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chemometrics

The science of relating measurements made on a chemical system or process to the state of the system via application of mathematical or statistical methods.

1. Note 1:Data treated by chemometrics are often multivariate.
2. Note 2:Although in some cases the mathematical and statistical techniques used in chemometric applications might be the same as those used in theoretical chemistry, it is important to emphasize that chemometrics should not involve theoretical calculations, but should deal primarily with the extraction of useful chemical information from measured data.
3. Note 3:Chemometrics is widely applied outside chemistry, e.g. in biology, metabolomics, engineering as well as sub-disciplines such as forensics, cultural studies etc.

Reference: [10]

autocorrelation

Correlation of a variable with itself over a successive time or space intervals (lags).

1. Note 1:If the mean of autocorrelated data is estimated the standard deviation of the mean depends on sampling mode.

autoscaling

Variance scaling of mean-centered data.

1. Note:See mean centering

categorical data

Data, values of which are one of a fixed number of nominal categories.

1. Note 1:Data in a contingency table is categorical.

contingency table

cross tabulation

Type of table in a matrix format that displays the multivariate frequency distribution of variables.

1. Note:The entries in the cells of a contingency table can be frequency counts or relative frequencies. See: categorical data

data matrix

Measurement results on a system arranged in a m×n matrix, with m objects and n variables.

1. Note 1:By convention a matrix is arranged with m rows and n columns.
2. Note 2:Objects are also called ‘samples’, but confusion with physical ‘test samples’ in analytical chemistry should be avoided.
3. Note 3:Variables are also called features, or explanatory variables.
4. Note 4:For multi-way data the data is arranged in a hypercube of m×n₁×n₂×…, where n₁, n₂ … are the kinds of explanatory variable.

data pre-processing

Manipulation of raw data prior to a specified data analysis treatment.

1. Note 1:The term “pre-processing” is preferred to the term “pre-treatment” to reduce confusion with physical sample preparation or treatment prior to experimental analysis.
2. Note 2:Aside from the three main categories of data pre-processing methods (meancentering, scaling and transformation), data pre-processing can refer to any other procedures carried out on the raw data, including mass binning and peak selection. In the case of multivariate images, this can also include region-of-interest selection and image filtering or binning.
3. Note 3:All data pre-processing methods imply some assumptions about the nature of the variability in the data set. It is important that these assumptions are understood and appropriate for the data set involved.
4. Note 4:More than one data pre-processing method can be applied to the same data set. The order of data pre-processing is important and can affect assumptions made on the nature of variance in the data set.

Reference: [11]

dynamic time warping

Process of synchronizing a data matrix so that it represents the same time shifts.

1. Note 1:The method is used in chromatography to align peaks by their retention times.

evaluation data

validation data

deprecated test data

deprecated test set

deprecated prediction data

deprecated prediction set

Data used to validate a model.

1. Note 1:Evaluation data should be independent of the data used to calibrate or train a model. See cross validation, training data.
2. Note 2:‘Test data’ is also used for data from an unknown sample, and should not be used for ‘evaluation data’.

explanatory variable

Variable that influences the value of a response variable, and is used to build models of the response variable.

exploratory data analysis

initial data analysis

EDA

Summary of the main characteristics of data, often using graphical methods.

1. Note 1: Exploratory data analysis is recommended before deciding an approach to chemometric modelling.

Reference: [12]

mean centering

centering

Data pre-processing in which the mean value of a variable is subtracted from data across all objects.

1. Note 1:Mean centering emphasises the differences between samples rather than differences between the samples and the variable’s origin (zero).
2. Note 2:Mean centering is generally recommended for principal component analysis, partial least squares and discriminant analysis of data, where relative values across the samples are more important than their absolute deviation from zero. Mean centering is not compatible with non-negativity constraints in, for example, multivariate curve resolution.
3. Note 3:Mean centering is generally applied with other data pre-processing methods. See scaling.

Reference: [11]

multiplicative scatter correction

MSC

multiplicative signal correction

Data pre-processing in which a constant and a multiple of a reference data set is subtracted from data.

1. Note 1:MSC is typically used in near-infrared spectrometry to remove effects of non-homogeneous particle size [13].

multivariate data

Data having two or more variables per object.

1. Note 1:The measurements results are often of the same kind of quantity.
2. Example 1:Absorbances measured at wavelengths in the range nm atond nm.
3. Example 2:Mass fractions of 10 elements measured by ICP-MS.

multi-way data

N-way data

Multivariate data having two or more kinds of explanatory variable per object.

1. Note 1:For two groups of explanatory variables the data is termed ‘three-way’.
2. Note 2:Models that decompose multi-way data include PARAFAC, Tucker3 model.
3. Example 1:Fluorescence intensities with excitation wavelength and emission wavelength representing the two variable axes, and the objects making the third direction.

noise

Response that gives no information.

normalization (in data pre-processing)

row scaling

Scaling method in which the scaling matrix consists of a single value for each object.

1. Note 1:The scaling value could be the value of a reference variable, the sum of selected variables or the sum of all variables for the sample.
2. Note 2:‘Normalization’ has many meanings in statistics (see www.cronistalascolonias.com.ar(statistics)). To resolve any ambiguity the nature of the scaling constant should be explained.

Reference: variance scaling, autoscaling [11]

random sampling

Sampling in which the sample locations are selected randomly from the whole population.

1. Note 1:The population defines the kind of quantity of the location. For example, in a time series, the quantity giving the location is time, in QSAR the location is a point in design space.

raw data

primary data

Data not yet subjected to analysis.

1. Note 1:Raw data can be an indication obtained from a measuring instrument or measuring system (VIM )

sampling

Selection of a subset of individuals from within a population to estimate characteristics of the whole population.

sampling error

The difference between an estimate of a parameter obtained from a sample and the population value.

1. Note 1:Unless the population values have been measured, sampling error cannot be directly estimated.

sampling unit

A defined quantity of material having a boundary which may be physical or temporal.

1. Note 1:Examples of physical boundaries are capsules, containers, and bottles.
2. Note 2:A number of sampling units may be gathered together, for example in a package or box.

scaling

weighting

Element-wise division of a data matrix by a scaling matrix.

1. Note:See variance scaling, autoscaling, normalization

smoothing

Transformation using an approximating function to capture important patterns in data, while removing noise or other fine-scale structures.

1. Note 1:Examples of smoothing functions are moving average, and Savitzky-Golay.

systematic sampling

Sampling in which individual samples are taken at equal intervals in location.

1. Note 1:In a time series, the quantity giving the location is time.
2. Note 2:The starting point may be assigned randomly within the first stratum.

take-it-or-leave-it data

TILI

Happenstance data that must be processed, or not processed, as is.

Reference: [14]

training data

training set

Data used for creating a model in supervised classification.

1. Note:See evaluation data.

transformation

Application of a deterministic mathematical function to each point in a set of data.

1. Note 1:Mathematically each data point z_i is replaced with the transformed value y_i=f(z_i), where f(.) is a mathematical function.
2. Note 2:Transforms may be applied so that the data appear to more closely meet the assumptions of a statistical inference procedure that is to be applied, or to improve the interpretability or appearance of graphs.
3. Note 3:Smoothing is an example of a transformation.
4. Example:f (x)=log(x).

variance scaling

Scaling in which the scaling matrix is the standard deviation of each variable across the objects.

1. Note 1:A variable occupies a column of the data matrix
2. Note 2:Variance scaling equalizes the importance of each variable in multivariate data.
3. Note 3:When used with mean centering variance scaling is known as autoscaling.

Reference: [11]

alias structure

List of combinations of effects that are aliased (confounded).

1. Note:See aliased effects

aliased effects

confounded effects

In a fractional-factorial design, effects for which the information obtained are identical.

1. Note 1:In a two-level design, the product of the coded levels for the aliased effects are equal.
2. Example 1:If there are four factors in a design: A, B, C, D then the main effect of A can be aliased with the three way effect B×C×D. So for the run that has B=–1, C=–1, D=+1, then A must=+1.

coded experimental design

coded design

Matrix of runs by factor levels in which each level is denoted by a code that represents the relative magnitude of the level.

1. Example 1:A two-level design is coded –1 and +1, a three level design is coded –1, 0 and +1.
2. Example 2:In a rotatable central composite design for 3 factors the coded levels are –√2, –1, 0, +1, +√2.

effect of a factor

effect

Coefficient of a term in a response model.

1. Note:See main effect, n^th-order effect, interaction effect.

design matrix

Matrix with rows representing individual experimental treatments (possibly transformed according to the assumed model) which can be extended by deduced levels of other functions of factor levels.

Reference: [16]

dummy factor

Factor that is known to have no effect on the response, used in an experimental design, to estimate repeatability standard deviation.

1. Example:A factor having levels ‘+’ singing the first verse of the National Anthem at the experiment, and ‘–’ singing the second verse of the National Anthem at the experiment.

experimental design

design of experiments

DoE

Efficient procedure for planning combinations of values of factors in experiments so that the data obtained can be analyzed to yield valid and objective conclusions.

1. Note 1:Experimental design is applied to determine the set of conditions that are required to obtain a product or process with desirable, often optimal properties. A characteristic of experimental design is that these conditions are determined in a statistically-optimal way.
2. Note 2:Response surface methodology is considered an important part of experimental design.
3. Note 3:An ‘experimental design’ (noun) usually refers to a table giving the levels of each factor for each run. See coded experimental design.

Reference: [17, 18]

factor (experimental design)

Input quantity in a model.

1. Note 1:The term has a different meaning when used in factor analysis.

factor level

level

Value of a factor in an experimental design.

1. Note 1:A design may be designated by the number of levels chosen for each factor, as in “two-level design”.
2. Note 2:When writing an experimental design the levels are usually coded. (See coded experimental design).

fractional-factorial design

deprecated: incomplete-factorial design

Experimental design obtained from a full factorial design in which experiments are systematically removed to fulfil stated statistical requirements.

1. Note 1:The aim of a fractional design is to reduce the number of experiments by confounding low-order effects (e.g. main effect, two-way interaction) with high order interactions, which are assumed to be small.
2. Note 2:A design, having L^k (see full factorial design) experiments, is fractionated to L^{k– p} experiments where p is an integer <k.
3. Note 3:The choice of design is governed by an alias structure.
4. Note 4:A fractional factorial design is incomplete, but all incomplete designs are not fractional factorial. See Plackett Burman design.

full-factorial design

Experimental design with all possible combinations of factor levels.

1. Note 1:If there are k factors, each at L levels, a full factorial design has L^k runs.

interaction effect

Effect of a factor where the term is the product of two or more factors.

1. Example 1:The yield of a synthesis is modelled in terms of the temperature T and concentration of a reactant c The estimated coefficient is the interaction effect of T and c.
2. Note:See main effect, n^th-order effect.

main effect

Effect of a factor where the term is a single factor.

1. Example 1:The yield of a synthesis is modelled in terms of the temperature T and concentration of a reactant c The estimated coefficients and are the main effects of T and c respectively.
2. Note:See nth-order effect, interaction effect.

model (experimental design)

Equation describing the response as a function of values of the factors.

1. Note 1:The model can be based on knowledge of the chemistry or physics of the system, but usually the model is empirical, being linear or quadratic with interaction terms.
2. Note 2:To obtain information about the significance of effects, data is usually mean centered and assessed against a coded experimental design.
3. Example 1:The yield of a synthesis is modelled in terms of the temperature T and concentration of a reactant c
4. Note:See n^th-orderinteraction effect, main effect

n^th-order effect

Effect of a factor where the term is a factor raised to the power n.

1. Example 1:The yield of a synthesis is modelled in terms of the temperature T and concentration of a reactant c The coefficients and are the second-order effects of T and c respectively.
2. Note:See main effect, interaction effect.

optimization

Minimization or maximization of a real function by systematically choosing the values of real or integer variables from within an allowed set.

Plackett-Burman design

Incomplete experimental design to estimate main effects for which each combination of factor levels for any pair of factors appears the same number of times.

1. Note 1:Plackett-Burman designs are typically given for two levels, with a number of experiments that is a multiple of 4 but not a power of 2. (The latter case is a fractional factorial design).
2. Note 2:For 4×N experiments, 4×N–1 main effects and the mean are estimated.
3. Note 3:If less than 4×N–1 factors are being studied, dummy factors are inserted which allow estimation of the repeatability standard deviation of the measurements.
4. Example:The coded experimental design for 4×N=12, where +1 and –1 represent the two levels of factors X₁ … X₁₁ is given below. The order of performing the runs should be randomised.

resolution of a design

resolution

One more than the smallest n^th-order interaction effect that some main effect is aliased with.

1. Note 1:Resolution is used to describe the extent to which fractional factorial designs create aliased effects.
2. Note 2:The resolution is written as a Roman numeral.
3. Note 3:Full factorial designs have no effects that are aliased and therefore have infinite resolution.
4. Example 1:For resolution III designs the main effects are aliased with two-factor interactions. For resolution IV designs no main effects are aliased with two-factor interactions, but two-factor interactions are aliased with each other. For resolution V designs no main effect or two-factor interaction is aliased with any other main effect or two-factor interaction, but two-factor interactions are aliased with three-factor interactions.

Reference: [19]

response

Measured or observed quantity in an experimental design.

response surface methodology

Experimental design in which the response is modelled in terms of one or more factorlevels.

1. Note 1:Response surface methodology is usually associated with optimization. The model used is typically a quadratic function leading to a maximum or minimum response in the factor space.
2. Note 2:The term ‘surface’ implies two factors and a single response, when a plot of the modelled response as a function of values of the factors leads to a surface in the three dimensional space. This can be generalized to any number of factors.

Reference: [20]

alternating least squares regression

alternating regression

ALS

Solution to the multivariate decomposition of a data matrix in which iteratively a solution of one output matrix is used to compute the second matrix, after the application of constraints.

1. Note:ALS is used to decompose multiple spectra (X) into concentration (C) and component spectra (S). X=C S^T_. Because spectra and concentrations cannot be negative at each iteration negative values are set to zero.

autoregression

A stochastic process in which future values are estimated based on a weighted sum of past values.

1. Note:A process called AR(1) is a first order process, meaning that the current value is based on the immediately preceding value. An AR(2) process has the current value based on the previous two values.

biplot

Combination plot of a scores plot as points and loadings plot as vectors for common factors.

1. Note 1:The plots are scaled to facilitate interpretation.
2. Note 2:Points in the scores plot (objects) that fall on a loadings vector are considered to be characterised by the variable associated with the vector.

bootstrapping

Estimation of parameters by multiple re-sampling from measured data to approximate its distribution.

1. Note 1:Multiple resamples of the original data allow calculation of the distribution of a parameter of interest, and therefore its standard error.
2. Example 1:The standard error of an estimate of parameter θ where B is the number of bootstrap samples, the i-th bootstrap estimate, and θ̅^* the mean value of the bootstrap estimates.
3. Note 2:Random sampling with replacement is used when the data are assumed to be from an independent and identically-distributed population.
4. Note 3:Bootstrapping is an alternative to cross validation in model validation.

Reference: [21, 22]

canonical variables

Linear combinations of data with the greatest correlation.

1. Note:See: canonical variate analysis.

canonical variate analysis

canonical analysis

Multivariate technique which finds linear combinations of two sets of data that are most highly correlated.

1. Note 1:The combinations with the greatest correlation, denoted U1 and V1 are known as the “first canonical variables”.
2. Note 2:The relationship between the canonical variables is known as the canonical function.
3. Note 3:The next canonical functions, U2 and V2 are then restricted so that they are uncorrelated with U1 and V1. Everything is scaled so that the variance equals 1.

common factor analysis

exploratory factor analysis

factor analysis

Factor analysis in which latent variables are calculated that maximise the correlation with observed variables.

1. Note 1:The common factors are not unique. Typically factors are rotated so that the factors are more easily interpreted in terms of the original variables.

core consistency diagnostic (CONCORDIA)

Method to assess the appropriateness of a PARAFAC model.

1. Note 1:An appropriate PARAFAC model is a model where the components primarily reflect low-rank, trilinear variation in the data.
2. Note 2:The principle of the method is to assess the degree of superdiagonality of the model.

Reference: [23]

correspondence factor analysis

correspondence analysis

Factor analysis applied to categorical data in which orthogonal factors are obtained from a contingency table.

cross validation

A re-sampling procedure that predicts the class or property of objects from a classification or regression model that is obtained without those observations.

1. Note 1:When a single object is removed, the procedure is known as leave-one-out cross validation. When n/G objects are deleted, the procedure is known as G-fold cross validation.
2. Note 2:The procedure is iterated leaving out all the objects in turn.
3. Note 3:The model is assessed by calculation of the root mean square error of prediction for continuous variables, and by the misclassification probability for classification.
4. Note 4:Use of independent evaluation data is preferred to cross validation, when there is concern about the independence of the objects in the data set.
5. Note 5:Cross-validation can be used with bootstrapping, one to optimize a model (e.g. how many PCs are appropriate) and the other for validation.

evolving factor analysis (EFA)

Factor analysis that follows the change or evolution of the rank of the data matrix as a function of an ordered variable.

1. Note 1:The ordering variable may be time. (see [24])
2. Note 2:The changing rank is calculated by principal-component analysis on an increasing data matrix.

Reference: [25]

factor (factor analysis)

deprecated component

deprecated pure component

Axis in the data space of a factor analysis model, representing an underlying dimension that contributes to summarizing or accounting for the original data set.

1. Note 1:In principal component analysis each factor is called a principal component. It is deprecated when used outside this context. To avoid confusion “principal component factor” is recommend by ISO.
2. Note 2:In multivariate curve resolution each factor is called a “pure component”. The terms “component” and “pure component” are deprecated as they may be confused with chemical components of the system.
3. Note 3:Each factor is associated with a set of loadings and scores, which occupies a column in the loadings and scores matrices respectively.

factor analysis

Matrix decomposition of a data matrix (X) into the product of a scores matrix (T) and the transpose of the loadings matrix (P^T).

1. Note 1:Hence X=TP^T+E, where E is a residual matrix.
2. Note 2:Factor analysis methods include common factor analysis (also called ‘factor analysis’) principal component analysis, and multivariate curve resolution.
3. Note 3:The number of factors selected in factor analysis is smaller than the rank of the data matrix.
4. Note 4:Factor analysis is equivalent to a rotation in data space where the factors form the new axes. This is not necessarily rotation that maintains orthogonality except in the case of PCA.
5. Note 5:The residual matrix contains data that are not described by the factor analysis model, and is usually assumed to contain noise.

Reference: [11]

G-fold cross validation

Cross validation of a data set of N objects in which N/G objects are removed at each iteration of the procedure.

1. Note 1:Objects 1 to N/G are removed on the first iteration, then objects N/G+1 to 2N/G after replacement of the first N/G objects, and so on.
2. Note 2:Because the perturbation of the model is larger than in leave-one-out cross validation, the prediction ability of the G-fold cross validation is less optimistic than obtained with leave-one-out cross validation.

latent variable

latent construct

hidden variable

Variable that is inferred through a mathematical model from other variables that are observed.

1. Note 1:The factors obtain from common factor analysis are termed latent variables.
2. Note 2:A distinction can be made between ‘hidden variable’, which is considered to be an actual variable that is buried in the effects of other variables and noise, and a ‘latent variable’ that is entirely hypothetical.

leave-one-out cross validation (LOOCV)

Cross validation in which one object is removed in each iteration of the procedure.

loadings

deprecated principal component spectrum

deprecated pure component spectrum

Projection of a factor onto the variables.

1. Note 1:‘Loadings’ (plural) refers to a column in the loadings matrix that relates to a particular factor. “loading” (singular) is the particular contribution of a variable in the original space to the factor.
2. Note 2:The loadings on a factor reflect the relationships between the variables on that factor. (See score)
3. Note 3:In principal component analysis the loadings are also the cosine angles between the variables and a particular factor.
4. Note 4:In multivariate curve resolution the term “pure component spectrum” is interchangeable with the term “loading” and is therefore deprecated. The term, in spectroscopy, may be confused with the spectrum for a pure material.

Reference: [11]

loadings plot

Plot of one loading against variable number, or two or three loadings against each other.

1. Note 1:Usually the loadings associated with the early factors (1, 2, 3) are plotted to reveal relationships among the variables.
2. Note 2:See: loadings plot, biplot

maximum likelihood principal component analysis (MLPCA)

Principal component analysis that incorporates information about measurement uncertainty to develop models that are optimal in a maximum likelihood sense.

Reference: [26].

mean squared error of prediction (MSEP)

mean squared error of estimation (MSEE)

In multivariate calibration the average of the squared deviation of estimated values from the values of evaluation data.

For Nevaluation data where c_i is an observed value and is the predicted value

1. Note 1:mean squared error of prediction is the square of root mean squared error of prediction.

multivariate curve resolution (MCR)

deprecated self-modelling curve resolution (SMCR)

deprecated self-modelling mixture analysis (SMMA)

Factor analysis for the decomposition of multicomponent data into a linear sum of chemically-meaningful components when little or no prior information about the composition is available.

1. Note 1:MCR factors are extracted by the iterative minimization of the residual matrix using an alternating least squares approach, while applying suitable constraints, such as non-negativity, to the loadings and scores. MCR can be performed on the data matrix with or without data pre-processing.
2. Note 2:MCR factors are not unique but are dependent on initial estimates, the number of factors to be resolved, constraints applied and convergence criteria. MCR factors are not required to be orthogonal.

non-linear iterative partial least squares (NIPALS)

Iterative decomposition of a data matrix to give principal components.

1. Note 1:Writing the model as X=TP^T+E, the first principal component is computed from a data matrix. The data explained by this PC are then subtracted from X and the algorithm applied again to residual data. The procedure is repeated until sufficient principal components are obtained.
2. Note 2:The algorithm is very fast if only a few principal components are required, because the covariance matrix is not computed.

nonlinear mapping (NLM)

Projection of objects defined in a multivariate space onto two- or three-dimensional space so that the distances between objects are preserved as well as possible.

1. Note 1:An often applied criterion for the mapping error (E) is the relative squared error between the true distance d_ij and mapped distance δ_ij.
2. Note 2:Several iterative optimization procedures can be applied to minimize E, such as steepest descent.

parallel factors analysis (PARAFAC)

canonical decomposition (CANDECOMP)

Decomposition of a three-way data matrix into the sum of sets of two-way loadings matrices.

1. Note 1:The PARAFAC model is also known as Canonical Decomposition (CANDECOMP).
2. Note 2:A representation of PARAFAC is where x_ijk is i,j,k –th element of the data matrix, and a_ir, b_jr, c_kr are the components of the loadings matrices. e_ijk is the i,j,k –th element of residual matrix.
3. Note 3:PARAFAC is a special case of the Tucker3 model (see Tucker tri-linear analysis) where the core matrix is the identity matrix, and r=s=t=R
4. Note 4:A schematic representation of the PARAFAC model is

Reference: [27]

prediction error sum of squares (PRESS)

sum of squared errors of prediction (SSEP)

residual sum of squares (RSS)

sum of squared residuals (SSR)

In multivariate calibration for a prediction set of N data where c_i is an observed value and is the predicted value.

1. Note:See root mean squared error of prediction

principal component– discriminant analysis (PC-DA)

Discriminant analysis on a multivariate data set that has been subject to principal component analysis.

1. Note 1:This procedure removes collinearity from the multivariate data and ensures that the new predictor variables, which are PCA scores, are distributed normally.

principal-component analysis (PCA)

Factor analysis in which factors are calculated that successively capture the greatest variance in the data set.

1. Note 1:The factors are orthogonal and are known as principal component factors.
2. Note 2:The factorization is written X=TP^T+E, where T is the scores matrix, P is the loadings matrix and E is a residual matrix. See non-linear iterative partial least squares.

Reference: [11]

principal-component factor

principal component (PC)

Orthogonal factors obtained in a principal-component analysis.

1. Note 1:The successive factors explain reducing fractions of the variance of the data set, and are written PC1, PC2 …
2. Note 2:ISO recommends use of the term principal-component factor.

Procrustes analysis

Comparison of shapes of multi-dimensional objects by a series of geometrical transformations to minimise the sum of squared distances between the transformed and target structures while maintaining the internal structure of the objects.

1. Note 1:For two objects defined by X and Y the manipulation orthogonal rotation/reflection matrix R Procrustes analysis minimises: ‖Y–XR ‖² subject to R^TR=RR^T=1
2. Note 2:Ordinary, or classical, Procrustes analysis is when an object is compared to one other object, which may be a reference shape. Generalized Procrustes analysis compares three or more shapes to an optimally-determined mean shape.

Reference: [28] p

Quantitative structure-activity relationship (QSAR)

QSPR

QSA/PR

Relationships between chemical structure, or structural-related properties, and target property of studied compounds.

1. Note 1:Typical target property is biological (or therapeutic) activity of a drug.
2. Note 2:Typical structural-related properties are Hammett electronic parameter, lipophilicity parameter, boiling and melting points, molecular weight and molar refractivity.
3. Note 3:Relationships are established by multivariate calibration.

Reference:[28] Ch

root mean square error of cross validation (RMSECV)

Root mean square error of prediction when the predicted data is obtained by cross validation.

root mean squared error of prediction (RMSEP)

root mean squared error of estimation (RMSPE)

standard error of prediction (SEP)

standard error of estimation

In multivariate calibration or classification for Nevaluation data where c_i is an observed value and is the predicted value

1. Note 1:RMSEP is related to the prediction error sum of squares (PRESS) by
2. Note 2:For completely independent, normally distributed evaluation data, RMSEP is a measure of the bias of the calibration.
3. Note 3:When prediction is by cross validation RMSEP may be termed root mean square error of cross validation.

simple-to-use interactive self-modelling mixture analysis (SIMPLISMA)

Interactive method to obtain concentrations and pure spectra from spectra of mixtures using directly-measured variables.

1. Note 1:The directly-measured variables are called ‘pure variables’ in the method.
2. Note 2:A data matrixD=C×P^T+E where C is a concentration matrix, P pure spectra of mixture components and E an error matrix. Pure spectra are estimated which allows projection of a concentration matrix C^* from which the data matrix can be reconstructed and compared with the measured spectra.

1. Note 3: Second derivatives of spectra can be used for modelling.

Reference: [30]

score

deprecated projections

deprecated pure-component concentration

Factor analysis projection of an object onto a factor.

1. Note 1:In PCA, the factors are orthogonal and the scores are an orthogonal projection of the objects onto a factor.
2. Note 2:The scores on a factor reflect the relationships between objects for that factor. (See loading).
3. Note 3:The term scores (plural) refers to a whole column in the scores matrix that relates to a particular factor. The term score (singular) is the projection of a particular object onto the factor.

Reference: [11]

scores plot

Plot of one score against object number, or two or three scores against each other.

1. Note 1:Usually the scores associated with the early factors (1, 2, 3) are plotted to reveal relationships among the objects.
2. Note:See: loadings plot, biplot

simulated annealing

Generic probabilistic meta-heuristic to locate a good approximation to the global optimum of a given function in a large search space, in which there is a slow decrease in the probability of accepting worse solutions as the solution space is explored.

1. Note 1:The function E(s) to be minimized is analogous to the internal energy of the system in that state. The goal is to bring the system, from an arbitrary initial state, to a state with the minimum possible energy. At each step, the heuristic considers some neighbouring state s’ of the current state s, and probabilistically decides between moving the system to state s’ or staying in state s. These probabilities ultimately lead the system to move to states of lower energy. Typically this step is repeated until the system reaches a state that is good enough for the application, or until a given computation budget has been exhausted.

Reference: [31], www.cronistalascolonias.com.ar

singular value decomposition

SVD

A factorization of an m×n matrix (M) such that M=UΣV^T, where U is an m×m matrix, Σ is a m×n matrix and V^T is a n×n matrix.

1. Note 1:If M is a data matrix with m objects and n variables, the matrix U is the scores matrix, the diagonal of Σ contain the square roots of the eigenvalues and V is the loadings matrix.

Tucker tri-linear analysis

Tucker3 model

Decomposition of a three-way data matrix into a three-way core matrix, and three, two-way loadings matrices.

1. Note 1:A representation of the Tucker3 model is where x_ijk is the data matrix, a_ir, b_js,c_kt are the loadings matrices, and z_rst is the core matrix. e_ijk is the residual matrix.
2. Note 2:A graphical representation of the Tucker3 model is
3. Note 3:See parallel factors analysis

artificial neural network (ANN)

Computing system made up of a number of simple, highly interconnected elements, which process information by their dynamic state response to external inputs.

1. Note 1:An ANN is composed of layers of nodes with an input layer accepting data, one or more hidden layers computed from earlier layers, and an output layer giving the results of the classification.
2. Note 2:Nodes are connected by non-linear functions that calculate the contribution (weight) of an earlier node to a later node.

Reference: Definition adapted from [32] and quoted in [33] (See: www.cronistalascolonias.com.ar~bolo/shipyard/neural/www.cronistalascolonias.com.ar)

backward chaining

back chaining

Inference method from hypothesis to data that supports the hypothesis

1. Note 1:Backward chaining is used in backward-propagation to train an artificial neural network.

backward propagation

back-propagation learning rule

back-propagation of errors

Supervised classification method for an artificial neural network