12.1.1 The Gram Schmidt Orthogonalization Process

Given linearly independent vectors Z₁,Z₂, . . . Z_k, there exist mutually perpendicular vectors U₁,U₂, ... U_k with the same linear span (Figure 12.2).

The Gram-Schmidt vectors are constructed sequentially by setting:

12.2

where ' denotes transpose of a vector. When calculating MD using the GSP, standardized values for the variables are used. Therefore, in the preceding set of equations Z₁, Z₂, . . . Z_k correspond to standardized values. From this set of equations, it is clear that the transformation process depends largely on the first variable.

Calculation of MD using Gram Schmidt's Orthogonalization Process

Suppose we have a sample of size n, and each sample contains observations on k variables. After standardizing the variables, we will have a set of standardized vectors. Let these vectors be as follows:

After performing GSP, we have orthogonal vectors as follows:

It easily follows that the mean of vectors U₁, U₂, . . . , U_k is zero. Let s₁, s₂, . . . s_k be standard deviations (s.d.s) of U₁,U₂, . . . ,U_k, respectively. Since we have a sample of size n, there will be n different MDs. The MD corresponding to the jth observation of the sample is computed using Equation (12.3).

12.3

where j = 1 . . . n. The values of MD obtained from Equations (12.1) and (12.3)> are exactly the same, and this can be proved easily.

The most important consideration for developing a multivariate measurement system using the Mahalanobis distance is to have a reference point. While it is easier to obtain a reference point for the scale with a single characteristic, it is not so easy to obtain a single reference point when we are dealing with multiple characteristics. Therefore, in MTS, the reference point corresponding to multiple variables is obtained with the help of a group of observations that are as uniform as possible and still enable us to distinguish their different patterns through the MD. These observations are modified in such a way that their center is located at the origin and the corresponding Mahalanobis distances are scaled so as to make the average distance of this group unity. The set of observations used for the reference is often referred to as the Mahalanobis space (MS) or reference group or normal group. Selection of this group is entirely at the discretion of the decision maker conducting the diagnosis. In fraud analytics, this group can be one where there are no incidences of fraud; in a medical diagnosis application, this group can be a set of people without any health problems; and in stock market predictions, this group could correspond to companies having average steady growth in a three-year period. The observations in this group are similar but not the same. Judicious selection of this group is extremely important for accurate diagnosis or predictions.

After developing the scale, the next step is the validation of the scale, which is done with the help of observations that are outside the Mahalanobis space. If the scale is good, the distances of these observations must be consistent with the decision maker's judgment. In other words, if an observation does not belong to the reference group, then it should have a larger distance. To quantify the diagnostics accuracy, we typically use a signal-to-noise (S/N) ratio metric. S/N ratio captures the magnitude of real effects after making some adjustment for uncontrollable variation (noise) by taking into account the correlation between the true or observed information (i.e., input signals) and the output of the diagnostic system (the Mahalanobis distance). In a multivariate context, S/N ratio can simply be defined as the measure of the accuracy of the diagnostics. A typical multidimensional predictive system that is used in MTS can be described using Figure 12.3.

As mentioned, the output or diagnosis accuracy should correlate well with the input signal, and S/N ratios measure this correlation. The diagnostics are conducted based on the information on the variables defining the system, and they should be “accurate” even in the presence of noise factors such as different places of measurement, operating conditions, and so forth.

If the accuracy of the diagnosis is satisfactory, then we identify a useful subset of variables that is sufficient for the measurement scale with acceptable diagnostic accuracy. Experience shows that, in many cases, the accuracy with useful variables is better than that with the original set of variables. However, in some cases, the accuracy with useful variables might be less—which might still be desirable, as it helps reduce the cost of inspection or measurement. In multidimensional systems, the total number of combinations to be examined would be of the order of several thousands; hence, it is not possible to examine all combinations. Here, we propose use of orthogonal arrays (OAs) to reduce the number of combinations to be tested. OAs are developed to estimate the effects of the variables by minimizing the number of combinations to be examined. They have been in use for quite a long time in the field of experimental design. In MTS, the variables are assigned to different columns of an OAs. Based on S/N ratios obtained from different variable combinations, important variables are identified. The future diagnosis is carried out only with these important variables.

12.3.1 The Role of Orthogonal Arrays

In robust engineering, the main role of OAs is to permit engineers to evaluate a product design with respect to robustness against noise and the cost involved. OA is an inspection device to prevent a poor design from going downstream.

Usually, these arrays are denoted as L_a(b^c)

1. a = number of experimental runs
2. b = number of levels of each factor
3. c = number of columns in the array
4. L denotes Latin square design

Arrays can have factors with many levels, although two- and three-level factors are most commonly encountered.

In the MTS, orthogonal arrays are used to select the variables of importance by minimizing the different combinations of the original set of variables. The variables are assigned to the different columns of the array. The presence and absence of the variables are considered the levels. Since the variables have only two levels, two-level arrays are used in MTS application. The importance of the variables is judged based on their ability to measure the degree of abnormality on the measurement scale. For each run of an OA, the MDs corresponding to the known abnormal conditions or the conditions outside the MS are computed. We need not consider the MDs corresponding to the reference group because we know that this group is healthy and, based on this group, scale is constructed.

For the purpose of illustration, consider the following example.

Let there be five variables X₁, X₂, X₃, X₄, and X₅. Let us allocate these variables in the first five columns of the L₈(2⁷) array, as shown in Table 12.1. In this table, “1” indicates the level corresponding to the presence of a variable and “2” indicates the level corresponding to the absence of a variable. Consider the first three runs of Table 12.1.

2nd Run: 1-1-1-2-2

In this run, the MS is constructed with the variable combination X₁-X₂-X₃. The correlation matrix is of the order 3 × 3. The MDs of abnormals are estimated with this matrix.

12.3.2 The Role of S/N Ratios in MTS

The S/N ratio tries to capture the magnitude of true information (i.e., signals) after making some adjustment for uncontrollable variation (i.e., noise).

In the case of multivariate diagnosis, the S/N ratio is defined as the measure of the accuracy of the measurement scale for predicting abnormal conditions. S/N ratio is expressed in decibel (dB) units. A higher value for the S/N ratio means lower error of prediction. Note that S/N ratios are calculated using abnormal conditions only, as it is required that the accuracy of the scale be tested based on these conditions.

In multidimensional applications, it is important to identify a useful set of variables that is sufficient to detect the abnormals. It is also important to assess the performance of the given system and the degree of improvement in the performance. In this method, S/N ratios are used to accomplish these objectives.

After obtaining the MDs for the known abnormal conditions corresponding to the various combinations of an OA, S/N ratios are computed for all these combinations to determine the useful set of variables. S/N ratios are important to improve the accuracy of the measurement scale and reduce the cost of diagnosis. The useful set of variables is obtained by evaluating the gain in the S/N ratio, which is the difference between the average S/N ratio when the variable is used in an OA and the average S/N ratio when the variable is not used in an OA. If the gain is positive, then the variable is useful.

12.3.3 Types of S/N Ratios

In MTS application, typically the following types of S/N ratios are used:

• Larger-the-better type
• Nominal-the-best type
• Dynamic type

When the true levels of abnormals are not known, the larger-the-better type of S/N ratios is used if all the observations outside the reference group are abnormals. This is because the MDs for abnormals should be higher. If the observations outside the reference group are a mixture of normals and abnormals, then the nominal-the-best type of S/N ratios is used. When the levels of abnormals are known, the dynamic type of S/N ratios is used. In some cases, it is quite difficult to obtain the levels of severity of the abnormals based on the knowledge of the person conducting the diagnosis. In such situations, if we know the abnormals with different degrees of severity (different categories), dynamic S/N ratios can be used by taking the average of the square root of the MDs (working average) in each category as known levels of abnormals.

Equations for S/N Ratios

Larger-the-Better Type

The procedure for calculating S/N ratios corresponding to a run of an OA is as follows:

Let there be t abnormal conditions. Let D₁², D₂², . . . , D_t² be MDs corresponding to the abnormal situations. The S/N ratio (for the larger-the-better criterion) corresponding to the qth run of an OA is given by the following:

12.4

Nominal-the-Best Type

The procedure for calculating S/N ratios corresponding to a run of an OA is as follows:

Let there be t abnormal conditions. Let D₁², D₂², . . . , D_t² be MDs corresponding to the abnormal situations. The S/N ratio (nominal-the-better type) corresponding to the qth run of an OA is calculated as follows:

where is the average of the D_is

12.5

Dynamic Type

An examples of the dynamic type is a weather forecasting system and the specific case of rainfall prediction. The procedure for calculating S/N ratios corresponding to a run of an OA is as follows:

Let there be t abnormal conditions. Let D₁², D₂², . . . , D_t² be the MDs corresponding to the abnormal conditions. Let M₁, M₂, . . . , M_t be the true levels of severity (rainfall values in the rainfall prediction system example).

For each run of the OA, construct the following ANOVA table:

Source	Degrees of Freedom	Sum of Squares
β	1	Sβ
e	T − 1	Se
Total	T	ST

In the preceding ANOVA table,

The S/N ratio corresponding to the qth run of OA is given by the following:

12.6

In the case of working average methods, true levels of severity (M) are replaced by the average of the square root of MDs.

12.3.4 Direction of Abnormals

One of the main reasons for using the Mahalanobis distance in multivariate diagnosis is its capability of distinguishing abnormalities from a “healthy” or “normal” group. Sometimes, the abnormal conditions arise out of extremely good situations. Therefore, it is important to identify the direction of the abnormalities to enable the distinction between “good” and “bad” abnormalities to be made. This makes the diagnosis process more effective. If the MD is calculated by using the inverse of the correlation matrix (the MTS method), such distinctions cannot be made. However, this can be accomplished if we use the Gram-Schmidt orthogonalization process to calculate the MD. This section outlines a procedure for identifying the direction of abnormals with suitable equations for different cases. The use of this procedure is demonstrated for a graduate student admission system.

The GSP vectors obtained from Equation (12.2) can be written as:

12.7

where c_2,1, c_2,1, . . . , c_k,k−1 are Gram-Schmidt vector coefficients.

First, a discussion of the direction of abnormals is presented in detail for a two-variable case. The same logic is extended for a higher number of variables.

In the case of two variables, the distribution of the points forms an elliptical shape. Since the MS is constructed based on this distribution, for simplicity, we can represent this distribution as the MS. It is a well-known fact that the elliptical shape remains unchanged after orthogonal transformation. Therefore, the distribution corresponding to the GSP vectors will also have an elliptical shape.

We know that the mean of the GSP vectors is located at the zero point. For the two-variable case, the mean of U₁ and U₂ is located at (0,0). All conditions (normal and abnormal) are above or below this point with the exception of the points that match the zero point (mean). Therefore, it is clear that the abnormals are above or below the zero point. Based on the position of the abnormals (above or below zero point) and the value of the MD, we can distinguish between “good” and “bad” abnormals. In the case of two variables, depending on the types of characteristics, we have the following four cases:

1. Both U₁ and U₂: larger-the-better type
2. U₁: smaller-the-better type; U₂: larger-the-better type
3. U₁: larger-the-better type; U₂: smaller-the-better type
4. Both U₁ and U₂: smaller-the-better type

For a two-variable case, the vectors U₁ and U₂ can be written as follows:

or

For all four of these cases, the rules for identifying the direction of abnormals are described in the following sections.

Case 1: Both U₁ and U₂—Larger-the-Better Type

An example where the characteristics U₁ and U₂ are of the larger-the-better type is a student admission system. In this system, U₁ could be “GPA” (grade point average) and U₂ could be “TOEFL exam score.” In this case, a condition corresponding to very low U₁ and U₂ is an abnormal, and a condition corresponding to very high U₁ and U₂ is also an abnormal.

For the jth condition to be a good-abnormal, the elements of U₁ and U₂ should be above zero (positive) and the corresponding MD should be larger than the threshold (T). These conditions can be mathematically represented as:

1. u_1j > 0 or z_1j > 0; u_2j > 0 or z_2j − c_2,1 u_1j > 0 or z_2j > c_2,1 u_1j and
2. MD_j > T

From Equation (12.3), we can write:

12.8

The pictorial representation of these rules is given in Figure 12.5.

For this case, the decision rule can be stated as follows: If elements of U₁ and U₂ corresponding to an abnormal condition are positive and MD is higher than the threshold (T), then the abnormal condition can be classified as “good” abnormality; otherwise, it is a “bad” abnormality.

Case 2: U₁—Smaller-the-Better Type and U₂—Larger-the-Better Type

An example of this situation is the banking system for granting loans.Here, U₁ could be “number of people in a household” and U₂ could be “income level of a family.” In this case, a condition corresponding to very high U₁ and very low U₂ is an abnormal, and a condition corresponding to very low U₁ and very high U₂ is also an abnormal.

For the jth condition to be a good-abnormal, the element of U₁ should be below zero (negative) and that of U₂ should be above zero (positive) and the corresponding MD should be larger than the threshold (T). These conditions can be mathematically represented as follows:

1. u_1j < 0 or z_1j < 0; u_2j > 0 or z_2j − c_2,1 u_1j > 0 or z_2j > c_2,1 u_1j and
2. MD_j > T

From Equation (12.3), we can write:

12.9

The pictorial representation of these rules is given in Figure 12.6.

For this case, the decision rule can be stated as follows: If the element of U₁ is negative and that of U₂ is positive and MD is higher than the threshold (T), then the abnormal condition can be classified as “good” abnormality; otherwise, it is “bad” abnormality.

Case 3: U₁—Larger-the-Better Type and U₂—Smaller-the-Better Type

An example of this situation is an inspection system, where the characteristic “tensile strength” (U₁) is of the larger-the-better type and the characteristic “number of occurrences of a particular defect” (U₂) is of the smaller-the-better type. In this case, a condition corresponding to very low U₁ and very high U₂ is an abnormal, and a condition corresponding to very high U₁ and very low U₂ is also an abnormal.

For the jth condition to be a good-abnormal, the element of U₁ should be above zero (positive) and that of U₂ should be below zero (negative) and the corresponding MD should be larger than threshold (T). These conditions can be mathematically represented as follows:

1. u_1j > 0 or z_1j > 0; u_2j < 0 or z_2j − c_2,1 u_1j < 0 or z_2j > c_2,1 u_1j and
2. MD_j > T

From Equation (12.3), we can write:

12.10

The pictorial representation of these rules is given in Figure 12.7.

For this case, the decision rule can be stated as follows: If the element of U₁ is positive and that of U₂ is negative and MD is higher than the threshold (T), then the corresponding abnormal condition can be classified as “good” abnormality; otherwise, it is a “bad” abnormality.

Case 4: Both U₁ and U₂ Are Smaller-the-Better Type

An example of a situation where both the characteristics are of the smaller-the-better type is a printed circuit board inspection system. In this example, the characteristics “number of occurrences of a particular defect” (U₁) and “line width reduction after etching process” (U₂) are of the smaller-the-better type. In this case, a condition corresponding to very high U₁ and U₂ is an abnormal, and a condition corresponding to very low U₁ and U₂ is also an abnormal.

For the jth condition to be a good-abnormal, the elements of U₁ and U₂ should be below zero (negative) and the corresponding MD should be larger than the threshold (T). These conditions can be mathematically represented as follows:

1. u_1j < 0 or z_1j < 0; u_2j < 0 or z_2j − c_2,1 u_1j < 0 or z_2j < c_2,1 u_1j and
2. MD_j > T

From Equation (12.3), we can write:

12.11

The pictorial representation of these rules is given in Figure 12.8.

For this case, the decision rule can be stated as follows: If the elements of U₁ and U₂ corresponding to an abnormal condition are negative and MD is higher than the threshold (T), then the abnormal condition can be classified as “good” abnormality; otherwise, it is a “bad” abnormality.

Decision Rule for Higher Dimensions

If there are k variables and the sample size is n, the conditions for the jth abnormal to be good are:

1. u_1j > 0, if U₁ is larger-the-better type (<0 if U₁ is smaller-the-better type)
2. u_2j > 0, if U₂ is larger-the-better type (<0 if U₂ is smaller-the-better type)
3. 
4. u_kj > 0, if U_k is larger-the-better type (<0 if U_k is smaller-the-better type)

and

or

otherwise, the abnormal condition is bad.

Though this procedure provides guidelines for identifying the direction of abnormals, it is left to the decision maker to decide the type of abnormal conditions. For example, in a student admission system, where the variables are mostly of the higher-the-better type, a student with very high scores on most of the variables can be considered a good-abnormal by the decision maker.

It is important to note that this procedure is applicable only to cases where we have knowledge about the type of variables (such as larger-the-better type or smaller-the-better type).

Graduate Admission System Example

The applicability of this procedure can be demonstrated using the example of a university student admission system. Suppose a student is admitted into the university based on the following three variables:

• Grade point average (GPA): This should be as high as possible on a 4.0 scale.
• GMAT score: This should be as high as possible (maximum score is 1600).
• SAT-Math score: This should be as high as possible (maximum score is 800).

The data corresponding to the reference group along with GSP vectors and MDs are shown in Table 12.2. Since these variables are of the larger-the-better type, this example falls into the first category. There are eight abnormal conditions A1 . . . A8. The abnormal data along with GSP vectors and MDs are given in Table 12.3.

From Table 12.3, it is clear that all of the abnormalities have high MDs. Based on MDs alone, we cannot distinguish between good and bad abnormalities. To make such distinctions, we need to look at GSP vectors. Assuming the threshold (T) is set at 3.0, A₇ and A₈ can be classified as “good” abnormalities because GSP vectors corresponding to these abnormalities have positive signs and their MDs are higher than threshold. The remaining abnormalities (A₁ to A₆) can be classified as “bad” abnormalities because not all corresponding GSP vectors have positive signs even though their MDs are higher than threshold.

12.5.1 Improvements Made Based on Recommendations from MTS Analysis

A new measurement system was developed by including the 11 attributes from the following:

• Survey
• Volatility within the client
• Volatility within the company

The company was convinced that the performance of the measurement system with these 11 attributes was much more meaningful and objective as compared to traditional subjective evaluations.

12.7.1 Construction of the Reference Group

For each of the five segments, the reference group was built based on the data set. For example, the reference group for S₁ was constructed by ­taking observations on 55 variables corresponding to that segment. With these reference groups, the corresponding MDs were calculated.

12.7.2 Validation of the Scale

The second stage in the MTS method is validation. The outside conditions for a given segment were chosen from the other segments. It was found that the abnormals, in all of the cases, had higher MDs; thus, the scale was validated.

12.7.3 Identification of Useful Variables

To identify a useful set from the 55 original variables, the L₆₄(2⁶³) orthogonal array (OA) was chosen for each segment analysis. The S/N ratios were computed based on the larger-the-better criterion, because prior information about abnormals was not available.

Table 12.8 provides the list of useful variables corresponding to all five segments. Since for each segment a suitable strategy was to be developed with a manageable number of variables, it was decided to restrict the number of variables per segment to 20. The selection of these variables was done based on the magnitude of gains in the S/N ratio.

With the useful set of variables, confirmation runs were conducted for all five segments. The results of the confirmation indicated that these variables were capable of recognizing the given patterns as effectively as in the case with all 55 variables. Figure 12.15 shows the recognition power of the useful variables in the respective car segments.

Decreasing the number of variables in the optimal system helped to reduce the complexity of the multidimensional systems and develop effective strategies to improve sales.

12.8.1 Apparatus

A diagram of the 127K27330 test fixture is shown in Figure 12.16. From left to right, the holding fixture consists of a hand shield, high-resolution encoder, hysteresis brake, customized brake coupling, the motor, and a Destaco clamp. The motor's output coupling engages the brake coupling, which turns the hysteresis brake, which turns the encoder. The Destaco clamp prevents any lateral movement.

12.8.2 Sensors

A PCB accelerometer (Model J352A78, 100 mV/g) was used to measure vibration. This accelerometer was stud mounted in the motor mount plate (see Figure 12.16).A 1-ohm resistor was used for current sensing. A Gurley (Model 8435H, 32,000 pulses/revolution) high-resolution encoder was used to measure motion and speed.

12.8.3 High-Resolution Encoder

Because the speed of the motor is approximately 1.5 rpm, a standard tachometer could not be used, since its output voltage would be negligible. A high-resolution encoder from Gurley (Model 8435H) was used for measuring motor speed. This encoder has a resolution of 32,000 pulses/revolution. A US Digital PC6-84-4 quadrature interpolator was used to increase the resolution of this encoder from 32,000 to 128,000 pulses/revolution.

A LabView example program called Buffered Counting of Events.vi served as the basis for inputting and interpreting the encoder pulse stream.

If, for example, there is an incoming pulse train and it is required that the pulses be counted and the count value accumulated after every 5 milliseconds (ms), here is what needs to be done:

1. Make one of the counters generate a pulse train of 5 ms (200 Hz) and connect the OUT of this counter (no. 1) to the GATE of the next counter (no. 2).
2. Connect the signal to the SOURCE of counter 2.
3. The pulse train generated by counter 1 goes high every 5 ms.

When it goes high for the first time, counter 2 starts counting the signal. It keeps counting until it sees another high at the gate, which would be after 5 ms. At this point, the count value of counter 2 is written to the buffer. Counter 2 is actually still counting. When it sees the next high on the gate, after 5 ms, again the counter value is written to the buffer. Therefore, the buffer would have monotonically increasing values representing the accumulated count of counter 1 after each 5-ms interval. We can do this buffered counting at any interval by changing the 200 Hz (5 ms) to some other value.

To summarize, the output of counter 1 is physically tied to the gate of counter 2. Counter 1 time-stamps the running count of counter 2.

12.8.4 Life Test

The life testing apparatus from the 127K1581 motor test was reused for the 127K27330 life test with a few modifications to the motor baseplate. A QuickBasic program called dc_motor.bas and an optomux/opto22 system provided electronic control. Ten new 127K27330 motors were used for the life test. The parameters were measured at intervals of 10,000 cycles starting from 0 cycles to 120,000 cycles during the life test.

12.8.5 Characterization

After every 10,000 cycles the 10 motors were taken off the test for characterization. The LabView characterization program is called hodaka_dc_motor.vi. The characterization program ran the motor at three different operating conditions: (1) motor voltage (mv) = 21.6 V, brake voltage (bv) = 4.95 V; (2) mv = 21.6 V, bv = 0 V; (3) mv = 12 V, bv = 4.95 V. These levels were specifically chosen to stress the motor and force a separation between new-motor performance and used-motor performance. The 4.95 V brake voltage corresponded to a 1.96-Nm (276.4-oz-in) load. In these tests, a personal computer, generic SA interface box, and National Instruments AT-MIO-16E-10 board were used for data acquisition.

Of the various parameters measured by hodaka_dc_motor.vi, the following parameters, in the three different operating conditions, were considered to be important:

1. Current
2. Vibration
3. Speed
4. Count—count measurements of counter 2

Some of the parameters could not be considered, because they were fixed while conducting the life test. Since speed measurements were derived from the counts by taking the derivative of the buffered counts, there was no need to consider both of them. Therefore, the parameter count was dropped from the analysis. We were left with nine variables: current (C), vibration (V), and speed (S) in three operating conditions. The variables were denoted as X₁, X₂, . . . , X₉. The data on these parameters was collected by using hodaka_dc_motor.vi.

After collecting data on these nine variables, the MTS method was used for data analysis.

12.8.6 Construction of the Reference Group or Mahalanobis Space

Motors 1 through 10 were all new when life testing was started. The presumption was that all new motors are good. Therefore, the parameters measured at 0 cycles (corresponding to these motors) were used to construct the Mahalanobis space for the healthy group. When data about the motors was combined, there were numerous observations corresponding to the parameters at 0 cycles. These parameters reached steady state after some point in time. To verify this fact, data on these parameters was plotted on separate graphs. Figure 12.17 shows some of these graphs. From this figure, it is clear that that the parameters attained steady state after some point in time.

To construct the Mahalanobis space (MS), it was not necessary to consider all of the data because the number of observations in the data was large. Therefore, a subset of the original data was used for constructing the MS. The subset selection was done in such a way that it contained an equal number of observations from the steady state and from the transient state. This helped in constructing a uniform MS. Based on this, the MDs corresponding to the observations in the MS were calculated.

12.8.7 Validation of the MTS Scale

After constructing the measurement scale, it was necessary to test the accuracy of the scale by measuring the MDs of some known conditions outside the MS. In this application, parameters measured after 10k, 20k . . . , 120k cycles were considered conditions outside the MS. Figure 12.18 shows the average MDs corresponding to these cycles. From this figure, we can say that the scale is good because abnormals have higher-value MDs. It is interesting to note that as the number of cycles increases the average MD also goes up, indicating that the motor performance deteriorates with an increase in the number of cycles.

12.8.8 Selection of Useful Variables

In the third stage of the analysis, useful parameters were selected by using orthogonal arrays and S/N ratios. Since there were nine parameters, the L₁₂(2¹¹) orthogonal array was used. Since we did not have prior knowledge about the abnormals, the larger-the-better type of S/N ratios was used in both methods. The results of MTS analysis are given in Table 12.9.

In Table 12.9, the variable combination X₁-X₂-X₃-X₅-X₆-X₈-X₉ is considered as a useful combination because the variables in this combination have positive gains.

A confirmation run was conducted for the useful variables in both methods. It was found that from abnormal MDs we could distinguish normals and abnormals. Figure 12.19 shows the difference between the original combination and the optimal combination in terms of distinction between normals and abnormals. From this figure, it is clear that the distinction was much better with the optimal combination.