Multidimensional method and system for statistical process control

• US 6,804,563 B1
• Filed: 03/21/2001
• Issued: 10/12/2004
• Est. Priority Date: 09/22/1998
• Status: Expired due to Fees

• Current Assignee: Daniel Lafaye De Micheaux
• Original Assignee: Daniel Lafaye De Micheaux
• Inventors: Lafaye de Micheaux, Daniel
• Primary Examiner(s): VOELTZ, EMANUEL T
• Application Number:
  US09/787,685
• Time in Patent Office: 1,301 Days
• Field of Search: 700/51, 700/108, 700/109, 700/121, 702/179, 702/183, 702/184, 702/187
• US Class Current: 700/51
• CPC Class Codes: G06Q 10/06 Resources, workflows, human or project management; Enterprise or organisation planning; Enterprise or organisation modelling

Abstract

The present invention relates to a method and to any system using such a method for performing statistical process control on the basis of taking indicators or measurements on the inputs, the outputs, and the control and operating parameters of said process, and which can be represented by observation points in frames of reference associating their values to their sampling indices. In the invention, the observed values are transformed so that the resulting values are compatible with the multidimensional Gaussian distribution model; said transformed observation points are situated in a multidimensional space, each dimension being associated with a measured magnitude; the out-of-control observation points situated in this way and concentrated in a particular direction are identified; and said direction is associated with a common cause for drift in said process, and each observation-and-anomaly pair is associated with indicators in order to propose zero, one, or more causes of anomaly that might relate to the observation made.

Claims

1. A statistical method of controlling an industrial process on the basis of readings of indicators or measurement of magnitudes characteristic of inputs, outputs, and control and operating parameters of said process, and capable of being represented by observation points in frames of reference associating their values to their sampling indices, said indicators or measurements being given by sensors or by manual input, in which:

a) the observed values are transformed so that the resulting values are compatible with the multidimensional Gaussian distribution model, and constitute data corresponding to the observation points used in the remainder of the method;

b) said observation points are situated in a multidimensional space, in which each dimension is associated with a measured magnitude;

c) amongst the observation points, points that are said to be “

under control” and

that correspond to proper operation of the process are distinguished from points which are said to be “

out of control”

;

d) the distribution center of the points under control is calculated as being the center of gravity of the observation points under control;

e) out of control observation points that are concentrated in some particular direction from the distribution center of the points under control are identified;

f) this direction is associated with a common cause for drift of said process;

g) each observation point and anomaly direction pair is associated with indicators in order to propose zero, one, or more causes of anomaly that are liable to be associated with the observation that has been made; and

h) when an anomaly is analyzed in this way, a warning is triggered and the drift detected in this way in the industrial process is remedied.

2. A statistical process control method according to claim 1, in which each observation and anomaly pair is associated with an indicator giving the intensity of the relationship between the observation and the anomaly, referred to as the “

cause variable”

, the indicator being obtained as the component of the observation point along the direction associated with the cause, said indicator being calculated as a scalar product of the observation vector multiplied by the direction vector for the direction of the cause, the scalar product matrix used as an indicator of the intensity of the relationship between an observation and an anomaly being the matrix V<sup>−

1</sup> where V<sup>−

1</sup> designates the matrix which is either the inverse of the covariance matrix V of the data under control when V is of full rank, or else the psuedo-inverse of V if V is not of full rank.

3. A statistical process control method according to claim 2, in which a proximity indicator is used between the observation and an anomaly, referred to as an “

angular proximity indicator”

, obtained by measuring the angle between a vector representing the observation and a vector representing the direction associate with the anomaly, the angular proximity indicator being the cosine of said angle, said cosine being calculated by dividing the intensity indicator by the norm of the vector representing the observation, which norm is defined by said matrix V<sup>−

1</sup>.

4. A statistical process control method according to claim 3, in which a function F is used to define the control limit Lα

of the angular proximity indicator beyond which an observation is allocated to said anomaly, said function F representing a distribution function of the cosine of the angle (x) between an arbitrary given direction of Rⁿ, where n≧

2, and a centered Gaussian random vector Rⁿ having the intensity matrix as its covariance matrix, said distribution function F of said random variable being given by the following formula;

$\begin{array}{cc}\mathrm{if}<br><br>\text{\hspace{1em}<br><br>}<br><br>x<<br><br>-1& F<br><br>\left(x\right)=0\\ \mathrm{if}<br><br>\text{\hspace{1em}<br><br>}<br><br>x><br><br>1& F<br><br>\left(x\right)=1\\ \mathrm{if}<br><br>\text{\hspace{1em}<br><br>}<br><br>x\ge <br><br>-1<br><br>\text{\hspace{1em}<br><br>}<br><br>\mathrm{and}<br><br>\text{\hspace{1em}<br><br>}<br><br>1& F<br><br>\left(x\right)=1-\frac{S(\mathrm{ArcCos}<br><br>\left(x\right)}{S<br><br>\left(\mathrm{\pi <br><br>}\right)}\end{array}$$\mathrm{where}<br><br>\text{\hspace{1em}<br><br>}<br><br>S\left(\mathrm{\theta <br><br>}\right)={\int <br><br>}_0^{\mathrm{\theta <br><br>}}<br><br>{\sin }^{n-2}<br><br>\left(t\right)<br><br>\text{\hspace{1em}<br><br>}<br><br><br><br>t<br><br>\text{\hspace{1em}<br><br>}<br><br>\mathrm{such}<br><br>\text{\hspace{1em}<br><br>}<br><br>\mathrm{that}<br><br>\text{\hspace{1em}<br><br>}<br><br>\text{\hspace{1em}<br><br>}<br><br>L<br><br>\text{\hspace{1em}<br><br>}<br><br>\mathrm{\alpha <br><br>}=F^{-1}<br><br>\left(\mathrm{\alpha <br><br>}\right),<br><br>\text{\hspace{1em}<br><br>}$

where α

represents the acceptable probability for a false alarm, α

preferably representing a value lying in the range 0.001% to 0.05%.

5. A statistical process control method according to claim 1, organized as:

a learning stage which analyzes a history of the process to contribute to identifying anomalies by performing the following steps;

A1;

prior transformation of the data;

A2;

separating measurements under control from measurements out of control;

A3;

identifying the types of special causes and the associated direction parameters by a method of automatically classifying observations that is adapted to identify observations that are likely to be associated with the same anomaly, and that enables observation points situated close to the same straight line coming from the distribution center of the observations under control to be grouped together;

A4;

creating and examining control charts on the special cause indicators; and

A5;

interpreting the special causes;

a control stage proper which, for each new observation of the process variables, diagnoses whether the observation depends on an anomaly, and if so which anomaly(ies) is/are probable, said stage comprising the following three steps;

S1;

prior to transformation of the data;

S2;

detecting and identifying anomalies by inspecting control charts; and

S3;

where necessary identifying a combination of causes or a new special cause and integrating that in the method.

6. A statistical process control method according to claim 3, in which the automatic classification method used is of the hierarchical ascending classification type accepting the absolute value of the cosine as a similarity index between two observation vectors and accepting the maximum binding criterion also known as the diameter criterion for directing the making up of groups.

7. A statistical process control method according to claim 5, in which the direction used in step A3 of the learning stage to represent an anomaly is the first inertial axis of the observation points associated with said anomaly, which axis passes through the distribution center of the under-control observations, said axis being determined by non-centered and non-reduced principal component analysis on the cloud of observation points associated with the anomaly, points previously centered on the distribution center of the under-control observation points.

8. A statistical process control method according to claim 5, in which control charts are used having probability thresholds with a logarithmic scale, the unilateral chart P_u being used when control relates to a single limit (as in the T² chart), the bilateral chart P_b being used when control relates to a lower limit and an upper limit, said charts providing easier interpretation and greater uniformity of presentation for charts associated with various probability distributions.