Small Error Probability Research Articles

Cloud inversion analysis of surrounding rock parameters for underground powerhouse based on PSO-BP optimized neural network and web technology

Aiming at the shortcomings of the BP neural network in practical applications, such as easy to fall into local extremum and slow convergence speed, we optimized the initial weights and thresholds of the BP neural network using the particle swarm optimization (PSO). Additionally, cloud computing service, web technology, cloud database and numerical simulation were integrated to construct an intelligent feedback analysis cloud program for underground engineering safety monitoring based on the PSO-BP algorithm. The program could conveniently, quickly, and intelligently carry out numerical analysis of underground engineering and dynamic feedback analysis of surrounding rock parameters. The program was applied to the cloud inversion analysis of the surrounding rock parameters for the underground powerhouse of the Shuangjiangkou Hydropower Station. The calculated displacement simulated with the back-analyzed parameters matches the measured displacement very well. The posterior variance evaluation shows that the posterior error ratio is 0.045 and the small error probability is 0.999. The evaluation results indicate that the intelligent feedback analysis cloud program has high accuracy and can be applied to engineering practice.

Research on Subsidence Prediction Method of Water-Conducting Fracture Zone of Overlying Strata in Coal Mine Based on Grey Theory Model

The development height and settlement prediction of water-conducting fracture zones caused by coal seam mining play an important role in the stability of overburden aquifers and the safety of roadways. Based on the engineering geological data of the J60 borehole in the Daliuta Coal Mine and the mining conditions of the 2−2 coal seam, China, this study established a similar material test model of mining overburden. The deformation characteristics of overlying strata in the mining process of coal seam were studied by using distributed optical fiber sensing technology, and the development height of water flowing fractured zone was determined. According to the equidistant sampling characteristics of Brillouin optical time domain reflection technology and the principle of the grey theory model, the settlement prediction model of the water-conducting fracture zone was established. By analyzing and comparing the prediction accuracy of the GM (1, 1) model, grey progressive model, and metabolic model, the optimal method for settlement prediction of the water-conducting fracture zone was discussed. The results show that, for the metabolic model, with the increase in the number of test sets and the decrease in the number of prediction sets, the mean square error ratio c and the small error probability p of the prediction accuracy evaluation parameters display a downward trend. The accuracy is related to the sudden change in the settlement of the water-conducting fracture zone caused by the breaking of the key stratum of the overlying rock. The optimal time of test sets selected for the best settlement prediction model is 7~8, and that of prediction sets selected is 5~6. For the GM (1, 1) model and the grey progressive model, the prediction accuracy of mining overburden subsidence is grade 4, which is not suitable for settlement prediction of water-flowing fractured zones.

Прогнозированиe трудоустройства студентов педагогического вуза на основе использования алгоритмов машинного обучения

One of the solutions to the problem, when not the best graduates enter the pedagogical profiles and after graduation are employed in the education system, is the prediction of professional orientation even at the stage of the student choosing their further professional trajectory. To solve this problem, the purpose of the study is to develop and experimentally prove the effectiveness of using a program for predicting the employment of students of a pedagogical university based on the introduction of various machine learning algorithms. Using a random selection of students, the collection and processing of their questionnaires (n=205) in 2011-2016 were carried out. Various machine learning algorithms were used to create the program: decision trees, logistic regression, and catboost. In the course of the experiment, the data of the questionnaires were loaded into the program for its training according to various algorithms, in order to ultimately obtain a finished intellectual product with the ability to predict the employment of graduates. In the final comparison, the program developed on the “decision trees” algorithm made only 2 out 19 questionnaires and 7 out 61, which was the best result - 89%. The implementation of this algorithm makes it possible to most accurately, with the least percentage of errors, identify students who will not be employed in the future according to their profile of study or not employed at all. Thus, the study developed an intelligent program that allows one to instantly process data and get an accurate forecast of employment with only a small probability of error.

Minimax detection of localized signals in statistical inverse problems

Abstract We investigate minimax testing for detecting local signals or linear combinations of such signals when only indirect data are available. Naturally, in the presence of noise, signals that are too small cannot be reliably detected. In a Gaussian white noise model, we discuss upper and lower bounds for the minimal size of the signal such that testing with small error probabilities is possible. In certain situations we are able to characterize the asymptotic minimax detection boundary. Our results are applied to inverse problems such as numerical differentiation, deconvolution and the inversion of the Radon transform.

Information Theoretic Signal Processing and Its Applications [Bookshelf

The roots of information theory are almost 100 years old and include early works by Fisher <xref ref-type="bibr" rid="ref1" xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">[1]</xref> , Hartley <xref ref-type="bibr" rid="ref2" xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">[2]</xref> , and others. According to a history of information theory <xref ref-type="bibr" rid="ref3" xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">[3]</xref> , motivated to understand how to draw information from experiments, Fisher <xref ref-type="bibr" rid="ref1" xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">[1]</xref> stated “the nature and degree of the uncertainty [must] be capable of rigorous expression.” Subsequently, he defined statistical information as the reciprocal of the variance of a statistical sample. However, it was Shannon’s work <xref ref-type="bibr" rid="ref4" xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">[4]</xref> that laid the mathematical foundations of information theory and revolutionized communications. Shannon developed two fundamental bounds, one on data compression and the other on transmission rate. He proved that even in the presence of noise, an arbitrarily small probability of error may be achieved as long as the transmission rate is below a quantity he defined as channel capacity.

Study on the Optimal Reaction Parameters of C4 Olefin Prepared by Ethanol Catalyzed Reaction Based on XGBoost Regression Prediction Algorithm

The production of C4 olefin by ethanol coupling catalytic reaction has a broad development prospect in the field of renewable energy. In this paper, the influencing factors of C4 olefin preparation process were studied, the influence of temperature and catalyst type on the reaction was analyzed, and the multivariate nonlinear regression fitting model was established under the assumption of ignoring the small probability of experimental error. Then, based on the grey correlation analysis model, the correlation degree between each variable and ethanol conversion rate and C4 olefin selection rate was studied.Then, using the catalyst parameter variables considered above, the relationship equations between four catalyst parameter variables, temperature and C4 olefin yield were calculated, and the overall constructor model of C4 olefin yield under the joint action of five variables was established by using the constructor method. Finally, using MPai data science platform software based on XGBoost regression prediction algorithm, the established constructor was cross-validated by 50% fold, and the optimal solution of reaction parameters was finally obtained.Finally, five experiments were selected to improve the experimental control group data and verify the accuracy of the theoretical value of the optimal solution. By changing the catalyst parameters and temperature, a relative control group was added to the experimental group, and the conclusion of the theoretical parameter optimal solution under unrestricted and temperature restrictions was experimentally verified.

Research on the Prediction Method of Monthly Hidden Danger Quantity in Coal Mine Based on BP Neural Network Periodic Combination Model

To better prevent the occurrence of hidden dangers of coal mine accidents and ensure the safety production of coal mine enterprises. This paper mines and analyses the pattern of historical monthly hidden danger quantity in the coal mine and constructs three models: the traditional backpropagation (BP) neural network model, the BP neural network based on the adaptive moment estimation optimization algorithm (Adam-BP) model, and the BP neural network prediction model with the introduction of monthly moderators (Month-Adam-BP). The experimental results show that the Adam-BP model can improve the prediction accuracy, in which the mean absolute percentage error (MAPE) improves by 8.93%, the root mean square error (RMSE) improves by 8.15%, the postdifference ratio C improves by 0.04, and the small error probability P improves by 0.12; the Month-Adam-BP model with the introduction of the monthly adjustment factor further improves the prediction accuracy, in which MAPE improves by 2.61%, RMSE improves by 5.41%, the postdifference ratio C improves by 0.06, and the small error probability P improves by 0.03. And the Month-Adam-BP model prediction accuracy reaches the level 2 standard with credible prediction effect; it can also be used to predict coal mines with periodic characteristics of hidden hazard data. Our prediction results show that the predicted number of hidden hazards in this coal mine for the next month is 29, which is an increase compared to the number of hidden dangers in the previous month. Thus, the coal mine safety managers need to strengthen the management of hidden hazards further to prevent accidents, which can better serve the standardization of coal mine safety production and ensure the smooth production of the coal mine.

Universal Fidelity Reduction of Quantum Operations from Weak Dissipation.

Quantum information processing is in real systems often limited by dissipation, stemming from remaining uncontrolled interaction with microscopic degrees of freedom. Given recent experimental progress, we consider weak dissipation, resulting in a small error probability per operation. Here, we find a simple formula for the fidelity reduction of any desired quantum operation, where the ideal evolution is confined to the computational subspace. Interestingly, this reduction is independent of the specific operation; it depends only on the operation time and the dissipation. Using our formula, we investigate the situation where dissipation in different parts of the system has correlations, which is detrimental for the successful application of quantum error correction. Surprisingly, we find that a large class of correlations gives the same fidelity reduction as uncorrelated dissipation of similar strength.

No Repetition

Stochastic sample-based estimators are among the most fundamental and universally applied tools in statistics. Such estimators are particularly important when processing huge amounts of data, where we need to be able to answer a wide range of statistical queries reliably, yet cannot afford to store the data in its full length. In many applications we need the sampling to be coordinated which is typically attained using hashing. In previous work, a common strategy to obtain reliable sample-based estimators that work within certain error bounds with high probability has been to design one that works with constant probability, and then boost the probability by taking the median over r independent repetitions. Aamand et al. (STOC'20) recently proposed a fast and practical hashing scheme with strong concentration bounds , Tabulation-1Permutation, the first of its kind. In this paper, we demonstrate that using such a hash family for the sampling, we achieve the same high probability bounds without any need for repetitions. Using the same space, this saves a factor r in time, and simplifies the overall algorithms. We validate our approach experimentally on both real and synthetic data. We compare Tabulation-1Permutation with other hash functions such as strongly universal hash functions and various other hash functions such as MurmurHash3 and BLAKE3, both with and without resorting to repetitions. We see that if we want reliability in terms of small error probabilities, then Tabulation-1Permutation is significantly faster.

Brain-inspired sensorimotor echolocation system for confident landmark recognition.

A landmark is a familiar target in terms of the echoes that it can produce and is important for echolocation-based navigation by bats, robots, and blind humans. A brain-inspired system (BIS) achieves confident recognition, defined as classification to an arbitrarily small error probability (PE), by employing a voting process with an echo sequence. The BIS contains sensory neurons implemented with binary single-layer perceptrons trained to classify echo spectrograms with PE and generate excitatory and inhibitory votes in face neurons until a landmark-specific face neuron achieves recognition by reaching a confidence vote level (CVL). A discrete random step process models the vote count to show the recognition probability can achieve any desired accuracy by decreasing PE or increasing CVL. A hierarchical approach first classifies surface reflector and volume scatterer target categories and then uses that result to classify two subcategories that form four landmarks. The BIS models blind human echolocation to recognize four human-made and foliage landmarks by acquiring suitably sized and dense audible echo sequences. The sensorimotor BIS employs landmark-specific CVL values and a 2.7° view increment to acquire echo sequences that achieve zero-error recognition of each landmark independent of the initial view.

Achievable Regions and Precoder Designs for the Multiple Access Wiretap Channels With Confidential and Open Messages

This paper investigates the secrecy achievable region of multiple access wiretap (MAC-WT) channels where, besides confidential messages, the users have also open messages to transmit. All these messages are intended for the legitimate receiver (or Bob for brevity) but only the confidential messages need to be protected from the eavesdropper (Eve). We first consider a discrete memoryless (DM) MAC-WT channel where both Bob and Eve jointly decode their interested messages. By using random coding, we find an achievable rate region, within which perfect secrecy can be realized, i.e., all users can communicate with Bob with arbitrarily small probability of error, while the confidential information leaked to Eve tends to zero. Due to the high implementation complexity of joint decoding, we also consider the DM MAC-WT channel where Bob simply decodes messages independently while Eve still applies joint decoding. We then extend the results in the DM case to a Gaussian vector (GV) MAC-WT channel. Based on the information theoretic results, we further maximize the sum secrecy rate of the GV MAC-WT system by designing precoders for all users. Since the problems are non-convex, we provide iterative algorithms to obtain suboptimal solutions. Simulation results show that compared with existing schemes, secure communication can be greatly enhanced by the proposed algorithms, and in contrast to the works which only focus on the network secrecy performance, the system spectrum efficiency can be effectively improved since open messages can be simultaneously transmitted.

General Strong Polarization

Arıkan’s exciting discovery of polar codes has provided an altogether new way to efficiently achieve Shannon capacity. Given a (constant-sized) invertible matrix M , a family of polar codes can be associated with this matrix and its ability to approach capacity follows from the polarization of an associated [0, 1]-bounded martingale, namely its convergence in the limit to either 0 or 1 with probability 1. Arıkan showed appropriate polarization of the martingale associated with the matrix ( G 2 = ( 1 1 0 1) to get capacity achieving codes. His analysis was later extended to all matrices M that satisfy an obvious necessary condition for polarization. While Arıkan’s theorem does not guarantee that the codes achieve capacity at small blocklengths (specifically in length, which is a polynomial in ( 1ε ) where (ε) is the difference between the capacity of a channel and the rate of the code), it turns out that a “strong” analysis of the polarization of the underlying martingale would lead to such constructions. Indeed for the martingale associated with ( G 2 ) such a strong polarization was shown in two independent works (Guruswami and Xia (IEEE IT’15) and Hassani et al. (IEEE IT’14)), thereby resolving a major theoretical challenge associated with the efficient attainment of Shannon capacity. In this work we extend the result above to cover martingales associated with all matrices that satisfy the necessary condition for (weak) polarization. In addition to being vastly more general, our proofs of strong polarization are (in our view) also much simpler and modular. Key to our proof is a notion of local polarization that only depends on the evolution of the martingale in a single time step. We show that local polarization always implies strong polarization. We then apply relatively simple reasoning about conditional entropies to prove local polarization in very general settings. Specifically, our result shows strong polarization over all prime fields and leads to efficient capacity-achieving source codes for compressing arbitrary i.i.d. sources, and capacity-achieving channel codes for arbitrary symmetric memoryless channels. We show how to use our analyses to achieve exponentially small error probabilities at lengths inverse polynomial in the gap to capacity. Indeed we show that we can essentially match any error probability while maintaining lengths that are only inverse polynomial in the gap to capacity.

The research of ARIMA, GM(1,1), and LSTM models for prediction of TB cases in China.

Background and objectiveTuberculosis (Tuberculosis, TB) is a public health problem in China, which not only endangers the population’s health but also affects economic and social development. It requires an accurate prediction analysis to help to make policymakers with early warning and provide effective precautionary measures. In this study, ARIMA, GM(1,1), and LSTM models were constructed and compared, respectively. The results showed that the LSTM was the optimal model, which can be achieved satisfactory performance for TB cases predictions in mainland China.MethodsThe data of tuberculosis cases in mainland China were extracted from the National Health Commission of the People’s Republic of China website. According to the TB data characteristics and the sample requirements, we created the ARIMA, GM(1,1), and LSTM models, which can make predictions for the prevalence trend of TB. The mean absolute error (MAE), root mean square error (RMSE), and mean absolute percentage error (MAPE) were applied to evaluate the effects of model fitting predicting accuracy.ResultsThere were 3,021,995 tuberculosis cases in mainland China from January 2018 to December 2020. And the overall TB cases in mainland China take on a downtrend trend. We established ARIMA, GM(1,1), and LSTM models, respectively. The optimal ARIMA model is the ARIMA (0,1,0) × (0,1,0)12. The equation for GM(1,1) model was X(k+1) = -10057053.55e(-0.01k) + 10153178.55 the Mean square deviation ratio C value was 0.49, and the Small probability of error P was 0.94. LSTM model consists of an input layer, a hidden layer and an output layer, the parameters of epochs, learning rating are 60, 0.01, respectively. The MAE, RMSE, and MAPE values of LSTM model were smaller than that of GM(1,1) and ARIMA models.ConclusionsOur findings showed that the LSTM model was the optimal model, which has a higher accuracy performance than that of ARIMA and GM (1,1) models. Its prediction results can act as a predictive tool for TB prevention measures in mainland China.

Dynamic Early-Warning Model of College Students’ Psychological Crisis Based on Characteristic Attribute

At present, the existing dynamic early-warning model of college students' psychological crisis cannot obtain psychological characteristics, and the error of psychological crisis prediction is large. The early-warning result of psychological crisis appears big deviation. A new dynamic early-warning model of college students' psychological crisis is designed based on characteristic attributes. In the process of modeling negative psychological crisis of college students, the fusion constraint fuzzy theory obtains the essence of college students' psychology from subjective and objective reasons, gives the different stages of college students' psychology, obtains the corresponding information entropy of the behavior hindrance in each stage of college students' psychology, and calculates the threshold value of the behavior tendency of college students' psychology. Based on the psychological state information of college students obtained by perception, the fuzzy comprehensive early-warning data model of feature attributes is constructed. According to the fuzzy comprehensive early-warning method and the early-warning index system, the early-warning factor set of college students' psychological crisis is constructed, and the dynamic early-warning model of college students' psychological crisis is constructed. Experimental results show that the mining results are consistent with the actual situation, and the grey relational degree, small error probability, and mean square ratio index are better than the present model.

Оптимизация параметров сигнала и дискретно-непрерывного канала с памятью для достижения его максимальной пропускной способности

The article is devoted to the definition of the bandwidth of the channel with memory. A discrete-continuous deterministic channel with fixed-band memory DCDCP was chosen as a model. The encoder forms code combinations of discrete in time and continuous in level symbols. At the output of the channel with memory, these combinations turn into implementations of continuous processes. The channel is deterministic, i.e. the parameters of the channel, the set of transmitted and, uniquely corresponding to it, the set of received combinations, are known both on the transmission and on the reception. Only permitted combinations are used for transmission. The energy of each of them and the energy of the difference of combinations are constant for the whole set. The transmitted combinations are separated by a protective time interval. The duration of the code combination is much longer than the protective interval, which is much larger than the channel memory. The energy of combinations at the output of the channel is maximum if the frequency response of the channel is also constant in the specified frequency band. As a result, the requirements for the form of information impulses are determined, which should have the form of reference functions. These requirements determine the optimal structural scheme of the transmitter: the encoder forms discrete in time and continuous in level pulses, the transmitter filter limits their spectrum to the channel frequency band, the power amplifier provides the necessary energy of each of them and the energy of the difference of the allowed combinations. Pulses in the form of reference functions can be formed if the encoder supplies a sequence of delta pulses to the input of the forming filter. When passing through the channel with the same THF, the shape of the combinations is not distorted. Their parameters at the input of the decoder correspond to the parameters laid down in the combination when they are formed in the transmission. Since the allowed combinations have a given energy and the energy of the combination difference, the optimal reception is realized using a set of correlators. There are different estimates of the Shannon limit, exceeding which allows you to get an arbitrarily small probability of error. To do this, the energy of the combination must grow in proportion to the logarithmus of the number of allowed combinations. The article calculated the exact value of this limit, coinciding with the smaller of the estimates. With a comprehensive optimization of the transmission rate and noise immunity of reception, the bandwidth of the memory channel is increased due to the fact that the excess ratio of bit energy to noise energy compared to the Shannon limit is converted into an information transfer rate. At the same time, the aspiration of the probability of error to zero with an arbitrarily large number of allowed combinations remains, although slower. On a discrete-continuous deterministic channel with memory with a given bandwidth, at the output of which a signal with a certain average bit power and additive white Gaussian noise with a given spectral energy density can be transmitted at a speed arbitrarily close to the ratio of the average power of the bit to the spectral energy density of the noise. The analysis of the singular case of transmission of information pulses of a given shape allows us to conclude that in the unrealizable case of receiving information impulses differentiated an infinite number of times, the probability of error can be made arbitrarily small at an arbitrarily high transmission speed.

Bit Error Probability Performance of Binary Dimmable Visible Light Communication Systems

This paper evaluates the practically achievable best error performance obtained from a dimmable visible light communication (VLC) system that conveys information using binary modulations, such as on-off keying (OOK). As compared to previous studies that address the ideal performance of dimmable VLC systems only over reliable channels with arbitrarily small error probability, this paper presents the best achievable error probability of various configurations with the dimming targets, signal quality, and data rate. The numerical results establish the ultimate decoding error performance targeted for dimmable VLC systems and are expected to provide a practical guideline for the enhanced VLC system design.

Prediction of Mumps Incidence Trend in China Based on Difference Grey Model and Artificial Neural Network Learning.

Background:We aimed to compare the prediction efficiency of back propagation (BP) network and grey model (GM) (1.1) for mumps infectious diseases and compare the application effect of the two models.Methods:By calculating the average incidence rate of mumps in January 2014 –2016, we conducted the modeling of the BP time series, GM (1,1) grey model and the combination models of them, and predicted the incidence rate in June 2016 in comparison with the actual one. We compared the quarterly incidence rate to test the two prediction models, and compared the advantages and disadvantages of these models.Results:R value of BP model was 68.45%, for GM (1,1) was 58.49%, and for combined forecasting model was 86.95%. We used the principal component analysis clustering method to control the samples, and found that the samples were close to the population mean. We found that the GM (1.1) model was more suitable for the prediction of mumps infection mode. We carried out dimension reduction analysis on the model data, and the accuracy of the data after dimension reduction is within the range of Da. For the discrete degree of the data in the combined model, matlab pipeline was used to verify the reliability of the data and results. By calculation after manifold optimization small error probability was P=0.875 and semi mean relative error 2.43%.Conclusion:BP, GM (1,1) is a better method for modeling the epidemic trend of mumps in China, but the efficiency of prediction is not as high as the combination of them.

Contact pressure of high-voltage DC power relay change and life prediction and structure optimization

To effectively predict the life of high voltage dc power relay, the authors design the structure of the relay to extending its life. The contact failure mechanism of relay products is analyzed by observing the appearance of several contact failures. The electromagnetic-thermal field coupling model is established and the state of heat flow and distribution of temperature increase between electrical and thermal coupling contacts are analyzed through simulations and experiments. An adaptive neural network control system is established to conduct sample training, obtain the best stiffness coefficient of the main spring, and optimize the design of the main spring structure composed of composite springs. The action voltage, release voltage, contact resistance, contact pressure, action time, and release time as characteristic parameters, and the relative error, correlation degree, variance ratio, and small error probability of each characteristic parameter, are calculated. The precision grade index of contact resistance and action time is defined as level 1. The two characteristic parameters are selected as the prediction variables to establish the life-prediction model based on gray theory. The relative error between the predicted and test life is 5.31%–5.6%, indicating that the life-prediction model has high accuracy. The test results show that the combined main spring structure can extend the life of a relay by 18.15%.

Blind Unwrapping of Modulo Reduced Gaussian Vectors: Recovering MSBs From LSBs

We consider the problem of recovering $n$ i.i.d samples from a zero mean multivariate Gaussian distribution with an unknown covariance matrix, from their modulo wrapped measurements, i.e., measurement where each coordinate is reduced modulo $\Delta$, for some $\Delta>0$. For this setup, which is motivated by quantization and analog-to-digital conversion, we develop a low-complexity iterative decoding algorithm. We show that if a benchmark informed decoder that knows the covariance matrix can recover each sample with small error probability, and $n$ is large enough, the performance of the proposed blind recovery algorithm closely follows that of the informed one. We complement the analysis with numeric results that show that the algorithm performs well even in non-asymptotic conditions.

Interdisciplinary Secondary-School Workshop

The paper describes a teaching unit of four hours with talented students aged 15-18. The workshop was designed as a problem-based sequence of tasks and was intended to deal with judging dice whether they are regular or loaded. We first introduced the students to the physics of free rotations of rigid bodies to develop the physics background of rolling dice. The highlight of this part was to recognise that cubes made from homogeneous material are the optimal form for six-sided objects leading to equal probabilities of the single faces. Experiments with all five regular bodies would lead to similar results; nevertheless, in our experiments we focused on regular cubes. This reinsures that the participants have their own experience with the context. Then, we studied rolling dice from the probabilistic point of view and – step-by-step – by extending tasks and simulations, we introduced the idea of the chi-squared test interactively with the students. The physics and the statistics part of the paper are largely independent and can be also be read separately. The success of the statistics part is best described by the fact that the students recognised that in some cases of loaded dice, it is easier to detect that property and in other cases one would need many data to make a decision with small error probabilities. A physical examination of the dice under inspection can lead to a quick and correct decision. Yet, such a physical check may fail for some reason. However, a statistical test will always lead to reasonable decision, but may require a large database. Furthermore, especially for smaller datasets, balancing the risk of different types of errors remains a key issue, which is a characteristic feature of statistical testing.&#x0D; Subject Classification: F90, K90, M50, R30