On the relationship between the Method of Least Squares and Gram-Schmidt orthogonalization

The method of Least Squares is due to Carl Friedrich Gauss. The Gram-Schmidt orthogonalization method is of much younger date. A method for solving Least Squares Problems is developed which automatically results in the appearance of the Gram-Schmidt orthogonalizers. Given these orthogonalizers an induction-proof is available for solving Least Squares Problems.

The Effect of Climate Variability on Main Components of Cow Milk in Iran

The main purpose of this study is to assess the relationship between six bioclimatic indices for cattle (temperature humidity (THI), environmental stress (ESI), equivalent temperature (ESI), heat load (HLI), modified heat load (HLInew) and respiratory rate predictor(RRP)) and fundamental milk components (fat, protein, and milk yield) considering uncertainty. The climate parameters used to calculate the climate indices were taken from the NASA-Modern Era Retrospective-Analysis for Research and Applications (NASA-MERRA) reanalysis from 2002 to 2010. Cow milk data were considered for the same period from April to September when cows use natural pasture, with possibility for cows to choose to stay in the barn or to graze on the pasture in the pasturing system. The study is based on a linear regression analysis using correlations as a summarizing diagnostic. Bootstrapping is used to represent uncertainty estimation through resampling in the confidence intervals. To find the relationships between climate indices (THI, ETI, HLI, HLInew, ESI and RRP) and main components of cow milk (fat, protein and yield), multiple liner regression is applied. The least absolute shrinkage selection operator (LASSO) and the Akaike information criterion (AIC) techniques are applied to select the best model for milk predictands with the smallest number of climate predictors. Cross validation is used to avoid over-fitting. Based on results of investigation the effect of heat stress indices on milk compounds separately, we suggest the use of ESI and RRP in the summer and ESI in the spring. THI and HLInew are suggested for fat content and HLInew also is suggested for protein content in the spring season. The best linear models are found in spring between milk yield as predictands and THI, ESI,HLI, ETI and RRP as predictors with p-value < 0.001 and R2 0.50, 0.49. In summer, milk yield with independent variables of THI, ETI and ESI show the highest relation (p-value < 0.001) with R2 (0.69). For fat and protein the results are only marginal. It is strongly suggested that new and significant indices are needed to control critical heat stress conditions that consider more predictors of the effect of climate variability on animal products, such as sunshine duration, quality of pasture, the number of days of stress (NDS), the color of skin with attention to large black spots, and categorical predictors such as breed, welfare facility, and management system. This methodology is suggested for studies investigating the impacts of climate variability/change on food quality/security, animal science and agriculture using short term data considering uncertainty or data collection is expensive, difficult, or data with gaps.

Applying Least Absolute Shrinkage Selection Operator and Akaike Information Criterion Analysis to Find the Best Multiple Linear Regression Models between Climate Indices and Components of Cow’s Milk

This study focuses on multiple linear regression models relating six climate indices (temperature humidity THI, environmental stress ESI, equivalent temperature index ETI, heat load HLI, modified HLI (HLI new), and respiratory rate predictor RRP) with three main components of cow’s milk (yield, fat, and protein) for cows in Iran. The least absolute shrinkage selection operator (LASSO) and the Akaike information criterion (AIC) techniques are applied to select the best model for milk predictands with the smallest number of climate predictors. Uncertainty estimation is employed by applying bootstrapping through resampling. Cross validation is used to avoid over-fitting. Climatic parameters are calculated from the NASA-MERRA global atmospheric reanalysis. Milk data for the months from April to September, 2002 to 2010 are used. The best linear regression models are found in spring between milk yield as the predictand and THI, ESI, ETI, HLI, and RRP as predictors with p-value < 0.001 and R2 (0.50, 0.49) respectively. In summer, milk yield with independent variables of THI, ETI, and ESI show the highest relation (p-value < 0.001) with R2 (0.69). For fat and protein the results are only marginal. This method is suggested for the impact studies of climate variability/change on agriculture and food science fields when short-time series or data with large uncertainty are available.