Johnson Curves Research Articles

Аппроксимация статистических данных заболеваемости коронавирусной инфекцией с учетом расслоения по сопутствующим диагнозам

The article considers the stratification of concomitant diagnoses of Covid-19 recovery statistics for the city of Irkutsk for 2020-2021. The previous study was conducted without taking into account such stratification. Various options for approximating real statistics by Gaussian and Lorentz functions, gamma distribution, and Johnson curves are considered. It is shown that the stratification of recovery statistics improves the approximation of Gaussian and Lorentz functions in comparison with integral statistics, and the construction of an approximation based on the Lorentz function always describes the real statistics better. Estimates of mathematical expectation and variance based on statistical data are consistent with estimates of these values based on the Gaussian approximation of statistics by the least squares method, i.e. the approaches are equivalent. At the same time, calculations of the Pearson Chi-squared criterion reject the hypothesis that empirical data correspond to the assumed theoretical distribution. Therefore, we cannot talk about finding the distribution function, but only about approximating statistics by certain types of curves. The fitting of empirical data by Gaussian and Lorentz functions was carried out using the least squares method. In general, the approximation error due to the stratification of statistics on concomitant diagnoses decreases from 6% to 3%.

Option Pricing by Willow Tree Method for Generalized Hyperbolic Lévy Processes

In this paper, a new approach is proposed to construct willow tree (WT) for generalized hyperbolic (GH) Lévy processes. There are two advantages of our proposed approach compared to the classical WT methods. Firstly, it avoids the moments matching from Johnson curve in the known WT construction. Secondly, the error of European option pricing is only determined by time partition ∆ t under some conditions. Since the moments of Lévy measure are removed from this algorithm, our approach improves the stability and accuracy of WT in option pricing. Numerical experiments support our claims. Moreover, the new approach can be extended to other Lévy processes if their characteristic functions are expressed by explicit forms.

Johnson curve model based CDF mapping method for estimating extreme values of the measured non-Gaussian wind pressures including the non-stationary effect investigation

The study about the extreme value estimation of non-Gaussian wind pressure signals is crucial for structural safety or health monitoring. This is because the long tail region of the non-Gaussian distribution will lead to the unexpected extreme value which results in structural damage or fatigue damage in the practical engineering. In a general view, the existing extremes value estimation method can be divided into two main categories: the CDF mapping method and the peak factor method. The former is relatively complicated but accurate, considered highly dependent on the accuracy of the non-Gaussian parent distribution fitting; The latter is relatively convenient but passable, only requiring the skewness and kurtosis of the standardized non-Gaussian processes used for the input of the polynomial transformation. In this work, firstly, a Johnson curve model-based CDF mapping method (JCCDF) is proposed to combine the advantages of both approaches. By employing the measured non-Gaussian wind pressures, the convenient and accurate JCCDF is verified with the other CDF mapping methods, and a simplified direct JCCDF-based extreme value variables mapping method is developed. Through observing the calculated extreme values in the long-tail region of different methods, it can be figured out that the results of the peak factor method are rather less, and the computed values based on the observed extreme method are close to that of the CDF mapping method. In the investigation about the influence of non-stationary characteristics on the extreme value estimation, it can be spotted that the computed extreme values in the long-tail region of all methods are getting greater with different degrees after considering the time-varying first and second-order statistical moment. This point reveals that the non-stationary characteristics need to be taken into account.

Robust willow tree method under Lévy processes

The cosine willow tree method (Ma et al., 2021) provides a first lattice framework to price options and calculate risk measures under various Lévy process models. However, it may fail under some circumstances due to the failure in moment matching by the Johnson curve. In this paper, we propose a more robust approach to construct a willow tree via numerical integration. Since the moment matching by the Johnson curve is avoided, the new approach provides a better convergence than the cosine willow tree method. Finally, numerical experiments are carried out to support our claims.

EFFICIENT RISK MEASURES CALCULATIONS FOR GENERALIZED CREDITRISK+ MODELS

Numerical calculations of risk measures and risk contributions in credit risk models amount to the evaluation of various forms of quantiles, tail probabilities and tail expectations of the portfolio loss distribution. Though the moment generating function of the loss distribution in the [Formula: see text] model is available in analytic closed form, efficient, accurate and reliable computation of risk measures (Value-at-Risk and Expected Shortfall) and risk contributions for the [Formula: see text] model poses technical challenges. We propose various numerical algorithms for risk measures and risk contributions calculations of the enhanced [Formula: see text] model under the common background vector framework using the Johnson curve fitting method, saddlepoint approximation method, importance sampling in Monte Carlo simulation and check function formulation. Our numerical studies on stylized credit portfolios and benchmark industrial credit portfolios reveal that the Johnson curve fitting approach works very well for credit portfolios with a large number of obligors, demonstrating high level of numerical reliability and computational efficiency. Once we implement the systematic procedure of finding the saddlepoint within an approximate domain, the saddlepoint approximation schemes provide efficient calculation and accurate numerical results. The importance sampling in Monte Carlo simulation methods are easy to implement, but they compete less favorably in accuracy and efficiency with other numerical algorithms. The less commonly used check function formulation is limited to risk measures calculations. It competes favorably in accuracy and reliability, but an extra optimization algorithm is required.

Efficient Risk Measures Calculations for Generalized CreditRisk+ Models

Numerical calculations of risk measures in credit risk models amount to evaluation of various forms of tail expectations of portfolio loss distribution. Though the moment generating function of the loss distribution in CreditRisk+ model is available in analytic closed form, efficient, accurate and reliable computation of risk measures (VaR and Expected Shortfall) and risk contributions for the CreditRisk+ model pose technical challenge. We propose various numerical algorithms for risk measures and risk contributions calculations of the enhanced CreditRisk+ model under the common background vector framework using the Johnson curve fitting method, saddle-point approximation method, curve fitting method, importance sampling in Monte Carlo simulation and check function formulation. Our numerical studies on stylized credit portfolios and benchmark industrial credit portfolios reveal that the Johnson curve fitting approach works very well for credit portfolios with a large number of obligors, demonstrating high level of numerical reliability and computational efficiency. The importance sampling in Monte Carlo simulation methods are easy to implement, but they compete less favorably in accuracy and reliability with other numerical algorithms. The less commonly used check function formulation is limited to risk measures calculations. It competes favorably in accuracy and reliability, but an external industrial optimization software is required.

Process Capability Indices under Non-Normality Conditions using Johnson Systems

Process capability indices (PCIs) quantify the ability of a process to produce on target and within specifications performances. Basic indices designed for normal processes gives flawed results for non-normal process. Numerous methods have been proposed for non-normal processes to estimate PCIs in which some of them are based on transformation methods. The Johnson system comprising three types that translate a continuous non-normal distribution to normal. The aim of this paper is to estimate four basic indices for non-normal process using Johnson system with single straightforward procedure. The efficacy of the proposed approach can be assessed for all three Johnson Curves (SB, SU, SL) but result for SU is presented in this paper. PCIs for a data set are estimated and percentiles are obtained by our proposed exact method based on selected Johnson density function which was earlier based on approximate methods without any prior knowledge of density function of non-normal process. We compare our results with other existing methods to estimate PCIs for non-normal process. From statistical analysis we have noted that this modification improve process capability indices.

Improving the accuracy of identifying the lognormal curve in the Johnson system

Multiple estimation methods provide discriminants to determine curves in the Johnson (1949) system, but few studies have been conducted to explore their accuracy. In this study, a simulation study was implemented to determine whether three discriminants can discriminate the types of Johnson curves correctly. The results show these discriminants failed to identify the lognormal curve in the system. The receiver operating characteristic curves and the Youden index (Youden 1950) were used to find the optimal cutoff points for these discriminants to identify the lognormal curve more precisely. A new discriminant was also proposed for the moment method (Johnson 1949) in this study.

Moment matching machine learning methods for risk management of large variable annuity portfolios

Variable annuity (VA) with embedded guarantees have rapidly grown in popularity around the world in recent years. Valuation of VAs has been studied extensively in past decades. However, most of the studies focus on a single contract. These methods can be extended neither to valuate nor to manage the risk of a large variable annuity portfolio due to the computational complexity. In this paper, we propose an efficient moment matching machine learning method to solve this problem. This method is proved to be a good candidate for risk management in terms of the speed of and the complexity of computing the annual dollar deltas, VaRs and CVaRs for a large variable annuity portfolio whose contracts are over a period of 25 years. There are two stages for our method. First, we select a small number of contracts and propose a moment matching Monte Carlo method based on the Johnson curve, rather than the well known nested simulations, to compute the annual dollar deltas, VaRs and CVaRs for each selected contract. Then, these computed results are used as a training set for well known machine learning methods, such as regression tree, neural network and so on. Afterwards, the annual dollar deltas, VaRs and CVaRs for the entire portfolio are predicted through the trained machine learning method. Compared to other existing methods Bauer et al. (2008); Gan (2013); Gan and Lin (2015), our method is very efficient and accurate. Finally, our test results support our claims.

Study on the surface topography considering grinding chatter based on dynamics and reliability

Surface grinding is always accompanied with chatter due to self-excited vibration. It often leads to an unexpected impact on the quality of the workpiece’s topography. However, the chatter is regarded as a harmonic vibration in most topography researches. This may lose preciseness when the relative vibration and the abrasive trajectory are taken into consideration. In order to study the relationship between the system’s dynamic characteristic and the workpiece’s topography, a two-DOF (degree of freedom) dynamic model with time-delay characteristic is established accordingly. Then, reliability analysis is introduced into chatter vibration by analyzing the fluctuations of dynamic parameters with two analysis methods, namely Monte Carlo (MC) and first-order second-moment (FOSM). With the above two reliability analysis methods, the calculations are carried out as follows: firstly, the non-Gaussian distribution of the grinding wheel based on Johnson Curves and filter techniques is established. Secondly, the results of the dynamic analysis are coupled into the grain trajectory equation. Thirdly, the influence of the wheel grinding parameters and dynamic parameters on the surface height is discussed by coupling the dynamic characteristics into the simulation model. Finally, the simulations and experiments are carried out on the impact of different feeding rates and sections on the workpiece to the surface heights. The comparisons verify the prediction of the simulation model. The obtained conclusions could be applied to optimize the workpiece’s topography by regulating the grinding parameters and dynamic parameters to weaken the chatter’s influence.

Travel Time Reliability for Urban Networks: Modelling and Empirics

The importance of travel time reliability in traffic management, control, and network design has received a lot of attention in the past decade. In this paper, a network travel time distribution model based on the Johnson curve system is proposed. The model is applied to field travel time data collected by Automated Number Plate Recognition (ANPR) cameras. We further investigate the network-level travel time reliability by connecting the network reliability measures such as the weighted standard deviation of travel time rate and the weighted skewness of travel time rate distributions with network traffic characteristics (e.g., the network density). The weighting is done with respect to the number of signalized intersections on a trip. A clear linear relation between the weighted average travel time rate and the weighted standard deviation of travel time rate can be observed for different time periods with time-varying demand. Furthermore, both the weighted average travel time rate and the weighted standard deviation of travel time rate increase monotonically with network density. The empirical findings of the relation between network travel time reliability and network traffic characteristics can be possibly applied to assess traffic management and control measures to improve network travel time reliability.

Analysis of Assessment Method about the Flight Technical Error Based on Johnson Curves

Ensured navigation performance and standardized functionality of RNP allow new procedures to be designed for airports where infrastructure or terrain makes it difficult or impossible to use some runways. Aircraft must be approved as being able to meet the RNP. Based on theory of statistical process control, this paper evaluates the flight technical error in optimal curve fitting of the distributions Johnson's. The concrete parameter selection, the selection process Johnson curves and a concrete model are given. The results of analysis can be used to the evaluation of flight technical error.

Silurian sequence stratigraphy of the Carnic Alps, Austria

Sequence stratigraphy provides an alternative approach to correlation that may also serve as a predictive framework for the interpretation of sea-level and sedimentological change. The main aim of this study is to apply sequence stratigraphic concepts to the biostratigraphically well constrained shallow to moderately deep shelf carbonates and basinal graptolitic shale facies of the Silurian successions in the Carnic Alps of Austria in order to correlate the sequence packages and sea-level changes established there with those identified in other areas of North America and Europe. Documenting local sea-level curves is essential for determining global eustasy. The sea-level curve for the Silurian of the Carnic Alps has been elaborated within a refined stratigraphic framework for the Silurian based on conodont and graptolite biozonation [Melchin, M.J., Cooper, R.A., Sadler, P.M., 2004. The Silurian Period. In: Gradstein, F.M., Ogg, J.G., Smith, A.G. (Eds), A Geologic Time Scale. Cambridge University Press, Cambridge, pp. 188–201.]. In particular, the minor and frequent sea-level changes within the Silurian of the Carnic Alps are of special interest as these stratigraphic intervals are poorly preserved and not well studied in other Silurian localities. The interpretation of the field and microfacies analysis indicates major sequence boundaries in the Llandovery (3), Wenlock (3), Ludlow (3) and Přídolí–Lochkovian (2) which may be correlated with coeval disconformities in the Appalachian Foreland Basin of eastern North America and/or in the Welsh Basin of the British Isles. The following times appear to represent relative sea-level highstand maxima in the Silurian of the Carnic Alps, as indicated by dark, graptolitic shales in deep shelf to basinal carbonate-dominated sections: a) early Aeronian, approximately the Coronograptus cyphus graptolite Zone/ Demirastrites triangulatus graptolite Zone), b) the early Telychian ( Oktavites spiralis graptolite Zone; Pterospathodus celloni conodont Superzone), c) late Telychian (lower Pterospathodus a. amorphognathoides conodont Zone); d) early to middle Sheinwoodian ( Kockelella ranuliformis to Ozarkodina sagitta rhenana conodont Zones; Monograptus riccartonensis graptolite Zones); e) mid-Wenlock ((?upper Kockelella walliseri conodont Zone; Cyrtograptus rigidus graptolite Zone); f) mid Homerian ( Ozarkodina bohemica conodont Zone; Gothograptus nassa graptolite Zone); g) near the Wenlock–Ludlow boundary ( Neodiversograptus nilssoni graptolite Zone); h) Polygnathoides siluricus conodont Zone; i) near the Ludlow–Přídolí boundary (upper Ozarkodina snajdri Interval Zone); j) lower Přídolí ( Monograptus parultimus graptolite Zone) and k) at the Silurian–Devonian boundary (earliest Lochkovian: Icriodus woschmidti woschmidti conodont Zone. Of these, the earlier (at least b–e) are well represented in the Appalachian Basin, as in Avalonian sections in Great Britain and in Baltica. Johnson [Johnson, M.E., 2006. Relationship of Silurian sea-level fluctuations to oceanic episodes and events. GFF 128, 115–121.] documented eight major highstands in global sea-level during the Silurian. The temporal resolution obtained in the Carnic Alps for local sea-level changes allows for refinement of the Telychian to Přídolí sea-level curve, as the stratigraphic successions are chronologically well-defined using both conodont and graptolite biostratigraphy and K-bentonite levels. Nearly all inferred deepenings in the Carnic Alps section, with the exception of that in the Polygnathoides siluricus conodont Zone, approximately match highstands recorded on the Silurian sea-level curves of Johnson [Johnson, M.E., 1996. Stable cratonic sequences and a standard for Silurian eustasy. In: Witzke, B.J., Ludvigson, G.A., Day, J. (Eds.), Paleozoic sequence stratigraphy – Views from the North American Craton. Geol. Soc. Am. Spec. Pap. vol. 306, Boulder, pp. 203–212., Johnson, M.E., 2006. Relationship of Silurian sea-level fluctuations to oceanic episodes and events. GFF 128, 115–121.]; however, the two Telychian highstands are not distinguished but rather are combined as highstand 4 on the Johnson curve, although they are recognized by Loydell [Loydell, D.K., 1998. Early Silurian sea-level changes. Geol. Mag. 135 (4), 447–471.], and the upper Kockelella walliseri conodont Zone deepening is not explicitly numbered by Johnson. These similarities suggest pervasive and probably eustatic events that are manifested in the Apulia Terrane [Cocks, L.R.M., Torsvik, T.H., 2002. Earth geography from 500 to 400 million years ago: a faunal and palaeomagnetic review. J. Geol. Soc. Lond. 159 (6), 631–644.], Laurentia and Avalonia during these time intervals.

Estimating Johnson Curve Population Distributions in MULTILOG

The shape of the latent trait distribution can be of considerable theoretical and methodological importance. A simulation study was performed to examine the distribution of the likelihood ratio statistic that was used to test for normality via Johnson curves, the power to detect deviations from normality, and the estimation properties of the item and latent trait distribution parameters. Except in conditions in which all items had high-difficulty parameters or were dichotomous, the distribution of the statistic used to test for normality could be approximated with a chi-square distributed with 2 degrees of freedom. In a variety of situations, the power was good enough to detect even small deviations from normality. Compared to assuming a normal distribution, allowing for Johnson curve latent trait distributions increased the standard errors of the marginal maximum likelihood item parameter estimates but reduced their bias in situations in which the latent trait was nonnormal.

An algorithm to determine the parameters of SU -curves in the johnson system of probabillity distributions by moment matching *

We describe an algorithm to fit an SU -Curve of the Johnson system by moment matching. The algorithm follows from a new parametrization, and reduces the problem to a root finding procedure that can be implemented efficiently using a bisection or a Newton-Raphson method. This allows the four parameters of the Johnson curve to be determined to any desired degree of accuracy, and is fast enough to be implemented in a real-time setting. A practical application of the method lies in the fact that many firms use the Johnson system to manage financial risk

Bayesian Discrimination Between Two Multivariate Normal Populations With Equal Covariance Matrices

Abstract Here Bayesian and classical estimates of P, the probability that a new observation belongs to one of two multivariate normal populations with equal covariance matrices, are compared. In particular, the regions of location of a new observation for which the estimates differ are shown. The Bayesian and classical allocation rule contours are given and their differences highlighted. The posterior moments of L, the log-density ratio for a new observation, are given, and the distribution of L is approximated using Johnson curves. The posterior distribution and interval for P are obtained by transformation. An example is given that demonstrates their use.

Bayesian Discrimination between Two Multivariate Normal Populations with Equal Covariance Matrices

Here Bayesian and classical estimates of P, the probability that a new observation belongs to one of two multivariate normal populations with equal covariance matrices, are compared. In particular, the regions of location of a new observation for which the estimates differ are shown. The Bayesian and classical allocation rule contours are given and their differences highlighted. The posterior moments of L, the log-density ratio for a new observation, are given, and the distribution of L is approximated using Johnson curves. The posterior distribution and interval for P are obtained by transformation. An example is given that demonstrates their use.

Generating ion implantation profiles in one and two dimensions. 1: density functions

In this, the first of two papers, the problem of constructing ion implantation profiles in one and two dimensions from depth-independent spatial moments is discussed. Comparisons are made between Pearson and Johnson curves, constructed from moments produced by a transport equation solver, and profiles obtained directly from Monte Carlo simulations. A set of such comparisons, using consistent input quantities, is performed over a range of ion - target mass ratios and energies. For projected range distributions of the ions B, P and As into a-Si, a single Johnson type describes the implants over the energy range 1 keV to 1 MeV. The description using Pearson curves requires two types (I and VI). Also, taking the Monte Carlo data as a reference, the Johnson curves are equivalent, if not superior, to the Pearson curves in terms of fit accuracy. For lateral distributions of the same ion types over the same energy range it is shown that if the depth-dependent lateral kurtosis is less than 3.0, then the Pearson type II (bounded), Johnson type (bounded) and the modified Gaussian (unbounded) curves prove acceptable representations. If the depth-dependent lateral kurtosis is greater than 3.0 then the Pearson type VII (unbounded) and Johnson type (unbounded) curves are good representations.

Generating ion implantation profiles in one and two dimensions. 2: depth-dependent moments and line-source responses

In this, the second of two papers, various techniques and models are proposed for the extraction of depth-dependent lateral moments from the depth-independent mixed moments produced by transport equation solvers. These depth-dependent moments are then compared with those obtained directly from Monte Carlo simulations. A set of such comparisons, using consistent input quantities, is performed over a range of ion - target mass ratios and energies. The depth-dependent moments are then combined with Pearson and/or Johnson curves to form two-dimensional ion implantation profiles. Comparisons are made between these line-source responses (LSRs) and LSRs obtained directly from Monte Carlo simulations into a-Si for the various models over a range of energies and ion types. Selection of appropriate models leads to LSRs for the ions B, P and As implanted into a-Si which are in good agreement with Monte Carlo simulations over three orders of magnitude of profile concentration. The techniques described will enable two-dimensional profile information to be stored and regenerated, quickly and efficiently, within process simulators so that rapid optimization of processing parameters may be achieved.

An influence method for outlier detection applied to time series traffic data

The work of Chernick et al. (1982) is extended to form a quantitative outlier detection statistic for use with time series data. The statistic is formed from the squared elements of the influence function matrix, where each element of the matrix gives the influence on the theoretical autocorrelation function at lag k (pk) of a pair of obser vations at time lag k. The approximate first four moments for the statistic are derived and, by fitting Johnson curves to these theoretical moments, critical points are also produced. The statistic is also used to form the basis of an adjustment procedure to treat outliers or estimate missing values in the time series. The nuclear power data of Chernick et al. and the traffic count data of the Department of Transport are used for practical illustration.