Bathtub Shape Research Articles

入浴姿勢の推定アルゴリズムの構築

Bathing in a bathtub filled with water is a custom common in Japan. Since comfort is important in bathing, comfortable bathtub design is desired. In the previous study, the algorithm to estimate reaction forces from bathtub was constructed. However, in order to evaluate the comfort of the bathtub in simulation completely, it was necessary to know bathing posture without experiments. The objective of this study was to construct the algorithm to estimate bathing posture. This algorithm determines the bathing posture that minimizes weighted summation of i) sum of joint torques of hip, knee, and ankle, ii) passive elastic joint torques of toes, and iii) inverse sign of the buoyancy. In the calculation, gravity, buoyancy, passive elastic joint torque and estimated reaction forces by algorithm constructed in previous study were considered. In order to examine this algorithm, a bathing experiment to obtain the data of bathing posture for 10 subjects and 4 bathtub shapes for each subject was conducted. It was found that the estimated bathing posture were consistent well with the measured bathing posture within 5.2% error.

Bathtub concept in health sciences: A comment

Bathtub shape has important connotations in public health and the concept could be reflected in two novel ways: (1) engineering (2) epidemiological model. Typical bath-tub curve is a combination of three failure rates, that is, decreasing, constant, and increasing failure rate, and different techniques are utilized to reduce these failures. This brief study highlights the concept of bath-tub, related theory, application, and how data modeling could be undertaken considering bath-tub failure rate.

Application of a hazard-based visual predictive check to evaluate parametric hazard models.

Parametric models used in time to event analyses are evaluated typically by survival-based visual predictive checks (VPC). Kaplan-Meier survival curves for the observed data are compared with those estimated using model-simulated data. Because the derivative of the log of the survival curve is related to the hazard--the typical quantity modeled in parametric analysis--isolation, interpretation and correction of deficiencies in the hazard model determined by inspection of survival-based VPC's is indirect and thus more difficult. The purpose of this study is to assess the performance of nonparametric hazard estimators of hazard functions to evaluate their viability as VPC diagnostics. Histogram-based and kernel-smoothing estimators were evaluated in terms of bias of estimating the hazard for Weibull and bathtub-shape hazard scenarios. After the evaluation of bias, these nonparametric estimators were assessed as a method for VPC evaluation of the hazard model. The results showed that nonparametric hazard estimators performed reasonably at the sample sizes studied with greater bias near the boundaries (time equal to 0 and last observation) as expected. Flexible bandwidth and boundary correction methods reduced these biases. All the nonparametric estimators indicated a misfit of the Weibull model when the true hazard was a bathtub shape. Overall, hazard-based VPC plots enabled more direct interpretation of the VPC results compared to survival-based VPC plots.

An Extension of Chen’s Family of Survival Distributions with Bathtub Shape or Increasing Hazard Rate Function

In this article we introduce an extension of Chen’s (2000) family of distributions given by Lehman alternatives (see Gupta et al., 1998) that is shown to present another alternative to the generalized Weibull and exponentiated Weibull families for modeling survival data. The extension proposed here can be seen as the extension to the Chen’s distribution as the exponentiated Weibull is to the Weibull. A detailed analysis of the density and hazard shapes is carried out. The new model is also seen to fit well to the flood data used in fitting the exponentiated Weibull model in Mudholkar and Hutson (1996).

The inverse Lindley distribution: a stress-strength reliability model with application to head and neck cancer data

In this article, we proposed one parameter, inverse Lindley distribution, with its fundamental properties such as quantiles, mode, stochastic ordering, entropy, and stress–strength reliability. The proposed distribution has upside-down bathtub shape for its failure rate function. The estimation of stress–strength reliability has been approached by both classical and Bayesian methods. Under Bayesian set-up, both non-informative (Jeffrey) and informative (gamma) priors are considered under symmetric (squared error) and asymmetric (entropy) loss functions. The Lindley’s approximation method is used for Bayesian computation. The performances of the estimators have been compared in terms of their mean squared errors using simulated samples. Two real data-sets, representing the survival times of head and neck cancer patients, are considered for demonstrating the applicability of the proposed model.

Beta Hazard Rate Distribution and Applications

We introduce a new finite range life distribution with a hazard rate function of the form of a nonstandard beta density function. The hazard rate function is either increasing, or has a bathtub shape with a long flat middle interval. The proposed model has a simple structure. Several datasets from the reliability literature with known bathtub shaped hazards were successfully fitted to this distribution.

Scanning multishot irradiations on a large-area glass substrate for Xe-Arc flash lamp crystallization of amorphous silicon thin-film

Applying an in-line Xe-arc flash lamp annealing (FLA) process for the fabrication of low-temperature polycrystalline silicon (LTPS) on a large-scale glass substrate, this study investigates the effects of scanning multishot irradiations on the thermo-mechanical deformation of the substrate and the microscopic phase-change of the amorphous silicon (a-Si) thin-film. A glass substrate with thin-film structures was preheated at 650 °C, exposed to sequential multiple flashes, and thus heated rapidly far beyond the glass softening temperature to achieve crystallization of a-Si. Temperature variations in the substrate were predicted by one-dimensional heat conduction model with radiation absorption, and the structural deformations were simulated through a FEM code and compared to the experimental observations. The maximum temperatures arisen in the substrate during the second or subsequent flash irradiations were substantially lower than that during the first irradiation owing to reduced absorptions of flash energies. These absorption reductions were resulted from phase-changes of silicon from amorphous to polycrystalline. Those temperatures, however, were still sufficiently high to cause a significant thermal warpage, resulting in the bathtub shape of the substrate. The critical mechanisms for the warpage were the structural shrinkage and the gravitational sagging. In addition, the microscopic structural characteristics of the LTPS fabricated by the scanning multishot FLA process were explored through optical and Raman spectroscopies, and transmission electron microscopy.

A-19 入浴姿勢における浴槽壁面反力の推定アルゴリズムの構築(映像分析,推定,構造)

In order to improve comfort in bathing, the relationship between the biomechanical load and the bathtub shape has to be clarified. For this purpose, many bathing experiments had to be conducted since it was necessary to measure the reaction forces acting on a human from the bathtub experimentally. The objective of this study was to construct an algorithm to estimate reaction forces from bathtun in order to calculate joint torque as biomechanical loads without bathing experiments. This algorithm calculates distribution of reaction forces which minimizes weighted summation of i) sum of squares of reaction forces, ii) joint torques of hip and knee joints and iii) friction force of the back, considering gravity, buoyancy and passive elastic joint torque. Weighting were expressed as a function of bathing posture including parameters determined for each person to reflect influence of differences among the individual. In order to examine this algorithm, a bathing experiment to obtain the data of bathing posture and reaction forces for 10 subjects and 24 posture conditions was conducted. It was found that the estimated reaction forces were consistent well with the measured values.

Log-location-scale-log-concave distributions for survival and reliability analysis

We consider a novel sub-class of log-location-scale models for survival and reliability data formed by restricting the density of the underlying location-scale distribution to be log-concave. These models display a number of attractive properties. We particularly explore the shapes of the hazard functions of these, LLSLC, models. A relatively elegant, if partial, theory of hazard shape arises under a further minor constraint on the hazard function of the underlying log-concave distribution. Perhaps the most useful LLSLC models are contained in a class of three-parameter distributions which allow constant, increasing, decreasing, bathtub and upside-down bathtub shapes for their hazard functions.

The Transmuted Inverse Exponential Distribution

This article introduces a two-parameter probability model which represents another generalization of the Inverse Exponential distribution by using the quadratic rank transmuted map. The proposed model is named Transmuted Inverse Exponential (TIE) distribution and its statistical properties are systematically studied. We provide explicit expressions for its moments, moment generating function, quantile function, reliability function and hazard function. We estimate the parameters of the TIE distribution using the method of maximum likelihood estimation (MLE). The hazard function of the model has an inverted bathtub shape and we propose the usefulness of the TIE distribution in modeling breast cancer and bladder cancer data sets.

Time-Weighted Multi-Touch Attribution and Channel Relevance in the Customer Journey to Online Purchase

We address statistical issues in attributing revenue to marketing channels and inferring the importance of individual channels in customer journeys toward an online purchase. We describe the relevant data structures and introduce an example. We suggest an asymmetric bathtub shape as appropriate for time-weighted revenue attribution to the customer journey, provide an algorithm, and illustrate the method. We suggest a modification to this method when there is independent information available on the relative values of the channels. To infer channel importance, we employ sequential data analysis ideas and restrict to data which ends in a purchase. We propose metrics for source, intermediary, and destination channels based on two- and three-step transitions in fragments of the customer journey. We comment on the practicalities of formal hypothesis testing. We illustrate the ideas and computations using data from a major UK online retailer. Finally, we compare the revenue attributions suggested by the methods...

A new upside-down bathtub shaped hazard rate model for survival data analysis

In medical, engineering besides demography and other applied disciplines, it is pronounced in some applications that the hazard rate of the data initially increased to a pick in the beginning age, declined abruptly till it stabilized. In statistics literature, such hazard rate is known as the upside-down bathtub shaped hazard rate and propound in the various survival studies. In this paper, we proposed a transmuted inverse Rayleigh distribution, which possesses the upside-down bathtub shape for its hazard rate. The fundamental properties such as mean, variance, mean deviation, order statistics, Renyi entropy and stress–strength reliability of the proposed model are explored here. Further, three methods of estimation namely maximum likelihood, least squares and maximum product spacings methods are used for estimating the unknown parameters of the transmuted inverse Rayleigh distribution, and compared through the simulation study. Finally, the applicability of the proposed distribution is shown for a set of real data representing the times between failures of the secondary reactor pumps.

A New Two-Parameter Lifetime Distribution with Bathtub, Up-Bathtub or Increasing Failure Rate Function

It is a common situation that the failure rate function has a bathtub shape for many mechanical and electronic components. A simple model based on the median of tree /or four identical independent random variables drown from the well known power distribution is presented for modeling this type of data. The failure rate can also be ups ide-down bathtub shaped or increasing. Most properties of the proposed distribution are investigated. Estimation procedures are intro duced as well as a graphical approach via probability plot. An application to a real data set is presented; and a simulation study is provid ed. Finally, some concluding remarks are presented.

Generalized q‐Weibull model and the bathtub curve

Purpose – The purpose of this paper is to analyze mathematical aspects of the q‐Weibull model and explore the influence of the parameter q.Design/methodology/approach – The paper uses analytical developments with graph illustrations and an application to a practical example.Findings – The q‐Weibull distribution function is able to reproduce the bathtub shape curve for the failure rate function with a single set of parameters. Moments of the distribution are also presented.Practical implications – The generalized q‐Weibull distribution unifies various possible descriptions for the failure rate function: monotonically decreasing, monotonically increasing, unimodal and U‐shaped (bathtub) curves. It recovers the usual Weibull distribution as a particular case. It represents a unification of models usually found in reliability analysis. Q‐Weibull model has its inspiration in nonextensive statistics, used to describe complex systems with long‐range interactions and/or long‐term memory. This theoretical backgrou...

Bathtub curves and pipe prioritization based on failure rate

PurposeThe purpose of this paper is to analyze conditional failure rates, and prioritize water pipelines for replacement based on their expected failure rate where pipes are grouped based on age and pipe type. Thus, predictions can be made on the expected number of breaks in future years.Design/methodology/approachThe time to failure of a pipe can be characterized by the stochastic properties of the population as a whole, from which the likelihood of component failure is derived. When the corresponding failure rate is plotted against time, a bathtub‐shaped curve results. The bathtub curve assists in determining maintenance schedules depending on the age of the pipe. Failure rates help determine whether the rates are more than an acceptable best practice threshold to signal replacement.FindingsDuctile iron pipes had the highest failure rates, followed by asbestos cement pipes; PVC and concrete cylinder pipes had the lowest failure rates, but because concrete cylinder pipes are very time‐consuming to repair and very expensive to install, only PVC pipes are recommended on the basis of this study. Cast iron pipes fit the bathtub shape; ductile iron and asbestos concrete were somewhat bathtub shaped, though the early phase period was not apparent; the bathtub curve for concrete cylinder was fully inverted; while PVC pipes showed zero probability of failure during the middle period. The shapes of bathtub curves drawn on conditional failure rates were similar to those for the failure rates. The bathtub curves indicate that the general failure performance of pipe materials is somewhat contrary to general principles in manufacturing.Practical implicationsAnalysis of failure serves a practical purpose for water utilities to allocate funds for pipe maintenance and prepare a schedule for pipe replacement, so as to provide the best quality services and safe drinking water to users of the utility.Social implicationsThe proper prioritization of water supply pipes for repair and replacement is of great social importance to the public at large, which expends considerable funds to maintain their drinking water supply.Originality/valueThe study of bathtub curves has not been seen before in the analysis of water supply pipes. A unique discovery is that the traditional shape of the bathtub curve is not always applicable for water supply pipes.

Estimation using hybrid censored data from a two-parameter distribution with bathtub shape

The problem of estimating unknown parameters of a two-parameter distribution with bathtub shape is considered under the assumption that samples are hybrid censored. The maximum likelihood estimates are obtained using an EM algorithm. The Fisher information matrix is obtained as well and the asymptotic confidence intervals are constructed. Further, two bootstrap interval estimates are also proposed for the unknown parameters. Bayes estimates are evaluated under squared error loss function. Approximate explicit expressions for these estimates are derived using the Lindley method as well as using the Tierney and Kadane method. An importance sampling scheme is then proposed to generate Markov Chain Monte Carlo samples which have been used to compute approximate Bayes estimates and credible intervals for the unknowns. A numerical study is performed to compare the proposed estimates. Finally, two data sets are analyzed for illustrative purposes.

A new bathtub curve model with a finite support

The failure rate with a bathtub shape usually increases very fast in the wear-out phase. In this case, the bathtub curve model with a finite support can better adapt the sharp change in failure rate. There are few models with the finite support. This paper presents such a model. However, the maximum likelihood estimator of the location parameter of such models sometimes converges to the largest observation of a dataset. An extended maximum spacing method is developed to estimate the parameters for the case where the maximum likelihood method fails. Three examples are included to illustrate the appropriateness of the proposed model and estimation method.

Evaluation and comparison of estimations in the generalized exponential-Poisson distribution

The uniformly minimum variance unbiased, maximum-likelihood, percentile and least-squares estimators of the probability density function and the cumulative distribution function are derived for the generalized exponential-Poisson distribution. This model has shown to be useful in reliability and lifetime data modelling, especially when the hazard rate function has a bathtub shape. Simulation studies are also carried out to show that the maximum-likelihood estimator is better than the uniformly minimum variance unbiased estimator (UMVUE) and that the UMVUE is better than others.

A Bayes comparison of Weibull extension and modified Weibull models for data showing bathtub hazard rate

A number of models have been proposed in the literature to model data reflecting bathtub-shaped hazard rate functions. Mixture distributions provide the obvious choice for modelling such data sets but these contain too many parameters and hamper the accuracy of the inferential procedures particularly when the data are meagre. Recently, a few distributions have been proposed which are simply generalizations of the two-parameter Weibull model and are capable of producing bathtub behaviour of the hazard rate function. The Weibull extension and the modified Weibull models are two such families. This study focuses on comparing these two distributions for data sets exhibiting bathtub shape of the hazard rate. Bayesian tools are preferred due to their wide range of applicability in various nested and non-nested model comparison problems. Real data illustrations are provided so that a particular model can be recommended based on various tools of model comparison discussed in the paper.

A MIXTURE OF EXPONENTIAL AND IFR GAMMA DISTRIBUTIONS HAVING AN UPSIDEDOWN BATHTUB-SHAPED FAILURE RATE

We consider a mixture of one exponential distribution and one gamma distribution with increasing failure rate. For the right choice of parameters, it is shown that its failure rate has an upsidedown bathtub shape failure rate. We also consider a mixture of a family of exponentials and a family of gamma distributions and obtain a similar result.