Ordinary Multiple Linear Regression Research Articles

Quantifying the Relationship between Adaptive Traits and Agro-climatic Conditions

Background & Aim: Durum wheat is an economically important and regularly eaten food for billions of people in the world. In the International Center for Agriculture Research in the Dry Areas (ICARDA), genbanks are using Focused Identification of the Germplasm Strategy (FIGS) to find out and quantify relationships between agro-climatic conditions and the presence of specific traits. Hence, the study is aimed to investigate the predictive value of various types of long-term agro-climatic variables on the future values of different traits. Method: Ordinary multiple linear regression with stepwise variable selection method on the complete data set, and multiple linear regression models with predictors selected by penalized methods with mean square error cross-validation as a model selection criterion, are used to analyze 238 durum wheat landraces. Each of the models are fitted on Days to Heading and Days to Maturity response variables with 57 predictor variables, independently. Ordinary least square and weighted least square estimation methods were used. Result: Findings implied that there is high multicollinearity among the predictor variables. It is found that there are some predictors which affect positively and some others affect negatively for both Days to Heading and Days to Maturity using both ordinary and shrinkage based models. It is revealed that the prediction from the lasso based model is not that much reasonable. Furthermore, for the Days to Heading showed that there seems better prediction as their predicted value increase continuously as a function of the actual values though there is considerable variability. Conclusion: In conclusion, inferences and predictions by the ordinary MLR models are not trusted due to the presence of multicollinearity, and violation of some model assumptions. However, predictions using the models with predictors selected by the shrinkage methods may be better as the effects of the variability on these methods are minimal. Moreover, the WLS methods might give more sensible predictions than the OLS estimation methods. Better predictions were found on the Days to Heading.

An improved quantum algorithm for ridge regression

Ridge regression (RR) is an important machine learning technique which introduces a regularization hyperparameter $\alpha$ to ordinary multiple linear regression for analyzing data suffering from multicollinearity. In this paper, we present a quantum algorithm for RR, where the technique of parallel Hamiltonian simulation to simulate a number of Hermitian matrices in parallel is proposed and used to develop a quantum version of $K$-fold cross-validation approach, which can efficiently estimate the predictive performance of RR. Our algorithm consists of two phases: (1) using quantum $K$-fold cross-validation to efficiently determine a good $\alpha$ with which RR can achieve good predictive performance, and then (2) generating a quantum state encoding the optimal fitting parameters of RR with such $\alpha$, which can be further utilized to predict new data. Since indefinite dense Hamiltonian simulation has been adopted as a key subroutine, our algorithm can efficiently handle non-sparse data matrices. It is shown that our algorithm can achieve exponential speedup over the classical counterpart for (low-rank) data matrices with low condition numbers. But when the condition numbers of data matrices is large to be amenable to full or approximately full ranks of data matrices, only polynomial speedup can be achieved.

A Study on Repeat Sales House Price Index Based on Penalized Quantile Regression

House price index (HPI) based on real transaction prices is commonly estimated by the repeat sales model which utilizes sales pairs traded over twice during the period of interest. Since the conventional repeat sales model is based on the ordinary multiple linear regression, it is affected considerably by outliers. In addition, it produces quite unstable HPI in case of insufficient sales pairs. To tackle these problems, a quantile regression based repeat sales model with an appropriate penalty function is suggested. The induced HPI from the proposed model shows much stability compared to the conventional methods. Moreover, the HPIs derived from the penalized quantile regressions for various quantile parameters provide us with valuable information which cannot be obtained from the existing repeat sales models based on ordinary multiple linear regression. As a result, the proposed method for constructing HPIs can be considered as an alternative to the conventional repeat sales price indices in that it provides very stable and informative HPIs.

Caries experience in 7-, 12-, and 15-year-old schoolchildren in the canton of Basel-Landschaft, Switzerland, from 1992 to 2011.

To investigate the changes in caries experience and prevalence among schoolchildren of the canton of Basel-Landschaft, Switzerland, over the course of 20 years. A random sample of either schoolchildren aged 7, 12, and 15 years (in 1992) or aged 12 and 15 years (in 1997) or their respective school classes (2001, 2006 and 2011) was selected so that approximately 10% of schoolchildren could be examined. The children's dmft and DMFT scores were determined according to the WHO methodology and analyzed using cluster-adjusted ordinary multiple linear regression modeling. For all age groups, the respective dmft/DMFT values decreased steadily from 1992 to 2006 but increased again in 2011. However, the observed differences were not statistically significant over the examination years from 2001 to 2011. Schoolchildren with a migrant background had approximately two- to threefold higher dmft/DMFT values. In the years from 1992 to 2001, a steady decline in caries was observed in all age groups of schoolchildren examined in the canton of Basel-Landschaft. However, in the subsequent 10 years, this decline has leveled off. The mean dmft/DMFT values are comparable to those in other parts of Switzerland. Migrants are a caries-risk group; the mean dmft/DMFT values were higher in schoolchildren with a migrant background than in the comparable Swiss children.

Food Safety Practices among Norwegian Consumers

An informed consumer can compensate for several potential food safety violations or contaminations that may occur earlier in the food production chain. However, a consumer can also destroy the work of others in the chain by poor food handling practices, e.g., by storing chilled ready-to-eat foods at abusive temperatures. To target risk-reducing strategies, consumer groups with high-risk behavior should be identified. The aim of this study was to identify demographic characteristics associated with high-risk food handling practices among Norwegian consumers. More than 2,000 randomly selected Norwegian consumers were surveyed, and the results were analyzed with a risk-based grading system, awarding demerit points for self-reported food safety violations. The violations were categorized into groups, and an ordinary multiple linear regression analysis was run on the summarized demerit score for each group and for the entire survey group as a whole. Young and elderly men were identified as the least informed consumer groups with the most unsafe practices regarding food safety. Single persons reported poorer practices than those in a relationship. People with higher education reported poorer practices than those with lower or no education, and those living in the capital of Norway (Oslo) reported following more unsafe food practices than people living elsewhere in Norway. Men reported poorer food safety practices than women in all categories with two exceptions: parboiling raw vegetables before consumption and knowledge of refrigerator temperature. These findings suggest that risk-reducing measures should target men, and a strategy is needed to change their behavior and attitudes.

Modified Ridge Regression Estimators

Ridge regression is a variant of ordinary multiple linear regression whose goal is to circumvent the problem of predictors collinearity. It gives up the Ordinary Least Squares (OLS) estimator as a method for estimating the parameters of the multiple linear regression model . Different methods of specifying the ridge parameter k were proposed and evaluated in terms of Mean Square Error (MSE) by simulation techniques. Comparison is made with other ridge-type estimators evaluated elsewhere. The new estimators of the ridge parameters are shown to have very good MSE properties compared with the other estimators of the ridge parameter and the OLS estimator. Based on our results from the simulation study, we may recommend the new ridge parameters to practitioners.

A Geographically Weighted Regression Analysis of Douglas-Fir Site Index in North Central Idaho

Abstract An analysis of Douglas-fir (Pseudotsuga menziesii [Mirb.] Franco var. glauca) site index (SI) was performed in north central Idaho using two forms of linear regression: standard multiple linear regression (MLR) and geographically weighted regression (GWR). The hypothesis was that the GWR model would provide better estimates of SI using edaphic, topographic, and climatic predictor variables than ordinary MLR. Elevation, volcanic ash depth, slope, and aspect were significantly correlated with Douglas-fir SI (R A 2 = 0.5). GWR accounted for an additional 29% of the variation in SI and reduced the error sum of squares by approximately 53%. The geographically weighted parameter estimates of the model intercept and elevation were nonstationary, which indicates that geographic location determines the influence of a variable on SI. Residual analysis from a MLR model showed large SI prediction error throughout the study area. This prediction error was significantly reduced using the GWR model. GWR shows promise for biological modeling.

A tutorial on multivariate calibration in atomic spectrometry techniques

Coupling multivariate regression methods to atomic spectrometry is an emerging field from which important advantages can be obtained. These include lower workloads, increased laboratory turnarounds, economy, higher efficiency in method development, and relatively simple ways to take account of complex interferences. In this paper four typical regression methods (ordinary multiple linear regression, principal components regression, partial least squares and artificial neural networks) are presented in a practice-oriented way. The main emphasis is placed on explaining their advantages, drawbacks, how to solve the latter and how atomic spectrometry can benefit from multivariate regression. Finally, a retrospective review considering the last sixteen years is made to present practical applications on: flame-, hydride generation-, electrothermal-atomic absorption spectrometry; inductively coupled plasma spectrometry and laser-induced breakdown spectrometry.

A review of statistical methods for the evaluation of aquatic habitat suitability for instream flow assessment

Habitat models serve three main purposes: First, to predict species occurrences on the basis of abiotic and biotic variables, second to improve the understanding of species-habitat relationships and third, to quantify habitat requirements. The use of statistical models to predict the likely occurrence or distribution of species based on relevant variables is becoming an increasingly important tool in conservation planning and wildlife management. This article aims to provide an overview of the current status of development and application of statistical methodologies for analysing the species-environment association, with a clear emphasis on aquatic habitat. It describes the main types of univariate and multivariate techniques available for analysis of species-environment association, and specifically focuses on the assessment of the strengths and weaknesses of the available statistical methods to estimate habitat suitability. A second objective of this article is to propose new approaches using existing statistical methods. A wide array of habitat statistical models has been developed to analyse habitat-species relationship. Generally, physical habitat is dependent on more than one variable (e.g. depth, velocity, substrate, cover) and several suitability indices must be combined to define a composite index. Multivariate approaches are more appropriate for the analysis of aquatic habitat as they inherently consider the interrelation and correlation structure of the environmental variables. Ordinary multiple linear regression and logistic regression are popular methods often used for modelling of species and their relationships with environment. Ridge regression and Principal component regression are particularly useful when the independent variables are highly correlated. More recent regression modelling paradigms like generalized linear models (GLMs) present advantages in dealing with non-normal environmental variables. Generalized additive models (GAMs) and artificial neural networks are better suited for analysis of non-linear relationships between species distribution and environmental variables. The fuzzy logic approach presents advantages in dealing with uncertainties that often exist in habitat modelling. Appropriate methods for analysis of multi-species data are also presented. Finally, the few existing comparative studies for predictive modelling are reviewed, and advantages and disadvantages of different methods are discussed. Copyright © 2006 John Wiley & Sons, Ltd.

Smoothness Priors and Nonlinear Regression

In applications, the linear multiple regression model is often modified to allow for nonlinearity in an independent variable. It is argued here that in practice it may often be desirable to specify a Bayesian prior that the unknown functional form is simple or uncomplicated rather than to parametize the nonlinearity. Discrete smoothness priors and continuous smoothness priors are defined and it is shown how posterior mean estimates can easily be derived using ordinary multiple linear regression modified with dummy variables and dummy observations. Relationships with spline and polynomial interpolation are pointed out. Illustrative examples of cost function estimation are provided.

Smoothness Priors and Nonlinear Regression

In applications, the linear multiple regression model is often modified to allow for nonlinearity in an independent variable. It is argued here that in practice it may often be desirable to specify a Bayesian prior that the unknown functional form is simple or uncomplicated rather than to parametize the nonlinearity. Discrete smoothness priors and continuous smoothness priors are defined and it is shown how posterior mean estimates can easily be derived using ordinary multiple linear regression modified with dummy variables and dummy observations. Relationships with spline and polynomial interpolation are pointed out. Illustrative examples of cost function estimation are provided.