A primer on model selection using the Akaike Information Criterion

A powerful investigative tool in biology is to consider not a single mathematical model but a collection of models designed to explore different working hypotheses and select the best model in that collection.In these lecture notes,the usual workflow of the use of mathematical models to investigate a biological problem is described and the use of a collection of model is motivated.Models depend on parameters that must be estimated using observations;and when a collection of models is considered,the best model has then to be identified based on available observations.Hence,model calibration and selection,which are intrinsically linked,are essential steps of the workflow.Here,some procedures for model calibration and a criterion,the Akaike Information Criterion,of model selection based on experimental data are described.Rough derivation,practical technique of computation and use of this criterion are detailed.

A Comparison of Four Methods of Estimating the Scale Parameter for the Exponential Distribution

In this paper, the estimators of the scale parameter of the exponential distribution obtained by applying four methods, using complete data, are critically examined and compared. These methods are the Maximum Likelihood Estimator (MLE), the Square-Error Loss Function (BSE), the Entropy Loss Function (BEN) and the Composite LINEX Loss Function (BCL). The performance of these four methods was compared based on three criteria: the Mean Square Error (MSE), the Akaike Information Criterion (AIC), and the Bayesian Information Criterion (BIC). Using Monte Carlo simulation based on relevant samples, the comparisons in this study suggest that the Bayesian method is better than the maximum likelihood estimator with respect to the estimation of the parameter that offers the smallest values of MSE, AIC, and BIC. Confidence intervals were then assessed to test the performance of the methods by comparing the 95% CI and average lengths (AL) for all estimation methods, showing that the Bayesian methods still offer the best performance in terms of generating the smallest ALs.

一项优化AJCC/UICC pTNM胃癌分期预后预测效能的多中心研究

背景与目的美国癌症联合会与国际抗癌联盟(AmericanJointCommitteeonCancer/UnionforInternationalCancerControl,A JCC/UICC)联合发布的第8版TNM病理分期(pathological tumor?node?metastasis,p TNM)较前进行了重要修改,以提高胃癌患者预后预测准确性。然而,该分期不同亚组间患者的预后仍存在一定的同质性。本研究旨在对比第8版和第7版AJCC/UICC pTNM分期对胃癌预后的预测效能,并纳入外部验证对现有分期进行优化。方法共纳入分析7911例就诊于中国3家大型医疗中心和10,208例美国流行病监测与最终治疗结果(Surveillance Epidemiology and End Results,SEER)数据库的患者临床资料。采用log-rank检验、线性趋势检验、似然比检验和赤池信息量准则(akaike information criterion,AIC)梯度评估第7、8版AJCC/UICC pTNM分期系统的同质性、辨别力和单调性,在此基础上优化分期并以SEER数据集作为外部验证。结果第7、8版分期系统在两组数据集中均存在明显分期偏移,且集中在III期患者。IIIA、IIIB和IIIC期各亚组内患者的生存率有显著差异,表明两个系统分期分层的同质性较差。我们根据中国患者数据构建一个新的改良版p TNM分期,上述分层同质性问题得到明显改善,进一步以SEER数据作为验证集同样得到了良好的结果。相较第7、8版分期系统,改良版p TNM分期在两组数据集中均有较高的log-rank、线性趋势、似然比卡方值和较低的AIC值,显示出更优的辨别力、同质性、单调性和预后预测效能。结论第8版AJCC/UICC pTNM分期系统优于第7版,但预后预测的同质性均较差。我们构建的改良版p TNM分期在两组大样本胃癌数据集中均显示出理想的分期分层和预后预测效能。

An unsupervised clustering method for nuclear magnetic resonance transverse relaxation spectrums based on the Gaussian mixture model and its application

To make the quantitative results of nuclear magnetic resonance(NMR) transverse relaxation(T;) spectrums reflect the type and pore structure of reservoir more directly, an unsupervised clustering method was developed to obtain the quantitative pore structure information from the NMR T;spectrums based on the Gaussian mixture model(GMM). Firstly, We conducted the principal component analysis on T;spectrums in order to reduce the dimension data and the dependence of the original variables. Secondly, the dimension-reduced data was fitted using the GMM probability density function, and the model parameters and optimal clustering numbers were obtained according to the expectation-maximization algorithm and the change of the Akaike information criterion. Finally, the T;spectrum features and pore structure types of different clustering groups were analyzed and compared with T;geometric mean and T;arithmetic mean. The effectiveness of the algorithm has been verified by numerical simulation and field NMR logging data. The research shows that the clustering results based on GMM method have good correlations with the shape and distribution of the T;spectrum, pore structure, and petroleum productivity, providing a new means for quantitative identification of pore structure, reservoir grading, and oil and gas productivity evaluation.

On the volatility of daily stock returns of Total Nigeria Plc: evidence from GARCH models, value-at-risk and backtesting

This study investigates the volatility in daily stock returns for Total Nigeria Plc using nine variants of GARCH models:sGARCH,girGARCH,eGARCH,iGARCH,aGARCH,TGARCH,NGARCH,NAGARCH,and AVGARCH along with value at risk estimation and backtesting.We use daily data for Total Nigeria Plc returns for the period January 2,2001 to May 8,2017,and conclude that eGARCH and sGARCH perform better for normal innovations while NGARCH performs better for student t innovations.This investigation of the volatility,VaR,and backtesting of the daily stock price of Total Nigeria Plc is important as most previous studies covering the Nigerian stock market have not paid much attention to the application of backtesting as a primary approach.We found from the results of the estimations that the persistence of the GARCH models are stable except for few cases for which iGARCH and eGARCH were unstable.Additionally,for student t innovation,the sGARCH and girGARCH models failed to converge;the mean reverting number of days for returns differed from model to model.From the analysis of VaR and its backtesting,this study recommends shareholders and investors continue their business with Total Nigeria Plc because possible losses may be overcome in the future by improvements in stock prices.Furthermore,risk was reflected by significant up and down movement in the stock price at a 99%confidence level,suggesting that high risk brings a high return.

Time-series analysis of monthly rainfall data for the Mahanadi River Basin, India

Time series analysis has two goals, modeling random mechanisms and predicting future series using historical data. In the present work, a uni-variate time series autoregressive integrated moving average （ARIMA） model has been developed for （a） simulating and forecasting mean rainfall, obtained using Theissen weights; over the Mahanadi River Basin in India, and （b） simula^ag and forecasting mean rainfall at 38 rain-gauge stations in district towns across the basin. For the analysis, monthly rainfall data of each district town for the years 1901-2002 （102 years） were used. Theissen weights were obtained over the basin and mean monthly rainfall was estimated. The trend and seasonality observed in ACF and PACF plots of rainfall data were removed using power transformation （a=0.5） and first order seasonal differencing prior to the development of the ARIMA model. Interestingly, the AR1MA model （1,0,0）（0,1,1）12 developed here was found to be most suitable for simulating and forecasting mean rainfall over the Mahanadi River Basin and for all 38 district town rain-gauge stations, separately. The Akaike Information Criterion （AIC）, good- ness of fit （Chi-square）, R2 （coefficient of determination）, MSE （mean square error） and MAE （mea absolute error） were used to test the validity and applicability of the developed ARIMA model at different stages. This model is considered appropriate to forecast the monthly rainfall for the upcoming 12 years in each district town to assist decision makers and policy makers establish priorities for water demand, storage, distribution, and disaster management.

Spatio-temporal epidemic type aftershock sequence model for Tangshan aftershock sequence

Shallow earthquakes usually show obvious spatio-temporal clustering patterns. In this study, several spatio-temporal point process models are applied to investigate the clustering characteristics of the well-known Tangshan sequence based on classical empirical laws and a few assumptions. The relative fit of competing models is compared by Akalke Information Criterion. The spatial clustering pattern is well characterized by the model which gives the best fit to the data. A simulated aftershock sequence is generated by thinning algorithm and compared with the real seismicity.

A New Generalized Weibull Model:Classical and Bayesian Estimation

Statistical distributions play a prominent role in applied sciences,particularly in biomedical sciences.The medical data sets are generally skewed to the right,and skewed distributions can be used quite effectively to model such kind of data sets.In the present study,therefore,we propose a new family of distributions suitable for modeling right-skewed medical data sets.The proposed family may be called a new generalized-X family.A special sub-model of the proposed family called a new generalized-Weibull distribution is discussed in detail.The maximum likelihood estimators of the model parameters are obtained.A brief Monte Carlo simulation study is conducted to evaluate the performance of these estimators.Finally,the proposed model is applied to the remission times of the stomach cancer patient’s data.The comparison of the goodness of fit results of the proposed model is made with the other competing models such as Weibull,Kumaraswamy Weibull,and exponentiated Weibull distributions.Certain analytical measures such as Akaike information criterion,Bayesian information criterion,Anderson Darling statistic,and Kolmogorov-Smirnov test statistic are considered to show which distribution provides the best fit to data.Based on these measures,it is showed that the proposed distribution is a reasonable candidate for modeling data in medical sciences and other related fields.

Characteristic Analysis and Harmonic Feature Identification of Micro-Vibration on Flywheels

To avoid the negative effects of disturbances on satellites,the characteristics of micro-vibration on flywheels are studied.Considering rotor imbalance,bearing imperfections and structural elasticity,the extended model of micro-vibration is established.In the feature extraction of micro-vibration,singular value decomposition combined with the improved Akaike Information Criterion(AIC-SVD)is applied to denoise.More robust and self-adaptable than the peak threshold denoising,AIC-SVD can effectively remove the noise components.Subsequently,the effective harmonic coefficients are extracted by the binning algorithm.The results show that the harmonic coefficients have great identification in frequency domain.Except for the fundamental frequency caused by rotor imbalance,the harmonics are also caused by the coupling of imperfections on bearing components.

Clustering of the Values of a Response Variable and Simultaneous Covariate Selection Using a Stepwise Algorithm

In supervised learning the number of values of a response variable can be very high. Grouping these values in a few clusters can be useful to perform accurate supervised classification analyses. On the other hand selecting relevant covariates is a crucial step to build robust and efficient prediction models. We propose in this paper an algorithm that simultaneously groups the values of a response variable into a limited number of clusters and selects stepwise the best covariates that discriminate this clustering. These objectives are achieved by alternate optimization of a user-defined model selection criterion. This process extends a former version of the algorithm to a more general framework. Moreover possible further developments are discussed in detail.

Typhoon hazard assessment at the site-specific scale based on the probability density evolution method and its application on the southeast coast of China

Herein, a typhoon hazard assessment method at the site-specific scale is proposed. This method integrates the nonlinear threedimensional wind field model and the probability density evolution method. At the site-specific scale, the track of a typhoon near the engineering site is approximated via a straight line. The wind field model is utilized to calculate the wind speed at the surface given the gradient wind field at the top of the boundary layer. A comparison between the simulated and observed wind histories for Typhoon Hagupit that made landfall in Guangdong Province demonstrates the fidelity of the wind field model. The probability density evolution method is utilized to calculate the propagation of the randomness from the basic random variables toward the extremities of the typhoon surface wind. To model the probability distribution of the basic random variables, several candidate distributions are considered to fit the observations. Akaike information criterion and Anderson-Darling distance are used for selecting the preferred probability distribution model. The adequacy of the probability density evolution method in assessing typhoon hazards is verified by comparing the results with those generated by Monte Carlo simulations. The typhoon wind hazards estimated by the present study are compared with those proposed by other studies and the design code, and the differences are analyzed and discussed. The results of the proposed method provide the reasonable probabilistic model for the assessment of the structural reliability and the improvement of community resilience in the typhoon-prone areas.

Proposal and validation of a modified staging system to improve the prognosis predictive performance of the 8th AJCC/ UICC pTNM staging system for gastric adenocarcinoma: a multicenter study with external validation

Background:The 8th edition of the American Joint Committee on Cancer/Union for International Cancer Control(AJCC/UICC)pathological tumor-node-metastasis(pTNM)staging system may have increased accuracy in predicting prognosis of gastric cancer due to its important modifications from previous editions.However,the homogeneity in prognosis within each subgroup classified according to the 8th edition may still exist.This study aimed to compare and analyze the prognosis prediction abilities of the 8th and 7th editions of AJCC/UICC pTNM staging system for gastric cancer and propose a modified pTNM staging system with external validation.Methods:In total,clinical data of 7911 patients from three high-capacity institutions in China and 10,208 cases from the Surveillance,Epidemiology,and End Results(SEER)Program Registry were analyzed.The homogeneity,discrimina-tory ability,and monotonicity of the gradient assessments of the 8th and 7th editions of AJCC/UICC pTNM staging system were compared using log-rank χ^(2),linear-trend χ^(2),likelihood-ratioχ2 statistics and Akaike information criterion(AIC)calculations,on which a modified pTNM classification with external validation using the SEER database was proposed.Results:Considerable stage migration,mainly for stage III,between the 8th and 7th editions was observed in both cohorts.The survival rates of subgroups of patients within stage IIIA,IIIB,or IIIC classified according to both editions were significantly different,demonstrating poor homogeneity for patient stratification.A modified pTNM staging system using data from the Chinese cohort was then formulated and demonstrated an improved homogeneity in these abovementioned subgroups.This staging system was further validated using data from the SEER cohort,and similar promising results were obtained.Compared with the 8th and 7th editions,the modified pTNM staging system displayed the highest log-rank χ^(2),linear-trend χ^(2),likelihood-ratio χ^(2),and lowest AIC values,indicating its superior discriminatory ability,monotonicity,homogeneity and prognosis prediction ability in both populations.Conclusions:The 8th edition of AJCC/UICC pTNM staging system is superior to the 7th edition,but still results in homogeneity in prognosis prediction.Our modified pTNM staging system demonstrated the optimal stratification and prognosis prediction ability in two large cohorts of different gastric cancer populations.

A homogenized approach to classify advanced gastric cancer patients with limited and adequate number of pathologically examined lymph nodes

Background:The prognosis of gastric cancer patients with a limited number of pathologically examined lymph nodes(eLN,<16)is dismal compared to those with adequately eLN(≥16),yet they are still classified within the same subgroups using the American Joint Committee on Cancer(AJCC)staging system.We aimed at formulating an easy-to-adopt and clinically reliable stratification approach to homogenize the classification for these two categories of patients.Methods:Patients staged according to the 8th AJCC pathological nodal(N)and tumor-node-metastasis(TNM)clas-sification were stratified into a Limited and Adequate eLN cohort based on their number of pathologically examined LNs.The statistical differences between the 5-year overall survival(OS)rates of both cohorts were determined and based on which,patients from the Limited eLN cohort were re-classified to a proposed modified nodal(N′)and TNM(TN′M)classification,by matching their survival rates with those of the Adequate eLN cohort.The prognostic perfor-mance of the N′and TN′M classification was then compared to a formulated lymph-node-ratio-based nodal classifica-tion,in addition to the 8th AJCC N and TNM classification.Results:Significant heterogeneous differences in 5-year OS between patients from the Limited and Adequate eLN cohort of the same nodal subgroups were identified(all P<0.001).However,no significant differences in 5-year OS were observed between the subgroups N0,N1,N2,and N3a of the Limited eLN cohort when compared with N1,N2,N3a,and N3b from the Adequate eLN cohort,respectively(P=0.853,0.476,0.114,and 0.230,respectively).A novel approach was formulated in which only patients from the Limited eLN cohort were re-classified to one higher nodal subgroup,denoted as the N′classification.This re-classification demonstrated superior stratifying and prognostic ability as compared to the 8th AJCC N and lymph-node-ratio classification(Akaike information criterion values[AIC]:12,276 vs.12,358 vs.12,283,respectively).The TN′M classification also demonstrated superior prognostic ability as compared to the 8th AJCC TNM classification(AIC value:12,252 vs.12,312).Conclusion:The proposed lymph node classification approach provides a clinically practical and reliable technique to homogeneously classify cohorts of gastric cancer patients with limited and adequate number of pathologically examined lymph nodes.

Retrieved lymph nodes from different anatomic groups in gastric cancer:a proposed optimal number,comparison with other nodal classification strategies and its impact on prognosis

Background:The optimal number of retrieved lymph nodes(LNs)in gastric cancer(GC)is still debatable and previ-ous studies proposing new classification alternatives mostly focused on the number of retrieved LNs without proper consideration on the anatomic nodal groups’location.Here,we assessed the impact of retrieved LNs from different nodal location groups on the survival of GC patients.Methods:Stage I-III gastric cancer patients who had radical gastrectomy were investigated.LN grouping was deter-mined according to the 13th edition of the JCGC.The optimal cut-off values of retrieved LNs in different LN groups(Group 1 and 2)were calculated,based on which a proposed nodal classification(rN)simultaneously accounting the optimal number and location of retrieved LNs was proposed.The performance of rN was then compared to that of LN ratio,log-odds of metastatic LNs(LODDs)and the 8th edition of the Union for International Cancer Control/American Joint Committee on Cancer(UICC/AJCC)N classification.Results:The optimal cut-off values for Group 1 and 2 were 13 and 9,respectively.The 5-year overall survival(OS)was higher for patients in retrieved Group 1 LNs>13(vs.Group 1 LNs≤13,63.2%vs.57.9%,P=0.005)and retrieved Group 2 LNs>9(vs.Group 2 LNs≤9,72.5%vs.60.7%,P=0.009).Patients staged as pN0-3b were sub classified using this Group 1 and 2 nodal analogy.The OS of pN0-N2 patients in retrieved Group 1 LNs>13 or Group 2 LNs>9 were superior to those in retrieved Group 1 LNs≤13 and Group 2 LNs≤9(All P<0.05);except for pN3 patients.The rN clas-sification was formulated and demonstrated better 5-year OS prognostication performance as compared to the LNR,LODDs,and the 8th UICC/AJCC N staging system.Conclusions:The retrieval of>13 and>9 LNs for Group 1 and Group 2,respectively,could represent an alternative lymph node retrieval approach in radical gastrectomy for more precise survival prognostication and minimizing staging migration,especially if>16 LNs is found to be difficult.

Bertalanffy-Pütter models for the first wave of the COVID-19 outbreak

The COVID-19 pandemics challenges governments across the world.To develop adequate responses,they need accurate models for the spread of the disease.Using least squares,we fitted Bertalanffy-Pütter(BP)trend curves to data about the first wave of the COVID-19 pandemic of 2020 from 49 countries and provinces where the peak of the first wave had been passed.BP-models achieved excellent fits(R-squared above 99%)to all data.Using them to smoothen the data,in the median one could forecast that the final count(asymptotic limit)of infections and fatalities would be 2.48 times(95%confidence limits 2.42-2.6)and 2.67 times(2.39-2.765)the total count at the respective peak(inflection point).By comparison,using logistic growth would evaluate this ratio as 2.00 for all data.The case fatality rate,defined as the quotient of the asymptotic limits of fatalities and confirmed infections,was in the median 4.85%(confidence limits 4.4%e6.5%).Our result supports the strategies of governments that kept the epidemic peak low,as then in the median fewer infections and fewer fatalities could be expected.

Ecogeographical variation of 12 morphological traits within Pinus tabulaeformis: the effects of environmental factors and demographic histories

Aims More data are needed about how genetic variation(GV)and envi-ronmental factors influence phenotypic variation within the natural populations of long-lived species with broad geographic distribu-tions.To fill this gap,we examined the correlations among envi-ronmental factors and phenotypic variation within and among 13 natural populations of Pinus tabulaeformis consisting of four demo-graphically distinct groups within the entire distributional range.Methods Using the Akaike’s information Criterion(AiC)model,we measured 12 morphological traits and constructed alternative candidate models for the relationships between each morphological trait and key climatic variables and genetic groups.We then compared the AiC weight for each candidate model to identify the best approximating model for ecogeographical variation of P.tabulaeformis.The partitioning of vari-ance was assessed subsequently by evaluating the independent vari-ables of the selected best models using partial redundancy analysis.Important Findings Significant phenotypic variation of the morphological traits was observed both within individual populations and among populations.Variation partition analyses showed that most of the phenotypic variation was co-determined by both GV and climatic factors.GV accounted for the largest proportion of reproductive trait variation,whereas local key climatic factors(i.e.actual evapotranspiration,AET)accounted for the largest proportion of phenotypic variation in the remaining investigated traits.Our results indicate that both genetic divergence and key environmental factors affect the phenotypic variation observed among populations of this species,and that reproductive and vegetative traits adaptively respond differently with respect to local environmental conditions.This partitioning of factors can inform those making predictions about phenotypic variation in response to future changes in climatic conditions(particularly those affecting AET).