For “MRI 1 ≥ 5.5” and “MRI 1 < 5.5,” two groups are formed.

The steps for calculating the threshold value of j: 3: {First region lesion size is 1 mm} attribute are as follows:

In Table 2.13, {First region lesion size is 1 mm} values (line: i: 1,…, 20) in j: 3 column are ranked from lowest to highest (when the attribute values are same, only one value is included in the ranking).

“First region lesion size is 1mm ≥ 0.5” and “First region lesion size is 1mm < 0.5,” and for these, two groups are formed.

The steps for calculating the threshold value of j: 4: {MRI 2} attribute are as follows:

In Table 2.13, {MRI 2} values (line: i: 1,…, 20) in j: 4 column are ranked from lowest to highest. (When the attribute values are same, only one value is included in the ranking.)

MRI 2sort={0,  6,  7,  9,  10,  11,  12,  13,  14,  17}                               MRI 2median=10                             MRI 2median+1=11                   MRI 2¯=10+112=10.5

“MRI 2 ≥ 10.5” and “MRI 2 < 10.5.” For these, two groups are formed.

The steps for calculating the threshold value of j: 5: {Second region lesion size is 1 mm} attribute are as follows:

In Table 2.13, {Second region lesion size is 1 mm} values (line: i: 1,…, 20) in j: 5 column are ranked from lowest to highest. (When the attribute values are the same, only one value is included in the ranking.)

Second region lesion size is1 mmsort:{0,  1,  2,  3}  Second region lesion size is1 mmmedian=1Second region lesion size is1 mmmedian+1=2Second region lesion size is1 mm¯ =1+22=1.5

“Second region lesion size is 1mm≥ 1.5” and “Second region lesion size is 1mm < 1.5.” For these, two groups are formed.

Steps (7–10) The Gini value of each attribute is calculated for the formation of decision tree from the sample MS dataset D(20 × 5). After all the attributes are split into two groups (see Table 2.14), calculation for Gini value for each attribute is conducted and the root of the tree is found (for more details see Section 6.5).

Table 2.14 lists the results derived from the splitting of the binary groups for the attributes (j = 1, 2, …, 6) belonging to sample MS dataset (RRMS and SPMS subgroups).

The results in Table 2.14 for each attribute in sample MS dataset D(20 × 6) will be used in the Gini calculation.

Calculation of Gini (EDSS) value for j: 1 (see Section 6.5; eq. (6.8)) is as follows.

Ginileft(EDSS)=1−[(1/12)2+(11/12)2]=0.17    Giniright(EDSS)=1−[(8/8)2+(0/8)2]=0  Gini(EDSS)=(12(0.17)+8(0))/20=0.102

Calculation of Gini (MRI 1) value for j: 2 (see Section 6.5; eq. (6.8)) is as follows:

 Ginileft(MRI  1)=1−[(5/12)2+(7/12)2]=0.51  Giniright(MRI  1)=1−[(4/8)2+(4/8)2]=0.5  Gini(MRI  1)=(12(0.51)+8(0.5))/20=0.506

Calculation of Gini (First region lesion size is 1 mm) value for j: 3 (see Section 6.5; eq. (6.8)) is as follows:

    Ginileft(First region lesion size is 1 mm)=1−[(3/6)2+(3/6)2]=0.5Giniright(First region lesion size is 1 mm)=1−[(6/14)2+(8/14)2]=0.51 Gini(First region lesion size is 1 mm)=(6(0.5)+14(0.51))/20=0.507

Calculation of Gini (MRI 2) value for j: 4 (see Section 6.5; eq. (6.8)) is as follows:

  Ginileft(MRI 2)=1−[(5/10)2+(5/10)2]=0.5Giniright(MRI 2)=1−[(4/10)2+(6/10)2]=0.48Gini(MRI 2)=(10(0.5)+10(0.48))/20=0.49

Calculation of Gini (Second region lesion size is 1 mm) value for j: 5 (see Section 6.5; eq. (6.8)) is as follows:

    Ginileft(Second region lesion size is 1 mm)=1−[(2/3)2+(1/3)2]=0.47Giniright(Second region lesion size is 1 mm)=1−[(7/17)2+(10/17)2]=0.51 Gini(Second region lesion size is 1 mm)=(3(0.47)+17(0.51))/20=0.504

The lowest Gini values are Gini(EDSS) = 0.102. The lowest value in the Gini value is the root of the decision tree. The decision tree obtained from the sample MS dataset is shown in Figure 2.11.

Figure 2.11: Decision tree obtained from sample MS dataset D(20 × 5).

The decision rule as obtained from sample MS dataset shown in Figure 2.11 is given in Rule 1:

Rule 1: If EDSS ≤ 5.75, the class is RRMS.

In Table 2.15, sample synthetic MS dataset D(8 × 6) in Rule 1 EDSS ≤ 5.75 (see Figure 2.11) is applied and obtained.

Table 2.15: Forming the sample synthetic MS dataset for EDSS ≤ 5.75.

Table 2.15 lists sample synthetic MS dataset D(8 × 6); if Rule 1 EDSS is ≤ 5.75 (see Figure 2.14), the class is RRMS (see Figure 2.11), which is applied and obtained.

In brief, sample MS dataset is chosen randomly from the MS dataset. CART algorithm is applied to the MS dataset. By applying this, we got the synthetic sample dataset. Sample synthetic dataset is composed of eight RRMS patients’ data (see Table 2.16; it is possible to increase or reduce the entry number in the synthetic MS dataset).

2.4.3Clinical psychology data: WAIS-R dataset

Mental activity and/ or mental disorder is a challenging topic in clinical psychology. There are several methods to analyze the cognitive ability of adults and to detect functions that they are not able to perform well. Among them one of the most expedient methods is the so-called WAIS given by David Wechsler in 1955 [28], following his preliminary work of 1939 [28–32]. The WAIS-R is a general intelligence test. It is defined as the global capacity of the individual to think rationally, to act in a purposeful way and to manage his/her environment in an effective manner. In line with this definition of intelligence as an aggregate of mental abilities or aptitudes, the WAIS-R includes 11 subtests that are divided into two parts, which are the verbal and the performance parts (Table 2.18 provides the details). The WAIS-R includes six verbal subtests and five performance subtests. The verbal tests are named as follows: information, comprehension, arithmetic, digit span, similarities and vocabulary. The performance subtests are as follows: picture arrangement, picture completion, block design, object assembly and digit symbol. The scores from this test are a verbal IQ, a performance IQ and a full-scale IQ (DM). The DM is a standard score, with a mean of 100 and a standard deviation of approximately 15. The parameters administered on the individuals on WAIS-R test are shown in Figure 2.12.

Table 2.16: If sample synthetic MS dataset is EDSS ≤ 5.75, the class RRMS is formed.

Figure 2.12: WAIS-R test parameters.

Table 2.17 provides elaborate explanations on the WAIS-R test parameters applied on the individuals. The parameters are the index, task, description and proposed abilities measured.

WAIS-R is used for the clinical psychology evaluation of the mental activity in individuals [30]. The attributes provided in Table 2.7 are determinants in making a decision whether the subjects are healthy or afflicted by the mental disorder [31, 32].

Now, let us explain in detail the synthetic WAIS-R data to be used in the algorithm applications for the next sections.

2.4.3.1Obtaining the synthetic WAIS-R dataset

WAIS-R test contains the analysis of the adults’ intelligence. WAIS-R dataset (600 × 21) is obtained from Majaz Moonis, Professor (M.D.) at the University of Massachusetts (UMASS), Department of Neurology and Psychiatry, Stroke Prevention Clinic Director, Worcester, USA.) In order to get synthetic WAIS-R dataset, CART algorithm was applied to the WAIS-R dataset. This test was administered on a total of 400 individuals, where 200 of these subjects were healthy, and the remaining 200 were afflicted by some disorder. The synthetic data have been formed in a homogenous manner. The detailed explanations about the WAIS-R dataset are provided in Table 2.18.

Synthetic WAIS-R dataset was obtained by applying CART algorithm to the WAIS-R dataset. A brief description of the contents in synthetic WAIS-R dataset is shown in Figure 2.13.

Figure 2.13: Obtaining synthetic WAIS-R data.

The synthetic WAIS-R dataset (400 × 21) is obtained by applying CART algorithm to the WAIS-R dataset (600 × 21) as shown in Figure 2.13.

The steps of getting the synthetic WAIS-R dataset (400 × 21) by applying CART algorithm (as shown in Figure 2.7; for more details see Section 6.5) on the WAIS-R dataset (600 × 21) are shown in Figure 2.13.

Table 2.17: The detailed descriptions of the WAIS-R test parameters [31, 32].

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

The average of xmedian and xmedian + 1 is calculated as xmedian+xmedian+12 (see eq. (2.1)).

Steps (7–10) The Gini value of each attribute is calculated for the formation of decision tree from the dataset (see Section 6.5). The lowest Gini value calculated is the root of the decision tree. The WAIS-R dataset is D(600 × 21), and on the basis of the decision tree obtained and the rules derived from this decision tree, the synthetic WAIS-R dataset D(400 × 21) is obtained. In our study, the synthetic WAIS-R dataset D(400 × 21) formed by the application of CART algorithm on the WAIS-R dataset is D(400 × 1).

Table 2.18: Synthetic WAIS-R dataset explanation.

The matrix dimension of the WAIS-R dataset is defined as follows (see Figure 2.5): For i=0,1,2,…,400 in x: i0, i1, . . . , i400; j=0, 1, . . . , 21 in y: j0, j1, . . . , j21 D(400 × 21) represents synthetic WAIS-R dataset.

The sample data values of synthetic WAIS-R dataset is listed in Table 2.19.

CART algorithm has been used to obtain the synthetic WAIS-R dataset D(400 × 21) from WAIS-R dataset D(600 × 21).

Let us now explain how the synthetic WAIS-R dataset D(400 × 21) is generated from WAIS-R dataset through a smaller scale of sample dataset by using CART algorithm as shown in Figure 2.7 (for more details see Section 6.5).

Example 2.3 Sample WAIS-R dataset D(10 × 10) (see Table 2.20) is chosen from WAIS-R dataset D(600 × 21). CART algorithm is applied and sample synthetic dataset is derived. The steps for this process are provided in detail.

Let us apply CART algorithm on the sample WAIS-R dataset D(10 × 10) (see Table 2.15).

Steps (1–6) The data belonging to each attribute in the WAIS-R dataset D(10 × 10) are ranked from lowest to highest (when the attribute values are same, only one value is included in the ranking; see Table 2.20). For each attribute in the WAIS-R 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 following the median value (see eq. (2.4(a))).

The average of xmedian and xmedian + 1 is calculated as xmedian+xmedian+12 (see eq. (2.1)).

The calculation of threshold value for each attribute (column; j: 1,2…,10) in the sample WAIS-R dataset D(10 × 10) listed in Table 2.20 is applied according to the following steps.

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

In Table 2.20, {Age} values (line: i: 1,…,10) in j: 1 column (see Table 2.20) are ranked from lowest to highest. (When the attribute values are same, only one value is included in the ranking.)

Agesort={17,  22,  23,  44,  53,  55,  56,  62,  63,  71}                            Agemedian=53                           Agemedian+1=55                          Age¯=53+552=54

For “Age ≥ 54” and “Age < 54,” two groups are formed.

The steps of the threshold value calculation for the j: 2: {Information} attribute are as follows.

Table 2.19: An excel table to provide an example of synthetic WAIS-R dataset D(400 x 21).

Table 2.20: Sample WAIS-R dataset chosen from WAIS-R dataset (with a dimension of D(10 x 10)).

In Table 2.20, {Information} values (line: i: 1,..., 10) in j: 2 column (see Table 2.20) are ranked from lowest to highest. (When the attribute values are same, only one value is included in the ranking.)

Informationsort={2,  3,  7,  10,  12,  14, 15}                  Informationmedian=10                Informationmedian+1=10               Information¯=10+122=11

For “Information ≥ 11” and “Information < 11,” two groups are identified.

The steps of the threshold value calculation for the j: 3: {Memory} attribute are as follows.

In Table 2.20, {Memory} values (line: i: 1,. . ., 10) in j: 3 column are ranked from lowest to highest. (When the attribute values are same, only one value is included in the ranking.)

Memorysort={3,  6,  7,  9,  11,  12,  13,  16}                     Memorymedian=9                   Memorymedian+1=11                 Memory¯=9+112=10

For “Memory ≥ 10” and “Memory < 10,” two groups are formed.

The steps of the threshold value calculation for the j: 4: {Logical Deduction} attribute are as follows.

In Table 2.20, {Logical deduction} values (line: i: 1,. . ., 10) in j: 4 column are ranked from lowest to highest. (When the attribute values are same, only one value is included in the ranking.)

Logical deductionsort={6,  8,  9,  13,  15,  16}                     Logical deductionmedian=9                 Logical deductionmedian+1=13                 Logical deduction¯=9+132=11

For “Logical deduction ≥ 11” and “Logical deduction < 11,” two groups are formed.

The steps of the threshold value calculation for the j: 5: {Ordering ability} attribute are as follows.

In Table 2.20, {Ordering ability} values (line: i: 1,. . ., 10) in j= 5 column are ranked from lowest to highest. (When the attribute values are same, only one value is included in the ranking.)

Ordering abilitysort:{8,  9,  10,  11,  13}                     Ordering abilitymedian=10                   Ordering abilitymedian+1=11                 Ordering ability¯=10+112=10.50

For “Ordering ability ≥ 10.50” and “Ordering ability < 10.50,” two groups are formed.

The steps of the threshold value calculation for the j: 6: {Three-dimensional modeling} attribute are as follows.

In Table 2.20, {Three-dimensional modeling} values (line: i: 1,. . ., 10) in j: 6 column are ranked from lowest to highest. (When the attribute values are same, only one value is included in the ranking.)

Three−dimensional modelingsort:{5,  6,  8,  9,  10,  12,  14,  15}                      Three−dimensional modelingmedian=9                       Three−dimensional modelingmedian+1=10Three−dimensional modeling¯ =9+1029.50

For “Three-dimensional modeling ≥ 9.50” and “Three-dimensional modeling < 9.50,” two groups are formed.

The steps of the threshold value calculation for the j: 7: {VIV} attribute are as follows.

In Table 2.20, {VIV} values (line: i: 1,. . ., 10) in j: 7 column are ranked from lowest to highest. (When the attribute values are same, only one value is included in the ranking.)

VIVsort:{0.83,  1,  1.16,  1.20,  1.25,  1.50,  1.83,  2.10,  2.80}                                        VIVmedian=1.25
         VIVmedian+1=1.50VIV¯=1.25+1.502=1.30

For “VIV ≥ 1.30” and “VIV < 1.30,” two groups are formed.

The steps of the threshold value calculation for the j: 8: {VIP} attribute are as follows.

In Table 2.20, {VIP} values (line: i : 1,. . ., 10) in j: 8 column are ranked from lowest to highest. (When the attribute values are same, only one value is included in the ranking).

VIVsort:{1.52,  1.56,  1.68,  1.76,  1.84,  1.92,  2.80}                     VIPmedian=1.7                   VIPmedian+1=1.84            VIP¯=1.76+1.842=1.80

For “VIP ≥ 1.80” and “VIP < 1.80,” two groups are formed.

The steps of the threshold value calculation for the j: 9: {VIT} attribute are as follows.

In Table 2.20, {VIT} values (line: i: 1,. . ., 10) in j: 9 column are ranked from lowest to highest. (When the attribute values are same, only one is included in the ranking.)

VITsort = {1.45, 1.46, 1.73, 1.74, 1.86, 1.98, 2.14, 2.20, 2.30, 2.72}

              VITmedian=1.86             VITmedian+1=1.98VIT¯=1.86+1.982=1.90

For “VIT ≥ 1.90” and “VIT < 1.90,” two groups are formed.

The steps of the threshold value calculation for the j: 10: {DM} attribute are as follows.

In Table 2.20, {DM} values (line: i : 1,. . ., 10) in j: 10 column are ranked from lowest to highest. (When the attribute values are same, only one value is included in the ranking.)

DMsort={−0.21, −0.11,  0,  0.06,  0.07,  0.10,  0.285}
               DMmedian=0.06             DMmedian+1=0.07DM¯=0.06+0.072=0.065

For “DM ≥ 0.065” and “DM < 0.065,” two groups are formed.

Steps (7–10) The Gini value of each attribute is calculated for the formation of decision tree of the sample WAIS-R dataset D(10 × 10). After being split into two groups, for each attribute, Gini value calculation is done and the root of the tree is found (for more details see Section 6.5).

Table 2.21 lists the results derived from the splitting of the binary groups for the attributes (j = 1, 2, . . .,6) belonging to sample WAIS-R dataset (patient and healthy class).

The results provided in Table 2.17 for each attribute in the sample WAIS-R dataset D(10 × 10) will be used for the Gini calculation.

The calculation of Gini(Age) for j: 1 (see Section 6.5; eq. (6.8)) is as follows:

Ginileft(Age)=1−[(1/5)2+(4/5)2]=0.32Ginileft(Age)=1−[(1/5)2+(4/5)2]=0.32  Gini(Age)=(5(0.32)+5(0))/10)=0.16

The calculation of Gini(Information) value for j: 1 (see Section 6.5; eq. (6.8)) is as follows:

Ginileft(Information)=1−[(0/4)2+(4/4)2]=0Giniright(Information)=1−[(6/6)2+(0/6)2]=0        Gini(Information)=(4(0)+6(0))/10=0

The calculation of Gini(Memory) value for j: 3 (see Section 6.5; eq. (6.8)) is as follows:

 Ginileft(Memory)=1−[(2/5)2+(3/5)2]=0.48Giniright(Memory)=1−[(4/5)2+(1/5)2]=0.32 Gini(Memory)=(5(0.48)+5(0.32))/10=0.4

The calculation of Gini(Logical deduction) value for j: 4 (see Section 6.5; eq. (6.8)) is as follows:

Ginileft(Logical deduction)=1−[(0/3)2+(3/3)2]=0

Table 2.21: Results derived from the splitting of the binary groups for the attributes that belong to sample WAIS-R dataset (patient and healthy class).

Giniright(Logical deduction)=1−[(6/7)2+(1/7)2]=0.26   Gini(Logical deduction)=(3(0)+7(0.26))/10=0.182

The calculation of Gini(Ordering ability) value for j: 5 (see Section 6.5; eq. (6.8)) is as follows:

  Ginileft(Ordering ability)=1−[(4/7)2+(3/7)2]=0.5Giniright(Ordering ability)=1−[(2/3)2+(1/3)2]=0.46 Gini(Ordering ability)=(7(0.5)−3(0.46))/10=0.49

The calculation of Gini(Three-dimensional modeling) value for j: 6 (see Section 6.5; eq. (6.8)) is as follows:

Ginileft(Three−dimensional modeling)=1−[(2/5)2+(3/5)2]=0.48Giniright(Three−dimensional modeling)=1−[(4/5)2+(1/5)2]=0.32Gini(Three−dimensional modeling)=(5(0.48)+5(0.32))/10=0.4
..................Content has been hidden....................

You can't read the all page of ebook, please click here login for view all page.
Reset
3.147.193.37