An hybrid detection system of control chart patterns using cascaded SVM and neural network–based detector

Abstract

Early detection of unnatural control chart patterns (CCP) is desirable for any industrial process. Most of recent CCP recognition works are on statistical feature extraction and artificial neural network (ANN)-based recognizers. In this paper, a two-stage hybrid detection system has been proposed using support vector machine (SVM) with self-organized maps. Direct Cosine transform of the CCP data is taken as input. Simulation results show significant improvement over conventional recognizers, with reduced detection window length. An analogous recognition system consisting of statistical feature vector input to the SVM classifier is further developed for comparison.

Appendices

Appendix 1: Formulation of features

Following are the mathematical formulations of the features used in this work, all of which have been taken from [4] with some minor modifications, in case of SRANGE, RVE and ABDPE.

• (a) AASBP: It is the average of absolute slope of straight lines passing through the consecutive points in terms of process standard deviation.

  $$ {\text{AASBP}} = {{\left( {\sum\limits_{{i = 1}}^{{N - 1}}{\left| {{\frac{{y_{{i + 1}} - y_{i} }}{{t_{{i + 1}} - t_{i} }}}}\right|} } \right)} \mathord{\left/ {\vphantom {{\left({\sum\limits_{{i = 1}}^{{N - 1}} {\left| {{\frac{{y_{{i + 1}} -y_{i} }}{{t_{{i + 1}} - t_{i} }}}} \right|} } \right)} {\left( {N -1} \right)\sigma }}} \right. \kern-\nulldelimiterspace} {\left( {N -1} \right)\sigma }} $$

  (15)

  Here, N is the number of points in the CCP run window, i.e., the length of run window. Y _i is the observation for ith sample point. Since in the CCP run window, the x-axis represents the sample numbers that are consecutive positive integers, difference between two consecutive time samples $\left({t_{i + 1} \;\rm {and}\;t_{i}} \right)$ is taken as 1. This applies to all of the following expressions. $\sigma$ is the standard deviation (note that it was used to generate the CCP, hence it is not the estimated process standard deviation $\hat{\sigma}$).

• (b) RVE: Ratio between variance of the observations and mean sum of squares of errors of the least-square (LS) line representing an overall pattern. In this study, instead of corrected sum of squares [4], uncorrected sums of squares of the observations and sample points are used.

  $$ {\text{RVE}} = {{\left( {{\frac{{s_{{y^{2} }} }}{{N - 1}}}}\right)} \mathord{\left/ {\vphantom {{\left( {{\frac{{s_{{y^{2} }}}}{{N - 1}}}} \right)} {\left( {{\frac{1}{{N - 2}}}\left( {s_{{y^{2}}} - {\frac{{s_{{yt}}^{2} }}{{s_{{t^{2} }} }}}} \right)} \right)}}}\right. \kern-\nulldelimiterspace} {\left( {{\frac{1}{{N -2}}}\left( {s_{{y^{2} }} - {\frac{{s_{{yt}}^{2} }}{{s_{{t^{2} }}}}}} \right)} \right)}}$$

  (16)

  where,

  $$ S_{{y^{2}}} = \sum\limits_{i}^{N} {\left({y_{i}} \right)^{2}} $$
  $$ S_{yt} = \sum\limits_{i}^{N} {y_{i} t_{i}} $$
  $$ S_{{t^{2}}} = \sum\limits_{i}^{N} {\left({t_{i}} \right)^{2}} $$

• (c) ALSPI: It is found from the area between the pattern and LS line per interval in terms of estimated process SD.

  $$ {\text{ALSPI}} = (area\;between\;pattern\;and\;LS\;line)/\left({N - 1} \right)\hat{\sigma} $$

  (17)

  It is calculated by summing the triangles and trapeziums that are formed by the LS line and overall pattern.

• (d) ASL: Average slope of the straight lines passing through six pair-wise combinations of midpoints in four equal segments. The midpoint of each segment can be calculated by,

  $$ y_{{mid}} = {{\left( {\sum\limits_{{i = i_{{sg}} }}^{{i_{{sg}}+ 7}} {y_{i} } } \right)} \mathord{\left/ {\vphantom {{\left({\sum\limits_{{i = i_{{sg}} }}^{{i_{{sg}} + 7}} {y_{i} } } \right)}{\left( {{\frac{N}{2}}} \right)}}} \right.\kern-\nulldelimiterspace} {\left( {{\frac{N}{4}}} \right)}} $$

  (18)

  and,

  $$ t_{{mid}} = {{\left( {\sum\limits_{{i = i_{{sg}} }}^{{i_{{sg}}+ 7}} {t_{i} } } \right)} \mathord{\left/ {\vphantom {{\left({\sum\limits_{{i = i_{{sg}} }}^{{i_{{sg}} + 7}} {t_{i} } } \right)}{\left( {{\frac{N}{2}}} \right)}}} \right.\kern-\nulldelimiterspace} {\left( {{\frac{N}{4}}} \right)}} $$

  (19)

  where, i _sg  = starting sample number of the segment. Since there will be four such segments, 6 $\left( {C_{2}^{4} } \right)$ straight lines can be generated.

  $${\text{ASL}} = {{\left( {\sum\nolimits_{\begin{subarray}{l}{i,j} \\ {i < j} \end{subarray} } {s_{{i,j}} } } \right)}\mathord{\left/ {\vphantom {{\left({\sum\nolimits_{\begin{subarray}{l} {i,j} \\ i < j\end{subarray} } {s_{{i,j}} } } \right)} 6}} \right.\kern-\nulldelimiterspace} 6}$$

  (20)

  where s _ij is the slope of the line joining the midpoint of ith and jth segment.

• (e) SRANGE: Range of slopes of straight lines passing through six pair-wise combinations of midpoints found previously.

  $$ {\text{SRANGE}} = {\text{maximum}}\left({S_{ij}} \right)-{\text{minimum}}\left({S_{ij}} \right) $$

  (21)

  where, i < j and i = 1, 2, 3 and j = 2, 3, 4. Here, SRANGE is modified by scaling it with estimated process SD $\left( {\hat{\sigma }} \right)$ i.e.,

  $$ {\text{SRANGE}}^{/} = {\text{SRANGE}}/\hat{\sigma} $$

  (22)

• (f) REPEPE: The CCP run window is divided into two segments in such a way that the two LS lines fitted on the segments lead to minimum pooled mean square error (PMSE). In this way, the optimal portioning point is found (i _opt ), such that

  $$ N/4 \le i_{opt} \le 3 N/4 $$

  (23)

  If the equations of the two LS lines are

  $$ y_{ls1} \left(i \right) = s_{1} i + c_{1} $$

  (24)
  $$ y_{ls2} \left(i \right) = s_{2} i + c_{2} $$

  (25)

  then, minimum

  $$ {\text{PMSE}} = {\frac{{\left({\left({i_{opt} - 1} \right)\sum\nolimits_{i = 1}^{{i_{opt}}} {\left({y_{ls1}\left(i \right) - y_{i}} \right)^{2} + \left({N - i_{opt} - 1} \right)\sum\nolimits_{{i = i_{opt} + 1}}^{N} {\left({y_{ls2} \left(i \right) - y_{i}} \right)^{2}}}} \right)}}{N - 2}} $$

  (26)

  Similarly, the mean square error (MSE) of the LS line (slope s) fitted with the overall pattern is calculated. So,

  $$ {\text{REPEPE}} = {\text{MSE}}/{\text{PMSE}} $$

  (27)

• (g) ABDPE: From partitioning as above, the average of slopes of the two LS line fitted to the two partitions is calculated and absolute slope difference with LS line fitted to overall pattern is found.

  $$ {\text{ABDPE}} = {\frac{{\left| {s - \left({s_{1} + s_{2}} \right)} \right|}}{{2\hat{\sigma}}}} $$

  (28)

Appendix 2: Simulation parameter values

See Tables 5, 6, 7

Table 5 CCP generating parameters

Table 6 Parameters for SVM classifier

Table 7 Parameters used in SOM classifier