QUADRATIC PLS REGRESSION

We treat here an extension of linear PLS regression to include regression on quadratic PLS components.The quadratic regression can be viewed as a natural extension of linear PLS regression to quadratic PLSaccording to the H-principle of mathematical modelling.The numerical implementation is treated indetail.It is shown that this approach can be used for models with large numbers of variables.Somemodelling strategies are discussed depending on the purpose of the modelling.Applications of thisapproach are treated.     

SHORT COMMUNICATION PLS FITS CLOSER THAN PCR

PLS1 regression is generally viewed as lying in between PCR and OLS regression.Proof is given thatthe coefficient of determination,R~2,for a PLS multivariate calibration model is at least as high as thatfor a PCR model with the same number of components.It appears that PLS can be linked to acorrelation-weighted polynomial regression of a constant response on the eigenvalues of the covariancematrix of the predictor variables.     

THE PERFORMANCE OF LEAST SQUARES AND ROBUST REGRESSION IN THE CALIBRATION OF ANALYTICAL METHODS UNDER NON-NORMAL NOISE DISTRIBUTIONS

By means of Monte Carlo simulations a comparison has been made between ordinary least squaresregression and robust regression. The robust regression procedure is based on the Huber estimate and iscomputed by means of the iteratively reweighted least squares algorithm. The performance of bothprocedures has been evaluated for estimation of the parameters of a calibration function and fordetermination of the concentration of unknown samples. The influence of the distributionalcharacteristics skewness and kurtosis has been studied, and the number of measurements used forconstructing the calibration curve has also been taken into account, Under certain conditions robustregression offers an advantage over least squares regression.     

A MULTIBLOCK PARTIAL LEAST SQUARES ALGORITHM FOR INVESTIGATING COMPLEX CHEMICAL SYSTEMS

The details of a general multiblock partial least squares(PLS)algorithm based on one originallypresented by Wold et al.have been developed and are completely presented.The algorithm can handlemost types of relationships between the blocks and constitutes a significant advancement in the modelingof complex chemical systems.The algorithm has been programmed in FORTRAN and has been testedon two simulated multiblock problems,a three-block and a five-block problem.The algorithm combinesthe score vectors for all blocks predicting a particular block into a new block.This new block is used topredict the predicted block in a manner analogous to the two-block PLS.In a similar manner if one blockpredicts more than one other block,the score vectors of all predicted blocks are combined to form a newblock,which is then predicted by the predictor block as in the two-block PLS.Blocks that both predictand are predicted are treated in such a way that both of these roles can be taken into account whencalculating interblock relationships.The results of numerical simulations indicate that the computerprogram is operating properly and that the multiblock PLS produces meaningful and consistent results.     

基于偏最小二乘法的玉米FPAR高光谱反演模型研究

以ASD FR便携式光谱仪与LI-191SA光量子仪对吉林中西部的玉米田进行多次观测,采集到123组有效数据,基于偏最小二乘法(PLS)对玉米FPAR进行高光谱反演。对可见光与近红外光谱(400～1 500nm)进行分析并建立反演模型,对FPAR预测效果进行验证,验证模型的R2为0.785,RMSE为0.117;同时进行了玉米FPAR与光谱反射率、反射率一阶导数之间的关系分析及植被指数与玉米FPAR之间的回归分析。研究结果表明,PLS方法建立的模型可有效地从玉米高光谱反射率数据反演出FPAR含量,反演结果精度较植被指数高。     

基于NDVI与偏最小二乘回归的荒漠化地区植被覆盖度高光谱遥感估测

以星载高光谱影像Hyperion为数据源,系统比较了NDVI与偏最小二乘回归（PLS）估测荒漠化地区植被覆盖度的能力,模型的建立（n=46）与独立检验所用样本（n=10）均为地面实测数据。研究结果表明,基于星载高光谱数据的NDVI与PLS模型可以有效地估测荒漠化地区植被覆盖度。相比于宽波段NDVI（RMSEP=10.5618）及基于803.3/671.02 nm计算的标准高光谱NDVI（RMSEP=8.3863）,选择特定高光谱波段（823.65/701.55 nm）构建的NDVI预测植被覆盖度的误差明显较低（RMSEP=6.5189）。基于高光谱所有波段原始反射率、一阶导数及包络线去除光谱的PLS回归模型表现,要明显优于仅利用两个波段信息的NDVI,其中基于原始反射率的PLS回归模型表现最佳,RMSEP为4.4998,约为因变量平均值的23%。     

PREDICTIVE ABILITY OF REGRESSION MODELS.PART Ⅰ:STANDARD DEVIATION OF PREDICTION ERRORS (SDEP)

The standard deviation of prediction errors(SDEP)is used to evaluate and compare the predictive abilityof some regression models,namely MLR,ACE and linear and non-linear PLS,the last being the bestone.The parameter is determined by a cross-validation approach as an average of several runs obtainedon forming groups in a random way.The variation in SDEP with the number of latent variables in PLSis also discussed.     

A TEST OF SIGNIFICANCE FOR PARTIAL LEAST SQUARES REGRESSION

Partial least squares (PLS) regression is a commonly used statistical technique for performingmultivariate calibration, especially in situations where there are more variables than samples. Choosingthe number of factors to include in a model is a decision that all users of PLS must make, but iscomplicated by the large number of empirical tests available. In most instances predictive ability is themost desired property of a PLS model and so interest has centred on making this choice based on aninternal validation process. A popular approach is the calculation of a cross-validated r~2 to gauge howmuch variance in the dependent variable can be explained from leave-one-out predictions. Using MonteCarlo simulations for different sizes of data set, the influence of chance effects on the cross-validationprocess is investigated. The results are presented as tables of critical values which are compared againstthe values of cross-validated r~2 obtained from the user's own data set. This gives a formal test forpredictive ability of a PLS model with a given number of dimensions.     

RECENT DEVELOPMENTS IN MULTIVARIATE CALIBRATION

With the goal of understanding global chemical processes,environmental chemists have some of the mostcomplex sample analysis problems.Multivariate calibration is a tool that can be applied successfully inmany situations where traditional univariate analyses cannot.The purpose of this paper is to reviewmultivariate calibration,with an emphasis being placed on the developments in recent years.The inverseand classical models are discussed briefly,with the main emphasis on the biased calibration methods.Principal component regression(PCR)and partial least squares(PLS)are discussed,along with methodsfor quantitative and qualitative validation of the calibration models.Non-linear PCR,non-linear PLSand locally weighted regression are presented as calibration methods for non-linear data.Finally,calibration techniques using a matrix of data per sample(second-order calibration)are discussed briefly.     

Diatom-based water chemistry reconstructions from northern Sweden: a comparison of reconstruction techniques

The diatom composition in surface sediments from 119 northern Swedish lakes was studied to examine the relationship with lake-water pH, alkalinity, and colour. Diatom-based predictive models, using weighted-averaging (WA) regression and calibration, partial least squares (PLS) regression and calibration, and weighted-averaging partial least squares (WA-PLS) regression and calibration, were developed for inferences of water chemistry conditions. The non-linear response between the diatom assemblages and pH and alkalinity was best modelled by weighted-averaging methods. The lowest prediction error for pH was obtained using weighted averaging, with or without tolerance downweighting. For alkalinity there was still some information in the residual structure after extracting the first weighted-averaging component, which resulted in a slight improvement of predictions when using a two component WA-PLS model. The best colour predictions were obtained using a two component PLS model. Principal component analysis (PCA) of the prediction errors, with some characteristics of the training set included as passive variables, was performed to compare the results for the different alkalinity predictive models. The results indicate that calibration techniques utilizing more than one component (PLS and WA-PLS) can improve the predictions for lakes with diatom taxa that have broad tolerances. Furthermore, we show that WA-PLS performs best compared with the other techniques for those lakes that have a high relative abundance of the most dominant taxa and a corresponding low sample heterogeneity.     

A partial least-squares regression approach to land use studies in the Suzhou-Wuxi-Changzhou region

In several LUCC studies, statistical methods are being used to analyze land use data. A problem using conventional statistical methods in land use analysis is that these methods assume the data to be statistically independent. But in fact, they have the tendency to be dependent, a phenomenon known as multicollinearity, especially in the cases of few observations. In this paper, a Partial Least-Squares (PLS) regression approach is developed to study relationships between land use and its influencing factors through a case study of the Suzhou-Wuxi-Changzhou region in China. Multicollinearity exists in the dataset and the number of variables is high compared to the number of observations. Four PLS factors are selected through a preliminary analysis. The correlation analyses between land use and in-fluencing factors demonstrate the land use character of rural industrialization and urbaniza-tion in the Suzhou-Wuxi-Changzhou region, meanwhile illustrate that the first PLS factor has enough ability to best describe land use patterns quantitatively, and most of the statistical relations derived from it accord with the fact. By the decreasing capacity of the PLS factors, the reliability of model outcome decreases correspondingly.     

DEPENDENCE BETWEEN FUSION TEMPERATURES AND CHEMICAL COMPONENTS OF A CERTAIN TYPE OF COAL USING CLASSICAL,NON-PARAMETRIC AND BOOTSTRAP TECHNIQUES

A statistical study of the dependence between various critical fusion temperatures of a certain kind ofcoal and its chemical components is carried out.As well as using classical dependence techniques(multiple,stepwise and PLS regression,principal components,canonical correlation,etc.)together withthe corresponding inference on the parameters of interest,non-parametric regression and bootstrapinference are also performed.     

基于栅格面积成分数据的土地利用格局解释模型稳健估计

针对最小二乘估计不能应用于栅格尺度以面积成分表征的土地利用格局驱动机理分析的难题,本文提出了一种利用偏最小二乘回归法稳健估计该类土地利用格局解释模型的方法。利用该方法能在解释变量间存在多重共线性的情况下,获得基于栅格面积成分数据的土地利用格局解释模型的稳健估计。本文推导了应用偏最小二乘回归分析的数据处理和建模估计过程,并运用该方法开展了针对黄淮海地区耕地、建设用地分布格局及其驱动因子的建模分析,得到了拟合优度高的估计结果。研究表明,偏最小二乘回归分析方法在开展栅格尺度以面积成分表征的土地利用格局驱动机理分析时具备高效与稳健的特征,适宜在类似研究中推广应用。     

偏最小二乘法在苏锡常地区土地利用研究中的应用

In several LUCC studies, statistical methods are being used to analyze land use data. A problem using conventional statistical methods in land use analysis is that these methods assume the data to be statistically independent. But in fact, they have the tendency to be dependent, a phenomenon known as multicollinearity, especially in the cases of few observations. In this paper, a Partial Least-Squares （PLS） regression approach is developed to study relationships between land use and its influencing factors through a case study of the Suzhou-Wuxi-Changzhou region in China. Multicollinearity exists in the dataset and the number of variables is high compared to the number of observations. Four PLS factors are selected through a preliminary analysis. The correlation analyses between land use and influencing factors demonstrate the land use character of rural industrialization and urbanization in the Suzhou-Wuxi-Changzhou region, meanwhile illustrate that the first PLS factor has enough ability to best describe land use patterns quantitatively, and most of the statistical relations derived from it accord with the fact. By the decreasing capacity of the PLS factors, the reliability of model outcome decreases correspondingly.     

PREDICTIVE ABILITY OF REGRESSION MODELS.PART Ⅱ:SELECTION OF THE BEST PREDICTIVE PLS MODEL

A procedure called GOLPE is suggested in order to detect those variables which increase the predictivityof PLS models.The procedure is based on evaluating the predictive power of a number of PLS modelsbuilt by different combinations of variables selected according to a factorial design strategy.Examplesare given of the efficiency of this variable selection procedure,which shows how these predictive PLSmodels are better than those obtained by all variables and better than the corresponding ordinaryregression models.     

基于高光谱的民勤土壤盐分定量分析

土壤盐渍化是重要的生态环境问题,严重影响着干旱、半干旱区的农牧业及经济发展。高光谱遥感技术能够提供地物的连续光谱信息,易于分析细微差别,在定量研究土壤盐分含量方面具有较大优势。民勤县位于甘肃省石羊河流域下游,水力资源匮乏,盐渍化问题十分严峻。本研究基于实验室光谱数据,通过建立模型定量分析土壤盐分含量。首先对原始数据进行连续统去除（cn）预处理,然后分别建立了土壤盐分含量的高光谱指数模型（NDSI）、偏最小二乘回归模型（PLS）、间隔偏最小二乘法模型（iPLS）和反向间隔偏最小二乘法模型（BiPLS）,考察各种模型对土壤盐分的预测能力。对比分析发现,使用全部波段信息建模的PLS模型优于仅使用两个波段信息的NDSI模型,而iPLS和BiPLS模型通过选择特征波段进行建模,结果均好于全谱PLS模型。其中,BiPLS模型波段选择的能力优于iPLS模型,得出的模型结果最好,预测相对偏差RPD达到2.02,决定系数R²和模拟值与预测值线性回归的斜率分别为0.76和0.92,模型可以近似地预测土壤盐分含量。结果说明特征波段选择方法能够从大量数据中提取有效信息,简化模型,并获取比NDSI模型和全谱PLS模型更优的预测结果。这些研究对于使用高光谱数据定量分析土壤盐渍化有一定的意义。     

Local least absolute deviation estimation of spatially varying coefficient models: robust geographically weighted regression approaches

The geographically weighted regression (GWR) has been widely applied to many practical fields for exploring spatial non-stationarity of a regression relationship. However, this method is inherently not robust to outliers due to the least squares criterion in the process of estimation. Outliers commonly exist in data sets and may lead to a distorted estimate of the underlying regression relationship. Using the least absolute deviation criterion, we propose two robust scenarios of the GWR approaches to handle outliers. One is based on the basic GWR and the other is based on the local linear GWR (LGWR). The proposed methods can automatically reduce the impact of outliers on the estimates of the regression coefficients and can be easily implemented with modern computer software for dealing with the linear programming problems. We then conduct simulations to assess the performance of the proposed methods and the results demonstrate that the methods are quite robust to outliers and can retrieve the underlying coefficient surfaces satisfactorily even though the data are seriously contaminated or contain severe outliers.     

基于模式优选的21世纪中国气候变化情景集合预估

未来气候变化情景预估是制定气候变化应对和适应策略的科学基础。本文利用参与耦合模式比较计划第五阶段（CMIP5）的30个气候模式的模拟数据,通过评估各模式对历史气候变化的模拟能力,筛选出模拟区域气候变化的最优模式组合,进而建立偏最小二乘回归（PLS）集合预估模型,据此利用最优模式模拟结果预估区域温度和降水变化情景。通过与历史数据的对比,研究发现本文基于最优模式建立的PLS集合预估模型不仅优于传统的多模式集合平均,而且也优于利用全部模式建立的PLS集合预估模型,体现了模式优选过程的重要性。本文基于优选模式的PLS集合预估模型预估结果表明：① 21世纪各区域温度将持续上升,且冬半年升温速率总体大于夏半年,北方地区升温速率总体高于南方地区;RCP 4.5排放情景下温度上升先快后慢,转折点出现在21世纪中期,RCP 8.5排放情景下,呈持续增加趋势,至21世纪末的升温幅度约为RCP 4.5情景的2倍。② 21世纪各区降水变化均呈显著增加趋势,并表现出高排放情景大于低排放情景,少雨区大于多雨区的特征,但是降水增加过程伴有明显的年代际波动。对比发现,传统的等权重集合平均全部模式（EMC）方法预估的中国夏季变暖速率高于冬季,且降水基本呈线性增加,有悖于全球变暖的基本特征及中国降水具有鲜明的年代际变化特征的基本认识。因而,本文预估的温度和降水变化特征均更符合中国气候变化的基本规律。     

PLS RESPONSE SURFACE OPTIMIZATION:THE CARSO PROCEDURE

The paper presents a procedure to obtain response surfaces with non-designed data.The method is basedon PLS modelling of the expanded X-matrix followed by transformation of the PLS loadings intopolynomial coefficients and detection of the co-ordinates of the best response within the experimentaldomain.The results are presented both graphically and numerically.The procedure is validated on anoptimization study of the yield of an organic reaction.     

A chironomid‐based model for inferring late‐summer hypolimnetic oxygen in southeastern Ontario lakes

Hypolimnetic oxygen depletion has been accelerated in many lakes due to cultural eutrophication. However, the extent and magnitude of environmental change is difficult to ascertain due to the lack of historical records. Larval Chironomidae (Diptera) are useful proxy indicators of oxygen, as they show a wide range of tolerances to oxygen conditions and their chitinous head capsules preserve well in lake sediments. Using paleolimnological techniques, chironomid assemblages from the surface sediments of 42 southeastern Ontario lakes were related to environmental conditions. Hypolimnetic oxygen conditions, measured as the average endofsummer hypolimnetic dissolved oxygen (AvgDO_(Summ)), explained the most variation in the chironomid assemblages, whereas dissolved inorganic carbon, the Anoxic Factor, max. depth and total phosphorus concentrations were also correlated with assemblage composition. Based on the relative abundances of 45 chironomid taxa, a robust, partial least squares (PLS) regression transfer function for AvgDO_(Summ) was constructed (r² = 0.74, r² _(jack) = 0.58, n = 40). This new transfer function should allow paleolimnologists to directly track past trends in hypolimnetic oxygen levels.