2.2.1Nominal attributes and numeric attributes

Nominal refers to names. A nominal attribute, therefore, can be names of things or symbols. Each value in a nominal attribute represents a code, state or category. For instance, healthy and patient categories in medical research are example of nominal attributes. The education level categories such as graduate of primary school, high school or university are other examples of nominal attributes. In studies regarding MS, subgroups such as relapsing remitting multiple sclerosis (RRMS), secondary progressive multiple sclerosis (SPMS) or primary progressive multiple sclerosis (PPMS) can be listed among nominal attributes. It is important to note that nominal attributes are to be handled distinctively from numeric attributes. They are incommensurable but interrelated. For mathematical operations, a nominal attribute may not be significant. In medical dataset, for example, the schools from which the individuals, on whom Wechsler Adult Intelligence Scale – Revised (WAIS-R) test was administered, graduated (primary school, secondary school, vocational school, bachelor’s degree, master’s degree, PhD) can be given as examples of nominal attribute. In economy dataset, for example, consumer prices, tax revenue, inflation, interest payments on external debt, deposit interest rate and net domestic credit can be provided to serve as examples of nominal attributes.

Definition 2.2.1.1. Numeric attribute is a measurable quantity; thus, it is quantitative, represented in real or integer values [7, 9, 10]. Numeric attributes can either be interval- or ratio-scaled.

Interval-scaled attribute is measured on a scale that has equal-size units, such as

Celsius temperature (C): (−40°C, −30°C, −20°C, 0°C, 10°C, 20 °C, 30°C, 40°C, 50°C, 60°C, 70°C, 80°C).

We can see that the distance from 30°C to 40°C is the same as distance from 70°C to 80°C.

The same set of values converted into Fahrenheit temperature scale is given in the following list:

Fahrenheit temperature (F): (−40°F, −22°F, −4°F, 32°F, 50°F, 68°F, 86°F, 104°F, 122°F, 140°F, 158°F, 176°F).

Thus, the distance from 30°C to 40°C [86°F – 104°F] is the same distance from 60°C to 70°C [140°F – 158°F].

Ratio-scaled attribute has inherent zero point. Consider the example of age. The clock starts ticking forward once you are born, but an age of “0” technically means you do not actually exist. Therefore, if a given measurement is ratio scaled, the value can be considered as one that is a multiple or ratio of another value. Moreover, the values are ordered so the difference between values, mean, median and mode can be calculated.

Definition 2.2.1.2. Binary attribute is a nominal attribute that has only two states or categories. Here, 0 typically shows that the attribute is absent and 1 means that the attribute is present. States such as true and false fall into this category. In medical cases, healthy and unhealthy states pertain to binary attribute type [11–13].

Definition 2.2.1.3. Ordinal attributes are those that yield possible values that enable meaningful order or ranking. A basic example is the one used commonly in clothes size: x-small, small, medium, large and x-large. In psychology, mild and severe refer to ranking a disorder [11–14].

Discrete attributes have either finite set of values or infinite but countable set of values that may or may not be represented as integers [7, 9, 10]. For example, age, marital status and education level each have a finite number of values; therefore, they are discrete. Discrete attributes may have numeric values, for example, age may be between 0 and 120. If an attribute is not discrete, it is continuous. Numeric attribute and continuous attribute are usually used interchangeably in literature. However, continuous values are real numbers, while numeric values can be either real numbers of integers. The income level of a person or the time it takes a computer to complete a task can be continuous [15–20].

Definition 2.2.1.4. Data Set (or dataset) is a collection of data. For i =0, 1,…, K in x: i0, i1,…, iK; j =0, 1,…, K in y: j0, j1,…, jL D(K × L) represents a dataset. x: i. is the entry on the line and y: j. is the matrix that represents the attribute in the column.

Figure 2.5 shows the dataset D(3 × 4) as depicted with row (x) and column (y) numbers.

2.3.1Real dataset

Real dataset refers to specific topic collected for a particular purpose. In addition, the real dataset includes the supporting documentation, definitions, licensing statements and so on. A dataset is a collection of discrete items made up of related data. It is possible to access to this data either individually or in combination. It is also possible that it is managed as an entire sample. Most frequently real dataset corresponds to the contents of a single statistical data matrix in which each column of the table represents a particular attribute and every row corresponds to a given member of the dataset. A dataset is arranged into some sort of data structure. In a database, for instance, a dataset might encompass a collection of business data (sales figures, salaries, names, contact information, etc.). The database can be considered a dataset per se, as can bodies of data within it pertaining to a particular kind of information, such as sales data for a particular corporate department.

In the applications and examples to be addressed in the following sections, dataset from economy field has been applied by having formed real data. Economy dataset, as real data, is selected from World Bank site (you can see http://data.worldbank.org/country/) [21] and used accordingly.

2.3.2Synthetic dataset

The idea of original fully synthetic data was generated by Rubin in 1993 [16]. Rubin originally designed this with the aim of synthesizing the Decennial Census long-form responses for the short-form households. A year later, in 1994, Fienberg generated the idea of critical refinement. In this refinement he utilized a parametric posterior predictive distribution (rather than a Bayes bootstrap) in order to perform the sampling. They together discovered a solution for how to treat synthetic data partially with missing data [14–17].

Creating synthetic data is a process of data anonymization. In other words, synthetic data are a subset of anonymized, or real, data. It is possible to use synthetic data in different areas to act as a filter for information that would otherwise compromise the confidentiality of particular aspects of the data. The particular aspects generally emerge in the form of patient individual information (i.e., individual ID, age, gender, etc.).

Using synthetic data has been recommended for data analysis applications for the following conveniences it offers:

In the applications to be addressed in the following sections, datasets from medicine have been applied forming synthetic data by CART algorithm (see Section 6.5) Matlab. There are two ways to deal with generating synthetic dataset from real dataset: one is to develop an algorithm to generate synthetic dataset from real dataset or another may use available software (SynTReN, R-package, ToXene, etc.) [18–20].

The synthetic datasets in our book have been obtained by the CART algorithm (for more details see Section 6.5) application on the real dataset.

For real dataset: i=0, 1, …, R in x: i0, i1,…, iR; j =0, 1, …, L in y: j0, j1,…, jL D(R × L) represents a dataset. x: i. is the entry on the line and y: j. is the matrix representing the attribute in the column.

For synthetic dataset, i=0, 1,…, K in x: i0, i1,…, iK; j=0, 1,…, K in y: j0, j1,…, jL D(K × L) represent a dataset. x: i. is the entry on the line and y: j. is the matrix representing the attribute in the column.

A brief description of the contents in synthetic dataset is shown in Figure 2.7.

The steps pertaining to the application of CART algorithm (Figure 2.6) to the real dataset D(K × L) for obtaining the synthetic dataset D(R × L) are provided as follows.

The steps of getting the synthetic dataset through the application of general CART algorithm are provided in Figure 2.7.

Steps (1–6) The data belonging to each attribute in the real dataset are ranked from lowest to highest (when the attribute values are same, only one value is included in the ranking). For each attribute in the real dataset, the average of median and the value following the median is calculated. xmedian represents the median value of the attribute and xmedian + 1 represents the value after the median value (see eq. (2.4(a))).

The average of xmedian and xmedian + 1 is calculated as $\frac{x_{\text{median}}+x_{\text{median+1}}}{2}$ see eq. (2.1)).

Steps (7–10) The Gini value of each attribute is calculated for the formation of decision tree from the real dataset (see Section 6.5). The lowest Gini value calculated is the root of the decision tree. Synthetic data are formed based on the decision tree obtained from the real dataset and the rules derived from the decision tree.

Let us now explain how the synthetic dataset is generated from real dataset through a smaller scale of sample dataset.

Example 2.1 Dataset D(10 × 2) consists of two classes (gender as female or male) and two attributes (weight in kilograms and height in centimeters; see Table 2.3) as CART algorithm has been applied and synthetic data need to be generated.

Now we can apply CART algorithm on the sample dataset D(10 × 2) (see Table 2.3)

Steps (1–6) The data that belong to each attribute in the dataset D(10 × 2) are ranked from lowest to highest (when the attribute values are the same, only one value is incorporated in the ranking; see Table 2.3). For each attribute in the dataset the average of median and the value following the median is computed. xmedian represents the median value of the attribute and xmedian + 1 represents the value after the median value (see eq. (2.4(a))).

Table 2.3: Dataset D(10 × 2).

The average of xmedian and xmedian + 1 is calculated as $\frac{x_{\text{median}}+x_{\text{median+1}}}{2}$ (see eq. (2.1)).

The threshold value calculation for each attribute (column; j: 1, 2) in the dataset D(10 × 2) in Table 2.3 is applied based on the following steps.

In the following, we discuss the steps of the threshold value calculation for the j: 1: {Weight} attribute.

In Table 2.3. {Weight} values (line: i : 1,. . ., 10) in j: 1 column is ranked from lowest to highest (when the attribute values are the same, only one value of is incorporated in the ranking).

For “Weight ≥ 76”, “Weight <76” two groups are formed.

The following are the steps for the calculation of threshold value for the j: 2: {Height} attribute.

In Table 2.3, {Height} values (line: i: 1,. . ., 10) in j: 2 column are ranked from lowest to highest. (When the attribute values are same, only one value is incorporated in the ranking.)

For “Height ≥ 181”, “Height <181,” two groups are formed.

Steps (7–10) The Gini value of each attribute is calculated for the formation of decision tree of the dataset D(10 × 2). For each attribute, Gini value calculation is done after being split into two groups, and the root of the tree is found (more details in Section 6.5.)

In Table 2.4, you can see the results derived from the splitting of the binary groups for the attributes (j = 1, 2) that belong to the sample real data (female, male class).

Table 2.4: Results derived from the splitting of the binary groups for the attributes that belong to dataset (female, male class).

The results provided in Table 2.4 for each attribute in sample dataset D(10 × 2) will be used for the Gini calculation.

The calculation of Gini (Weight) for j: 1 (see Section 6.5, eq. (6.8)):

The calculation of Gini (Height) value for j: 1 (see Section 6.5, eq. (6.8)):

The lowest Gini value is Gini(Weight) = 0. The lowest value in the Gini values is the root of the decision tree. The decision tree obtained from the dataset is shown in Figure 2.8.

The decision rule as obtained from dataset in Figure 2.8 is given as Rules 1 and 2:

Rule 1: If Weight ≥76, class is male.

Rule 2: If Weight <76, class is female.

In Table 2.5, synthetic dataset D(3 × 10) in Rule 1 Weight ≥ 76 and in Rule 2 Weight <76 (see Figure 2.8) are applied and obtained accordingly.

Table 2.5: Forming the sample synthetic dataset for
Weight ≥ 76 and Weight <76.

In Table 2.6, synthetic dataset D(3 × 2) is obtained by applying Rules 1 and 2 (see Figure 2.8).

Table 2.6: Forming the sample synthetic dataset for Weight ≥76, then class is male and Rule 2: Weight <76, then class is female.

CART algorithm is applied to the real dataset. On applying this, we have received synthetic dataset. Synthetic dataset is composed of two entries from the male class and one entry (row) from the female class (it is possible to increase or reduce the entry number in the synthetic dataset).

2.4.1Economy data: Economy (U.N.I.S.) dataset

As the first set of real dataset, we will use, in the following sections, some data related to USA (abbreviated by the letter U), New Zealand (N), Italy (I) and Sweden (S) economies for U.N.I.S. countries’ economies. Data belonging to their economies from 1960 to 2015 are defined based on the attributes given in Table 2.7 (Economy dataset; http://data.worldbank.org) [21].

The matrix dimension of economy (U.N.I.S.) dataset is defined as follows (see Figure 2.9).

For i=0, 1,…, 228 in x: i0, i1,…, i228; j=0, 1,…, 18 in y: j0, j1,…, j18, × D(228 18) represents economy dataset.

In Table 2.7, the economic growth and parameters thereof are explained as to U. N.I.S. countries for the period between 1960 and 2015.

The sample data values that belong to these economies can be presented as in Table 2.8 (see the World Bank website [21]).

2.4.2MS data and content (neurology and radiology data): MS dataset

MS is a demyelinating disease of central nervous system (CNS). The disease is brought about by an autoimmune reaction that results from a complex interaction of genetic and environment factors [22–25]. MS may affect any site in the CNS; nevertheless, it is possible to recognize six types regarding the disease (Table 2.9).

Table 2.7: U.N.I.S. economies dataset explanation.

Table 2.8: An excel table to provide an example of economy (U.N.I.S.) data.

Table 2.9: Six types of MS [23].

One of the most important tools in the diagnosis of MS is MRI. Using this tool, specialists examine the current formation of the plaques that form in the MS disease.

MRI: This is a very important tool that can reveal the inflammated or harmed tissue sites in the diagnosis of MS. This study makes use of the MRI of the patients who have been diagnosed with MS based on Mc Donald criteria [24]. The dataset has been formed by getting the measurement as to the lesions from three sites based on the MRI [25]. These sites are the brain stem, corpus callosum-periventricular region and upper cervical region (see Figure 2.9).

EDSS: EDSS is a scale that is capable of measuring eight regions, partially of the CNS known as the functional system. This scale measures the freedom of movement first by using the temporary numbness in the face and fingers or visual impairments and walking distance [26]. The ranking of each item from these systems is conducted based on the disability status, and to this functional system, mobility as well as daily life limitations are added. Twenty ranks within EDSS are provided with their corresponding definitions in Table 2.10 [26, 27].

Table 2.10: Description of EDSS [26, 27].

Now let us provide definitions for the MS dataset (synthetic MS data) with details, which are used in the applications of the algorithms in the following sections.

2.4.2.1Obtaining synthetic MS dataset

The MS dataset consist of EDSS and MRI data belonging the MS dataset (139 × 112) which constitute 120 MS patients (with RRMS, SPMS, PPMS subgroups) and 19 healthy individuals. Level of disability in MS patients was determined by using the Extended Disability Status Scale (EDSS). Besides EDSS, MRIs of the patients were also used. Magnetic resonance images are obtained with a 1.5 Tesla device (Magnetom, Siemens Medical Systems, Erlangen, Germany). The brainstem, corpus callosum-periventricular region, including the upper cervical lesions in the three regions were included in the information. The MRIs have been evaluated by radiologists. MRIs of the patients were obtained from Hacettepe University Radiology Department and Primer Magnetic Resonance Imaging Center. The MS data have been obtained from Rana Karabudak, Professor (MD), the Director of Neuroimmunology Unit in Neurology Department, Ankara, Turkey.

A brief description of the contents in synthetic MS dataset is shown in Figure 2.10.

Synthetic MS dataset (304 × 112) is obtained by applying CART algorithm to the MS dataset (139 × 112) as shown in Figure 2.10.

For the synthetic MS dataset D(304 × 12) addressed in the following application sections regarding neurology and radiology data, there are three clinical definite MS subgroups (RRMS, PPMS and SPMS) and a healthy group to be handled [25]. From the synthetic data belonging to these patients, the following subgroups of 76 patients with RRMS, 76 with PPMS, 76 with SPMS and 76 healthy individuals have been formed homogenously. Dataset is composed of MRI for the three sites of the brain with lesion size and lesion count and EDSS belonging to the MS patients.

Table 2.11 lists the relevant data for MRI for the three sites of the brain with lesion size and lesion count and EDSS belonging to the MS patients.

The matrix dimension of the MS dataset is defined as follows (see Figure 2.10).

For i=0, 1,…, 304 in x: i0, i1,…, i304; j=0, 1,…, 112 in y: j0, j1,…, j112, D(304 × 112) represents synthetic MS dataset.

Table 2.11: Synthetic MS dataset explanation.

The sample data values of synthetic MS dataset belonging to the individuals are listed in Table 2.12.

The first region represents the brain stem lesion count (MRI 1) and lesion size, while the second region represents corpus callosum lesion count (MRI 2) and lesion size. The third region represents upper cervical region lesion count (MRI 3) and lesion size. The number of lesions belonging to the brain stem is (0 mm, 1 mm, …, 16 mm, …) and for corpus callosum the number of lesions is (0mm, 2.5 mm,…, 8mm,…, 27 mm,…). The number of lesions for the upper cervical region is (0 mm, 2.5mm,…, 8mm,…, 15 mm, …).

CART algorithm has been used to obtain the synthetic MS dataset D(139 × 121) from MS dataset D(304 × 121).

Let us now explain how synthetic MS dataset D(304 × 112) is generated from MS dataset through a smaller scale of sample dataset by using CART algorithm shown in Figure 2.7 (for more details see Section 6.5).

Example 2.2 Sample MS dataset D(20 × 5) (see Table 2.13) is chosen from MS dataset D(304 × 112). CART algorithm is applied and sample synthetic dataset is derived. The steps for this process are provided in detail.

Let us apply the CART algorithm to the sample MS dataset below D(20 × 5) (see Table 2.14).

The steps of obtaining the sample synthetic MS dataset through the application of CART algorithm on the sample MS dataset D(20 × 5) are shown in Figure 2.7 (for more details see Section 6.5). The steps for the algorithm are given in detail.

Steps (1–6) The data belonging to each attribute in the sample MS dataset D(20× 5) (see Table 2.14) are ranked from lowest to highest (when the attribute values are same, only the one value is included in the ranking). For each attribute in the sample MS dataset, the average of median and the value following the median are calculated. xmedian represents the median value of the attribute and xmedian + 1 represents the value after the median value (see eq. (2.4(a))).

The average of xmedian and xmedian + 1 is calculated as (see eq. (2.1)). The threshold value calculation for each attribute (column; j: 1,2…,6) in the sample MS dataset D(20 × 5) listed in Table 2.7 is applied according to the following steps.

Table 2.12: An excel table to provide an example of synthetic MS data.

Table 2.13: Sample MS dataset D(20 × 5) chosen from the MS dataset.

Table 2.14: Results derived from the splitting of the binary groups for the attributes that belong to sample MS dataset (RRMS and SPMS subgroups).

The steps of the threshold value calculation for the j: 1: {EDSS} attribute are as follows:

In Table 2.13, {EDSS} values (line: i: 1,…, 20) in j: 1 column are ranked from lowest to highest (when the attribute values are same, only one value is included in the ranking).

$\begin{array}{l}{\text{EDSS}}_{\text{sort}}=\left\{\mathrm{3.3.5},4,5,5.5,6,6.5,7,7.5,8\right\}\\ {\text{EDSS}}_{\text{median}}=5.5\left(\text{see}\text{eq}\text{.}\left(2.4\left(\text{a}\right)\right)\right)\\ {\text{EDSS}}_{\text{median+1}}=6\\ \overline{\text{EDSS}}=\frac{5.5+6}{2}=5.75\left(\text{see}\text{eq}\text{.}\left(2.1\right)\right)\end{array}$

Two groups are formed for “EDSS ≥ 5.75” and “EDSS < 5.75.”

The steps for calculating the threshold value of j: 2: {MRI 1} attribute are as follows:

In Table 2.13, {MRI 1} values (line: i: 1,…, 20) in j: 2 column are ranked from lowest to highest. (When the attribute values are same, only one value is included in the ranking.)

$\begin{array}{l}{\text{MRI1}}_{\left(\text{sort}\right)}=\left\{0,1,2,3,4,5,6,7,8,9,10\right\}\\ {\text{MRI1}}_{\text{median}}=5\\ {\text{MRI1}}_{\text{median+1}}=6\\ \overline{\text{MRI1}}=\frac{5+6}{2}=5.5\end{array}$