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approach which partitions the raw data matrix into model and residual matrices,
and ordinary MV can be applied to these two derived matrices. he covariate adjust-
ment process is accomplished through by estimating conditional correlations. For
a discrete covariate, a correlation matrix for variables is decomposed into within-
andbetween-componentmatrices.Whenthecovariateiscontinuous,theconditional
correlation is equivalent to the partial correlation under the assumption of joint nor-
mality.
Data with Missing Values
15.8.3
herelativity of a statistical graph (Chen,
)is the main concept used in seriation
algorithms to construct meaningful clustered matrices. his property can be used to
develop aweighted pattern methodtoimpute the missing values. heinitial proxim-
ity measurements for rows and columns with missing values can be computed with
pair-wisecompleteobservations, andthenimputedvaluesareestimatedandupdated
iteratively for the subsequent proximity calculations and imputation.
Modeling Proximity Matrices
15.8.4
Many statistical modeling procedures try to visually explore the high-dimensional
pattern embedded in a proximity matrix that records pair-wise similarity or dissim-
ilarity for a set of objects through a low-dimensional projection. Multidimensional
scaling, hierarchical clustering analysis, and factor analysis are three such statistical
techniques. Four types of matrices are usually involved in the modeling processes of
these statistical procedures. heinput proximity matrix is transformed into a dispar-
itymatrix priortofixing thestatistical modelthat summarizesthe information inthe
output distance matrix. A stress (residual) matrix is calculated toassess the goodness
of fit for the modeling. Such a study aims to create a comprehensive diagnosis envi-
ronment for statistical methods through various types of matrix visualization for the
numerical matrices involved in the modeling process.
Conclusion
15.9
Matrix visualization is the color order-based representation of data matrices. It is
beneficial to employ human vision to explore the structure in a matrix in the pur-
suit of further appropriate mathematical operations and statistical modeling. A good
matrix visualization environment helps us to gain comprehensive insights into the
underlying process. Rather than rely solely on numerical characteristics, it is sug-
gestedthatmatrixvisualizationshouldbeperformedasapreliminarystepinmodern
exploratory data analysis, and research into and applications of matrix visualization
continue to be of much interest.
Matrix visualization displays provide five levels of information: (
) raw scores for
every sample/variable combination; (
) an individual sample score vector across all
