Abstract
In analytical chemistry, multivariate calibration is applied when substituting a time-consuming reference measurement (based on e.g. chromatography) with a high-throughput measurement (based on e.g. vibrational spectroscopy). An average error term, of the response variable, is often used to evaluate the performance of a calibration model. However, indirect relationships, between the response and explanatory variables, may be used for calibration. In such cases, model validity cannot necessarily be determined solely by the average error term. One should also consider the use of the models, as well as the validity of the indirect relationships in future samples. If the analyte of interest is partly quantified from signals of interfering compounds, then these interfering compounds will play a hidden role in the calibration. This hidden role may affect future use of the calibration model as strong covariance relationships between analyte estimates and interfering compounds may be imposed. Hence, such model cannot detect changes in the relationship between the analyte and interfering compounds. The problem is called the cage of covariance. This paper discusses the concept cage of covariance and possible consequences of applying models exposed to this issue.
Highlights
The desire to rapidly extract large amounts of sample information is natural
In a classical least squares perspective, spectroscopic measurements, Xðn  mÞ, of multi-component samples are viewed as the outer product of the component concentration profiles, Cðn  qÞ, the pure component signals at unitary concentration, Sðm  qÞ, and an error term, Eðn  mÞ, where n is the number of samples, m is the number of measured variables and q is the number of chemical components
This study shows the importance of considering the rank of the subspace of explanatory variables used for prediction, the covariance structures of response variables as well as their estimates when regressing multiple response variables (e.g. fatty acid (FA)) onto the same explanatory variables
Summary
The desire to rapidly extract large amounts of sample information is natural. this desire has recently led to misuse of vibrational spectroscopy returning misleading results. Examples from food research are e.g. prediction of fatty acid (FA) composition in bovine milk [1] or determination of biochemical quality parameters in fermented cocoa [2] In both studies [1,2], more than 30 response variables (reference variables) were estimated from the explanatory variables (spectroscopic measurements). In a classical least squares perspective, spectroscopic measurements, Xðn  mÞ, of multi-component samples are viewed as the outer product of the component concentration profiles, Cðn  qÞ, the pure component signals at unitary concentration, Sðm  qÞ, and an error term, Eðn  mÞ, where n is the number of samples, m is the number of measured variables and q is the number of chemical components Independent information of the q components is not directly available from X if rðXÞ < q
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