Search results for: logarithms

Authors: Ahmed M. Mansour, Ali M. El-Sawy, Ahmed T. Sayed

Abstract:

Logarithms reduce products to sums and powers to products; they play an important role in signal processing, communication and information theory. They are primarily used for hardware calculations, handling multiplications, divisions, powers, and roots effectively. There are three commonly used bases for logarithms; the logarithm with base-10 is called the common logarithm, the natural logarithm with base-e and the binary logarithm with base-2. This paper demonstrates different methods of calculation for log2 showing the complexity of each and finds out the most accurate and efficient besides giving in- sights to their hardware design. We present a new method called Floor Shift for fast calculation of log2, and then we combine this algorithm with Taylor series to improve the accuracy of the output, we illustrate that by using two examples. We finally compare the algorithms and conclude with our remarks.

Keywords: logarithms, log2, floor, iterative, CORDIC, Taylor series

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Authors: G. L. C. Yap

Abstract:

Modelling realized volatility with high-frequency returns is popular as it is an unbiased and efficient estimator of return volatility. A computationally simple model is fitting the logarithms of the realized volatilities with a fractionally integrated long-memory Gaussian process. The Gaussianity assumption simplifies the parameter estimation using the Whittle approximation. Nonetheless, this assumption may not be met in the finite samples and there may be a need to normalize the financial series. Based on the empirical indices S&P500 and DAX, this paper examines the performance of the linear volatility model pre-treated with normalization compared to its existing counterpart. The empirical results show that by including normalization as a pre-treatment procedure, the forecast performance outperforms the existing model in terms of statistical and economic evaluations.

Keywords: Gaussian process, long-memory, normalization, value-at-risk, volatility, Whittle estimator

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Authors: Tala Toonsi, Marah Alagha, Lina Alnowaiser, Hala Rajab

Abstract:

This report is about the Photomath app, which is an AI application that uses image recognition technology, specifically optical character recognition (OCR) algorithms. The (OCR) algorithm translates the images into a mathematical equation, and the app automatically provides a step-by-step solution. The application supports decimals, basic arithmetic, fractions, linear equations, and multiple functions such as logarithms. Testing was conducted to examine the usage of this app, and results were collected by surveying ten participants. Later, the results were analyzed. This paper seeks to answer the question: To what level the artificial intelligence features are accurate and the speed of process in this app. It is hoped this study will inform about the efficiency of AI in Photomath to the users.

Keywords: photomath, image recognition, app, OCR, artificial intelligence, mathematical equations.

Procedia PDF Downloads 131

Authors: Shin Woo Kim, Eui Ju Lee

Abstract:

The recent development of biofuel production technology facilitates the use of bioethanol and biodiesel on automobile. Bioethanol, especially, can be used as a fuel for gasoline vehicles because the addition of ethanol has been known to increase octane number and reduce soot emissions. However, the wide application of biofuel has been still limited because of lack of detailed combustion properties such as auto-ignition temperature and pollutant emissions such as NOx and soot, which has been concerned mainly on the vehicle fire safety and environmental safety. In this study, the combustion characteristics of gasoline/ethanol fuel were investigated both numerically and experimentally. For auto-ignition temperature and NOx emission, the numerical simulation was performed on the well-stirred reactor (WSR) to simulate the homogeneous gasoline engine and to clarify the effect of ethanol addition in the gasoline fuel. Also, the response surface method (RSM) was introduced as a design of experiment (DOE), which enables the various combustion properties to be predicted and optimized systematically with respect to three independent variables, i.e., ethanol mole fraction, equivalence ratio and residence time. The results of stoichiometric gasoline surrogate show that the auto-ignition temperature increases but NOx yields decrease with increasing ethanol mole fraction. This implies that the bioethanol added gasoline is an eco-friendly fuel on engine running condition. However, unburned hydrocarbon is increased dramatically with increasing ethanol content, which results from the incomplete combustion and hence needs to adjust combustion itself rather than an after-treatment system. RSM results analyzed with three independent variables predict the auto-ignition temperature accurately. However, NOx emission had a big difference between the calculated values and the predicted values using conventional RSM because NOx emission varies very steeply and hence the obtained second order polynomial cannot follow the rates. To relax the increasing rate of dependent variable, NOx emission is taken as common logarithms and worked again with RSM. NOx emission predicted through logarithm transformation is in a fairly good agreement with the experimental results. For more tangible understanding of gasoline/ethanol fuel on pollutant emissions, experimental measurements of combustion products were performed in gasoline/ethanol pool fires, which is widely used as a fire source of laboratory scale experiments. Three measurement methods were introduced to clarify the pollutant emissions, i.e., various gas concentrations including NOx, gravimetric soot filter sampling for elements analysis and pyrolysis, thermophoretic soot sampling with transmission electron microscopy (TEM). Soot yield by gravimetric sampling was decreased dramatically as ethanol was added, but NOx emission was almost comparable regardless of ethanol mole fraction. The morphology of the soot particle was investigated to address the degree of soot maturing. The incipient soot such as a liquid like PAHs was observed clearly on the soot of higher ethanol containing gasoline, and the soot might be matured under the undiluted gasoline fuel.

Keywords: gasoline/ethanol fuel, NOx, pool fire, soot, well-stirred reactor (WSR)

Procedia PDF Downloads 185

Authors: M. A. C. S. Sampath Fernando, James M. Curran, Renate Meyer

Abstract:

Model assessment, in the Bayesian context, involves evaluation of the goodness-of-fit and the comparison of several alternative candidate models for predictive accuracy and improvements. In posterior predictive checks, the data simulated under the fitted model is compared with the actual data. Predictive model accuracy is estimated using information criteria such as the Akaike information criterion (AIC), the Bayesian information criterion (BIC), the Deviance information criterion (DIC), and the Watanabe-Akaike information criterion (WAIC). The goal of an information criterion is to obtain an unbiased measure of out-of-sample prediction error. Since posterior checks use the data twice; once for model estimation and once for testing, a bias correction which penalises the model complexity is incorporated in these criteria. Cross-validation (CV) is another method used for examining out-of-sample prediction accuracy. Leave-one-out cross-validation (LOO-CV) is the most computationally expensive variant among the other CV methods, as it fits as many models as the number of observations. Importance sampling (IS), truncated importance sampling (TIS) and Pareto-smoothed importance sampling (PSIS) are generally used as approximations to the exact LOO-CV and utilise the existing MCMC results avoiding expensive computational issues. The reciprocals of the predictive densities calculated over posterior draws for each observation are treated as the raw importance weights. These are in turn used to calculate the approximate LOO-CV of the observation as a weighted average of posterior densities. In IS-LOO, the raw weights are directly used. In contrast, the larger weights are replaced by their modified truncated weights in calculating TIS-LOO and PSIS-LOO. Although, information criteria and LOO-CV are unable to reflect the goodness-of-fit in absolute sense, the differences can be used to measure the relative performance of the models of interest. However, the use of these measures is only valid under specific circumstances. This study has developed 11 models using normal, log-normal, gamma, and student’s t distributions to improve the PCR stutter prediction with forensic data. These models are comprised of four with profile-wide variances, four with locus specific variances, and three which are two-component mixture models. The mean stutter ratio in each model is modeled as a locus specific simple linear regression against a feature of the alleles under study known as the longest uninterrupted sequence (LUS). The use of AIC, BIC, DIC, and WAIC in model comparison has some practical limitations. Even though, IS-LOO, TIS-LOO, and PSIS-LOO are considered to be approximations of the exact LOO-CV, the study observed some drastic deviations in the results. However, there are some interesting relationships among the logarithms of pointwise predictive densities (lppd) calculated under WAIC and the LOO approximation methods. The estimated overall lppd is a relative measure that reflects the overall goodness-of-fit of the model. Parallel log-likelihood profiles for the models conditional on equal posterior variances in lppds were observed. This study illustrates the limitations of the information criteria in practical model comparison problems. In addition, the relationships among LOO-CV approximation methods and WAIC with their limitations are discussed. Finally, useful recommendations that may help in practical model comparisons with these methods are provided.

Keywords: cross-validation, importance sampling, information criteria, predictive accuracy

Procedia PDF Downloads 359