Normal Cumulative Distribution Function Research Articles

Optimized dispersion Higuchi fractal dimension and its refined composite multi-scale version for signal analysis

Higuchi fractal dimension (HFD), as a classic nonlinear dynamic metric, which is commonly used to detect signal dynamic changes. However, it is difficult for HFD to process signal outliers. To address this issue, dispersion HFD (DHFD) is proposed, which improves the signal complexity representation ability of HFD by introducing the normal cumulative distribution function and round function in dispersion entropy. Nevertheless, the parameter selection of DHFD can affect the complexity value. Therefore, an optimized dispersion HFD (ODHFD) is proposed, which solves the threshold selection problem of DHFD and can more effectively reflect the complexity of the signal. In addition, an optimized refined composite DHFD (ORCMDHFD) has been proposed, which can more comprehensively reflect the complexity information for the signal at multiple scales. The simulation experiment results show that DHFD has a smaller standard deviation than HFD when calculating white noise signal complexity, and DHFD have the least dependence on signal length compared to other metrics, as well as RCMDHFD has the best separability for simulated noise signals. Actual experiments have shown that ODHFD and ORCMDHFD is superior to other entropy and fractal dimension metrics in distinguishing ship radiated noise and mechanical fault signals, and has broad application prospects in the field of signal analysis.

Fault Diagnosis of Wind Turbine Gearbox Based on Modified Hierarchical Fluctuation Dispersion Entropy of Tan-Sigmoid Mapping.

Vibration monitoring and analysis are important methods in wind turbine gearbox fault diagnosis, and determining how to extract fault characteristics from the vibration signal is of primary importance. This paper presents a fault diagnosis approach based on modified hierarchical fluctuation dispersion entropy of tan-sigmoid mapping (MHFDE_TANSIG) and northern goshawk optimization-support vector machine (NGO-SVM) for wind turbine gearboxes. The tan-sigmoid (TANSIG) mapping function replaces the normal cumulative distribution function (NCDF) of the hierarchical fluctuation dispersion entropy (HFDE) method. Additionally, the hierarchical decomposition of the HFDE method is improved, resulting in the proposed MHFDE_TANSIG method. The vibration signals of wind turbine gearboxes are analyzed using the MHFDE_TANSIG method to extract fault features. The constructed fault feature set is used to intelligently recognize and classify the fault type of the gearboxes with the NGO-SVM classifier. The fault diagnosis methods based on MHFDE_TANSIG and NGO-SVM are applied to the experimental data analysis of gearboxes with different operating conditions. The results show that the fault diagnosis model proposed in this paper has the best performance with an average accuracy rate of 97.25%.

New simple bounds for standard normal distribution function

This paper presents new simple lower and upper bounds for the cumulative normal distribution function, Φ ( z ) . The accuracy and closeness of the proposed bounds to the exact Φ ( z ) are investigated based on the maximum absolute error and the mean absolute error. It is found that the maximum absolute error of the proposed lower bound is 8.55 × 10 − 3 and it is 4.1 × 10 − 4 for the upper bound. In addition, based on 5001 values between z = 0 and z = 5 with step 0.001, we found that the mean absolute error is 3.27 × 10 − 3 for the lower bound and it is 1.1 × 10 − 4 for the upper bound and these two values decrease with increasing the z value.

Incipient fault detection and condition assessment in DFIGs based on external leakage flux sensing and modified multiscale poincare plots analysis

Abstract Although doubly fed induction generators (DFIG) are widely used, difficulties in early fault detection and severity assessment for inter-turn short-circuit (ITSC) faults are highly prominent. In this manuscript, a novel incipient fault detection and state assessment method based on external leakage flux sensing and modified multiscale poincare plots (MMSPP) is proposed. An external leakage flux sensor is placed on the axial-end position of the generator to monitor the presence and evolution of ITSC faults. Multiscale poincare mapping is a novel nonlinear tool that is further developed and modified using the normal cumulative distribution function and multiscale computing methods to capture the behavior evolution and changes in the generator’s external leakage flux signals. The healthy indicator is based on the analysis of elliptical orbit features extracted from MMSPP, established by support vector data description with parameter optimization. The effectiveness of the proposed method was implemented and verified under an experimental environment on a 100 kW DFIG platform to detect incipient inter-turn short circuits (mainly considering the two-turn ITSC) and evaluate the performance degradation (three- to eight-turn ITSC) with 0%, 50%, and 100% different load conditions. The experimental results showed the viability of the approach and fault indicator for incipient fault detection and condition assessment of the wind generator’s low inter-turn insulation faults and for relative quantification of ITSC fault severity.

Using Simpson’s Rule to Evaluate the Normal Cumulative Distribution Function in an Algorithmic Option Pricing Problem

Algorithmic problems are frequently used to help ensure academic honesty during assessments in quantitative courses, asthey allow instructors to easily generate multiple versions of a test, quiz, or assignment with questions that are similar but whoseinput variables and correct answers differ. In general, instructors who create an algorithmic problem must program its solutionby entering an algebraic expression using the input variables. While this is usually straightforward enough, it can be problematicwhen the solution requires a more advanced function that has no algebraic representation. One such challenge occurs withfinancial option pricing problems based on the Black-Scholes model, which relies on the normal cumulative distribution function(CDF). Although students can easily obtain normal CDF values from a calculator or spreadsheet function, no such function isavailable within testing software, presenting an obstacle for instructors attempting to author algorithmic problems on optionpricing. I demonstrate how to overcome this difficulty by using Simpson’s Rule to numerically integrate the underlying normalprobability density function, thereby generating a highly accurate approximation of the CDF that can be programmed into thesoftware and used in the Black-Scholes formula.

An improved detection method of GNSS faults with fractional information divergence

Early detection of faults guarantees the integrity of navigation systems. The standard receiver autonomous integrity monitoring (RAIM) incorporated with the Kullback-Leibler divergence (KLD) has been proposed to improve detection performance. However, the KLD-RAIM cannot ensure a high detection rate in the presence of small or growing faults. This paper broadens the existing KLD-RAIM to further enhance the detection ability for satellite navigation-based aviation applications. The Cumulative residual KL information (CRKL) replaces the KLD to quantify the dissimilarity between the current and normal cumulative distribution functions for the CRKL-RAIM construction. Then CRKL is extended to fractional orders using Riemann–Louville (RL) calculus to form fractional CRKL-RAIM for detection sensitivity enhancement. Simulations using static GNSS station data were conducted to compare four methods: the standard RAIM, existing KLD-RAIM, the proposed CRKL-RAIM, and fractional CRKL-RAIM. When dealing with small step faults, the detection sensitivity increases progressively, with the fractional CRKL-RAIM capable of detecting 10-meter faults. Regarding growing faults, the standard RAIM fails to detect and KLD-RAIM barely detects faults around 600 s with 50% missed detection rate. By contrast, CRKL-RAIM and fractional CRKL-RAIM detect faults around 100 and 200 s in advance, respectively, without any missed detection events.

Refining analytic approximation based estimation of mixed multinomial probit models by parameter selection

Applying analytic approximations for computing multivariate normal cumulative distribution functions has led to a substantial improvement in the estimability of mixed multinomial probit models, both in terms of accuracy and especially in terms of computation time. This paper makes a contribution by presenting a possible way to improve the accuracy of estimating mixed multinomial probit model covariances based on the idea of parameter selection using cross-validation. Comparisons to the MACML approach indicate that the proposed parameter selection approach is able to recover covariance parameters more accurately, even when there is a moderate degree of independence between the random coefficients. The approach also estimates parameters efficiently, with standard errors tending to be smaller than those of the MACML approach, which can be observed by means of a real data case.

A Fast Multipoint Expected Improvement for Parallel Expensive Optimization

The multipoint expected improvement (EI) criterion is a well-defined parallel infill criterion for expensive optimization. However, the exact calculation of the classical multipoint EI involves evaluating a significant amount of multivariate normal cumulative distribution functions, which makes the inner optimization of this infill criterion very time consuming when the number of infill samples is large. To tackle this problem, we propose a novel fast multipoint EI criterion in this work. The proposed infill criterion is calculated using only univariate normal cumulative distributions; thus, it is easier to implement and cheaper to compute than the classical multipoint EI criterion. It is shown that the computational time of the proposed fast multipoint EI is several orders lower than the classical multipoint EI on the benchmark problems. In addition, we propose to use cooperative coevolutionary algorithms (CCEAs) to solve the inner optimization problem of the proposed fast multipoint EI by decomposing the optimization problem into multiple subproblems with each subproblem corresponding to one infill sample and solving these subproblems cooperatively. Numerical experiments show that using CCEAs can improve the performance of the proposed algorithm significantly compared with using standard evolutionary algorithms. This work provides a fast and efficient approach for parallel expensive optimization.

A distribution-free smoothed combination method to improve discrimination accuracy in multi-category classification.

Results from multiple diagnostic tests are combined in many ways to improve the overall diagnostic accuracy. For binary classification, maximization of the empirical estimate of the area under the receiver operating characteristic curve has widely been used to produce an optimal linear combination of multiple biomarkers. However, in the presence of a large number of biomarkers, this method proves to be computationally expensive and difficult to implement since it involves maximization of a discontinuous, non-smooth function for which gradient-based methods cannot be used directly. The complexity of this problem further increases when the classification problem becomes multi-category. In this article, we develop a linear combination method that maximizes a smooth approximation of the empirical Hyper-volume Under Manifolds for the multi-category outcome. We approximate HUM by replacing the indicator function with the sigmoid function and normal cumulative distribution function. With such smooth approximations, efficient gradient-based algorithms are employed to obtain better solutions with less computing time. We show that under some regularity conditions, the proposed method yields consistent estimates of the coefficient parameters. We derive the asymptotic normality of the coefficient estimates. A simulation study is performed to study the effectiveness of our proposed method as compared to other existing methods. The method is illustrated using two real medical data sets.

Bivariate structural-fire fragility curves for simple-span overpass bridges with composite steel plate girders

This study demonstrates the development of bivariate fragility curves for a fire-exposed simple-span overpass bridge prototype with composite steel plate girders. The fire and resulting heat transfer to the girders are modeled using the computationally efficient Modified Discretized Solid Flame (MDSF) model, developed previously by the authors. Several input parameters to model the thermo-structural response of the girders (particularly the material strength and applied loading) are stochastically selected for Monte Carlo Simulation via Latin Hypercube Sampling. The thermo-structural response of the composite steel girders is calculated using uncoupled, reduced-form finite element analyses. The structural-fire response of the prototype bridge girder is iteratively calculated for a large suite (∼1,000 iterations) of fire scenarios and then categorized into escalating damage levels based on the maximum deflection reached during the fire event. The damage from each fire scenario is correlated to two measures of fire hazard intensity: the peak heat release rate, and the total thermal energy imparted along the girder span. Bivariate fragility curves that correlate the two intensity measures to each damage level via a cumulative normal distribution function are obtained for the prototype bridge with span length varying from 12.2 to 42.7 m. An illustrative example uses these fragility curves to assess fire-induced damage for the two overpass spans in the MacArthur Maze interchange in Oakland, CA that collapsed due to a 2007 tanker truck fire.

Development of Integrated Choice and Latent Variable (ICLV) Models Using Matrix-Based Analytic Approximation and Automatic Differentiation Methods on TensorFlow Platform

This study further explores the multinomial probit-based integrated choice and latent variable (ICLV) models. The LDLT matrix-based analytic approximation methods, including Mendell and Elston (ME) method, bivariate ME (BME) method, and two-variate bivariate screening (TVBS) method, were adapted to calculate the multivariate cumulative normal distribution (MVNCD) function in the ICLV model because of the better performances in accuracy and computational time. Integrated with the composite marginal likelihood (CML) estimation approach, the ICLV model based on high-dimensional integration can be estimated accurately within a reasonable time. In this study, some three-alternative and four-alternative ICLV models are simulated to examine their abilities to recover model parameters. It is found that the parameter estimates and standard error estimates are acceptable for both models and the computational time is expected to decrease using tensor data structures on the TensorFlow platform. For the four-alternative ICLV models, the TVBS method has the highest level of accuracy. The BME method is also a good alternative to TVBS if computational time is of great concern. The application of the automatic differentiation (AD) technique in the model can free researchers from coding analytical gradients of log-likelihood functions and thereby greatly reduce the workload of researchers.

Dispersion entropy-based Lempel-Ziv complexity: A new metric for signal analysis

Lempel-Ziv complexity (LZC) is one of the most important metrics for detecting dynamic changes in non-linear signals, but due to its dependence on binary conversion, LZC tends to lose some of the effective information of the time series, while the noise immunity is not guaranteed and cannot be applied to the detection of real-world signals. To address these limitations, we have developed a dispersion entropy-based LZC (DELZC) based on the normal cumulative distribution function (NCDF) and dispersion permutation patterns. In DELZC, the time series are first processed by NCDF to increase the number of classes and thus reduce the loss of information, and in addition, the dispersive entropy (DE) in terms of the ordinal number of the permutation pattern is considered to replace the binary conversion of LZC, thus improving the ability to capture the dynamic changes in the time series. In signal analysis using a set of time series, several easy-to-understand concepts are used to demonstrate the superiority of DELZC over other three LZC metrics in detecting the dynamic variability of the signals, namely LZC, dispersion LZC (DLZC) and permutation LZC (PLZC). The synthetic signal experiments demonstrate the superiority of DELZC in detecting the dynamic changes of time series and characterizing the complexity of signal, and also have lower noise sensitivity. Moreover, DELZC has the best performance in diagnosing four states of rolling bearing fault signals and classifying five types of ship radiation noise signals, with higher recognition rates than LZC, PLZC and DLZC.

Mixed Multinomial Probit Model Accommodating Flexible Covariance Structure and Random Taste Variation: An Application to Commute Mode Choice Behavior

This paper developed a mixed multinomial probit (MMNP) model with alternative error specification and random coefficients (for both generic variables and personal attributes) to accommodate flexible covariance structure and taste variation. The MMNP model can be efficiently estimated with analytic approximations of multivariate normal cumulative distribution functions, which avoid defects of simulation-based integration in the mixed multinomial logit (MMNL) model. The integral dimension of the MMNL model increases as random coefficients increase, but it only depends on the number of available alternatives in the MMNP model. Simulation experiments and empirical analysis of Shanghai commuters’ mode choice behavior were undertaken to examine the performance of MMNP models. Both simulation results and empirical results show that MMNP models can well accommodate flexible covariance structures and taste variation reflected through random coefficients being associated with both generic and personal variables. Empirical results indicate that the MMNP model performs better than traditional discrete choice models, such as the multinomial logit, the cross-nested logit, MMNL, and multinomial probit models. Random coefficients of “in-vehicle time of car” and “number of companions” indicate taste heterogeneity and the identifiability of random coefficients associated with both generic and personal attributes. Pairwise positive correlations between car/taxi, bus/metro, and bus/bus and metro are to be expected. However, the positive correlation between the car and metro modes may be unique to the Chinese city, Shanghai, because of the developed metro system. Unequal error variances reflect heterogeneities in unspecified factors in commute modes’ utilities. The MMNP model will offer an alternative efficient way to accommodate taste heterogeneity and flexible error covariance structure in discrete choice models. Compared with the MMNL model, the MMNP model can accommodate more random coefficients without increasing computational complexity.

Interpretability of Neural Networks with Probability Density Functions

AbstractIt is an interesting topic to interpret artificial neural networks (ANNs) by considering some change various approaches. This paper explores the relationship between the input and output units of the simplest ANN, a single layer perceptron for the binary classification problem, from the probability point of view. If the feature variables of datasets follow independent normal distribution and outputs are activated by sigmoid function or smooth Relu function, we advocate that the probability density function (pdf) of the output variable is an exponential family distribution. Furthermore, by introducing an intermediate variable, the pdf of the output variable can be written as a linear combination of three normal distributions with same spread but different centers. Based on these results, the probability of the predicted class label can be written as a standard normal cumulative distribution function (cdf). The originality of this paper comes with interesting theoretical results to provide ANNs with a new description of the relationship between input variables to output variables, which can enable ANNs to be understood from a new perspective. Extensive experiments based on one artificial synthesized dataset and ten real‐world benchmark datasets validate the reasonability of those results.

A new very simply explicitly invertible approximation for the standard normal cumulative distribution function

<abstract><p>This paper proposes a new very simply explicitly invertible function to approximate the standard normal cumulative distribution function (CDF). The new function was fit to the standard normal CDF using both MATLAB's Global Optimization Toolbox and the BARON software package. The results of three separate fits are presented in this paper. Each fit was performed across the range $ 0 \leq z \leq 7 $ and achieved a maximum absolute error (MAE) superior to the best MAE reported for previously published very simply explicitly invertible approximations of the standard normal CDF. The best MAE reported from this study is 2.73e–05, which is nearly a factor of five better than the best MAE reported for other published very simply explicitly invertible approximations.</p></abstract>

888Discriminating between risk discriminators: OPERA, AUC, and polygenic variance

Abstract Focus of Presentation Epidemiological risk estimates are often adjusted for age (and, if necessary, sex) by design or analysis, and for other factors. Consequently, risk estimates do not pertain to the crude risk factor, but to its population residual after adjusting for age, sex and other covariates. For disease risk, the change in Odds PER Adjusted standard deviation (OPERA) estimates the risk gradient on an appropriate scale; log(OPERA) is the natural measure to compare estimates. Findings Under a multiplicative risk model for a normally distributed (adjusted) risk factor, log(OPERA) = the difference in mean between cases and controls. The area under the receiver operating curve (AUC) = □(log(OPERA)/√2), where □ is the standard normal cumulative distribution function. The risk discrimination from combining risk factors can be predicted from their OPERAs. The polygenic standard deviation estimated from pedigree data = log(OPERA). The OPERA for knowing all familial risk factors can be calculated. Conclusions/Implications We present examples from the breast cancer literature where the wrong conclusions can be made by not using the OPERA concept. We give examples of the value of OPERA estimates in predicting the risk discrimination of their combination and demonstrate why the better one predicts the disease the harder it is to predict it better. Key messages OPERA overcomes problems about interpreting risk factors, and combinations of risk factors, in a way not apparent using changes in AUC. OPERA also puts an upper bound on the role of genetic factors in explaining differences in risk across the population.

Closed-form pricing formula for foreign equity option with credit risk

Since credit risk in the over-the-counter (OTC) market has undoubtedly become very important issue, credit risk has to be considered when the options in the OTC market are priced. In this paper, we consider the valuation of foreign equity options with credit risk. In order to derive a closed-form pricing formula of this option, we adopt the partial differential equation (PDE) approach and use the Mellin transform method to solve the PDE. Specifically, triple Mellin transforms are used, and the pricing formula is presented as 3-dimensional normal cumulative distribution functions. Finally, we verify that our closed-form formula is accurate by comparing it with the numerical result from the Monte-Carlo simulation.

Performance-based seismic assessment of shield tunnels by incorporating a nonlinear pseudostatic analysis approach for the soil-tunnel interaction

The purpose of this study is to present an efficient and reasonable methodology to evaluate the seismic fragility of shield tunnels based on investigating the seismic performance of segments and joints in detail. The finite element model of segments and joints is established using appropriate constitutive models and elements. A pseudostatic soil-tunnel interaction analysis approach based on the multimodal adaptive pushover of the 1D free-field model is introduced to improve the computational efficiency. The seismic performances of a shield tunnel, including the plastic strain of the segments, joint deformation and bolt yielding, are investigated by pseudostatic analysis, and the entire failure process is summarized. Therefore, a framework for seismic fragility assessment of shield tunnels is proposed. The scaling approach substituting the pseudostatic soil-tunnel analysis for traditional incremental dynamic analysis is adopted to consider the uncertainty of input motion. A Monte Carlo simulation is carried out to consider the uncertainties of soil and tunnel properties. The damage state levels are divided based on the crucial events during the failure process. Finally, the seismic fragility curves are fitted with normal and log-normal cumulative distribution functions. And the effect of the sampling size is discussed. Moreover, the seismic fragility curves taking PGA as IM are also obtained and compared with the existing seismic fragility curves in previous studies.

A simple approximation for the bivariate normal integral

A simple approximation for the bivariate normal cumulative distribution function based on the error function is derived. We compare the accuracy of our approximation with those of several existing approximation methods and the command in MATLAB. Our approximation performs quite well in the central portion and extreme tails of the distribution. We also apply the proposed method to evaluate the likelihood function of the binary endogenous regressor probit (BERP) model. The simulations indicate that the numerical stability and the accuracy of the proposed approach is promising when conducting the maximum likelihood estimation of the BERP model.

Pemodelan Indeks Pembangunan Manusia (IPM) Menggunakan Analisis Regresi Probit

Ordinal probit regression analysis is non-linear regression analysis that used to find affected independent variables for ordered categorical dependent variable and regression model in this analysis used Normal cumulative distribution function. Parameter estimation in this model used Maximum Likelihood Estimation (MLE) method. This model has been applied to Human Development Index (HDI) in Borneo Island in 2017 case study. HDI is the most important measurement in improving the human development quality in all cities/regencies in Indonesia. Some factors that affected to IPM, they are Life Expectancy (X1), School Expectancy (X2), Spending per Capita (X3), Average School Duration (X4), and Labour Force Participation Rate (X5). Based on research that was performed by researcher, resulted two factors affecting to HDI, those are Life Expetancy and Average School Duration. This model has classification accuracy of 89,29%, APER (Apparent Error Rate) value of 10,71%, and AIC (Akaike Information Criterion) value of 39,75; this model was very good because prediction value is almost approaching to observation value (actual value).