Grubbs Test Research Articles

Моделювання аномалій щільності потоку нейтронів в автоматизованій системі контролю ядерної безпеки об’єкта «Укриття»

Analysis of the dynamics of neutron flux density (NFD) from fuel-containing masses (FCM) in the «Shelter» Object shows the presence of values that exceed the average values for different observation periods. Identification of such values by the criterion of «anomaly / non-anomaly» will allow excluding uninformative events from the array of observations. Or, in the case of the anomaly confirmation, it will allow forming effective actions for decision-making in order to eliminate the consequences of such events. To solve the problem of detecting anomalous measurements, now there is utilized the theory of statistical solutions which is based on the use of parametric methods. The utilization of these methods requires a priori information about the nature of the distribution of the measured process and its parameters. In order to find an effec-tive solution to the problem of detecting and eliminating anomalous measurements, it is necessary to know the statistical characteristics of normal and anomalous components. This paper proposes statistical criteria for estimating anomalies in time series of NFD which have different approaches to the formation of observation intervals, power and reliability of anomaly detection. Depending on the type of distribution of the array of observations, a set of criteria is proposed. These criteria are most expedient to use when checking the anomaly of series levels for the exponential distribution, the Poisson and Weibull distribution. The capacity of the criteria has been evaluated depending on the sample size. The article defines the accuracy of determination of anomalies by criteria with the known values of the mean and dispersion in the studied sample. As the result, it is recommended to use the Grubbs test to study the anomaly of the sample levels with n>700, and the Dixon and Smolyak-Titarenko criteria for the samples with n<50. The utilization of optimal criteria depending on the characteristics of the studied samples will increase the mathematical significance of the obtained results and, as a result, will improve the quality of management decisions and nuclear safety on the «Shelter» Object as a whole.

A Comparative Analysis of Different Outlier Detection Techniques in Cognitive Radio Networks with Malicious Users

In a cognitive radio (CR), opportunistic secondary users (SUs) periodically sense the primary user’s (PU’s) existence in the network. Spectrum sensing of a single SU is not precise due to wireless channels and hidden terminal issues. One promising solution is cooperative spectrum sensing (CSS) that allows multiple SUs’ cooperation to sense the PU’s activity. In CSS, the misdetection of the PU signal by the SU causes system inefficiency that increases the interference to the system. This paper introduces a new category of a malicious user (MU), i.e., a lazy malicious user (LMU) with two operating modes such as an awakened mode and sleeping mode. In the awakened mode, the LMU reports accurately the PU activity like other normal cooperative users, while in the sleeping mode, it randomly reports abnormal sensing data similar to an always yes malicious user (AYMU) or always no malicious user (ANMU). In this paper, statistical analysis is carried out to detect the behavior of different abnormal users and mitigate their harmful effects. Results are collected for the different hard combination schemes in the presence of the LMU and opposite categories of malicious users (OMUs). Simulation results collected for the error probability, detection probability, and false alarm at different levels of the signal-to-noise ratios (SNRs) and various contributions of the LMUs and OMUs confirmed that out of the many outlier detection tests, the median test performs better in MU detection by producing minimum error probability results in the CSS. The results are further compared by keeping minimum SNR values with the mean test, quartile test, Grubbs test, and generalized extreme studentized deviate (GESD) test. Similarly, performance gain of the median test is examined further separately in the AND, OR, and voting schemes that show minimum error probability results of the proposed test as compared with all other outlier detection tests in discarding abnormal sensing reports.

Effect of Soluble Salt Loss via Spring Water on Irrigation-Induced Landslide Deformation

Landslide exposes the previously blocked groundwater discharge. High concentrations of soluble salt form salt sinters that can be observed near discharge passages. Based on existing laboratory investigation results of soil leaching and shearing reported in the literature, the effect of the soluble salt loss via spring water on irrigation-induced landslide deformation was studied under large-scale conditions. During our field investigation of landslides in the Heitai terrace of the Yellow River’s upper reaches in Gansu Province, China, 35 spring outlets were found, and the Heitai terrace was divided into five subareas, based on the difference in spring flow. Deformation data for the terrace were obtained by small baseline subset technology (SBAS-InSAR). These data were analyzed in combination with the amount of soluble salt loss, to explore the correlation between the deformation of the landslide and the soluble salt loss in the loess irrigation area. The results showed that the cumulative deformation and the loss of soluble salt were increasing continuously in the terrace. Although the increasing intensity of each subarea was different, the changing intensity of the two during the corresponding monitoring period was highly consistent. The statistical analysis revealed a strong positive correlation between the accumulated loss of soluble salt via spring water and the accumulated displacement of the terrace edge (p < 0.01). After the slope k between the two was tested by the Grubbs test and t-test, the k was no abnormality (α = 0.05) and difference (Sig > 0.05), further providing the basis for confirming the existence of this positive correlation. When the loss of soluble salt in rock and soil increased gradually, the accumulated deformation of the terrace edge also increased continuously. The findings of this study are of great significance for understanding the formation mechanism of landslides and the identifying landslide revival in irrigation areas of the Loess Plateau.

Iterative-learning error compensation for autonomous parking of mobile manipulator in harsh industrial environment

In wide-area and multi-sites manufacturing scenarios, the mobile manipulator suffers from inadequate autonomous parking performance due to the harsh industrial environment. Instead of struggling to model various errors or calibrate multiple sensors, this paper resolves the above challenge by proposing an iterative-learning error compensation scheme that consists of offline pre-regulation and online compensation, which can improve the compensation efficiency and accommodate the error fluctuations caused by environmental fluctuations. Integrating an improved Monte-Carlo localization and eye-in-hand vision technique, an effective measurement system is firstly developed to accurately obtain the parking data without requiring superfluous facilities or cumbersome measurement. Then, after removing the data outliers utilizing the Grubbs test, offline pre-regulation is achieved to give a suitable initial value and increase the compensation convergence. To reduce the time-varying systematic errors and parking error fluctuations, online compensation is presented by offering an efficacious estimation of environmental fluctuations using fuzzy logic rules and providing an adaptive iterative-learning law. Finally, the feasibility and effectiveness of the presented compensation method are validated by extensive experiments implemented on a self-developed mobile manipulator.

Advanced methods of earnings management: monotonic trends and change-points under spotlight in the Visegrad countries

Research background: Enterprises manage earnings in an effort to balance their profit fluctuations to provide increasingly consistent earnings in every reporting period. Earnings management is legal and very effective method of accounting techniques and may be used to obtain specific objectives of the enterprises involving the manipulation of accruals. Therefore, there is a need to analyze it in the context of group of countries, while the issue of their detection in the new ways appears. 
 Purpose of the article: The analysis of annual earnings before interest and taxes (EBIT) of 5,640 enterprises from the Visegrad Four during the period 2009?2018 confirms that the development of earnings management in these countries is not a randomness. Thus, the aim of this article is to determine the existence of positive trend in earnings management and to detect the change-point in its development for each Visegrad country.
 Methods: Grubbs test, Mann-Kendall trend test and Buishand test were used as appropriate statistical methods. Mann-Kendall test identifies significant monotonic trend occurrence in earnings manipulation in every country. Buishand test indicates significant years, which divides the development of EBIT into two homogenous groups with individual central lines.
 Findings & Value added: Based on the statistical analysis applied, we rejected randomness in the managing of earning, but we determined the trend of its increasing. The positive earnings manipulation was not homogenous in the analyzed period, however, a change-point was defined. Year 2014 was identified as a break-point for Slovak, Polish and Hungarian enterprises considering the earnings manipulation. Year 2013 was detected as a change-point in Czech enterprises. The methodical approach used may be very helpful for researchers from other countries to determine, detect and understand earnings management as well as for the investors to make decisions based on a specificities of an individual country.

Alter Ego Only Four Times? The Case Study of Business Profits in the Visegrad Group

The paper studies a new point of view and the approach to profit as an inherent part of business finance as well as a symbol of every healthy economy. The fundamental function of the profit is a stimulus; it means initial motivator of the business activity. The profit provides core resources for survival at the business start and after the stabilization, it is the synonym for progress. The aim of this paper is to detect significant change-points in times series of EBITDA during the analysed period in every country of the Visegrad Group to recognize the progress years in the monotonic development. We use a method of homogeneity test of time series that delivers significant robust results. We observe the variable EBITDA to eliminate different tax, interest and depreciation policies of these emerging countries. The original research of this article is based on empirical results of business profits of the sample of 3,853 enterprises covered by the broad theoretical review. Firstly, we identify missing values; and detect the outliers by Z-score and Grubbs test. EBITDA of 1,058 Slovak enterprises, 688 Czech enterprises, 1,376 Polish enterprises and 731 Hungarian enterprises is analysed during the period from 2010 to 2018. We eliminate the inconsistent observations and construct average values of EBITDA. Secondly, we prove normality by Jarque-Bera test, and support it by Shapiro-Wilk test, Anderson-Darling test, Lilliefors test to deliver reliable results. Thirdly, we find an independency of distribution that confirm randomness by the Box-Pierce test. And finally, we identify the years that affect heterogeneity of EBITDA in the countries of the Visegrad Four. We uncover some really surprising results. For all countries in the Visegrad Four, the year 2013 is detected as a change-point at a significance level of 0.05. This significant year shifts EBITDA between two homogeneous series with corresponding central lines and recognizes the similar annual development within the groups. In addition, we discuss the results to the areas and factors affecting the business risk. The adjustable area represented by the business dynamism has no significant impact on the development of EBITDA. The uncontrollable macroeconomic factors such as a GDP, unemployment rate, inflation rate, average monthly gross wage, and Ease of doing business index demonstrate the same development of Slovak, Czech, Polish and Hungarian enterprises. We connect our gained results to the undisputed influence of these factors and its derived components on monotonic development of EBITDA. Despite the fact, that the countries are not economically interconnected as they used to be in the past, in has to be underlined that their mutual relations are still very narrow and close and that might be the reason, why identical results are achieved in the countries with divergent development.

Environment Monitoring for Anomaly Detection System Using Smartphones.

Currently, the popularity of smartphones with networking capabilities equipped with various sensors and the low cost of the Internet have opened up great opportunities for the use of smartphones for sensing systems. One of the most popular applications is the monitoring and the detection of anomalies in the environment. In this article, we propose to enhance classic road anomaly detection methods using the Grubbs test on a sliding window to make it adaptive to the local characteristics of the road. This allows more precision in the detection of potholes and also building algorithms that consume less resources on smartphones and adapt better to real conditions by applying statistical outlier tests on current threshold-based anomaly detection methods. We also include a clustering algorithm and a mean shift-based algorithm to aggregate reported anomalies on data to the server. Experiments and simulations allow us to confirm the effectiveness of the proposed methods.

Improve of Footwear Comfort Sensation with Material Packages and Knitted Fabrics

The main goal of this paper was to analyse the hygienic properties of textile packages used for the construction of shoe uppers. Distance fabrics with varied hygienic properties were the basis of these packages. The discomfort indexes, which describe changes in footwear microclimate, were calculated according to the moisture absorbance capacity and temperature changes in the immediate surrounding of the foot skin surface. The experiment was done for a group of grain leather uppers, where the Grubbs test (a = 0.05) gave positive information about the outliers, describing such parameters as the water vapour permeability and water vapour coefficient. The phase changes of the shoe microclimate were detected via temperature and relative humidity sensors during simulation of the shoes used via an elliptical trainer for a group of 7 men. Statistically significant differences between the packages’ upper – lining confirmed the possibility of monitoring the circulation of biophysical mediums inside a footwear volume. The appropriate choice of package materials could raise the comfort conditions for users. For certain material configurations the microclimate conditions described by the discomfort index were improved.

Improvement in Heart Rate Variability Following Spinal Adjustment: A Case Study in Statistical Methodology for a Single Office Visit

Introduction: Statistical analysis is typically applied at the group level. The present study analyzes data during a single office visit as a novel approach providing real-time feedback to the clinician and patient regarding efficacy of an intervention. In this study, heart rate variability (HRV) was analyzed before versus after a chiropractic spinal adjustment. Methods: The patient is an adult female who signed a consent form for the study. HRV was measured twice before a chiropractic adjustment and once afterwards using app-based technology. The three HRV values (two pre and one post) were then statistically analyzed using an online calculator for outliers using Grubbs test. Results: The two pre-adjustment HRV (rMSSD) readings were consistently low: pre 1 = 16.0 milliseconds [ms] and pre 2 = 16.2 ms. The low HRV was an indicator that the patient’s nervous system was not functioning optimally. The patient’s atlas (C1) vertebra was palpated to be slightly out of alignment. These two findings (low HRV and vertebral misalignment) indicated the presence of a chiropractic subluxation (of the atlas vertebra). The subluxation was adjusted and within minutes the HRV increased (improved) to 27.5 ms. This improvement was calculated to be a statistically significant outlier (p < 0.05). Conclusion: This study is an example of how statistical methods can be applied to the level of an individual patient during one office visit to assess neurological effectiveness of a chiropractic adjustment. Since this is a case study, the results may not apply to all patients. Therefore, further studies in other patients, and for longer follow-up times, are reasonable next steps.

A new driving condition identification method for heavy duty vehicles based on HHMM

Aiming to improve active safety of heavy duty vehicles, a new dynamic driving condition identification method is developed in this paper through incorporating Hierarchical Hidden Markov Models (HHMM) into the rollover warning system for heavy duty vehicles to assist the driver to be aware of the driving conditions. The corresponding data under typical driving conditions are first collected and then put into test with Student's t-test method and Grubbs's test method (T-G test method). The outliers filtered by T-G test from the data are detected and eliminated. K-Means algorithm, used to set up the rollover threshold value and Baum-Welch algorithm for optimising the proposed rollover warning model, are discussed in detail. Computer simulations under different driving conditions are carried out to verify the optimised HHMM. The simulation results demonstrate that the proposed driving condition identification method can effectively identify the driving status with a high accuracy under a variety of driving conditions and could be used for real-time rollover warning control.

Aplicación de la estadística robusta en la identificación de valores “OUTLIERS” en análisis de residuos de clorpirifos en maní. programa elaborado en Hp-Basic utilizando la calculadora gráfica HP Prime

En el presente trabajo se ha aplicado la estadística robusta para la identificación de valores “outliers” en la determinación del factor recobrado en el análisis de residuos de clorpirifos en maní por cromatografía de gases con detector ECD. Para ello fue elaborado un programa en lenguaje HP-BASIC, basado en la estadística robusta y específicamente en el algoritmo de Huber y el cual fue editado en la calculadora gráfica de última generación HP PRIME Graphing Calculator. El programa fue ejecutado ingresando los datos obtenidos de los resultados de la evaluación del factor de recobrado para el método de análisis de clorpirifos en una muestra de referencia interna de maní, obteniéndose como resultado un estimado de los parámetros de la población y la identificación de 3 “outliers”. Este programa fue validado con uno de referencia elaborado en Excel por el Comité de Métodos Analíticos de la Royal Society of Chemistry, observándose que la desviación de los parámetros de la población (m,s) fueron menores que la precisión iterativa, lo que indica que no hubo diferencias significativas entre los dos programas. Por otro lado, estos resultados se compararon con los obtenidos por las pruebas clásicas de Dixon y de Grubbs, concluyendo que el método de Huber es el más sensible. El menos sensible es el test de Dixon, dado que su aplicación no detectó ningún outlier, mientras que el test de Grubbs identificó los dos valores más pequeños.

An Assessment of GCM Performance at a Regional Scale Using a Score-Based Method

A multicriteria score-based method was developed to assess the performances of 18 general circulation models (GCMs) in the study region from 1970 to 2005. The results indicate the following. (1) GCMs simulate temperature better than rainfall. The temporal and spatial distributions of simulated temperature performed well compared with those from the observations. In comparison to temperature, the spatial distribution of simulated precipitation performed poorly. Most of the GCMs underestimated temperature and overestimated precipitation. (2) The Grubbs test was used to detect anomalous moving changes in the rank score (RS) results; the inm-cm4 and ipsl-cm5b-lr models were rejected when simulating temperature, while the bnu-esm and canesm2 models performed poorly when simulating precipitation. (3) Adding or removing any criterion does not significantly influence the RS result, which indicates that the multicriteria score-based method is robust. The advantages of using multicriteria score-based method to assess GCMs performance were demonstrated, and this method also provides a more comprehensive assessment when compared with the single-criterion method. The multicriteria method could replace other criteria as the research requirements and could be easily extended to different study regions; the results could be used for better informed regional climate change impact analyses.

Searching for Key Factors in Enterprise Bankrupt Prediction: A Case Study in Slovak Republic

Abstract The issue of enterprise in bankrupt or financial health as a whole is still very actual topic not only in Slovakia but also in abroad. Works dealing with the enterprise in bankruptcy have already appeared in the 1930s of the 20th century. Bankrupt of enterprise affect all subject in relationship with this enterprise. Financial experts were looking for the ways for enterprise bankrupt prediction. This article is based on the searching for key factors that could indicate the enterprise in bankrupt in Slovak conditions. This article tries to work with financial variables from the area of financial health assessment of enterprise and works with the sample of Slovak enterprises. This sample includes 8,522 financial statements of enterprises in 2016. According to several relevant decisions rules, for example, the value of equity or equity debt ratio, enterprises are divided into two categories – bankrupt enterprises and creditworthy enterprises. Subsequently, this article tries to find statistically significant financial variables that could indicate involving enterprises in these two categories and works with several statistical methods for searching significant relationship between variables and the tightness of relations between them. As a main statistical method, Pearson´s correlation coefficient is used, which is supported by correlation matrices. In addition, it is necessary to test an existence of outliers in the sample of enterprises. Existence of outliers is tested by the Grubbs test of outliers.

The application of selected statistical tests in the detection and removal of outliers in water engineering data based on the example of piezometric measurements at the Dobczyce dam over the period 2012-2016

Due to their size, water construction belongs to the largest and heaviest engineering structures. Ensuring the safe operation of such facilities requires continuous monitoring. Among the basic forms of monitoring in such facilities one should list continuous seasonal piezometric measurements, which are obligatory elements of general control measurements aimed at ensuring safety when using the facility. The latter is directly related to guaranteeing the safety of people living and working in an area exposed to destruction in the event of a possible disaster involving the building. From the perspective of increasing the safety of the hydrotechnical facility, optimal conditions occur when the water levels in piezometers oscillate around a constant value as this signalizes that filtration processes in the body and the surface of a dam are stable. Various factors may disturb measurements of water levels in open piezometers or water pressure in closed piezometers. These factors may take the form of systematic, random or obvious errors. Thus, before analyzing this type of data, the largest errors (outliers) should be removed from the sample as they could significantly affect the outcomes of analysis and lead to a false interpretation. In such a situation, it is necessary to apply respective statistical tests, which allow verification of whether a particular portion of data may be treated as a set of outliers at a given significance level α. In this work the following statistical tests were used to identify and remove outliers: Q-Dixon test, Grubbs test, Hampel test, Rosner test, Iglewicz and Hoaglin test, Tietjen-Moore test and quartile test. The scope of the empirical analysis is focused on piezometric measurements at the Dobczyce dam over the period 2012-2016.

A Model Approach to Infer the Quality in Agricultural Sprayers Supported by Knowledge Bases and Experimental Measurements

This paper presents a method to infer the quality of sprayers based on data collection of the drop spectra and their physical descriptors, which are used to generate a knowledge base to support decision-making in agriculture. The knowledge base is formed by collected experimental data, obtained in a controlled environment under specific operating conditions, and the semantics used in the spraying process to infer the quality in the application. The electro-hydraulic operating conditions of the sprayer system, which include speed and flow measurements, are used to define experimental tests, perform calibration of the spray booms and select the nozzle types. Using the Grubbs test and the quartile-quartile plot an exploratory analysis of the collected data was made in order to determine the data consistency, the deviation of atypical values, the independence between the data of each test, the repeatability and the normal representation of them. Therefore, integrating measurements to a knowledge base it was possible to improve the decision-making in relation to the quality of the spraying process defined in terms of a distribution function. Results shown that the use of advanced models and semantic interpretation improved the decision-making processes related to the quality of the agricultural sprayers.

Single-Molecule Studies of Unlabeled Full-Length p53 Protein Binding to DNA.

p53 is an antitumor protein that plays an important role in apoptosis, preserving genomic stability and preventing angiogenesis, and it has been implicated in a large number of human cancers. For this reason it is an interesting target for both fundamental studies, such as the mechanism of interaction with DNA, and applications in biosensing. Here, we report a comprehensive study of label-free, full length p53 (flp53) and its interaction with engineered double-stranded DNA in vitro, at the single-molecule level, using atomic force microscopy (AFM) imaging and solid-state nanopore sensing. AFM data show that dimeric and tetrameric p53 bind to the DNA in a sequence-specific manner, confirming previously reported relative binding affinities. The statistical significance is tested using both the Grubbs test and stochastic simulations. For the first time, ultralow noise solid-state nanopore sensors are employed for the successful differentiation between bare DNA and p53/DNA complexes. Furthermore, translocation statistics reflect the binding affinities of different DNA sequences, in accordance with AFM data. Our results thus highlight the potential of solid-state nanopore sensors for single-molecule biosensing, especially when labeling is either not possible or at least not a viable option.

Data Transformation Technique to Improve the Outlier Detection Power of Grubbs’ Test for Data Expected to Follow Linear Relation

Grubbs test (extreme studentized deviate test, maximum normed residual test) is used in various fields to identify outliers in a data set, which are ranked in the order ofx1≤x2≤x3≤⋯≤xn (i=1,2,3,…,n). However, ranking of data eliminates the actual sequence of a data series, which is an important factor for determining outliers in some cases (e.g., time series). Thus in such a data set, Grubbs test will not identify outliers correctly. This paper introduces a technique for transforming data from sequence bound linear form to sequence unbound form(y=c). Applying Grubbs test to the new transformed data set detects outliers more accurately. In addition, the new technique improves the outlier detection capability of Grubbs test. Results show that, Grubbs test was capable of identifing outliers at significance level 0.01 after transformation, while it was unable to identify those prior to transforming at significance level 0.05.

Quality assessment of Liuwei Wuling tablet by HPLC fingerprint and quantitative analysis

Reversed phase high performance liquid chromatography with diode array detector was employed for simultaneous determination of six components and specific chromatograms analysis in Liuwei Wuling tablets with gradient elution of acetonitrile and water containing 0.1% phosphoric acid as mobile phase. The results showed that six components containing specnuezhenide, phillyrin, schisandrin, schisantherin A, schizandrin A and schizandrin B were separated well under the analytical condition. The average recoveries ranged from 98.96% to 100.5% with RSD less than 2. 0%. Twenty-five common peaks were selected as the specific chromatograms of Liuwei Wuling tablets with schisantherin A as the reference peak. Similarities calculated by cosine of angle, correlation coefficient and peak area ratio similarity (PAR) were all above 0. 95, indicating a good similarity between the reference and twenty batches of samples. Grubbs test and cluster analysis indicated that the established HPLC fingerprints and HPLC quantitative analysis can be used efficiently in the quality control of Liuwei Wuling tablets.

Comparative performance of four single extreme outlier discordancy tests from Monte Carlo simulations.

Using highly precise and accurate Monte Carlo simulations of 20,000,000 replications and 102 independent simulation experiments with extremely low simulation errors and total uncertainties, we evaluated the performance of four single outlier discordancy tests (Grubbs test N2, Dixon test N8, skewness test N14, and kurtosis test N15) for normal samples of sizes 5 to 20. Statistical contaminations of a single observation resulting from parameters called δ from ±0.1 up to ±20 for modeling the slippage of central tendency or ε from ±1.1 up to ±200 for slippage of dispersion, as well as no contamination (δ = 0 and ε = ±1), were simulated. Because of the use of precise and accurate random and normally distributed simulated data, very large replications, and a large number of independent experiments, this paper presents a novel approach for precise and accurate estimations of power functions of four popular discordancy tests and, therefore, should not be considered as a simple simulation exercise unrelated to probability and statistics. From both criteria of the Power of Test proposed by Hayes and Kinsella and the Test Performance Criterion of Barnett and Lewis, Dixon test N8 performs less well than the other three tests. The overall performance of these four tests could be summarized as N2≅N15 > N14 > N8.

Robust collaborative spectrum sensing in the presence of deleterious users

Collaborative spectrum sensing has attracted significant research attention in the last few years and is widely accepted as a viable approach to improve spectrum sensing reliability. Fusing data from multiple opportunistic users (OUs) in order to produce reliable sensing results implies a reliance on the OU to provide correct information. In the presence of malfunctioning or selfish users, performance of collaborative spectrum sensing deteriorates significantly. In this study, the authors propose mechanisms for the detection and suppression of such deleterious OUs (DOUs) for hard and soft decision fusion. More specifically, a credibility-based mechanism for hard decision fusion using a hard decision combining beta reputation (HDC-BR) system is introduced. The authors proposed method does not require knowledge of the total number of deleterious users in advance. In HDC-BR, the fusion centre assigns and updates weights to each user's decisions based on an individual user credibility score, which is calculated using the BR system. The presence of DOUs in soft decision-based collaborative spectrum sensing has even more adverse effects on system performance. The authors also propose a scheme for the case of soft decision fusion to detect and eliminate falsified user observations at the fusion centre using a modified Grubbs test; they refer to it as soft-decision combining-modified Grubbs (SDC-MG). They compare the performance of the proposed methods with malicious user detection schemes proposed in the literature as well as with the case where no DOU suppression scheme is implemented, and conclude that SDC-MG performs much better than HDC-BR in a low signal-to-noise ratio regime.