Tables, Rows, and Columns

The tables are formally called relations, referring to the mathematical set theory used in the original work on relational databases.1 In database theory, rows are called tuples and columns are called attributes of a tuple. The focus of this book is practical, so the common terms table, row, and column are used. The detail of using the SQL language to perform these operations is left to a later chapter of this book. 

A table is a collection of data in rows and columns. As with tables in a scientific publication, each row typically represents some entity, such as a molecule, and each column represents some attribute of the entity, such as the name, molecular weight, ionization potential, or other theoretical or experimental data measurement. A table in a publication is laid out for clarity to the reader. Spreadsheet programs typically include ways to control the layout and look of the table. Display and layout features are irrelevant in a relational database. 

A table in a relational database is intended to provide a consistent way to organize large amounts of data, constrain the data in meaningful ways, and extend the tables when new data becomes available. It does not contain any formatting or display information. Programs that access the database provide any display or formatting of the data in the table. 

Client programs are discussed in later chapters of this book. The structured query language (SQL) designed for creating, selecting, deleting, and updating the database is discussed in Chapter 3. 

A relational table has a name, chosen when it is created. Although any name is possible, the name typically reflects the nature or source of the data contained in the table. Each column must also have a name. Consider Table 2.1, called EPA since it was constructed from data provided by the Environmental Protection Agency.2 This table is readily understandable to any chemist. Each row contains information about one compound and each column contains a molecular attribute or property. In order to make it part of a relational database, a minimum of two things must be specified for each column the column name and the column data type. In this example, the column names are Name, Formula, MW, logP, and MP corresponding to the compound name, molecular formula, molecular weight, octanol-water partition coefficient, and melting point. The column name in a relational table is arbitrary but is usually representative of the data contained in the column. 

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C08-0046. Predict the location in the periodic table (row and column) of element 111. 

A major disadvantage of a matrix representation for a molecular graph is that the number of entries increases with the square of the number of atoms in the molecule. What is needed is a representation of a molecular graph where the number of entries increases only as a linear function of the number of atoms in the molecule. Such a representation can be obtained by listing, in tabular form only the atoms and the bonds of a molecular structure. In this case, the indices of the row and column of a matrix entry can be used for identifying an entry. In essence, one has to distinguish each atom and each bond in a molecule. This is achieved by a list of the atoms and a list of the bonds giving the coimections between the atoms. Such a representation is called a connection table (CT). 

The characteristic of a relational database model is the organization of data in different tables that have relationships with each other. A table is a two-dimensional consti uction of rows and columns. All the entries in one column have an equivalent meaning (c.g., name, molecular weight, etc. and represent a particular attribute of the objects (records) of the table (file) (Figure 5-9). The sequence of rows and columns in the tabic is irrelevant. Different tables (e.g., different objects with different attributes) in the same database can be related through at least one common attribute. Thus, it is possible to relate objects within tables indirectly by using a key. The range of values of an attribute is called the domain, which is defined by constraints. Schemas define and store the metadata of the database and the tables. 

For a sample of polystyrene in benzene, experimental values of Kcj/R are entered in the body of Table 10.2. The values are placed at the intersection of rows and columns labeled c and 9, respectively. In the following example... 

In Table 7.7, all the transition wavenumbers have been arranged in rows and columns so that the differences between wavenumbers in adjacent columns correspond to vibrational level separations in the lower (ground) electronic state and the differences between adjacent rows to separations in the upper electronic state. These differences are shown in parentheses. The variations of the differences (e.g. between the first two columns), are a result of uncertainties in the experimental measurements. 

Periodic function A physical or chemical property of elements that varies periodically with atomic number, 152 Periodic Table An arrangement of the elements in rows and columns according to atomic numbers such that elements with similar chemical properties foil in the same column,... 

The rows and columns of Mendeleev s table are meant to reflect the periodic function asserted to exist by this periodic law. 

A database (or data base) is a collection of data that is organised so that its contents can easily be accessed, managed, and modified by a computer. The most prevalent type of database is the relational database which organises the data in tables multiple relations can be mathematically defined between the rows and columns of each table to yield the desired information. An object-oriented database stores data in the form of obj ects which are organised in hierarchical classes that may inherit properties from classes higher in the tree structure. 

Six repeat titrations went into every mean and standard deviation listed (for a total of 4 x 4 X 6 = 96 measurements). The data in Table 2.17 are for batches A, B, C, and D, and the calculated means respectively standard deviations are for methods 1, 2, 3, and 4. The means for both rows and columns are given. The lowest mean in each row is given in bold. The overall... 

Periodic table of the elements with all the elements included in their proper rows and columns. 

Armed with these conditions, we can correlate the rows and columns of the periodic table with values of the quantum numbers it and /. This correlation appears in the periodic table shown in Figure 8. Remember that the elements are arranged so that Z increases one unit at a time from left to right across a row. At the end of each row, we move down one row, to the next higher value of It, and return to the left side to the next higher Z value. Inspection of Figure reveals that the ribbon of elements is cut after elements 2, 10, 18, 36, 54, and 86. 

For this qualitative problem, use the periodic table to determine the order of orbital filling. Locate the element in a block and identify its row and column. Move along the ribbon of elements to establish the sequence of filled orbitals. 

Today we work confidently with the rows and columns of the periodic table. Yet less than 150 years ago, only about half of all elements known today had been discovered, and these presented a bewildering collection of chemical and physical properties. The discoveiy of the patterns that underlie this apparent randomness is a tale of inspired chemical detective work. 

The predictions made by Mendeleev provide an excellent example of how a scientific theory allows far-reaching predictions of as-yet-undiscovered phenomena. Today s chemists still use the periodic table as a predictive tool. For example, modem semiconductor materials such as gallium arsenide were developed in part by predicting that elements in the appropriate rows and columns of the periodic table should have the desired properties. At present, scientists seeking to develop new superconducting materials rely on the periodic table to identify elements that are most likely to confer superconductivity. 

Some analytical instruments produce a table of raw data which need to be processed into the analytical result. Hyphenated measurement devices, such as HPLC linked to a diode array detector (DAD), form an important class of such instruments. In the particular case of HPLC-DAD, data tables are obtained consisting of spectra measured at several elution times. The rows represent the spectra and the columns are chromatograms detected at a particular wavelength. Consequently, rows and columns of the data table have a physical meaning. Because the data table X can be considered to be a product of a matrix C containing the concentration profiles and a matrix S containing the pure (but often unknown) spectra, we call such a table bilinear. The order of the rows in this data table corresponds to the order of the elution of the compounds from the analytical column. Each row corresponds to a particular elution time. Such bilinear data tables are therefore called ordered data tables. Trilinear data tables are obtained from LC-detectors which produce a matrix of data at any instance during the... 

The eigenvectors extracted from the cross-product matrices or the singular vectors derived from the data matrix play an important role in multivariate data analysis. They account for a maximum of the variance in the data and they can be likened to the principal axes (of inertia) through the patterns of points that represent the rows and columns of the data matrix [10]. These have been called latent variables [9], i.e. variables that are hidden in the data and whose linear combinations account for the manifest variables that have been observed in order to construct the data matrix. The meaning of latent variables is explained in detail in Chapters 31 and 32 on the analysis of measurement tables and contingency tables. 

In the previous subsection, we have described S and L as containing the coordinates of the rows and columns of a data table in factor-space. Below we show that, in some cases, it is possible to graphically reconstruct the data table and the two cross-product matrices derived from it. It is not possible, however, to reconstruct at the same time the data and all the cross-products, as will be seen. We distinguish between three types of reconstructions. 

One can also state that the log double-centered biplot shows interactions between the rows and columns of the table. In the context of analysis of variance (ANOVA), interaction is the variance that remains in the data after removal of the main effects produced by the rows and columns of the table [12], This is precisely the effect of double-centering (eq. (31.49)). 

Atmospheric data from Table 31.1, after double-closure. The weights w are proportional to the row- and column-sums of the original data table. They are normalized to unit sum. 

Double-closure is the joint operation of dividing each element of the contingency table X by the product of its corresponding row- and column-sums. The result is multiplied by the grand sum in order to obtain a dimensionless quantity. In this context the term dimensionless indicates a certain synunetry in the notation. If x were to have a physical dimension, then the expressions involving x would appear as dimensionless. In our case, x represents counts and, strictly speaking, is dimensionless itself. Subsequently, the result is transformed into a matrix Z of deviations of double-closed data from their expected values ... 

In the following section on the analysis of contingency tables we will relate the distances of chi-square in terms of contrasts. In the present context we use the word contrast in the sense of difference (see also Section 31.2.4). For example, we will show that the distance of chi-square from the origin 5, can be related to the amount of contrast contained in row i of the data tables, with respect to what can be expected. Similarly, the distance 5 can be associated to the amount of contrast in column j, relative to what can be expected. In a geometrical sense, one will find rows and columns with large contrasts at a relatively large distance from the origin of and S", respectively. The distance of chi-square 5- then represents the amount of contrast between rows i and i with respect to the difference between their expected values. Similarly, the distance of chi-square indicates the amount... 

The two plots can be superimposed into a biplot as shown in Fig. 32.7. Such a biplot reveals the correspondences between the rows and columns of the contingency table. The compound Triazolam is specific for the treatment of sleep disturbances. Anxiety is treated preferentially by both Lorazepam and Diazepam. The latter is also used for treating epilepsy. Clonazepam is specifically used with epilepsy. Note that distances between compounds and disorders are not to be considered. This would be a serious error of interpretation. A positive correspondence between a compound and a disorder is evidenced by relatively large distances from the origin and a common orientation (e.g. sleep disturbance and Triazolam). A negative correspondence is manifest in the case of relatively large distances from the origin and opposite orientations (e.g. sleep disturbance and Diazepam). 

The three latent vectors account for respectively 86, 13 and 1% of the interaction. The next two columns in Tables 32.11 and 32.12 show the distances 5 and 8 of rows and columns from the origin of space and their contributions y and to the interaction ... 

The final column in Tables 32.11 and 32.12 lists the precisions 7t and with which the rows and columns are represented in the plane spanned by the first two latent vectors ... 

Figure 32.8 shows the biplot constructed from the first two columns of the scores matrix S and from the loadings matrix L (Table 32.11). This biplot corresponds with the exponents a = 1 and p = 1 in the definition of scores and loadings (eq. (39.41)). It is meant to reconstruct distances between rows and between columns. The rows and columns are represented by circles and squares respectively. Circles are connected in the order of the consecutive time intervals. The horizontal and vertical axes of this biplot are in the direction of the first and second latent vectors which account respectively for 86 and 13% of the interaction between rows and columns. Only 1% of the interaction is in the direction perpendicular to the plane of the plot. The origin of the frame of coordinates is indicated... 

The log-linear model (LLM) is closely related to correspondence factor analysis (CFA). Both methods pursue the same objective, i.e. the analysis of the association (or correspondence) between the rows and columns of a contingency table. In CFA this can be obtained by means of double-closure of the data in LLM this is achieved by means of double-centring of the logarithmic data. 

For the same reason as for double-closure, double-centring always reduces the rank of the data matrix by one, as a result of the introduction of a linear dependence among the rows and columns of the data table. 

Fig. 32.11. Log-linear model (LLM) biplot computed from the data in Table 32.10. Conventions are the same as in Fig. 32.10. The areas of circles (representing years) and of squares (representing categories) are made proportional to the row- and column-totals in Table 32.10.

Fig. 37.5. Biplot obtained from correspondence factor analysis of the data in Table 37.8 [43], Circles refer to compounds. Squares relate to observations. Areas of circles and squares are proportional to the marginal sums of the rows and columns in the table. The horizontal and vertical components represent 40 and 31 %, respectively, of the interaction in the data.

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