility and to remove the background phase is proposed. Phase induced from simulated spheroids was fitted and compared to their true magnetic moments, an important concept for the proposed method. A cylindrical phantom and its simulation, a phantom with straws filled with Gd-DTPA, and a simulated head model were used to study systematic errors due to some confounding factors. The feasibility for in vivo applications was demonstrated from an actual human head. Susceptibility and remaining phase after removing the background phase were measured in all cases. ResultsSimulations show that magnetic moments of various spheroids and phantoms can be accurately quantified from images, regardless of the partial volume effect. All measured susceptibility values are within ±0.16 ppm of −9.4 ppm for agarose and 0.05 ppm of 1 ppm for Gd-DTPA. Most residual phase is within ±0.1 rad from the phantom center. Susceptibilities close to −9.4 ppm are also obtained for the simulated and actual head. Correspondingly, the remaining phase has a mean value less than two standard deviations. ConclusionThe proposed method from phantom studies can reliably remove the background phase without overcorrections. The in vivo example demonstrates the feasibility of the method.

Generalisation of the identity method for determination of high-order moments of multiplicity distributions with a software implementation

The incomplete particle identification limits the experimentally-available phase space region for identified particle analysis. This problem affects ongoing fluctuation and correlation studies including the search for the critical point of strongly interacting matter performed on SPS and RHIC accelerators. In this paper we provide a procedure to obtain nth order moments of the multiplicity distribution using the identity method, generalising previously published solutions for n=2 and n=3. Moreover, we present an open source software implementation of this computation, called Idhim, that allows one to obtain the true moments of identified particle multiplicity distributions from the measured ones provided the response function of the detector is known.

High-probability minimax probability machines

In this paper we focus on constructing binary classifiers that are built on the premise of minimising an upper bound on their future misclassification rate. We pay particular attention to the approach taken by the minimax probability machine (Lanckriet et al. in J Mach Learn Res 3:555–582, 2003), which directly minimises an upper bound on the future misclassification rate in a worst-case setting: that is, under all possible choices of class-conditional distributions with a given mean and covariance matrix. The validity of these bounds rests on the assumption that the means and covariance matrices are known in advance, however this is not always the case in practice and their empirical counterparts have to be used instead. This can result in erroneous upper bounds on the future misclassification rate and lead to the formulation of sub-optimal predictors. In this paper we address this oversight and study the influence that uncertainty in the moments, the mean and covariance matrix, has on the construction of predictors under the minimax principle. By using high-probability upper bounds on the deviation between true moments and their empirical counterparts, we can re-formulate the minimax optimisation to incorporate this uncertainty and find the predictor that minimises the high-probability, worst-case misclassification rate. The moment uncertainty introduces a natural regularisation component into the optimisation, where each class is regularised in proportion to the degree of moment uncertainty. Experimental results would support the view that in the case of with limited data availability, the incorporation of moment uncertainty can lead to the formation of better predictors.

On the aerosol and cloud phase function expansion moments for radiative transfer simulations

AbstractThe accuracy of the Henyey‐Greenstein (HG) approximation in radiative transfer process is systematically investigated for aerosol and cloud particles. For small‐size aerosols, the phase function moment by the HG approximation is close to that of the true phase function; therefore, the differences in four‐stream radiative transfer calculations are very small whether using the HG approximation or the true phase function moments. However, for large‐size aerosols, liquid cloud droplets and ice crystals, the HG approximation produces very different phase function moment result compared to the true phase function. In case of small optical depth, the single layer four‐stream radiative transfer calculations show that the HG approximation can produce relative errors larger than 20% while the errors are generally less than 5% by using true phase function moments. By applying the four‐stream radiative transfer scheme to a multilayer atmosphere with aerosol and cloud, the differences in flux error are generally less than 2 Wm−2 by using either the HG approximation or true phase function moments, but can be over 10 Wm−2 for ice cloud case. The aerosol/cloud instantaneous radiative forcing has been analyzed in climate simulation. Compared to the four‐stream radiative transfer scheme, the two‐stream radiative transfer scheme overestimates the aerosol radiative flux in low‐latitude regions and underestimates the upward flux for the high‐latitude regions. By using the HG approximation, the difference in aerosol radiative forcing between the four‐stream and the two‐stream radiative transfer schemes is enhanced; the enhancement can be up to 0.5 Wm−2. The parameterization schemes for aerosol and cloud optical properties must be extended to contain higher‐order phase function moments, if higher‐order stream radiative transfer algorithms are to be used.

Model-Free Risk-Neutral Moments and Proxies

Estimation of risk-neutral (RN) moments is of great interest to both academics and practitioners. We study 1) the model-free measure of RN moments by Bakshi, Kapadia and Madan (2003); 2) RN moments that are used in the VIX and SKEW index by the Chicago Board Options Exchange; 3) nonparametric RN moments that are calculated as the difference of implied volatilities across moneyness levels; and 4) the level, slope and curvature of the implied volatility smirk. More specifically, we investigate the estimation procedure by examining the consequence of directly using raw option data versus applying various smoothing methods to the option data. In the simulation study, we study estimation errors arise from integration truncation, discreteness of strike prices and asymmetric truncation. We show that applying smoothing methods reduces the estimation errors of true moments but the size and direction of estimation errors are largely unquantifiable. In the empirical study, we find that applying smoothing methods increases the Kendall and Spearman rank correlations among RN moment estimates. We conduct a case study that examines the relationship between RN skewness and future realised stock returns from 1996 to 2014. We show that a strategy that is to long the quintile portfolio with the lowest RN skewness stocks yields a negative and significant Fama-French Five-Factor alpha. This finding is robust across all RN skewness measures.

Learning from Leibniz: Whitehead (and Russell) on Mind, Matter and Monads

Whitehead's system may be interpreted as a majestic attempt at recasting Leibniz's theory of monads in terms of sounder ontological categories. After a brief introductory section on the sources of Whitehead's knowledge of Leibniz's philosophy, the paper explains why Whitehead turned to Leibniz for metaphysical inspiration. Attention then shifts to Whitehead's understanding of the problems involved with Leibniz's theory of monads and his alternative explanation of monadic causation. Whitehead's endeavour to install windows in Leibniz's monads may not be entirely convincing, but there are philosophical gems scattered here and there in his analyses – true moments of insight that repay close examination.

Effectiveness of Linear Extrapolation in Model-Free Implied Moment Estimation

This study proposes a new methodology with which the effectiveness of linear extrapolation (LE) of Jiang and Tian (2005) for the implied moment estimators of Bakshi et al. (2003) can be evaluated, even when the true moments are unknown. Using S&P 500 index options data and truncation sensitivity functions, this study suggests that although LE is effective for all three estimators, the implied skewness and kurtosis estimators can remain sensitive to truncation, i.e., the complete unavailability of option prices for a part of deep-out-of-the-money or deep-in-the-money region of strike price domain, even when LE is employed to mitigate truncation.

Validity conditions for moment closure approximations in stochastic chemical kinetics.

Approximations based on moment-closure (MA) are commonly used to obtain estimates of the mean molecule numbers and of the variance of fluctuations in the number of molecules of chemical systems. The advantage of this approach is that it can be far less computationally expensive than exact stochastic simulations of the chemical master equation. Here, we numerically study the conditions under which the MA equations yield results reflecting the true stochastic dynamics of the system. We show that for bistable and oscillatory chemical systems with deterministic initial conditions, the solution of the MA equations can be interpreted as a valid approximation to the true moments of the chemical master equation, only when the steady-state mean molecule numbers obtained from the chemical master equation fall within a certain finite range. The same validity criterion for monostable systems implies that the steady-state mean molecule numbers obtained from the chemical master equation must be above a certain threshold. For mean molecule numbers outside of this range of validity, the MA equations lead to either qualitatively wrong oscillatory dynamics or to unphysical predictions such as negative variances in the molecule numbers or multiple steady-state moments of the stationary distribution as the initial conditions are varied. Our results clarify the range of validity of the MA approach and show that pitfalls in the interpretation of the results can only be overcome through the systematic comparison of the solutions of the MA equations of a certain order with those of higher orders.

Location‐only and use‐availability data: analysis methods converge

Location‐only and use‐availability data: analysis methods converge

Effect of parallel segmented flow chromatography on the height equivalent to a theoretical plate II – Performances of 4.6 mm × 30 mm columns packed with 2.6 μm Accucore-C18 superficially porous particles

The reduced trans-column (or long-range eddy dispersion) height equivalent to a theoretical plate (HETP) of short and wide 4.6mm×30mm columns packed with 2.6μm Accucore-C18 superficially porous particles was measured under conventional (no split flow) and parallel segmented (outlet and inlet) flow chromatography. The overall reduced HETP was derived from the true moments of the recorded concentration profiles. The longitudinal diffusion HETP term was measured at a very small flow rate (0.05mL/min). The solid–liquid mass transfer resistance was derived from the shell diffusivity, using the composite Garnett–Torquato model of effective diffusion in a heterogeneous system made of a dense packing of core–shell particles immersed in a continuous matrix (the eluent). The trans-channel and short-range interchannel eddy dispersion HETP terms were assumed to be equal to the calculated h data after solving the Navier–Stokes equation and simulating the advection–diffusion transport process. Experimental results confirmed that the optimum efficiency of these short columns was increased by a factor of about two. The ratio of the detection sensitivities on the PSFC stream and on a regular stream increased from 1 to 1.45 when the retention factor decreases from about 10 to 0.5. These phenomena are due to a strong reduction of the trans-column eddy dispersion HETP term. The system loses about 60% of the sample mass when only outlet skimming is carried out when the flow rate ratio of 55% is applied, as was done in this work. It loses about 50% of the sample when inlet/outlet segmentation is applied. In gradient elution, the peak capacity is increased by only 15%, due to post-column band spreading, which should imperatively be minimized when the outlet flow is split.

A quasi-maximum likelihood method for estimating the parameters of multivariate diffusions

A quasi-maximum likelihood procedure for estimating the parameters of multi-dimensional diffusions is developed in which the transitional density is a multivariate Gaussian density with first and second moments approximating the true moments of the unknown density. For affine drift and diffusion functions, the moments are exactly those of the true transitional density and for nonlinear drift and diffusion functions the approximation is extremely good and is as effective as alternative methods based on likelihood approximations. The estimation procedure generalises to models with latent factors. A conditioning procedure is developed that allows parameter estimation in the absence of proxies.

Theoretical and experimental impact of the bed aspect ratio on the axial dispersion coefficient of columns packed with 2.5 μm particles

The impact of the ratio of the column diameter to the average particle size (or bed aspect ratio) on the column performance was investigated from theoretical and experimental viewpoints. The experiments were conducted for two series of 100mm long columns, 2.1, 3.0, and 4.6mm in diameter, packed with 2.5μm fully porous particles of Bridge Ethylene Hybrid (BEH) for one series and Charged Surface Hybrid (CSH) for the other. The heights equivalent to a theoretical plate (HETP) of two low molecular weight compounds, uracil (non-retained, k=0) and naphthalene (retained, k=2.5), were determined from the true moments of the recorded peak profiles. The results showed a systematic decrease of the column performance for uracil at high flow velocities with decreasing column inner diameter, in agreement with the theoretical predictions of the variation of the trans-column eddy dispersion HETP term with decreasing bed aspect ratio. This result is consistent with the increasing volume fraction of the wall region of the column, in which the average linear velocity of the mobile phase over a distance of 5 particle diameters from the column wall is about 10% larger than in the bulk center of the column (infinite diameter column). For the retained compound, the discrepancies are levelled out due to the longer average residence time and larger particle diffusivities of retained compounds, which allow a more efficient relaxation of the radial concentration gradients. Further improvements of the performance of the larger I.D. columns (3.0 and 4.6mm I.D.) may be achieved by decreasing the harmful effect of this trans-column velocity bias by injecting and/or collecting the sample molecules in a wide central zone of the column. For 2.1mm I.D. columns, this approach would prove useful only when HPLC instruments providing a lower extra-column band broadening contribution will become available. Finally, the further minimization of the trans-column eddy dispersion HETP term and the design of new, better inlet/outlet column endfitting/frit assemblies requires newer and more accurate models of eddy dispersion in packed columns than those previously provided by Gunn and Giddings and the numerical calculation of band profiles using original functions to account for the distribution and collection of the sample molecules at the inlet and outlet of the column.

Robust formulations for clustering-based large-scale classification

Chebyshev-inequality-based convex relaxations of Chance-Constrained Programs (CCPs) are shown to be useful for learning classifiers on massive datasets. In particular, an algorithm that integrates efficient clustering procedures and CCP approaches for computing classifiers on large datasets is proposed. The key idea is to identify high density regions or clusters from individual class conditional densities and then use a CCP formulation to learn a classifier on the clusters. The CCP formulation ensures that most of the data points in a cluster are correctly classified by employing a Chebyshev-inequality-based convex relaxation. This relaxation is heavily dependent on the second-order statistics. However, this formulation and in general such relaxations that depend on the second-order moments are susceptible to moment estimation errors. One of the contributions of the paper is to propose several formulations that are robust to such errors. In particular a generic way of making such formulations robust to moment estimation errors is illustrated using two novel confidence sets. An important contribution is to show that when either of the confidence sets is employed, for the special case of a spherical normal distribution of clusters, the robust variant of the formulation can be posed as a second-order cone program. Empirical results show that the robust formulations achieve accuracies comparable to that with true moments, even when moment estimates are erroneous. Results also illustrate the benefits of employing the proposed methodology for robust classification of large-scale datasets.