Orthogonal and Linear Regressions and Pencils of Confocal Quadrics

Regression analysis is a statistical technique for investigating and modeling the relationship between variables. The purpose of this study was the trivial presentation of the equation for orthogonal regression (OR) and the comparison of classical linear regression (CLR) and OR techniques with respect to the sum of squared perpendicular distances. For that purpose, the analyses were shown by an example. It was found that the sum of squared perpendicular distances of OR is smaller. Thus, it was seen that OR line has appeared to present a much better fit for the data than CLR line. Depending on those results, the OR is thought to be a regression technique to obtain more accurate results than CLR at simple linear regression studies.

This paper discusses robust estimation for structural errors-in-variables (EV) linear regression models. Such models have important applications in many areas. Under certain assumptions, including normality, the maximum likelihood estimates for the EV model are provided by orthogonal regression (OR) which minimizes the orthogonal distance from the regression line to the data points instead of the vertical distance used in ordinary regression. OR is very sensitive to contamination and thus efficient robust procedures are needed. This paper examines the theoretical properties of bounded influence estimators for univariate Gaussian EV models using a generalized $M$-estimate approach. The results include Fisher consistency, most $B$-robust estimators and the OR version of Hampel's optimality problem.

In this work the performances of several field calibration methods for low-cost sensors, including linear/multi linear regression and supervised learning techniques, are compared. A cluster of either metal oxide or electrochemical sensors for nitrogen monoxide and carbon monoxide together with miniaturized infra-red carbon dioxide sensors was operated. Calibration was carried out during the two first weeks of evaluation against reference measurements. The accuracy of each regression method was evaluated on a five months field experiment at a semi-rural site using different indicators and techniques: orthogonal regression, target diagram, measurement uncertainty and drifts over time of sensor predictions. In addition to the analyses for ozone and nitrogen oxide already published in Part A [1], this work assessed if carbon monoxide sensors can reach the Data Quality Objective (DQOs) of 25% of uncertainty set in the European Air Quality Directive for indicative methods. As for ozone and nitrogen oxide, it was found for NO, CO and CO2 that the best agreement between sensors and reference measurements was observed for supervised learning techniques compared to linear and multilinear regression.

International sensitivity index calibrations based on the W.H.O. recommended method depend on orthogonal regression analysis. As this is not readily available in statistical packages, comparison has been made with simple linear regression analysis in a study of coagulometer effects on the International Normalized Ratio (INR) at 155 European centres. Sets of seven lyophilized normal and 20 lyophilized artificially depleted abnormal plasmas were provided with five coumarin test plasmas and two European Concerted Action on Anticoagulation reference thromboplastins (low International Sensitivity Index (ISI) human and high ISI rabbit). Local ISI based on the artificially depleted lyophilized plasmas using conventional orthogonal regression gave good correction for local coagulometer effects on the human reagent and minimal correction with the rabbit reagent INR. Results were considerably worse after attempts at correction using calibration based on linear regression analysis with both reagents. The results indicate that calibration of coagulometer prothrombin time systems using simple linear regression is not appropriate.

The reference method for $${\hbox {PM}}_{{10}}$$ and $${\hbox {PM}}_{2.5}$$ measurement in Europe is active sampling with subsequent gravimetric analysis. Also tapered element oscillating microbalance (TEOM) monitors are widely used across Europe for the assessment of particulate matter (PM), even though they are known to be biased due to the loss of semi-volatile materials attributed to sample heating. In our work we correct TEOM $${\hbox {PM}}_{{10}}$$ measurements at eight different monitoring sites located in different surroundings and different climates across Slovenia to replicate reference measurements. We use simultaneous reference and TEOM measurements from a period of seven years (2011–2017) and assess the advantage of custom defined correction factors (with linear and orthogonal regression) compared to default correction factors. We further try to improve the corrections by adding meteorological parameters as inputs and training a linear model (lasso) and nonlinear model (random forest). Random forest and lasso models also enable us to evaluate the impact of different meteorological parameters or chemical compounds on TEOM measurements. Our results show that TEOM measurements can be efficiently corrected using correction factors defined with two linear regressions (for summer and winter) at most locations across continental Slovenia. Koper, which lies at the sea side, was the most problematic measurement site. There the measurements were the most affected by the meteorological situation, and they could not be successfully corrected. When examining the effect of the chemical composition at two different locations, levels of $${\hbox {NH}}_4^+$$ , $${\hbox {NO}}_3^{-}$$ , and organic carbon exhibited major impact on the discrepancies between reference and TEOM measurements.

A simplified method of International Normalized Ratio (INR) derivation using linear regression of certified INR plotted against local prothrombin time (PT) results has been compared with INR from conventional orthogonal regression. Linear regression assumes error only with the local PT results whereas orthogonal regression assumes error with both reference and local results. The reliability of local INR derivation using lyophilized plasmas has been assessed in a collaborative study. INR from conventional fresh plasma International Sensitivity Index (ISI) calibrations have been compared with INR from calibrations with two types of lyophilized plasma, artificially depleted and coumarin. Although calibration slopes differed with the two types of analysis and the different lyophilized plasmas, both gave reasonable approximations to fresh plasma ISI calibrations. With orthogonal regression the overall percentage INR deviation was 5.25% with the artificially depleted plasmas and 6.85% for the results with lyophilized coumarins. With the linear regression, deviation was 8.40% for the artificially depleted plasmas and 5.05% for coumarin-treated patients' lyophilised-plasma. The simpler regression method appears to be worthy of further study as the present report has demonstrated that if the calibrant plasmas are accurately certified with the thromboplastin International Reference Plasma (IRP) results approximate to the conventionally determined INR using the manual PT technique. Coagulometers require further assessment.

Five common linear regression methods were evaluated for their ability to determine the correct values of slope and intercept of a known function after random errors were added to x and y. The error variances were controlled to simulate research problems commonly studied by linear regression. The total error of each method was assessed by the absolute value of the bias in the estimate of slope. Whenever differences among methods were observed, the mean of the slope determined by two reciprocal techniques performed as well as or better than orthogonal regression, regression of y upon x, or x upon y. All the methods studied appeared to perform equally well when x and y errors were heteroscedastic or when the data set was small (n = 7). Regression of y upon x was equal or superior to other methods when n = 7 or n = 20 and y and x errors were homoscedastic. When the data set was large (n = 50) and the error in x greater than that in y, the standard method (regression of y upon x) was inferior to all other methods. It is suggested that linear regression by the traditional method of y upon x (a method present in many hand-held calculators) is appropriate in the majority of clinical situations, but when n is large and errors in x are much larger than those in y, orthogonal regression or the averaging method may be preferable.

Abstract. Global warming has enduring consequences in the ocean, leading to increased sea surface temperatures (SSTs) and subsequent environmental impacts, including coral bleaching and intensified tropical storms. It is imperative to monitor these trends to enable informed decision-making and adaptation. In this study, we comprehensively examine the methods for extracting long-term temperature trends, including STL, seasonal-trend decomposition procedure based on LOESS (locally estimated scatterplot smoothing), and the linear regression family, which comprises the ordinary least-squares regression (OLSR), orthogonal regression (OR), and geometric-mean regression (GMR). The applicability and limitations of these methods are assessed based on experimental and simulated data. STL may stand out as the most accurate method for extracting long-term trends. However, it is associated with notably sizable computational time. In contrast, linear regression methods are far more efficient. Among these methods, GMR is not suitable due to its inherent assumption of a random temporal component. OLSR and OR are preferable for general tasks but require correction to accurately account for seasonal signal-induced bias resulting from the phase–distance imbalance. We observe that this bias can be effectively addressed by trimming the SST data to ensure that the time series becomes an even function before applying linear regression, which is named “evenization”. We compare our methods with two commonly used methods in the climate community. Our proposed method is unbiased and better than the conventional SST anomaly method. While our method may have a larger degree of uncertainty than combined linear and sinusoidal fitting, this uncertainty remains within an acceptable range. Furthermore, linear and sinusoidal fitting can be unstable when applied to natural data containing significant noise.

Orthogonal regression is one of the standard linear regression methods to correct for the effects of measurement error in predictors. We argue that orthogonal regression is often misused in errors-in-variables linear regression because of a failure to account for equation errors. The typical result is to overcorrect for measurement error, that is, overestimate the slope, because equation error is ignored. The use of orthogonal regression must include a careful assessment of equation error, and not merely the usual (often informal) estimation of the ratio of measurement error variances. There are rarer instances, for example, an example from geology discussed here, where the use of orthogonal regression without proper attention to modeling may lead to either overcorrection or undercorrection, depending on the relative sizes of the variances involved. Thus our main point, which does not seem to be widely appreciated, is that orthogonal regression, just like any measurement error analysis, requires ...

Comparing the in vitro digestibility of leaves from tropical trees when using the rumen liquor from cattle, sheep or goats

A well-accepted tool for method validation is a method comparison study. Results are usually assessed on a scatter plot of which the fitting line is calculated by several approaches, for example, ordinary (vertical) linear regression (OLR), orthogonal regression (OR), Deming regression (DR), Passing-Bablok method (PBR) or standardized principal component regression (SPCR). DR was applied in its general form (gDR), requiring information of the imprecision of at least two different quantities and as simple DR (sDR) with imprecision information of only one quantity. The equation of the regression line calculated by these concepts varies depending on range of measurement, analytical variation and on imprecision ratio (s AY /s AX ). There is still a global debate about which statistical concept is the most adequate for validating purposes. Various paired random samples with a size of 100 were simulated in 5000 replicates and evaluated with different regression models. The behavior of the slope and intercept of the regression lines were compared under various conditions. Two extreme ranges of measurement and several variance ratios in the absence and presence of bias were studied. The results clearly demonstrated that DR is the only model which can be applied without any precautions under conditions which usually occur in method comparison studies, and therefore should be preferred in laboratory medicine. Other models require restrictions with regard to range of measurement and/or imprecision profile. Differences of the concentrations at different positions of the measurement interval calculated with regression coefficients of both DRs did not deviate more than the permissible bias. Therefore, the advantage of using gDR does not justify its greater disadvantages in comparison with sDR.

A well-known approach to linear least squares regression is that which involves minimizing the sum of squared orthogonal projections of data points onto the best fit line. This form of regression is known as orthogonal regression, and the linear model that it yields is known as the major axis. A similar method, reduced major axis regression, is predicated on minimizing the total sum of triangular areas formed between data points and the best fit line. Either of these methods is appropriately applied when both x and y are measured, a typical case in the natural sciences. In comparison to classical linear regression, equation derivation for the slope of the major axis and reduced major axis lines is a nontrivial process. For this reason, derivations are presented herein drawing from previous literature with as few steps as possible to enable an easily accessible understanding. Application to eruption data for Old Faithful geyser, Yellowstone National Park, Wyoming and Montana, USA enables a teaching opportunity for choice of model.

This study uses Las Vegas near-road measurements of carbon monoxide (CO) and nitrogen oxides (NOx) to test the consistency of onroad emission constraint methodologies. We derive commonly used CO to NOx ratios (ΔCO:ΔNOx) from cross-road gradients and from linear regression using ordinary least squares (OLS) regression and orthogonal regression. The CO to NOx ratios are used to infer NOx emission adjustments for a priori emissions estimates from EPA’s MOtor Vehicle Emissions Simulator (MOVES) model assuming unbiased CO. The assumption of unbiased CO emissions may not be appropriate in many circumstances but was implemented in this analysis to illustrate the range of NOx scaling factors that can be inferred based on choice of methods and monitor distance alone. For the nearest road estimates (25m), the cross-road gradient and ordinary least squares (OLS) agree with each other and are not statistically different from the MOVES-based emission estimate while ΔCO:ΔNOx from orthogonal regression is significantly higher than the emitted ratio from MOVES. Using further downwind measurements (i.e., 115m and 300m) increases OLS and orthogonal regression estimates of ΔCO:ΔNOx but not cross-road gradient ΔCO:ΔNOx. The inferred NOx emissions depend on the observation-based method, as well as the distance of the measurements from the roadway and can suggest either that MOVES NOx emissions are unbiased or that they should be adjusted downward by between 10% and 47%. The sensitivity of observation-based ΔCO:ΔNOx estimates to the selected monitor location and to the calculation method characterize the inherent uncertainty of these methods that cannot be derived from traditional standard-error based uncertainty metrics.

Orthogonal regression is one of the prominent approaches for linear regression used to adjust the estimate of predictor errors. It can be considered as a least square regression with orthogonal constraints. It can maintain more discriminative constraints in the projection subspace than the least square regression and can avoid trivial solutions. In contrast to basic linear regression, orthogonal regression involves a computation error in both the answer and the predictor. Only the response variable contains the estimated error in simple regression. Orthogonal regression has also been utilized as the variable error occurs. Based on the data properties, specific models of orthogonal regression can be selected depending on whether there are calculation errors and/or equation errors. This article presents a comprehensive review of various variants of orthogonal regressions. The comparisons are drawn among the various variants of orthogonal regressions by considering various characteristics. The use of orthogonal regressions in various domains is also studied. Finally, various future directions are also presented.

AIMS: Errors in reporting International Normalised Ratios (INR) may be corrected by assignment of a System International Sensitivity Index (System ISI). This 57 centre study tests the validity of several...