Simple Closed-form Formulas Research Articles

Quick assessment for systematic test statistic inflation/deflation due to null model misspecifications in genome-wide environment interaction studies.

Gene-environment (GxE) interaction is one potential explanation for the missing heritability problem. A popular approach to genome-wide environment interaction studies (GWEIS) is based on regression models involving interactions between genetic variants and environment variables. Unfortunately, GWEIS encounters systematically inflated (or deflated) test statistics more frequently than a marginal association study. The problematic behavior may occur due to poor specification of the null model (i.e. the model without genetic effect) in GWEIS. Improved null model specification may resolve the problem, but the investigation requires many time-consuming analyses of genome-wide scans, e.g. by trying out several transformations of the phenotype. It is therefore helpful if we can predict such problematic behavior beforehand. We present a simple closed-form formula to assess problematic behavior of GWEIS under the null hypothesis of no genetic effects. It requires only phenotype, environment variables, and covariates, enabling quick identification of systematic test statistic inflation or deflation. Applied to real data from the Alzheimer’s Disease Neuroimaging Initiative (ADNI), our formula identified problematic studies from among hundreds GWEIS considering each metabolite as the environment variable in GxE interaction. Our formula is useful to quickly identify problematic GWEIS without requiring a genome-wide scan.

Separate analysis of cell‐edge and cell‐centre user performance for irregular massive MIMO network with interference cancellation

How much performance gain for the user equipment (UE) located at cell-edge can be obtained, from the interference cancellation (IC) by exploiting extra degrees of freedom in multi-user massive multiple-input multiple-output (MIMO), needs to be investigated. Also, it is necessary to reveal the impact of this IC on the performance of cell-centre UE (CCU). In this study, the authors separately analyse the coverage probability of cell-edge UE (CEU) and CCU under the IC in irregular network utilising tools from stochastic geometry. Specifically, the closed-form expressions for the upper bound of CEU coverage probability and the lower bound of coverage probability for CCU are given out. For large threshold scenario, the authors show the potential of massive MIMO with IC on achieving the cell-centre-like experience in a more simple closed-form formula. Analytical and numerical results show that a reasonable balance between the CEU performance enhancement and the CCU performance degradation under the IC can be reached. Moreover, slight influence of cell-edge area on the coverage probability for both CEU and CCU is observed. Finally, it is found that the coverage probability of CEU is more sensitive to the variation of density of UEs and the base stations than that of CCU.

Coverage Analysis of Inter-Tier Interference Cancellation for Massive MIMO HetNet With Repulsion

The inter-tier interference cancellation (IC) by employing zero-forcing precoding method for two-tier heterogeneous network under considering repulsion between base stations (BSs) with adopting the β-Ginibre point process (β-GPP) model is investigated here. The distribution of the number of users that request IC under β-GPP model is first derived. Then, the approximation for coverage probability of the typical user with inter-tier IC is given out under massive multiple-input multiple-output (MIMO) assumption. Numerical results validate the accuracy of the approximation, and show that, compared with Poisson point process model, the β-GPP model is more accurate in prediction of the coverage probability in actual network with inter-tier IC. Furthermore, when the number of antennas in macro BS and that of users all approach infinity with fixed ratio, the coverage probability when the user is served by the macro BS can be simplified to a simple closed-form formula, which reveals the potential of massive MIMO to improve user performance under the case of large signal-to-interference ratio threshold.

Pricing Options Under Simultaneous Stochastic Volatility and Jumps: A Simple Closed-Form Formula Without Numerical/Computational Methods

We overcome the limitations of the previous literature in the European options pricing. In doing so, we provide a closed-form formula that doesn't require any numerical/computational methods. The formula is as simple as the classical Black-Scholes pricing formula. In addition, we simultaneously include jumps and stochastic volatility.

Closed-form estimations of the bistable region in metal cutting via the method of averaging

Machine tool vibrations in turning processes are analyzed by taking into account the nonlinearity of the cutting force characteristics. Unstable limit cycles are computed for the governing nonlinear delay-differential equation in order to determine the bistable technological parameter region where stable stationary cutting and large-amplitude machine tool vibrations coexist. Simple closed-form formulas are derived for the amplitude of limit cycles and for the size of the bistable region considering a general cutting force characteristics. The analytical results are determined by the method of averaging, which can be used to treat the nonlinearities without their third-order approximation. The results are confirmed by numerical continuation and using Melnikov’s integral.

Closed-form model to determine the co-axial probe reactance of an equilateral triangular patch antenna

AbstractA simple closed-form analytical formula is proposed to compute the probe reactance of an equilateral triangular patch antenna. The variation of the probe reactance with the variation of antenna dimension, substrate electrical parameters, and probe location is examined thoroughly. The computed values employing the present model show excellent agreement with experimental and simulation results.

Evaluation of Macro Diversity Gain in Long Range ALOHA Networks

Macro reception diversity is a capability allowing a packet to be received by several base stations. In this letter, we propose a simple but accurate analytical model to estimate the macro diversity gain in pure and slotted ALOHA wireless networks. The proposed model takes into account Rayleigh fading, shadowing, and capture effect. We obtain simple closed-form formulas for the network packet loss rate. We find that systems with macro reception diversity can benefit from shadowing. With path loss exponent 4, packet loss rate constraint 10% and 8 dB shadowing, the capacity of pure ALOHA with macro diversity is at least 2 times as much as that in systems in which a device transmits only to the best base station.

A Simple Closed-Form Formula for Pricing Basket Options

Options involving more than one underlying, often called <i>basket options</i>, are quite common, but valuation presents a mathematical problem. Although returns for a single stock can be assumed to follow a lognormal diffusion, leading to a variety of possible standard pricing models, sums, averages, or differences among lognormals are not lognormal. A number of approaches to approximating the returns distribution for an underlying basket have been proposed. Among the more successful is Levy’s technique of fitting an approximating lognormal for the basket by calculating and matching the first two moments of the basket’s return distribution. A problem arises, however, when the component assets have heterogeneous volatilities. Kan proposes a new technique that breaks the valuation formula into several pieces, each of which can be well approximated by moment matching. The accuracy of the Kan model over a wide range of parameter values is shown to be better than that of alternative models, and computation speed is nearly as good as the Levy model, which is much less accurate in general. <b>TOPICS:</b>Options, statistical methods

Design Formulas for Planar Fabry–Pérot Cavity Antennas Formed by Thick Partially Reflective Surfaces

Simple closed-form formulas are proposed to estimate the broadside radiation performance of planar Fabry–Perot-cavity (FPC) antennas formed by an arbitrary thick partially reflective-surface (PRS). The new formulas also estimate the attenuation and propagation constant of the leaky waves excited in the FPC when covered by such a thick PRS. When the PRS thickness vanishes the derived formulas reduce to those previously derived for FPC formed by infinitesimally thin PRS. Notably, we demonstrate that the same fundamental limit for the figure of merit defined as the pattern bandwidth-directivity product holds, as that for FPC with thin PRS, under the approximation that the phase variation of the PRS scattering parameters is not too rapid. The accuracy of the proposed formulas is verified with comparisons to numerical full-wave simulations.

Maximizing Expected Achievable Rates for Block-Fading Buffer-Aided Relay Channels

In this paper, the long-term average achievable rate over block-fading buffer-aided relay channels is maximized using a hybrid scheme that combines three essential transmission strategies, which are decode-and-forward, compress-and-forward, and direct transmission. The proposed hybrid scheme is dynamically adapted based on the channel state information. The integration and optimization of these three strategies provide a more generic and fundamental solution and give better achievable rates than the known schemes in the literature. Despite the large number of optimization variables, the proposed hybrid scheme can be optimized using simple closed-form formulas that are easy to apply in practical relay systems. This includes adjusting the transmission rate and compression when compress-and-forward is the selected strategy based on the channel conditions. Furthermore, in this paper, the hybrid scheme is applied to three different models of the Gaussian block-fading buffer-aided relay channels, depending on whether the relay is half or full duplex and whether the source and the relay have orthogonal or non-orthogonal channel access. Several numerical examples are provided to demonstrate the achievable rate results and compare them to the upper bounds of the ergodic capacity for each one of the three channel models under consideration.

Concise SER Analysis of Differential Quaternary Phase Shift Keying with Gray Coding Over Composite Multipath Shadowing Channels

To effectively assess the symbol error rate (SER) performance achieved by π/4 differential quaternary phase shift keying (π/4-DQPSK) with Gray coding over Rician-Nakagami fading channels, a simple closed-form asymptotic SER formula is derived, which is much simpler than the existing exact SER expression. From the proposed simple closed-form formula, we can obtain some useful insights into the achievable error rate performance for the DQPSK modulation with Gray coding for various channel conditions in the high signal-to-noise ratio (SNR) regions.

Error Probability Analysis and Applications of Amplitude-Coherent Detection in Flat Rayleigh Fading Channels

This paper presents the detector design and symbol error rate (SER) analysis of $M$ -ary amplitude shift keying over multipath fading channels using amplitude-coherent detection (ACD). The optimum detector is derived using the maximum likelihood criterion, and then it is used to derive two efficient low-complexity suboptimal detectors. The probability distribution function of the decision variables and the SER of the two suboptimal detectors are expressed using simple closed-form analytical formulas when single receiving antenna is used. The SER with receiver diversity is obtained using Monte Carlo simulation. The obtained analytical and simulation results reveal that ACD can provide reliable SER as compared with noncoherent detection. Moreover, we present an efficient blind channel estimation algorithm using ACD and hybrid modulation frame structures.

Label-noise resistant logistic regression for functional data classification with an application to Alzheimer's disease study.

Alzheimer's disease (AD) is usually diagnosed by clinicians through cognitive and functional performance test with a potential risk of misdiagnosis. Since the progression of AD is known to cause structural changes in the corpus callosum (CC), the CC thickness can be used as a functional covariate in AD classification problem for a diagnosis. However, misclassified class labels negatively impact the classification performance. Motivated by AD-CC association studies, we propose a logistic regression for functional data classification that is robust to misdiagnosis or label noise. Specifically, our logistic regression model is constructed by adopting individual intercepts to functional logistic regression model. This approach enables to indicate which observations are possibly mislabeled and also lead to a robust and efficient classifier. An effective algorithm using MM algorithm provides simple closed-form update formulas. We test our method using synthetic datasets to demonstrate its superiority over an existing method, and apply it to differentiating patients with AD from healthy normals based on CC from MRI.

A Tale in Three Cities: Comparison between GVV, SVI and IRV (Presentation Slides)

This presentation was based on the work of Jian Sun and Qiankun Niu, and presented at Morgan Stanley Global Modeling Meeting by Qiankun Niu on August 27, 2014. We thank Peter Carr for his invitation.Market Model is classified as the approach to model the implied volatility surface directly, which based on the observation that vallina options are liquidity traded in the market and the prices are driven by supply and demand factors. No-arbitrage imposes strict and complicated shape constraints on the implied volatility surface. Existed solution for this issue is to use another assistant quantity but sacrifice the simplicity of market models. To the best of our knowledge, a full arbitrage-free market model for the evolution of the implied volatility surface has yet to be developed.This work is an extension of the recent paper Carr and Sun (2013). The main thrust of this paper is to first follow the market model approach to establish a no arbitrage model named IRV model with simple closed form formula. Then, we will demonstrate the consistency between our IRV model and other two empirical models: SVI model in Gatheral and Jacquier (2014) and GVV model in Arslan et al. (2009). Potential application will be discussed in later sections. Applying the new model to the S&P 500 index options market and swaption market, the calibration results show that the model proposed in this paper works very well.

A closed-form formula for the conditional moments of the extended CIR process

In this paper, we derive a simple closed-form formula for the γth conditional moment of a variance process, based on the extended Cox–Ingersoll–Ross (ECIR) process, for any real number γ. The closed-form formula presented here is a result of successfully finding an exact solution for the partial differential equation, based on the ECIR process. We conduct Monte Carlo simulations to illustrate the accuracy of the current closed-form formula.

An Efficient Method to Analyze Lock Range in Ring Oscillators With Multiple Injections

In this brief, a simple method is proposed to analyze the lock range (LR) of ring oscillators with multiple injection signals by utilizing the generalized Adler equation. An exact formula is introduced for the LR in the case of a single injection. The proposed method is employed to study the asymmetry in high and low LRs. To describe the dependence of the oscillator LR to injection phases in the case of multiple injections, a simple closed-form formula and an empirical formula are derived in weak injection and strong injection, respectively. All of the proposed mathematical findings comprise the results of recent works that have been done in this area and are verified by circuit simulation.

Analytical Formulas for Tradeoff Among Channel Loss, Length, and Frequency of &lt;inline-formula&gt; &lt;tex-math notation="LaTeX"&gt;$RC$ &lt;/tex-math&gt;&lt;/inline-formula&gt;- and &lt;inline-formula&gt; &lt;tex-math notation="LaTeX"&gt;$LC$ &lt;/tex-math&gt;&lt;/inline-formula&gt;-Dominant Single-Ended Interconnects for Fast Equalized Link Tradeoff Estimation

This paper presents simple and intuitive closed-form formulas for tradeoff among channel loss, length, and frequency of single-ended interconnects to aid fast equalized link tradeoff estimation. Based on a transfer function model for single-ended interconnects, we derived two closed-form tradeoff formulas for $RC$ - and $LC$ -dominant interconnects. The formulas’ accuracies and computation time improvements are verified by comparison with the SPICE simulation and the measurement results. According to our formulas, the loss is roughly a linear function of the product of the length and the square root of the frequency in $RC$ -dominant interconnects. In $LC$ -dominant interconnects, the channel loss due to skin effect is roughly proportional to the product of the length and the square root of the frequency, while the channel loss due to dielectric loss is approximately proportional to the product of the length and the frequency. From the channel loss of our formulas, an equalized eye size can be easily estimated. Therefore, designers can use our tradeoff formulas easily and quickly to estimate link-level tradeoff among the equalized eye size, the data rate, and the channel length in designing an equalized link through $RC$ - and $LC$ -dominant single-ended interconnects.

On the short-maturity behaviour of the implied volatility skew for random strike options and applications to option pricing approximation

In this paper, we propose a general technique to develop first- and second-order closed-form approximation formulas for short-maturity options with random strikes. Our method is based on a change of numeraire and on Malliavin calculus techniques, which allow us to study the corresponding short-maturity implied volatility skew and to obtain simple closed-form approximation formulas depending on the derivative operator. The numerical analysis shows that these formulas are extremely accurate and improve some previous approaches for two-asset and three-asset spread options such as Kirk’s formula or the decomposition method presented in Alòs et al. [Energy Risk, 2011, 9, 52–57]. This methodology is not model-dependent, and it can be applied to the case of random interest rates and volatilities.

Analytical formula for residual current density excited in the process of gas ionization by a few-cycle laser pulse in the low-intensity limit

This work is devoted to analytical study of excitation of the residual current density (RCD) in the process of gas ionization by a few-cycle laser pulse. The RCD remains in the laser-produced plasma after the passage of the laser pulse and is as an initial push leading to excitation of the plasma oscillations which can radiate terahertz waves. We derive simple closed-form analytical formula for RCD for relatively small peak intensity of few-cycle laser pulse, which corresponds to small final degree of ionization. The dependences of the RCD on laser pulse parameters are discussed.

A Simple and Effective Closed-Form GN Model Correction Formula Accounting for Signal Non-Gaussian Distribution

The GN model of nonlinear fiber propagation has been shown to overestimate the variance of nonlinearity due to the signal Gaussianity approximation, leading to maximum reach predictions for realistic optical systems, which may be pessimistic by about 5% to 15%, depending on fiber type and system setup. Analytical corrections have been proposed, which, however, substantially increase the model complexity. In this paper, we provide a simple closed-form GN model correction formula, derived from the EGN model, which we show to be quite effective in correcting for the GN model tendency to overestimate nonlinearity. The formula also permits to clearly identify the correction dependence on key system parameters, such as span length and loss.