Improved statistical features-based control chart patterns recognition using ANFIS with fuzzy clustering

Abstract

Various types of abnormal control chart patterns can be linked to certain assignable causes in industrial processes. Hence, control chart patterns recognition methods are crucial in identifying process malfunctioning and source of variations. Recently, the hybrid soft computing methods have been implemented to achieve high recognition accuracy. These hybrid methods are complicated, because they require optimizing algorithms. This paper investigates the design of efficient hybrid recognition method for widely investigated eight types of X-bar control chart patterns. The proposed method includes two main parts: the features selection and extraction part and the recognizer design part. In the features selection and extraction part, eight statistical features are proposed as an effective representation of the patterns. In the recognizer design part, an adaptive neuro-fuzzy inference system (ANFIS) along with fuzzy c-mean (FCM) is proposed. Results indicate that the proposed hybrid method (FCM-ANFIS) has a smaller set of features and compact recognizer design without the need of optimizing algorithm. Furthermore, computational results have achieved 99.82% recognition accuracy which is comparable to published results in the literature.

Appendices

Appendix 1

The mathematical expressions are summarized here for eight different types of statistical features.

• 1. Mean: Mean is the average value of all points. Here $X_{i}$ is the input vector for fully developed pattern and n is the window size. The general expression for the mean is given as

  $${\text{Mean}} = \frac{{\mathop \sum \nolimits_{i = 1}^{n} X_{i} }}{n}$$

  (5)

• 2. Standard Deviation: The standard deviation is found by general formula given as

  $${\text{Standard}}\;{\text{Deviation}} = \sqrt {\frac{{\mathop \sum \nolimits_{i = 1}^{n} \left( {X_{i} - \mu } \right)^{2} }}{n - 1}}$$

  (6)

  where X_i is the individual measurement and µ is the mean and n is the total number of samples points or window size.

• 3. Skewness: Skewness is the estimate of symmetry of the shape of the distribution it may either zero or positive or negative. The estimate of the data points from X₁ to X_n is given by equations

  $$\gamma = \frac{{n\mathop \sum \nolimits_{i = 1}^{n} \left( {X_{i} - \mu } \right)^{3} }}{{\left[ {\left( {n - 1} \right)\left( {n - 2} \right)s^{3} } \right]}}$$

  (7)

  where X_i is individual µ is mean and s is sample standard deviation and n is the total number of points or window size [37].

• 4. Mean Square Value: Mean square value is calculated using following formula

  $${\text{MSV}} = \frac{1}{n + 1}\mathop \sum \limits_{i = 1}^{n} X_{i}^{2}$$

  (8)

  where X_i is individual values and n is the total number of points or window size.

• 5. CUSUM: CUSUSM is the cumulative SUM chart values. The last statistical value of CUSUM is taken as feature in this study. The general formula for upper and lower CUSUM statistics is given as:

  $$C_{i}^{ + } = \hbox{max} \left[ {0,x_{i} - \left( {\mu_{0} + K} \right) + C_{i - 1}^{ + } } \right]$$

  (9a)
  $$C_{i}^{ - } = \hbox{max} \left[ {0,\left( {\mu_{0} - K} \right) - x_{i} + C_{i - 1}^{ - } } \right]$$

  (9b)

  where starting values of $C_{i}^{ + } ,C_{i}^{ - }$ are taken zero.

• 6. Autocorrelation: The average autocorrelation exist when the signals are dependent on previous part of signals. The autocorrelation is calculated by following equation:

  $$A_{xx} \left[ m \right] = \frac{1}{N + 1 - m}\left[ {x_{0} x_{1} + x_{1} x_{1 + m} + \ldots x_{N - m} x_{N} } \right]$$

  (10)

• 7. Kurtosis: Kurtosis measure the peakness of the distribution. The following formula gives the kurtosis for any distribution

  $$k = \frac{{E\left[ {\left( {X - \mu } \right)^{4} } \right]}}{{\sigma^{4} }} - 3$$

  (11)

  The factor 3 is used for normal distribution in order to get k = 0.

• 8. Slope: The first-order line fitting to the data points is given by equation

  $$Y = mX + C$$

  (12)

  where C y-intercept and m is the slope of the line.

  This can be calculated by the following equation

  $$m = \frac{{\mathop \sum \nolimits_{i = 1}^{n} (X_{i} - \bar{X})\left( {Y_{i} - \bar{Y}} \right)}}{{\mathop \sum \nolimits_{i = 1}^{n} \left( {X_{i} - \bar{X}} \right)^{2} }} .$$

  (13)

  The slope m is used as feature in this study.

Appendix 2

• 1. Normal pattern: Normal pattern is the pattern for stable processes. The basic equation is given by

  $$y_{t } = \mu + {\rm N}_{t}$$

  (14)

  where N_t is the random generation and µ is the mean.

• 2. Trend up/down: The trend up equation is given by

  $$y_{t } = \mu + {\rm N}_{t} \pm \gamma_{1} t$$

  (15)

  where the N_t is the random generation and $\gamma_{1}$ is slope of the trend. The $\pm$ sign is used for trend up and down patterns, respectively.

• 3. Shift up/down: The shift up/down equation is given by

  $$y_{t } = \mu + {\rm N}_{t} \pm \gamma_{2}$$

  (16)

  where ${\rm N}_{t}$ is the random generation and $\gamma_{2}$ is the shift magnitude. The $\pm$ sign is used for shift up and down patterns, respectively.

• 4. Cyclic pattern: The basic equation for cyclic pattern is given by

  $$y_{t } = \mu + {\rm N}_{t} \pm \gamma_{3} \sin \left( {\frac{2\pi t}{{\gamma_{4} }}} \right)$$

  (17)

  where ${\rm N}_{t}$ is the random generation and $\gamma_{3}$ is the amplitude and $\gamma_{4}$ is the frequency of signals.

• 5. Systematic pattern: The basic equation for systematic departure is given by

  $$y_{t } = \mu + {\rm N}_{t} \pm \gamma_{5} \left( { - 1} \right)^{t}$$

  (18)

  where ${\rm N}_{t}$ is the random generation and $\gamma_{5}$ is the systematic departure parameters.

• 6. Stratification pattern: The basic equation for stratification is given by

  $$y_{t } = \mu + \gamma_{6} {\rm N}_{t}$$

  (19)

  where ${\rm N}_{t}$ is the random generation and $\gamma_{6}$ is the stratification parameter.

Appendix 3

• 1. Accuracy Overall accuracy formula is given by

  $${\text{Accuracy}} = \frac{{{\text{TP}} + {\text{TN}}}}{N}*100\%$$

  (20)

• 2. Sensitivity: The measure of actual positive is called sensitivity and is given by formula

  $${\text{Sensitivity}} = \frac{\text{TP}}{{{\text{TP}} + {\text{FN}}}}*100\%$$

  (21)

• 3. Specificity: The measure of actual negative is called specificity.

  $${\text{Specificity}} = \frac{\text{TN}}{{{\text{TN}} + {\text{FP}}}}*100\%$$

  (22)

  where TP, TN, FN, FP represent abbreviations of True Positive, True Negative, False Negative and False Positive, respectively. Sensitivity indicates how well the methods/algorithm performs on the positive class and specificity indicates how well the algorithm performs on the negative class. N represents total number of patterns.

• 4. Root-mean-square error (RMSE): The RMSE equates the anticipated output value y, and the actual output of the FIS $\hat{y}$. Let N represent the number of control chart patterns for training. The RMSE for training can be expressed as:

  $${\text{RMSE}} = \sqrt {\frac{{\mathop \sum \nolimits_{i = 1}^{N} y_{i} - \hat{y}_{i} }}{N}}$$

  (23)