In this simple case it will just correspond to a scaling and a permutation.

A categorical scale whose points have no inherent order. Ordinal A scale whose points do have an inherent order but intervals between consecutive points on the scale are not necessarily equal.

Interval A scale with equal intervals between consecutive points and an arbitrary zero point. Celsius or Fahrenheit temperature scales.

Ratio A scale with equal intervals between consecutive points and an absolute zero point. This table places measurement scales according to their expressive power.

The only meaningful comparison for nominal measurements is whether or not they are the same. Ordinal measurements are more expressive, being able to be placed in order.

In addition to having an ordinal quality, the difference between interval measurements is meaningful. A ratio scale is the most expressive: This study treats the data of the UBS textual apparatus as purely nominal.

The readings of a variation unit differ, but it is meaningless to apply the mathematical operations of division or subtraction to them. One may argue that this data is ordinal because the UBS editors give priority to the first reading of each variation unit.

Also, if the readings of each variation unit were somehow to be arranged in order, perhaps using a genealogical principle, then the apparatus data could be treated as ordinal.

Encoding nominal data Nominal data is encoded prior to analysis, often by a one-to-one mapping of states to numerical labels. Binary data has only two states, the first signifying presence and the second absence of some trait.

By convention, presence is represented by 1 and absence by 0. Multistate data has two or more states. It can be encoded by assigning a unique numeral to each state, normally starting with 1 and proceeding with 2, 3, etc. These numerals have no inherent significance, serving merely as labels to distinguish different states.

Being nominal data, the only comparison that makes sense is whether two labels are the same; it does not make sense to compare the magnitudes or order of the numerical labels that represent the states of such data.

It is always possible to convert multistate nominal data into a binary form by resolving each variable into a group of one or more binary variables.

Missing data Parts of the data set may be missing, as when a witness does not cover a part of the biblical text, is fragmentary, or has an ambiguous reading. If the reading of a witness at a particular variation unit is not known then the corresponding element of the data matrix is not defined.

Missing data is different to the state of binary data that signifies absence. In the latter, the state is defined; in the former it is not. To illustrate, consider a piece of text in which some witnesses have the definite article and others do not.

In a binary representation, presence of the article is labelled as 1 and absence as 0. The state of a witness which has a lacuna at the relevant place is not defined. The corresponding cell of the data matrix is therefore labelled as NA. Missing data causes problems when it comes to analysis.

Ideally, the state of every witness would be defined for every variation unit under examination. This is seldom the case in practice, so a strategy for dealing with missing data is required.

One approach, which I call the exclusive strategy, specifies a reference witness, eliminates all variation units i. The result is a reduced data matrix which is completely free of missing data. Another approach, which I call the inclusive strategy, sets the condition that each pair of witnesses in the desired data matrix must share a minimum number of defined variation units.

Through an iterative process, witnesses that cause this condition to be violated are eliminated until the processed data matrix satisfies the condition. The result is a reduced data matrix which retains more witnesses than would be the case if the exclusive strategy were used.

However, witnesses within the data matrix may still have undefined variation units. A reference witness can also be specified when using the inclusive strategy, in which case columns of the original data matrix for which the reference witness are not defined are eliminated prior to beginning the iterative reduction process.

In this way, the reference witness is guaranteed to be included in the resulting data matrix. Encoding an apparatus The first step in converting the information of an apparatus into a form suitable for multivariate analysis is to construct a data matrix. By convention, rows of a data matrix are devoted to cases i.

Consequently, the first task is to isolate the reading of every witness for every variation unit.With respect to the quantum matrix model, x and p are related via Fourier tranform, call it F, as follows: F x = p F where F^{-1} = F P Thus perhaps finding the n by n matrix counterpart for F one.

It helps, of course, if the students understand the connection between applying a matrix to a column vector and taking a linear combination of the columns of the matrix.

permalink embed. The adjugate matrix is found by first finding the transpose matrix, finding the cofactors of the transpose matrix, and finally applying alternating signs to the transpose matrix.

Buy Matrix analysis and applied linear algebra on kaja-net.com FREE SHIPPING on qualified orders. Improving Speed Performance of Loops Looping over very large data sets can become slow in R. However, this limitation can be overcome by eliminating certain operations in loops or avoiding loops over the data intensive dimension in an object altogether.

Tutorial on time series analysis in Excel. Includes examples and software for moving average, exponential smoothing, Holt and Holt-Winters, ARIMA (Box-Jenkins).

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Non-negative matrix factorization - Wikipedia