Arccosine Function Research Articles

An improved reptile search algorithm based on mathematical optimization accelerator and elementary functions

The reptile search algorithm (RSA) is a dynamic and effective meta-heuristic algorithm inspired by the behavior of crocodiles in nature and the way of hunting prey. Unlike other crawler search algorithms, it uses four novel mechanisms to update the location of the solutions, such as walking at high or on the belly, and hunting in a coordinated or cooperative manner. In this algorithm, the total number of iterations is divided into four intervals, and different position-updating strategies are used to make the algorithm easily fall into the local optimum. Therefore, an improved reptile search algorithm based on a mathematical optimization accelerator (MOA) and elementary functions is proposed to improve its search efficiency and make it not easily fall into local optimum. MOA was used to realize the switching of RSA’s four searching modes by introducing random perturbations of six elementary functions (sine function, cosine function, tangent function, arccosine function, hyperbolic secant function and hyperbolic cosecant function), four mechanisms are distinguished by random number instead of the original RSA algorithm’s inherent four mechanisms by iteration number, which increases the randomness of the algorithm and avoids falling into local optimum. The random perturbations generated by elementary functions are added to the variation trend of parameter MOA to improve the optimization accuracy of the algorithm. To verify the effectiveness of the proposed algorithm, 30 benchmark functions in CEC2017 were used for carrying out simulation experiments, and the optimization performance was compared with BAT, PSO, ChOA, MRA and SSA. Finally, two practical engineering design problems are optimized. Simulation results show that the proposed sechRSA has strong global optimization ability.

An Index for the Degree and Directionality of Asymmetry for Square Contingency Tables with Ordered Categories

For square contingency tables with ordered categories, an index based on Cressie and Read's power divergence (or Patil and Taillie's diversity index) has been proposed in order to measure the degree of departure from symmetry. Although there are two types of maximum asymmetry (i.e., whether (1) all the observations concentrate in the top-right cell in the table, or (2) they concentrate in the bottom-left cell), the existing index cannot distinguish the two directions of maximum asymmetry. This paper proposes a directional index based on an arc-cosine function in order to simultaneously represent the degree and directionality of asymmetry.The proposed index would be useful for comparing degrees of asymmetry for several square contingency tables. Numerical examples show the utility of the proposed index using some datasets. We evaluate the usefulness of the proposed index by applying it to real data of the clinical study. The proposed index provides analysis results that are easier to interpret than the existing index.

Prediction of wax deposit thickness in waxy crude oil pipelines using improved GM(1,1) model

In this paper, the GM(1,1) model with function arccosx transformation and GM(1,1) model with function transformation are established by using arccosine function transformation method and aarccosx function transformation method, and the GM(1,1) model with function cosx2 transformation is established by using function transformation theory, and GM(1,1) model with function cosx2+c transformation is established by using translational transformation theory on the basis of this model. The prediction accuracy of GM(1,1) model, GM(1,1) model with function arccosx transformation, GM(1,1) model with function aarccosx transformation, GM(1,1) model with function cosx2 transformation, and GM(1,1) model with function cosx2+c transformation are compared by modeling with the field pipeline data and the indoor loop data. The influence of a value in GM(1,1) model with function aarccosx transformation on prediction accuracy is discussed, and the influence of c value in GM(1,1) model with function cosx2+c transformation on prediction accuracy is discussed. With the increase of a and c values, the average relative error show a trend of decreasing and then increasing, by comparing the average relative errors under different a and c values, the optimal a value and c value and the optimal prediction accuracy are obtained. The results show that the GM(1,1) model with function cosx2+c transformation in the indoor loop has an average relative error of 0.6490% when c=0.114, which is the minimum average relative error compared to other models and achieves the highest prediction accuracy. The GM(1,1) model with function cosx2+c transformation in the field pipeline has an average relative error of 1.94156% when c=−0.555, which is the minimum average relative error compared to other models and achieves the highest prediction accuracy. Among the five models, only the GM(1,1) model with function cosx2+c transformation has fitted and predicted values that are closer to the actual thickness values in the indoor loop experimental data and the field pipeline data, and the predicted values are more consistent with the actual conditions in the field pipeline. This paper verifies the feasibility of using the GM(1,1) model with function cosx2+c transformation to predict the wax deposition thickness of the pipe wall, and provides a reference for subsequent research on accurate prediction of wax deposition thickness.

Prediction and evaluation of surface roughness with hybrid kernel extreme learning machine and monitored tool wear

In the modern manufacturing industry, surface roughness is a critical parameter to characterize surface quality. The accurate prediction of surface roughness is of great significance for data-driven intelligent manufacturing. However, it's hard to accurately predict surface roughness in the complex machining process, because of the existence of some uncontrollable factors, such as tool wear. To address the aforementioned issue, a novel hybrid kernel extreme learning machine with Gaussian and arc-cosine kernel function (RBF_Arc_HKELM) was proposed to predict surface roughness. Then an optimized whale optimization algorithm was introduced to improve the prediction accuracy. Considering that tool wear is an indirect quantity and changes dynamically with the cutting process, a novel tool wear monitoring framework with attention mechanism, weighted feature averaging, and deep learning models was proposed. Afterward, the basic cutting parameters combined with the monitored tool wear were fed into the trained RBF_Arc_HKELM model for surface roughness estimation. Finally, surface roughness was evaluated by the established RBF_Arc_HKELM model. To verify the validity and performance of the established models, milling experiments were conducted under different cutting parameter combinations and tool wear levels, and some other intelligent algorithms were also used for surface roughness prediction and tool wear monitoring. Compared with kernel extreme learning machine with RBF (RBF_KELM), support vector regression (SVR), Gaussian process regression (GPR), and light gradient boosting machine (LightGBM), in terms of mean absolute error (MAE), the prediction accuracy of RBF_Arc_HKELM is improved by 17.82 %, 15.36 %, 14.16 %, and 6.26 %, respectively. These results indicated that the proposed model has great leverage in validity and accuracy. Moreover, compared with the measured tool wear as the input, the satisfactory prediction results of surface roughness were also obtained with the monitored tool wear as the input of the RBF_Arc_HKELM model with a drop in prediction accuracy of only 8.66 %.

Optimization of twice-daily direct flyover data collection for satellite observations at uncertain locations

This study investigates a means to achieve twice-daily direct flyovers of a ground target whose location is unknown at the time of launch. The initial orbital parameters were chosen such that the expected cost of maneuvering to observe the ground target twice per day could be minimized. Previous work on this problem has only included one direct flyover daily and has left viewing a second pass during the day as a happenstance event. A two-stage stochastic programming model was developed and solved using the sample-average approximation (SAA) method along with the L-shaped method. As adding a second direct flyover daily results in a nonlinear model that includes an arccosine function, we proposed an approximation approach for the arccosine function as part of a quadratic program and evaluated the accuracy of our approach. Our results showed that for some scenarios, the maneuver costs were too high to allow for twice-daily direct flyovers, but in other scenarios, the cost was much more reasonable. However, by selecting initial orbital parameters that allowed for the potential of maneuvering for twice-daily direct flyovers, once the scenario had been realized, the mission manager could still decide whether the satellites should be maneuvered to achieve twice-daily observations.

Design of PLC-based system for linearity output voltage of AC–DC converter

This paper presents the PLC-Based system for controlling the SCR power module which is AC–DC converter proposes to linearity output voltage with trigger signal. The technique converted the trigger signal by arccosine function and analog module in the PLC. Converted signal from PLC system is fed to SCR power module instant the traditional signal. Proposed of this paper describes the PLC hardware configuration, compute/math and trigonometric functions with Studio 5000 software. In addition, the interfacing, parameters configuration, and monitoring are considered for testing. The technique with PLC-based, devices configuration and experimental results have been presented will be useful for electric power supply in the industry with more and more demand to apply the PLC system to develop the traditional industry to semi or automatic control in the industry.

On a new distribution based on the arccosine function

This note focuses on a new one-parameter unit probability distribution centered around the inverse cosine and power functions. A special case of this distribution has the exact inverse cosine function as a probability density function. To our knowledge, despite obvious mathematical interest, such a probability density function has never been considered in Probability and Statistics. Here, we fill this gap by pointing out the main properties of the proposed distribution, from both the theoretical and practical aspects. Specifically, we provide the analytical form expressions for its cumulative distribution function, survival function, hazard rate function, raw moments and incomplete moments. The asymptotes and shape properties of the probability density and hazard rate functions are described, as well as the skewness and kurtosis properties, revealing the flexible nature of the new distribution. In particular, it appears to be “round mesokurtic” and “left skewed”. With these features in mind, special attention is given to find empirical applications of the new distribution to real data sets. Accordingly, the proposed distribution is compared with the well-known power distribution by means of two real data sets.

An index to simultaneously analyze the degree and directionality of departure from global marginal homogeneity in square contingency tables

For square contingency tables with ordered categories, an index based on Kullback–Leibler information (or Shannon entropy) has been proposed in order to measure the degree of departure from global marginal homogeneity. Although there are two types of maximum global marginal inhomogeneity [i.e., whether (1) all observations concentrate in the lower-left triangle cells in the table, or whether (2) they concentrate only in the upper-right triangle cells], the existing index cannot distinguish the two directions of global marginal inhomogeneity. This study proposes a directional index based on an arc-cosine function in order to simultaneously analyze the degree and directionality of departure from global marginal homogeneity. The proposed index would be useful for comparing degrees of departure from global marginal homogeneity for several types of tables. Numerical examples show the utility of the proposed index using two datasets, in which the existing index has the same value. We evaluate the useful of the proposed index by applying it to real data of clinical study, and consider that the proposed index produces results that are easier to interpret than the existing index.

Directional Index for Measuring Global Symmetry in Square Tables

An index based on the power divergence, which represents the degree of deviation from global symmetry, has been proposed for square tables with ordinal categories. However, this index cannot distinguish between the two directions of global asymmetry (i.e., whether (1) all observations concentrate in the lower triangular cells of the table, or whether (2) they concentrate in the upper triangular cells). This study proposes a directional index based on an arc–cosine function that can distinguish direction. The proposed index is valuable for comparing degrees of deviation from global symmetry for several types of tables. Numerical examples show the utility of the proposed index using two datasets, in which the existing index has the same value. We evaluate the utility of the proposed index by applying to real data of clinical study. We consider that the proposed index gives results easier to interpret than the existing index.

Development of two-frame shadow moiré profilometry by iterative self-tuning technique

Shadow moire fringes have a complex intensity distribution, which makes the existing arcsine function or arccosine function that is used in random phase-shift extraction algorithms unstable in applications. We propose a high-precision algorithm to determine the random phase shift in a robust way. The idea consists of constructing three consecutive fringe patterns by the addition of two background terms suppressed fringe patterns. Then, an iterative self-tuning phase-shifting algorithm is developed to extract the measurement phase in a pointwise manner. Due to the use of an iterative procedure and tangent function, the present method can evaluate the phase shift accurately and robustly and can be implemented easily in many applications. In addition, the proposed method provides a solution for the development of the two-frame random shadow moire technique. We present simulation and optical experiments to demonstrate the correctness of the proposed method. The results show that the proposed method performs better than other methods.

A new instantaneous frequency extraction method for nonstationary response signals in civil engineering structures

Civil engineering structures in essence belong to time-varying and nonlinear structures and the resultant dynamic responses are usually nonstationary signals. A new method is proposed to extract instantaneous frequency of such nonstationary signals. The new method combines the extended analytical mode decomposition, normalization scheme, and improved arccosine function algorithm. In this method, a multicomponent response signal is first decomposed into several mono-components signals by the extended analytical mode decomposition theorem, and then the extracted mono-components are demodulated into purely frequency modulated signals by a repeated normalization scheme. After that, the improved arccosine function is presented to compute the instantaneous phase and instantaneous frequency. A numerical example and an experimental study of a cable with time-varying tension force are provided to illustrate the working of the method. The results demonstrate that the proposed method can extract instantaneous frequencies from multicomponent signals more accurately than Hilbert–Huang transform.

Random phase-shifting algorithm by constructing orthogonal phase-shifting fringe patterns.

The intensity distribution of fringe patterns becomes nonsinusoidal in real testing environments. Thus, the performance of existing phase shift extraction algorithms, which usually compute the desired phase shift by arccosine function or arcsine function, may be affected. In the presented paper, we report an arctangent-function-based technique to solve this disturbance. First, two orthogonal fringe patterns are constructed through subtraction and addition of two background-removed images. Second, the unequal amplitude between two new fringe patterns is eliminated using a normalization process. Third, the phase shift is determined by computing the norms of the two new images. The proposed method is fast and can be implemented easily in many applications. We verify the algorithm performance and robustness using both simulated and experimental data, indicating the high accuracy of the presented method.

Single frame profilometry with rapid phase demodulation on colour-coded fringes

Digital fringe profilometry is a non-contact surface profiling technique with huge potential at real-time dynamic whole-field measurement. However, this technique is usually bottlenecked at the phase demodulation and unwrapping during fringe analysis. This paper proposes a single frame profilometry system that used direct arccosine function demodulation on colour-coded sinusoidal fringes to simplify the fringe analysis process. Since the range of arccosine function output is restricted from 0 to π, the intensity gradient was used along with arccosine function to demodulate the fringe intensity levels into wrapped phase map (0–2π). The projected fringes were coloured in red, green and blue according to the De Bruijn's sequence. The fringe order was identified directly from the colours of three consecutive fringes by matching to the De Bruijn's sequence to unwrap the wrapped phase map into continuous phase map. The phase differences between the continuous phase maps of reference plane and object surface were then obtained and related to the equipment setup position using trigonometry to rebuild the 3D model. The proposed method was tested experimentally by reconstructing three physical objects. Although the reconstructed surface contained phase errors due to gamma non-linearity, the geometrical shapes of the objects can be reconstructed with reasonable accuracy and consistency. The percentage deviations of dimensions in x, y and z-axis were 1.24%, −1.96% and −2.30% respectively. Meanwhile the uncertainties of dimensions in x, y and z-axis were ±0.15%, ±0.24% and ±1.07% respectively at 95% confidence level.

Comparative analysis of nanometric inspection methods in fringeless speckle pattern interferometry.

In digital speckle pattern interferometry, fringeless speckle pattern interferograms are obtained when the object field deformation is insufficient to produce local phase variations higher than 2π. Therefore, the use of the well-known phase recovery algorithms based on fringe processing is not adequate. In this work, distinct algorithms based on the application of a straightforward arccosine function to a filtered interferogram and the correlation of intensity images and implicit smoothing splines are proposed, analyzed, and compared for the fast inspection of nanometric displacement fields, avoiding the acquisition of several images. In addition, three different methods for the normalization of fringeless speckle pattern interferograms are proposed. The Structural Similarity Index is used to assess the performance of the tested methods by means of numerical simulations under different illuminations, signal-to-noise ratios, phase excursions, and mean speckle size conditions. The analysis shows that the phase recovered by the methods based on the arccosine function and correlation are appropriate for a fast inspection solution. The implicit smoothing spline outperforms other methods in almost all conditions.

Three-frame self-calibration phase shift algorithm using the Gram-Schmidt orthonormalization approach.

Affected by the height dependent effects, the phase-shifting shadow moiré can only be implemented in an approximate way. In the technique, a fixed phase step around π/2 rad between two adjacent frames is usually introduced by a grating translation in its own plane. So the method is not flexible in some situations. Additionally, because the shadow moiré fringes have a complex intensity distribution, computing the introduced phase shift from the existing arccosine function or arcsine function-based phase shift extraction algorithm always exhibits instability. To solve it, we developed a Gram-Schmidt orthonormalization approach based on a three-frame self-calibration phase-shifting algorithm with equal but unknown phase steps. The proposed method using the arctangent function is fast and can be implemented robustly in many applications. We also do optical experiments to demonstrate the correction of the proposed method by referring to the result of the conventional five-step phase-shifting shadow moiré. The results show the correctness of the proposed method.

Research optimization on logistics distribution center location based on adaptive particle swarm algorithm

Logistics distribution center location is an important task in the network optimization of a logistics system. A novel approach of adaptive particle swarm optimization (APSO) algorithm with nonlinear inertia weight and time-varying acceleration coefficient is proposed to achieve distribution center location problem. Mathematic model of logistics distribution center location is established according to assumptions and constraints. Then, average particle spacing as guidance factor of population diversity is employed to establish nonlinear inertia weight and arccosine function is adopted to design nonlinear symmetric acceleration coefficient in order to control the balance of global search and local search for distribution center location problem. To demonstrate the practicality and effectiveness of the proposed algorithm, an application example is provided. Experiments show that APSO can slove the location problem of distribution center with better performance..

Improving flood forecasting capability of physically based distributed hydrological models by parameter optimization

Abstract. Physically based distributed hydrological models (hereafter referred to as PBDHMs) divide the terrain of the whole catchment into a number of grid cells at fine resolution and assimilate different terrain data and precipitation to different cells. They are regarded to have the potential to improve the catchment hydrological process simulation and prediction capability. In the early stage, physically based distributed hydrological models are assumed to derive model parameters from the terrain properties directly, so there is no need to calibrate model parameters. However, unfortunately the uncertainties associated with this model derivation are very high, which impacted their application in flood forecasting, so parameter optimization may also be necessary. There are two main purposes for this study: the first is to propose a parameter optimization method for physically based distributed hydrological models in catchment flood forecasting by using particle swarm optimization (PSO) algorithm and to test its competence and to improve its performances; the second is to explore the possibility of improving physically based distributed hydrological model capability in catchment flood forecasting by parameter optimization. In this paper, based on the scalar concept, a general framework for parameter optimization of the PBDHMs for catchment flood forecasting is first proposed that could be used for all PBDHMs. Then, with the Liuxihe model as the study model, which is a physically based distributed hydrological model proposed for catchment flood forecasting, the improved PSO algorithm is developed for the parameter optimization of the Liuxihe model in catchment flood forecasting. The improvements include adoption of the linearly decreasing inertia weight strategy to change the inertia weight and the arccosine function strategy to adjust the acceleration coefficients. This method has been tested in two catchments in southern China with different sizes, and the results show that the improved PSO algorithm could be used for the Liuxihe model parameter optimization effectively and could improve the model capability largely in catchment flood forecasting, thus proving that parameter optimization is necessary to improve the flood forecasting capability of physically based distributed hydrological models. It also has been found that the appropriate particle number and the maximum evolution number of PSO algorithm used for the Liuxihe model catchment flood forecasting are 20 and 30 respectively.

FPGA-based hardware implementation of arctangent and arccosine functions for the inverse kinematics of robot manipulator

Purpose– The inverse kinematics in robot manipulator have to handle the arctangent and arccosine function. However, the two functions are complicated and need much computation time so that it is difficult to be realized in the typical processing system. The purpose of this paper is to solve this problem by using Field Programmable Gate Array (FPGA) to speed up the computation power.Design/methodology/approach– The Taylor series expansion method is firstly applied to transfer arctangent and arccosine function to a polynomial form. And Look-Up Table (LUT) is used to store the parameters of the polynomial form. Then the behavior of the computation algorithm is described by Very high-speed IC Hardware Description Language (VHDL) and a co-simulation using ModelSim and Simulink is applied to evaluate the correctness of the VHDL code.Findings– The computation time of arctangent and arccosine function using by FPGA need only 320 and 420 ns, respectively, and the accuracy is <0.01°.Practical implications– Fast computation in arctangent and arccosine function can speed up the motion response of the real robot system when it needs to perform the inverse kinematics function.Originality/value– This is the first time such to combine the Taylor series method and LUT method in the computation the arctangent and arccosine function as well as to implement it with FPGA.

Time Series Forecasting Using a Hybrid Adaptive Particle Swarm Optimization and Neural Network Model

AbstractFor time series forecasting, the problem that we often encounter is how to increase the prediction accuracy as much as possible with the irregular and noise data. This study proposes a novel multilayer feedforward neural network based on the improved particle swarm optimization with adaptive genetic operator (IPSO- MLFN). In the proposed IPSO, inertia weight is dynamically adjusted according to the feedback from particles’ best memories, and acceleration coefficients are controlled by a declining arccosine and an increasing arccosine function. Further, a crossover rate which only depends on generation and does not associate with the individual fitness is designed. Finally, the parameters of MLFN are optimized by IPSO. The empirical results on the container throughput forecast of Shenzhen Port show that forecasts with IPSO-MLFN model are more conservative and credible.

1-Norm character of phase shifting interferograms and its application in phase shift extraction

Aiming at the situation that the fringe number in the interferogram is less than one, a novel phase shift extraction algorithm is proposed. By introducing the 1-norm of a sequence of phase-shifting interferograms and analyzing its character, which changes in the cosine form with the phase shift, the phase shift can be extracted from an arccosine function easily. Specially, so long as the fringe number in the interferogram is less than 0.5, both the simulation calculation and experimental research show that the phase shift with high precision can be obtained directly and rapidly in the proposed algorithm.