Relative Errors

We consider a fundamental open problem in parametric Bayesian theory, namely the validity of the formal Edgeworth expansion of the posterior density. While the study of valid asymptotic expansions for posterior distributions constitutes a rich literature, the validity of the formal Edgeworth expansion has not been rigorously established. Several authors have claimed connections of various posterior expansions with the classical Edgeworth expansion, or have simply assumed its validity. Our main result settles this open problem. We also prove a lemma concerning the order of posterior cumulants which is of independent interest in Bayesian parametric theory. The most relevant literature is synthesized and compared to the newly-derived Edgeworth expansions. Numerical investigations illustrate that our expansion has the behavior expected of an Edgeworth expansion, and that it has better performance than the other existing expansion which was previously claimed to be of Edgeworth-type.

This paper presents results showing that the error involved in using the double saddlepoint distribution function approximations of Skovgaard (1987, J. Appl. Probab. 24 875 887) are uniformly bounded. Particular attention is paid to distributions of sufficient statistics arising from generalized linear models. This work is intended in part to validate the use of the Markov Chain Monte Carlo by Kolassa and Tanner (1994, J. Amer. Statist. Assoc. 89 697 702) using these conditional distribution function approximations.

Saddlepoint Approximations have long been used to approximate densities and distribution functions of random variables with known cumulant generating function defined on an open interval about the origin. This approximation has very desirable asymptotic properties when approximating densities and tail probabilities for sums of random variables, and also often performs remarkably well for small sample sizes, including samples of one. Calculating the saddlepoint approximation requires calculating the Legendre transform of the log of the cumulant generating function. In some cases this cumulant generating function may be unavailable; in other cases the Legendre transform is difficult to calculate analytically. This paper discusses modifications to the saddlepoint approximation necessary when the cumulant generating function is replaced by a similar but more tractable function whose Legendre transform can be given explicitly. Calculations for the logistic distribution are presented to illustrate the case of a known but intractable cumulant generating function, and an example involving an overdispersed binomial model is presented to illustrate the case of an unavailable cumulant generating function. An application to a random effects logistic linear model is discussed. 1. The Problem. Consider the following problem: X lt X^ X3,..., X n ,... are independent and identically distributed random variables. Assume that their common distribution is continuous, with probability density function/^*) and cumulant generating function K(t) = log E[e tX ], defined on some convex set icR. Without loss of generality assume further that E[XH = 0. I wish to approximate the density function/ rt Cs) and distribution function F n (s) of 5= ι analytically, and am faced with three options: a. Convolute the density n times. b. Use the classical Edgeworth series approximation to/ rt (x).

The asymptotic distribution of the pairwise log likelihood ratio is a linear combination of independent chi-square random variables with coefficients depending on the elements of the Godambe information. Adjusted versions of the pairwise log likelihood statistic have been proposed, but they still depend on the Godambe information matrix. Approximated p-values for testing a composite hypothesis may be obtained by refferring the observed value of such statistics to a critical value. The asymptotic theory can be used to approximate the desired quantile, but the approximation may be inaccurate. In this work we provide a nonparametric saddlepoint statistic derived from the pairwise score function. This statistic enjoys some desirable properties: it is asymptotically chi-square distributed and the approximation has a relative error of second order. Thereby our proposal claims a high level of accuracy with no need to estimate the Godambe information.

The class of composite likelihood functions provides a flexible and powerful toolkit to carry out approximate inference for complex statistical models when the full likelihood is either impossible to specify or unfeasible to compute. However, the strenght of the composite likelihood approach is dimmed when considering hypothesis testing about a multidimensional parameter because the finite sample behavior of likelihood ratio, Wald, and score-type test statistics is tied to the Godambe information matrix. Consequently inaccurate estimates of the Godambe information translate in inaccurate p-values. In this paper it is shown how accurate inference can be obtained by using a fully nonparametric saddlepoint test statistic derived from the composite score functions. The proposed statistic is asymptotically chisquare distributed up to a relative error of second order and does not depend on the Godambe information. The validity of the method is demonstrated through simulation studies.

Assume we use a binary floating-point arithmetic and that RN is the round-to-nearest function. Also assume that c is a constant or a real function of one or more variables, and that we have at our disposal a correctly rounded implementation of c, sayĉ = RN(c). For evaluating x • c (resp. x/c or c/x), the natural way is to replace it by RN(x •ĉ) (resp. RN(x/ĉ) or RN(ĉ/x)), that is, to call functionĉ and to perform a floatingpoint multiplication or division. This can be generalized to the approximation of n/d by RN(n/d) and the approximation of n • d by RN(n •d), wheren = RN(n) andd = RN(d), and n and d are functions for which we have at our disposal a correctly rounded implementation. We discuss tight error bounds in ulps of such approximations. From our results, one immediately obtains tight error bounds for calculations such as x * pi, ln(2)/x, x/(y + z), (x + y) * z, x/sqrt(y), sqrt(x)/y, (x + y)(z + t), (x + y)/(z + t), (x + y)/(zt), etc. in floatingpoint arithmetic.

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Floating-point numbers have an intuitive meaning when it comes to physics-based numerical computations, and they have thus become the most common way of approximating real numbers in computers. The IEEE-754 Standard has played a large part in making floating-point arithmetic ubiquitous today, by specifying its semantics in a strict yet useful way as early as 1985. In particular, floating-point operations should be performed as if their results were first computed with an infinite precision and then rounded to the target format. A consequence is that floating-point arithmetic satisfies the ‘standard model’ that is often used for analysing the accuracy of floating-point algorithms. But that is only scraping the surface, and floating-point arithmetic offers much more.In this survey we recall the history of floating-point arithmetic as well as its specification mandated by the IEEE-754 Standard. We also recall what properties it entails and what every programmer should know when designi...

Load Forecasting is powerful tool to make important decisions such as to purchase and generate the electric power, load switching, development plans and energy supply according to the demand. The important factors for forecasting involve short, medium and long term forecasting. Factors in short term forecasting comprises of whether data, customer classes, working, non-working days and special event data, while long term forecasting involves historical data, population growth, economic development and different categories of customers.In this paper we have analyzed the load forecasting data collected from one grid that contain the load demands for day and night, special events, working and non-working days and different hours in day. We have analyzed the results using Machine Learning techniques, 10 fold cross validation and stratified CV. The Machines Learning techniques used are LDA, QDA, SVM Polynomial, Gaussian, HRBF, MQ kernels as well as LDA and QDA. The errors methods employed against the techniques are RSE, MSE, RE and MAPE as presented in the table 2 below. The result calculated using the SVM kernel shows that SVM MQ gives the highest performance of 99.53 %.

Load Forecasting is powerful tool to make important decisions such as to purchase and generate the electric power, load switching, development plans and energy supply according to the demand. The important factors for forecasting involve short, medium and long term forecasting. Factors in short term forecasting comprises of whether data, customer classes, working, non-working days and special event data, while long term forecasting involves historical data, population growth, economic development and different categories of customers.In this paper we have analyzed the load forecasting data collected from one grid that contain the load demands for day and night, special events, working and non-working days and different hours in day. We have analyzed the results using Machine Learning techniques, 10 fold cross validation and stratified CV. The Machines Learning techniques used are LDA, QDA, SVM Polynomial, Gaussian, HRBF, MQ kernels as well as LDA and QDA. The errors methods employed against the techniques are RSE, MSE, RE and MAPE as presented in the table 2 below. The result calculated using the SVM kernel shows that SVM MQ gives the highest performance of 99.53 %.

We consider a fundamental open problem in parametric Bayesian theory, namely the validity of the formal Edgeworth expansion of the posterior density. While the study of valid asymptotic expansions for posterior distributions constitutes a rich literature, the validity of the formal Edgeworth expansion has not been rigorously established. Several authors have claimed connections of various posterior expansions with the classical Edgeworth expansion, or have simply assumed its validity. Our main result settles this open problem. We also prove a lemma concerning the order of posterior cumulants which is of independent interest in Bayesian parametric theory. The most relevant literature is synthesized and compared to the newly-derived Edgeworth expansions. Numerical investigations illustrate that our expansion has the behavior expected of an Edgeworth expansion, and that it has better performance than the other existing expansion which was previously claimed to be of Edgeworth-type.

We give the perturbation bounds for the eigenprojections of a Hermitian matrix H = G JG' , where G has full column rank and J is nonsingular, under the perturbations of the factor G. Our bounds hold, for example, when G is given with elementwise relative error. Our bounds contain relative gaps between the eigenvalues and may thus be much less pessimistic than the standard norm estimates.

We obtain eigenvalue perturbation results for a factorised Hermitian matrix H = GJ G * where J 2 = I and G has full row rank and is perturbed into G + δG, where δG is small with respect to G. This complements the earlier results on the easier case of G with full column rank. Applied to square factors G our results help to identify the so-called quasidefinite matrices as a natural class on which the relative perturbation theory for the eigensolution can be formulated in a way completely analogous to the one already known for positive definite matrices.

We consider tests or confidence intervals on one of the components of multivariate M-estimates. We obtain marginal tail area approximations for the one-dimensional test statistic based on the appropriate component of the M-estimate for both standardized and Studentized versions. The result is proved under conditions which allow the application to finite sample situations such as the bootstrap and involves a careful discretization with saddlepoints being used for each neighbourhood. These results are used to obtain second order relative error results on the accuracy of the Studentized and the tilted bootstrap. The tail area approximations are applied to a Poisson regression model and shown to have very good accuracy.

When a unique M-estimate exists, its density is obtained as a corollary to a more general theorem which asserts that under mild conditions the intensity function of the point process of solutions of the estimating equations exists and is given by the density of the estimating function standardized by multiplying it by the inverse of its derivative. We apply the results to give a result for Huber's proposal 2 applied to regression and scale estimates. We also give a saddlepoint approximation for the density and use this to give approximations for tail areas for smooth functions of the M-estimates. * X θ = X θ -1 X θ when the inverse exists and take it as infinite elsewhere. If we restrict attention to a sequence of sets in for which the density of * X θ exists, then a limit of this density at x = 0 is shown to equal the limit of intensity functions defined on the same sequence of sets. The conditions imposed here differ from those of and his result is a special case of ours. Jensen and

Power grid is a common platform connecting all the generating stations and demand centres (loads), where the energy is generated and consumed instantaneously. The demand varies from minute to minute, and the generation needs to be adjusted accordingly to meet the demand. However, with large intermittent renewable power plants coming online, this balancing becomes even more complicated. Although there is a growing emphasis on predicting the renewable generation, prediction of demand also can help in grid-level energy management. In this paper, we compare the prevalent demand forecasting practice with the model developed using multivariable regression technique. This simple model shows an improvement of 3% over the present demand prediction scenario, with respect to the mean absolute error. In future, use of more precise data for model training and addition of further variables may increase the accuracy level. This model does not need any large computational set-up or capacity buildin...

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In this paper, we consider the estimation of the unknown parameters of the multiple chirp signal model in presence of additive error. The chirp signals are quite common in many areas of science and engineering, specially sonar, radar, audio signals etc. The observed signals are usually corrupted by noise. In different signal processing applications it is observed that the errors may be heavy tailed. In this paper it is assumed that the additive errors have mean zero but may not have finite variance and are independent and identically distributed. We consider the least squares estimators and the approximate least squares estimators which maximize a periodogram like function. It has been observed that both the estimators are strongly consistent. The asymptotic distribution of the least squares estimators is obtained under the assumption that the additive errors are from a symmetric stable distribution. The approximate least squares estimators have the same asymptotic distribution as t...

We consider a resampling scheme for parameters estimates in nonlinear regression models. We provide an estimation procedure which recycles, via random weighting, the relevant parameters estimates to construct consistent estimates of the sampling distribution of the various estimates. We establish the asymptotic normality of the resampled estimates and demonstrate the applicability of the recycling approach in a small simulation study and via example.

This paper deals with the accuracy of complex division in radix-two floating-point arithmetic. Assuming that a fused multiply-add (FMA) instruction is available and that no underflow/overflow occurs, we study how to ensure high relative accuracy in the componentwise sense. Since this essentially reduces to evaluating accurately three expressions of the form ac + bd, an obvious approach would be to perform three calls to Kahan's compensated algorithm for 2×2 determinants. However, in the context of complex division, two of those expressions are such that ac and bd have the same sign, suggesting that cheaper schemes should be used here (since cancellation cannot occur). We first give a detailed accuracy analysis of such schemes for the sum of two nonnegative products, providing not only sharp bounds on both their absolute and relative errors, but also sufficient conditions for the output of one of them to coincide with the output of Kahan's algorithm. By combining Kahan's algorithm with this particular scheme, we then deduce two new division algorithms. Our first algorithm is a straight-line program whose componentwise relative error is always at most 5u + 13u 2 with u the unit roundoff; we also provide examples of inputs for which the error of this algorithm approaches 5u, thus showing that our upper bound is essentially the best possible. When tests are allowed we show with a second algorithm that the bound above can be further reduced to 9 2 u+9u 2 , and that this improved bound is reasonably sharp.

We provide a detailed analysis of Kahan's algorithm for the accurate computation of the determinant of a 2 × 2 matrix. This algorithm requires the availability of a fused multiply-add instruction. Assuming radix-β, precision-p floating-point arithmetic with β even, p ≥ 2, and barring overflow or underflow we show that the absolute error of Kahan's algorithm is bounded by (β + 1)/2 ulps of the exact result and that the relative error is bounded by 2u with u = 1 2 β 1−p the unit roundoff. Furthermore, we provide input values showing that i) when β/2 is odd-which holds for 2 and 10, the two radices that matter in practice-the absolute error bound is optimal; ii) the relative error bound is asymptotically optimal, that is, for such input the ratio (relative error)/2u has the form 1 − O(β −p). We also give relative error bounds parametrized by the relative order of magnitude of the two products in the determinant, and we investigate whether the error bounds can be improved when adding constraints: When the products in the determinant have opposite signs, which covers the computation of a sum of squares, or when Kahan's algorithm is used for computing the discriminant of a quadratic equation.

We obtain eigenvalue perturbation results for a factorised Hermitian matrix H = GJ G * where J 2 = I and G has full row rank and is perturbed into G + δG, where δG is small with respect to G. This complements the earlier results on the easier case of G with full column rank. Applied to square factors G our results help to identify the so-called quasidefinite matrices as a natural class on which the relative perturbation theory for the eigensolution can be formulated in a way completely analogous to the one already known for positive definite matrices.

This paper proposes an extension of the log periodogram regression in perturbed long memory series that accounts for the added noise, also allowing for correlation between signal and noise, which represents a common situation in many economic and financial series. Consistency (for d < 1) and asymptotic normality (for d < 3/4) are shown with the same bandwidth restriction as required for the original log periodogram regression in a fully observable series, with the corresponding gain in asymptotic efficiency and faster convergence over competitors. Local Wald, Lagrange Multiplier and Hausman type tests of the hypothesis of no correlation between the latent signal and noise are also proposed.

Two-component mixture distributions with one component a point mass and the other a continuous density may be used as priors for Bayesian inference when sparse representation of an underlying signal is required. We show how saddlepoint approximation in such models can yield highly accurate quantiles for posterior distributions, and illustrate this numerically, using wavelet regression with point mass/Laplace and point mass/normal prior distributions.

Predicting densities of nonconventional solvents like deep eutectic solvents (DESs) as a function of temperature is of considerable importance in the development and design of new processes utilizing these solvents. Because of the nature of bonding existing between the salt and the hydrogen bond donor, conventional methods result in very large deviations. In this study, the density of DESs based on three different salts was estimated using empirical method. Nine different salts:hydrogen bond donor combinations were selected to test this method. The densities of all DESs were measured at a temperature range (298.15-368.1 K). The critical properties of salt and hydrogen bond donor were estimated using the Modified Lydersen-Joback-Reid method, while that of the mixture were calculated using Lee-Kesler equation. The Rackett equation modified by Spencer and Danner was employed to predict the DES density. The values of measured and predicted densities were compared and the average of absolute relative error percentage (ARPE) for all DESs was found to be 1.9%. The effect of salt to HBD molar ratio on ARPE in predicted DESs densities was also investigated.

This manuscript presents an approximation to the distribution function of a smooth transformation of a random vector, conditional on the event that values of other smooth transformations of the same random vector lie in a small rectangle. This approximation is used to extend the application of standard saddlepoint conditional tail area approximations in circumstances beyond continuous and lattice cases currently justified in the literature. We consider application to two examples, finite sampling and score testing in logistic regression, where conditioning on a rectangle is essential.

We consider tests or confidence intervals on one of the components of multivariate M-estimates. We obtain marginal tail area approximations for the one-dimensional test statistic based on the appropriate component of the M-estimate for both standardized and Studentized versions. The result is proved under conditions which allow the application to finite sample situations such as the bootstrap and involves a careful discretization with saddlepoints being used for each neighbourhood. These results are used to obtain second order relative error results on the accuracy of the Studentized and the tilted bootstrap. The tail area approximations are applied to a Poisson regression model and shown to have very good accuracy.

This manuscript presents an approximation to the distribution function of a smooth transformation of a random vector, conditional on the event that values of other smooth transformations of the same random vector lie in a small rectangle. This approximation is used to extend the application of standard saddlepoint conditional tail area approximations in circumstances beyond continuous and lattice cases currently justified in the literature. We consider application to two examples, finite sampling and score testing in logistic regression, where conditioning on a rectangle is essential.

We solved the wind-influenced projectile motion problem with the same initial and final heights and obtained exact analytical expressions for the shape of the trajectory, range, maximum height, time of flight, time of ascent, and time of descent with the help of the Lambert W function. It turns out that the range and maximum horizontal displacement are not always equal. When launched at a critical angle, the projectile will return to its starting position. It turns out that a launch angle of 90° maximizes the time of flight, time of ascent, time of descent, and maximum height and that the launch angle corresponding to maximum range can be obtained by solving a transcendental equation. Finally, we expressed in a parametric equation the locus of points corresponding to maximum heights for projectiles launched from the ground with the same initial speed in all directions. We used the results to estimate how much a moderate wind can modify a golf ball's range and suggested other possible applications.

Load Forecasting is powerful tool to make important decisions such as to purchase and generate the electric power, load switching, development plans and energy supply according to the demand. The important factors for forecasting involve short, medium and long term forecasting. Factors in short term
forecasting comprises of whether data, customer classes, working, non-working days and special event data, while long term forecasting involves historical data, population growth, economic development and different categories of customers.In this paper we have analyzed the load forecasting data collected from one grid that contain the load demands for day and night, special events, working and non-working days
and different hours in day. We have analyzed the results using Machine Learning techniques, 10 fold cross validation and stratified CV. The Machines Learning techniques used are LDA, QDA, SVM Polynomial, Gaussian, HRBF, MQ kernels as well as LDA and QDA. The errors methods employed against the
techniques are RSE, MSE, RE and MAPE as presented in the table 2 below. The result calculated using the SVM kernel shows that SVM MQ gives the highest performance of 99.53 %.

Predicting densities of nonconventional solvents like deep eutectic solvents (DESs) as a function of temperature is of considerable importance in the development and design of new processes utilizing these solvents. Because of the nature of bonding existing between the salt and the hydrogen bond donor, conventional methods result in very large deviations. In this study, the density of DESs based on three different salts was estimated using empirical method. Nine different salts:hydrogen bond donor combinations were selected to test this method. The densities of all DESs were measured at a temperature range (298.15–368.1 K). The critical properties of salt and hydrogen bond donor were estimated using the Modified Lydersen–Joback–Reid method, while that of the mixture were calculated using Lee–Kesler equation. The Rackett equation modified by Spencer and Danner was employed to predict the DES density. The values of measured and predicted densities were compared and the average of absolute relative error percentage (ARPE) for all DESs was found to be 1.9%. The effect of salt to HBD molar ratio on ARPE in predicted DESs densities was also investigated.▶ An empirical method is introduced to predict deep eutectic solvents densities as function of temperature. ▶ The correlation showed high prediction quality for all tested DESs. ▶ The density of the DESs lies between the densities of the corresponding salt and HBD used in their synthesis. ▶ The relative difference of densities between the DES and the HBD directly affects the quality of density prediction.