CN107122611A - Penicillin fermentation process quality dependent failure detection method - Google Patents
Penicillin fermentation process quality dependent failure detection method Download PDFInfo
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Abstract
The present invention relates to a kind of penicillin fermentation process quality dependent failure detection method, step is：The data under the normal operating operating mode of multiple batches are collected, are divided into process data collection and qualitative data collection, and is deployed and is standardized, new course of normal operation data set and qualitative data collection is obtained；Process data and qualitative data to normal operating carry out many subspace canonical variable analyses, obtain five sub-spaces and five kinds of statistic variables, and calculate the threshold value of monitoring statisticss amount；Real-time running data is collected, is standardized, obtains new process data collection and qualitative data collection；Many subspace canonical variable analyses are performed to real-time running data, threshold condition is exceeded by the monitoring statisticss amount for analyzing five sub-spaces, failure detection result is obtained.The present invention further considers covariance information based on canonical variable analysis, carries out five sub-spaces mappings to initial data, the fine detection of failure can be realized, so as to judge whether quality is influenceed by procedure fault.
Description
Technical field
The invention belongs to industrial process fault detection technique field, it is related to interval industrial process fault detection method, specifically
Ground is said, relate to a kind of penicillin fermentation process quality dependent failure detection method.
Background technology
In order to adapt to the market demand that modern society is fast changing, modern industry production is just progressively being relied on for counsel in many product of production
The batch industrial production processes of kind, small lot and high added value.Compared to continuous process, batch industrial production processes mechanism is answered
Miscellaneous, operation complexity is high, and product quality is easily by the shadow of the uncertain factors such as environmental condition, raw material quality, status of equipment
Ring.Typical batch industrial production processes have pharmacy, fermentation and semiconductor machining etc., the industrial mistake of the interval that is seen everywhere in life
The product of journey production.Therefore, the quality of the security of interval industrial process production, stability and product is increasingly closed
Note.
As highly integrated, large-scale modern industry system is increasingly formed, substantial amounts of industrial processes service data is obtained
To store so that the fault detection technique extensive use based on data-driven.Researcher proposes a series of based on data drive
Dynamic batch industrial production processes fault detection method, including pivot analysis (PCA), independent component analysis (ICA), offset minimum binary
(PLS), canonical variable analysis (CVA) etc..Wherein, canonical variable analysis (CVA) is that Juricek B C et al. were proposed in the last few years
A kind of novel fault detection method (bibliography：Juricek B C,Seborg D E,Larimore W E.Fault
detection using canonical variate analysis[J].Industrial&engineering
chemistry research,2004,43(2):458-474), attracted wide attention in industrial process fault diagnosis field.
Different with quality variable influence according to failure process variable, the failure mode during actual industrial production is divided into 3
Kind, i.e.,：Process, which breaks down, to be caused abnormal quality, process to break down not cause abnormal quality (due to the effect of suppressor)
And process fault-free abnormal quality (quality is influenceed by other factors such as equipment).Existing fault detection method majority is concentrated on
Fault detection problem (i.e. how fast and effeciently discovery procedure failure), but for the related procedure fault detection of quality (i.e.
Detect and judge whether the failure has an impact to quality after procedure fault) research it is relatively fewer, the latter is raw for interval industry
Production process has great Research Significance.For traditional fault detection method based on CVA, although achieve it is preliminary into
Work(application, but it has a drawback in that：(1) due to not differentiating between process data space and qualitative data space, it is impossible to distinguish
Whether failure causes product quality abnormal；(2) data overall space is analyzed during statistical modeling, monitor mode is inadequate
Finely, fault detect effect is reduced.
The content of the invention
The present invention be directed to presence in interval industrial process fault detect can not distinguish failure whether cause product quality it is abnormal,
There is provided a kind of simple, accurate effective penicillin fermentation process quality dependent failure detection side for the low deficiency of fault detect effect
Method, this method can differentiate whether procedure fault causes product quality abnormal, can finely monitor, and improve fault detect effect.
In order to achieve the above object, the invention provides a kind of penicillin fermentation process fault detection method, containing following
Step：
(1) data in penicillin fermentation process under the normal operating operating mode of multiple batches are collected, process data is divided into
Collect X0(I × J × K) and qualitative data collection Y0(I × M × K), wherein, I represents batch number, and J represents process variable number, and M is represented
Quality variable number, K represents number of samples；By process data collection X0(I × J × K) and qualitative data collection Y0(I × M × K) is carried out
Expansion and standardization, obtain new course of normal operation data set X (IK × J) and qualitative data collection Y (IK × M)；
(2) many sons are carried out to new course of normal operation data set X (IK × J) and qualitative data collection Y (IK × M) empty
Between canonical variable analyze, obtain procedure quality correlation subspaces Φxy, non-related processing principal component subspaceNon-related processing
Residual error subspaceUncorrelated quality principal component subspaceUncorrelated quality residual subspaceFive sub-spaces, are calculated
Procedure quality correlation subspaces ΦxyStatisticNon-related processing principal component subspaceStatisticNon-related processing
Residual error subspaceStatistic SPEex, uncorrelated quality principal component subspaceStatisticUncorrelated quality residual
SpaceStatistic SPEeyFive kinds of statistics, and the threshold values of the algorithm calculating monitoring statisticss amount using Density Estimator；
(3) real-time running data, including process data collection are collectedWith qualitative data collectionIt is standardized and obtains new
Process data collection X*With qualitative data collection Y*；
(4) on the basis of step (3), many subspace canonical variables is performed to real-time running data and analyzed, by dividing
The monitoring statisticss amount for analysing five sub-spaces exceeds threshold values situation, obtains failure detection result.
Further, in the step (1), by process data collection X0(I × J × K) and qualitative data collection Y0(I×M×K)
First carry out deploying to obtain process data collection X by batch1(I × JK) and qualitative data collection Y1(I × MK), calculates mistake after expansion respectively
Journey data set average mean (X1) and standard deviation std (X1) and qualitative data collection average mean (Y1) and standard deviation std (Y1),
The data under normal operative employee's condition are standardized by formula (1), (2), formula (1), (2) expression formula it is as follows：
X2=(X1-mean(X1))/std(X1) (1)
Y2=(Y1-mean(Y1))/std(Y1) (2)
After above-mentioned formula (1), (2) standardization, deployed according still further to variable, you can obtain new normal operating
Process data collection X (IK × J) and qualitative data collection Y (IK × M).
Further, in the step (2), the step of obtaining five sub-spaces is：To new course of normal operation data
Collect X (IK × J) and qualitative data collection Y (IK × M) and carry out many subspace canonical variable analyses, covariance matrix and cross covariance battle array
It can be obtained by formula (3), the expression formula of formula (3) is as follows：
Singular value decomposition is carried out by formula (4), the expression formula of formula (4) is as follows：
Σxx -1/2ΣxyΣyy -1/2=U Σ VT (4)
Canonical variable larger to correlation k before extracting, constitutes vector u, v：
In formula, x and y is matrix X and matrix Y row vector, U respectivelykArranged for the preceding k of matrix U, VkArranged for the preceding k of matrix V；
It is procedure quality correlation subspaces Φ by the subspaces of vectorial u, vxy；
To procedure quality correlation subspaces ΦxyConstructing statistic is
Vectorial u is projected, obtained
And then obtain estimating residual errorEstimate residual error exCorresponding data matrix is：
By estimation residual error exThe subspace opened is non-related processing subspace Φex；
According to formula (9) to non-related processing subspace ΦexPivot analysis is carried out, the expression formula of formula (9) is as follows：
In formula,Represent TexReturn to non-related processing subspace ΦexProjection matrix,Represent pivot residual error square
Battle array, TexIt is k to represent column vector numberexPivot sub-matrix, kexRepresent pivot number, PexRepresent loading matrix；
ByThe subspace opened is non-related processing principal component subspaceByThe subspace opened is uncorrelated mistake
Journey residual error subspace
According to non-related processing principal component subspaceScore vectorAnd residual vector
Obtain non-related processing principal component subspaceStatisticAnd non-related processing residual error subspaceStatistic SPEex
For：
In formula, ΛexRepresent kexThe eigenvalue cluster of individual pivot into diagonal matrix；
Vector v is projected, obtained
And then obtain estimating residual errorEstimate residual error eyCorresponding data matrix is：
By estimation residual error eyThe subspace opened is uncorrelated quality subspace Φey；
According to formula (14) to non-related processing subspace ΦeyPivot analysis is carried out, the expression formula of formula (14) is as follows：
In formula,Represent TeyReturn to uncorrelated quality subspace ΦeyProjection matrix,Represent pivot residual error square
Battle array, TeyIt is k to represent column vector numbereyPivot sub-matrix, keyRepresent pivot number, PeyRepresent loading matrix；
ByThe subspace opened is uncorrelated quality principal component subspaceByThe subspace opened is uncorrelated
Quality residual subspace
According to non-related processing principal component subspaceScore vectorAnd residual vectorObtain uncorrelated quality principal component subspaceStatisticAnd uncorrelated quality residual
SubspaceStatistic SPEeyFor：
In formula, ΛeyRepresent keyThe eigenvalue cluster of individual pivot into diagonal matrix.
Further, in step (3), according to formula (17), (18) to the process data collection of real time executionAnd mass number
According to collectionBe standardized, formula (17), (18) expression formula it is as follows：
Further, in step (4), by analyzing the monitoring statisticss amount of five sub-spaces beyond threshold condition acquisition event
When hindering testing result, if the procedure quality correlation subspaces Φ at continuous multiple momentxyStatisticMore than threshold value, then show
The process variable related to quality breaks down in penicillin fermentation process；If non-related processing principal component subspaceStatistics
AmountOr non-related processing residual error subspaceStatistic SPEexBeyond threshold value, then show in penicillin fermentation process only
It is related to the process variable unrelated with quality to break down；If uncorrelated quality principal component subspaceStatisticOr it is uncorrelated
Quality residual subspaceStatistic SPEeyBeyond threshold value, then show the fault impact matter occurred in penicillin fermentation process
Quantitative change amount, it is off quality.
Compared with prior art, the beneficial effects of the present invention are：
The above-mentioned fault detection method that the present invention is provided, based on canonical variable analysis, extracts the correlation of quality and process,
Monitored while implementation process data space is with qualitative data space, can differentiate whether procedure fault causes product quality different
Often.The above-mentioned fault detection method that the present invention is provided, introduces pivot analysis, raw quality data and process data is divided into
Five sub-spaces, realize the fine monitoring of fault detect, help to improve failure detection result.
Brief description of the drawings
Fig. 1 is the flow chart of penicillin fermentation process quality dependent failure detection method of the embodiment of the present invention.
Fig. 2 is the Subspace partition figure of penicillin fermentation process quality dependent failure detection method of the embodiment of the present invention.
Fig. 3 is the flow chart of penicillin fermentation of the embodiment of the present invention.
Fig. 4 a-c are traditional CVA batch process fault detection method fault detect figures under failure of the embodiment of the present invention 2.
Fig. 5 a-e are the penicillin fermentation process quality dependent failure detection figure under failure of the embodiment of the present invention 2.
Fig. 6 a-c are traditional CVA batch process fault detection method fault detect figures under failure of the embodiment of the present invention 4.
Fig. 7 a-e are the penicillin fermentation process quality dependent failure detection figure under failure of the embodiment of the present invention 4.
Fig. 8 a-b are the quality variable and the comparison figure of normal quality variable under failure of the embodiment of the present invention 2 and failure 4.
Embodiment
Below, the present invention is specifically described by exemplary embodiment.It should be appreciated, however, that not entering one
In the case of step narration, element, structure and features in an embodiment can also be advantageously incorporated into other embodiment
In.
Referring to Fig. 1, Fig. 2, present invention is disclosed a kind of penicillin fermentation process quality dependent failure detection method, containing with
Lower step：
(1) data in penicillin fermentation process under the normal operating operating mode of multiple batches are collected, process data is divided into
Collect X0(I × J × K) and qualitative data collection Y0(I × M × K), wherein, I represents batch number, and J represents process variable number, and M is represented
Quality variable number, K represents number of samples；By process data collection X0(I × J × K) and qualitative data collection Y0(I × M × K) is carried out
Expansion and standardization, obtain new course of normal operation data set X (IK × J) and qualitative data collection Y (IK × M)；
(2) many sons are carried out to new course of normal operation data set X (IK × J) and qualitative data collection Y (IK × M) empty
Between canonical variable analyze, obtain procedure quality correlation subspaces Φxy, non-related processing principal component subspaceNon-related processing
Residual error subspaceUncorrelated quality principal component subspaceUncorrelated quality residual subspaceFive sub-spaces, are calculated
Procedure quality correlation subspaces ΦxyStatisticNon-related processing principal component subspaceStatisticNon-related processing
Residual error subspaceStatistic SPEex, uncorrelated quality principal component subspaceStatisticUncorrelated quality residual
SpaceStatistic SPEeyFive kinds of statistics, and the threshold values of the algorithm calculating monitoring statisticss amount using Density Estimator；
(3) real-time running data, including process data collection are collectedWith qualitative data collectionIt is standardized and obtains new
Process data collection X*With qualitative data collection Y*；
(4) on the basis of step (3), many subspace canonical variables is performed to real-time running data and analyzed, by dividing
The monitoring statisticss amount for analysing five sub-spaces exceeds threshold values situation, obtains failure detection result.
The above-mentioned detection method of the present invention includes two stages of off-line modeling and on-line monitoring, by pivot analysis by original matter
Amount data and process data are divided into five sub-spaces, by monitoring whether five sub-spaces statistics exceed threshold value, judge blue or green
Whether the failure in mycin fermentation process is related to quality, so that the size that failure judgement influences on product quality, realizes failure
The precise controlling of detection, improves failure detection result.
With reference to a simulation example and with reference to accompanying drawing to penicillin fermentation process quality dependent failure of the present invention
Detection method, which is made, to be further illustrated.
Illustrated by taking penicillin fermentation system as an example.Penicillin is a kind of antibiotic for being widely used in clinic, and it is sent out
Ferment process is a kind of typical biochemical reaction.Referring to Fig. 3, under certain fermentation condition, such as suitable culture medium, pH value, temperature
Degree, air mass flow, stirring etc., penicillin fermentation bacterium grow and synthetic antibiotic., it is necessary to constantly empty in whole process
Air-flow and stirring, maintain certain temperature and tank pressure, pass through and add the pH value that acid, alkali control zymotic fluid, in addition it is also necessary to interval or
It is continuous to add the substrate such as glucose and ammonium salt, promote the production of penicillin to cover other feed liquids.Penicillin is mould
Plain bacterium secondary metabolite, in different fermentation periods, growth, breeding, the aging of existing thalline itself, there is the conjunction of penicillin again
Into and hydrolysis.
Pensim2.0 emulation platforms be by Illinois technical college (Illinois Institute of Technology,
IIT professor Cinar) lead control research group exploitation penicillin fermentation process simulation software, due to the software have compared with
Good practicality and validity, has become more influential penicillin simulation software in the world.In simulation process, it can adopt
Collect 18 kinds of variables, choose 16 kinds of measurands and carry out fault detect, wherein quality variable there are two kinds, and measurand is as shown in table 1.
Table 1
| Variable sequence number | Name variable | Variable sequence number | Name variable |
| x1 | Ventilation rate | x9 | PH value |
| x2 | Power of agitator | x10 | Temperature of reactor |
| x3 | Bottoms stream rate of acceleration | x11 | Reactor heat production |
| x4 | Feed supplement temperature | x12 | Acid stream rate of acceleration |
| x5 | Concentration of substrate | x13 | Alkali stream rate of acceleration |
| x6 | Dissolved oxygen saturation degree | x14 | Cold water stream rate of acceleration |
| x7 | Culture volume | y1 | Cell concentration |
| x8 | CO2Concentration | y2 | Penicillin concn |
In table 1, x1To x14It is process measurement variable, y1And y2It is mass measurement variable.
By the setting to primary condition, normal penicillin fermentation data set and failure penicillin fermentation can be obtained
Data set.The normal primary condition of penicillin fermentation sets as shown in table 2.
Table 2
| Variable | Excursion | Variable | Excursion |
| Culture medium concentration (g/L) | 14-18 | The heat (kcal) of generation | 0 |
| Oxyty (mmol/L) | 1-1.2 | Ventilation rate (1h-1) | 8-9 |
| Microorganism concn (g/L) | 0 | Power of agitator (W) | 29-31 |
| Penicillin concn (g/L) | 0 | Medium feed flow velocity (1h-1) | 0.039-0.045 |
| Fermentation volume (L) | 100-104 | Medium feed temperature (K) | 295-296 |
| CO2Concentration (mmol/L) | 0.5-1 | Fermentation jar temperature (K) | 295-301 |
| PH value | 4.5-5.5 |
Comprising 3 failed operation mode in above-mentioned penicillin simulation process, failure modalities list is shown in Table 3.
Table 3
| NO. | Failure-description |
| 1 | Ventilation rate slope (step) failure |
| 2 | Power of agitator slope (step) failure |
| 3 | Bottoms stream rate of acceleration slope (step) failure |
The penicillin fermentation process quality dependent failure detection method of above-mentioned penicillin fermentation system, contains following steps：
Step one：The data under the normal operating operating mode of multiple batches in penicillin fermentation process are collected, process is divided into
Data set X0(I × J × K) and qualitative data collection Y0(I × M × K), wherein, I represents batch number, and J represents process variable number, M
Representation quality variable number, K represents number of samples；By process data collection X0(I × J × K) and qualitative data collection Y0(I×M×K)
First carry out deploying to obtain process data collection X by batch1(I × JK) and qualitative data collection Y1(I × MK), calculates mistake after expansion respectively
Journey data set average mean (X1) and standard deviation std (X1) and qualitative data collection average mean (Y1) and standard deviation std (Y1),
The data under normal operative employee's condition are standardized by formula (1), (2), formula (1), (2) expression formula it is as follows：
X2=(X1-mean(X1))/std(X1) (1)
Y2=(Y1-mean(Y1))/std(Y1) (2)
After above-mentioned formula (1), (2) standardization, deployed according still further to variable, you can obtain new normal operating
Process data collection X (IK × J) and qualitative data collection Y (IK × M).
Step 2：New course of normal operation data set X (IK × J) and qualitative data collection Y (IK × M) is carried out more sub
Space canonical variable analysis, obtains procedure quality correlation subspaces Φxy, non-related processing principal component subspaceNon-related processing
Residual error subspaceUncorrelated quality principal component subspaceUncorrelated quality residual subspaceFive sub-spaces, are calculated
Procedure quality correlation subspaces ΦxyStatisticNon-related processing principal component subspaceStatisticNon-related processing
Residual error subspaceStatistic SPEex, uncorrelated quality principal component subspaceStatisticUncorrelated quality residual
SpaceStatistic SPEeyFive kinds of statistics, and the threshold values of the algorithm calculating monitoring statisticss amount using Density Estimator.It has
Body step is：
Many subspace typical cases are carried out to new course of normal operation data set X (IK × J) and qualitative data collection Y (IK × M)
Variable analysis, covariance matrix and cross covariance battle array can be obtained by formula (3), and the expression formula of formula (3) is as follows：
Singular value decomposition is carried out by formula (4), the expression formula of formula (4) is as follows：
Σxx -1/2ΣxyΣyy -1/2=U Σ VT (4)
Canonical variable larger to correlation k before extracting, constitutes vector u, v：
In formula, x and y is matrix X and matrix Y row vector, U respectivelykArranged for the preceding k of matrix U, VkArranged for the preceding k of matrix V；
It is procedure quality correlation subspaces Φ by the subspaces of vectorial u, vxy；
To procedure quality correlation subspaces ΦxyConstructing statistic is
Vectorial u is projected, obtained
And then obtain estimating residual errorEstimate residual error exCorresponding data matrix is：
By estimation residual error exThe subspace opened is non-related processing subspace Φex；
According to formula (9) to non-related processing subspace ΦexPivot analysis is carried out, the expression formula of formula (9) is as follows：
In formula,Represent TexReturn to non-related processing subspace ΦexProjection matrix,Represent pivot residual error square
Battle array, TexIt is k to represent column vector numberexPivot sub-matrix, kexRepresent pivot number, PexRepresent loading matrix；
ByThe subspace opened is non-related processing principal component subspaceByThe subspace opened is uncorrelated mistake
Journey residual error subspace
According to non-related processing principal component subspaceScore vectorAnd residual vector
Obtain non-related processing principal component subspaceStatisticAnd non-related processing residual error subspaceStatistic SPEex
For：
In formula, ΛexRepresent kexThe eigenvalue cluster of individual pivot into diagonal matrix；
Vector v is projected, obtained
And then obtain estimating residual errorEstimate residual error eyCorresponding data matrix is：
By estimation residual error eyThe subspace opened is uncorrelated quality subspace Φey；
According to formula (14) to non-related processing subspace ΦeyPivot analysis is carried out, the expression formula of formula (14) is as follows：
In formula,Represent TeyReturn to uncorrelated quality subspace ΦeyProjection matrix,Represent pivot residual error square
Battle array, TeyIt is k to represent column vector numbereyPivot sub-matrix, keyRepresent pivot number, PeyRepresent loading matrix；
ByThe subspace opened is uncorrelated quality principal component subspaceByThe subspace opened is uncorrelated matter
Measure residual error subspace
According to non-related processing principal component subspaceScore vectorAnd residual vector
Obtain uncorrelated quality principal component subspaceStatisticAnd uncorrelated quality residual subspaceStatistic SPEey
For：
In formula, ΛeyRepresent keyThe eigenvalue cluster of individual pivot into diagonal matrix.
Step 3：Collect real-time running data, including process data collectionWith qualitative data collectionAccording to formula (17),
(18) to the process data collection of real time executionWith qualitative data collectionIt is standardized, formula (17), the expression of (18)
Formula is as follows：
New process data collection X is obtained after formula (17), (18) standardization*With qualitative data collection Y*。
Step 4：On the basis of step 3, many subspace canonical variables are performed to real-time running data and analyzed, by dividing
The monitoring statisticss amount for analysing five sub-spaces exceeds threshold values situation, obtains failure detection result.
During by analyzing the monitoring statisticss amount of five sub-spaces beyond threshold condition acquisition failure detection result, if continuous many
The procedure quality correlation subspaces Φ at individual momentxyStatisticMore than threshold value, then show in penicillin fermentation process with quality
Related process variable breaks down, and influences larger to product quality variable.If non-related processing principal component subspaceStatistics
AmountOr non-related processing residual error subspaceStatistic SPEexBeyond threshold value, then show in penicillin fermentation process only
What it is related to the process variable unrelated with quality to break down, on product quality without influence or influence very little；If uncorrelated
Quality principal component subspaceStatisticOr uncorrelated quality residual subspaceStatistic SPEeyBeyond threshold value, then
Show the fault impact quality variable occurred in penicillin fermentation process, and influence is serious, it is off quality, and seriously not
It is qualified.
In the present embodiment, using traditional CVA batch processes fault detection method and above-mentioned penicillin fermentation mistake of the invention
Cheng Zhiliang dependent failures detection method carries out fault detect to penicillin fermentation system.
Illustrate failure detection result by taking failure in 4 as an example, fault type is as shown in table 4.
Table 4
| Failure sequence number | Fault type | Failure introduces the time |
| 1 | Bottoms stream rate of acceleration+0.0001/h slopes change | 100h |
| 2 | The Spline smoothing of bottoms stream rate of acceleration+20% | 100h |
| 3 | The Spline smoothing of ventilation rate+10% | 100h |
| 4 | Power of agitator+0.03/h slopes change | 100h |
The detection time for calculating above-mentioned 4 kinds of failures respectively is included in table 5, and fault sample recall rate is included in into table 6.For table 5
In " _ " represent that the statistic without departing from threshold value, that is, does not have failure detection time；For " _ " in table 6 represent that inspection is not present
Extracting rate.
Table 5
Table 6
It can be obtained according to table 5- tables 6, traditional CVA fault detection methods and penicillin fermentation process quality phase of the present invention
Failure can be detected by closing fault detection method, and by contrast, the detection time of the latter is quicker, and failure recall rate is more
Height, overall Detection results are better than the former.
For traditional CVA batch process fault detection methods, when its 3 statistics are statisticStatisticSystem
Any one statistic measured in SPE exceeds threshold value, illustrates now to break down, but fine area can not be carried out to failure
Point, that is, it is merely able to deterministic process and breaks down, it is impossible to which whether confrontation amount produces influence to the failure of judgement now.For penicillin hair
Ferment procedure quality dependent failure detection method, can pass through procedure quality correlation subspaces statisticWhether deterministic process failure
Quality is influenceed, if, the monitoring statisticss amount at continuous multiple moment in real-time monitoringMore than threshold value, show that event occurs for the process
Barrier, and the procedure fault influences larger to product quality, should give attention., whereas if this monitoring statisticss amountNormally, only
There is non-related processing principal component subspaceStatisticOr non-related processing residual error subspaceStatistic SPEexIt is super
Go out control limit, show that the process breaks down, but the procedure fault is on product quality is without what influence or influences very little.
Explained the situation below according to two examples.
Traditional CVA fault detection methods result figure of failure 2 is drawn on Fig. 4 a-c, penicillin fermentation process quality related
Fault detection method result be drawn on Fig. 5 a-e and failure 4 traditional CVA fault detection methods result figure be drawn on Fig. 6 a-c,
Penicillin fermentation process quality dependent failure detection method result is drawn in Fig. 7 a-e.
It can be seen from Fig. 4-Fig. 7 batch process is carried out using penicillin fermentation process quality dependent failure detection method
Fault detect, process data space and qualitative data space can be monitored simultaneously, can differentiate whether procedure fault influences production
Quality.
Analyzed below for each failure：
Failure 2：
Fig. 4 a-c are the testing results using tradition CVA fault detection methods, it can be seen that statistic exceeds threshold value, are judged
It is faulty to occur, but can not judge whether this failure produces influence to product quality.And use penicillin fermentation process
Quality dependent failure detection method, as shown in Fig. 5 a-e, it is seen that procedure quality correlation subspaces statisticBeyond threshold value, explanation
Procedure fault influences quality, should now send alarm, stops production process, in order to avoid cause unnecessary loss.
Failure 4：
Fig. 6 a-c are the testing results using tradition CVA fault detection methods, it can be seen that statistic exceeds threshold value, are judged
It is faulty to occur, but can not judge whether this failure produces influence to product quality.Fig. 7 a-e are using penicillin hair
The testing result of ferment procedure quality dependent failure detection method, it can be seen that procedure quality correlation subspaces statisticDo not examine
Failure is measured, when illustrating that this failure occurs, failure is existed only in exporting in incoherent input subspace, and quality variable does not have
It is affected, although now process breaks down, does not influence quality, then production process can not stops, it is to avoid former material
The waste of material.
For failure 2 and failure 4, the quality variable of the production process of failure is compared with normal quality variable
Compared with as shown in figure 8 a-b.In figure, thick line is the quality measurement variation tendency line of normal data, the matter of failure 2 and failure 4
Amount measurement data variation tendency is marked in Fig. 8 a-b respectively.
Penicillin fermentation process quality dependent failure detection method of the present invention is verified, it can be seen from Fig. 8 a, 8b
The quality variable of failure 2 differs larger with normal quality variable, and the quality variable of failure 4 is consistent with normal quality, with
Conclusion is consistent.
Embodiment provided above is only of the invention with illustrating for convenience, not limiting the scope of the invention,
Technical scheme category of the present invention, person of ordinary skill in the field makees various simple deformations and modification, all should include
In above claim.
Claims (5)
1. a kind of penicillin fermentation process quality dependent failure detection method, it is characterised in that contain following steps：
(1) data in penicillin fermentation process under the normal operating operating mode of multiple batches are collected, process data collection X is divided into0
(I × J × K) and qualitative data collection Y0(I × M × K), wherein, I represents batch number, and J represents process variable number, M representation qualities
Variable number, K represents number of samples；By process data collection X0(I × J × K) and qualitative data collection Y0(I × M × K) is deployed
And standardization, obtain new course of normal operation data set X (IK × J) and qualitative data collection Y (IK × M)；
(2) many subspace allusion quotations are carried out to new course of normal operation data set X (IK × J) and qualitative data collection Y (IK × M)
Type variable analysis, obtains procedure quality correlation subspaces Φxy, non-related processing principal component subspaceNon-related processing residual error
SubspaceUncorrelated quality principal component subspaceUncorrelated quality residual subspaceFive sub-spaces, calculating process
Quality correlation subspaces ΦxyStatisticNon-related processing principal component subspaceStatisticNon-related processing residual error
SubspaceStatistic SPEex, uncorrelated quality principal component subspaceStatisticUncorrelated quality residual subspaceStatistic SPEeyFive kinds of statistics, and the threshold values of the algorithm calculating monitoring statisticss amount using Density Estimator；
(3) real-time running data, including process data collection are collectedWith qualitative data collectionIt is standardized and obtains new mistake
Journey data set X*With qualitative data collection Y*；
(4) on the basis of step (3), many subspace canonical variables is performed to real-time running data and analyzed, pass through analysis five
The monitoring statisticss amount of sub-spaces exceeds threshold values situation, obtains failure detection result.
2. penicillin fermentation process quality dependent failure detection method as claimed in claim 1, it is characterised in that the step
(1) in, by process data collection X0(I × J × K) and qualitative data collection Y0(I × M × K) is first carried out deploying to obtain process by batch
Data set X1(I × JK) and qualitative data collection Y1(I × MK), calculates process data collection average mean (X after expansion respectively1) and mark
Quasi- difference std (X1) and qualitative data collection average mean (Y1) and standard deviation std (Y1), by formula (1), (2) to normal operating
Data under operating mode are standardized, formula (1), (2) expression formula it is as follows：
X2=(X1-mean(X1))/std(X1) (1)
Y2=(Y1-mean(Y1))/std(Y1) (2)
After above-mentioned formula (1), (2) standardization, deployed according still further to variable, you can obtain new course of normal operation
Data set X (IK × J) and qualitative data collection Y (IK × M).
3. penicillin fermentation process quality dependent failure detection method as claimed in claim 1 or 2, it is characterised in that described
In step (2), the step of obtaining five sub-spaces is：To new course of normal operation data set X (IK × J) and qualitative data
Collect Y (IK × M) and carry out many subspace canonical variable analyses, covariance matrix and cross covariance battle array can be obtained by formula (3), public
The expression formula of formula (3) is as follows：
<mrow>
<mfenced open = "{" close = "">
<mtable>
<mtr>
<mtd>
<mrow>
<msub>
<mi>&Sigma;</mi>
<mrow>
<mi>x</mi>
<mi>x</mi>
</mrow>
</msub>
<mo>=</mo>
<mfrac>
<mn>1</mn>
<mrow>
<mi>I</mi>
<mi>K</mi>
<mo>-</mo>
<mn>1</mn>
</mrow>
</mfrac>
<msup>
<mi>X</mi>
<mi>T</mi>
</msup>
<mi>X</mi>
</mrow>
</mtd>
</mtr>
<mtr>
<mtd>
<mrow>
<msub>
<mi>&Sigma;</mi>
<mrow>
<mi>y</mi>
<mi>y</mi>
</mrow>
</msub>
<mo>=</mo>
<mfrac>
<mn>1</mn>
<mrow>
<mi>I</mi>
<mi>K</mi>
<mo>-</mo>
<mn>1</mn>
</mrow>
</mfrac>
<msup>
<mi>Y</mi>
<mi>T</mi>
</msup>
<mi>Y</mi>
</mrow>
</mtd>
</mtr>
<mtr>
<mtd>
<mrow>
<msub>
<mi>&Sigma;</mi>
<mrow>
<mi>x</mi>
<mi>y</mi>
</mrow>
</msub>
<mo>=</mo>
<mfrac>
<mn>1</mn>
<mrow>
<mi>I</mi>
<mi>K</mi>
<mo>-</mo>
<mn>1</mn>
</mrow>
</mfrac>
<msup>
<mi>X</mi>
<mi>T</mi>
</msup>
<mi>Y</mi>
</mrow>
</mtd>
</mtr>
</mtable>
</mfenced>
<mo>-</mo>
<mo>-</mo>
<mo>-</mo>
<mrow>
<mo>(</mo>
<mn>3</mn>
<mo>)</mo>
</mrow>
</mrow>
Singular value decomposition is carried out by formula (4), the expression formula of formula (4) is as follows：
Σxx -1/2ΣxyΣyy -1/2=U Σ VT (4)
Canonical variable larger to correlation k before extracting, constitutes vector u, v：
<mrow>
<mfenced open = "{" close = "">
<mtable>
<mtr>
<mtd>
<mrow>
<mi>u</mi>
<mo>=</mo>
<msup>
<mrow>
<mo>&lsqb;</mo>
<msub>
<mi>u</mi>
<mn>1</mn>
</msub>
<mo>,</mo>
<msub>
<mi>u</mi>
<mn>2</mn>
</msub>
<mo>,</mo>
<mo>...</mo>
<mo>,</mo>
<msub>
<mi>u</mi>
<mi>k</mi>
</msub>
<mo>&rsqb;</mo>
</mrow>
<mi>T</mi>
</msup>
<mo>=</mo>
<msubsup>
<mi>U</mi>
<mi>k</mi>
<mi>T</mi>
</msubsup>
<msubsup>
<mi>&Sigma;</mi>
<mrow>
<mi>x</mi>
<mi>x</mi>
</mrow>
<mrow>
<mo>-</mo>
<mn>1</mn>
<mo>/</mo>
<mn>2</mn>
</mrow>
</msubsup>
<mi>x</mi>
</mrow>
</mtd>
</mtr>
<mtr>
<mtd>
<mrow>
<mi>v</mi>
<mo>=</mo>
<msup>
<mrow>
<mo>&lsqb;</mo>
<msub>
<mi>v</mi>
<mn>1</mn>
</msub>
<mo>,</mo>
<msub>
<mi>v</mi>
<mn>2</mn>
</msub>
<mo>,</mo>
<mo>...</mo>
<mo>,</mo>
<msub>
<mi>v</mi>
<mi>k</mi>
</msub>
<mo>&rsqb;</mo>
</mrow>
<mi>T</mi>
</msup>
<mo>=</mo>
<msubsup>
<mi>V</mi>
<mi>k</mi>
<mi>T</mi>
</msubsup>
<msubsup>
<mi>&Sigma;</mi>
<mrow>
<mi>y</mi>
<mi>y</mi>
</mrow>
<mrow>
<mo>-</mo>
<mn>1</mn>
<mo>/</mo>
<mn>2</mn>
</mrow>
</msubsup>
<mi>y</mi>
</mrow>
</mtd>
</mtr>
</mtable>
</mfenced>
<mo>-</mo>
<mo>-</mo>
<mo>-</mo>
<mrow>
<mo>(</mo>
<mn>5</mn>
<mo>)</mo>
</mrow>
</mrow>
In formula, x and y is matrix X and matrix Y row vector, U respectivelykArranged for the preceding k of matrix U, VkArranged for the preceding k of matrix V；
It is procedure quality correlation subspaces Φ by the subspaces of vectorial u, vxy；
To procedure quality correlation subspaces ΦxyConstructing statistic is
<mrow>
<msubsup>
<mi>T</mi>
<mi>s</mi>
<mn>2</mn>
</msubsup>
<mo>=</mo>
<msup>
<mi>u</mi>
<mi>T</mi>
</msup>
<mi>u</mi>
<mo>-</mo>
<mo>-</mo>
<mo>-</mo>
<mrow>
<mo>(</mo>
<mn>6</mn>
<mo>)</mo>
</mrow>
</mrow>
Vectorial u is projected, obtained
<mrow>
<mover>
<mi>x</mi>
<mo>^</mo>
</mover>
<mo>=</mo>
<msubsup>
<mi>&Sigma;</mi>
<mrow>
<mi>x</mi>
<mi>x</mi>
</mrow>
<mrow>
<mn>1</mn>
<mo>/</mo>
<mn>2</mn>
</mrow>
</msubsup>
<msub>
<mi>U</mi>
<mi>k</mi>
</msub>
<msubsup>
<mi>U</mi>
<mi>k</mi>
<mi>T</mi>
</msubsup>
<msubsup>
<mi>&Sigma;</mi>
<mrow>
<mi>x</mi>
<mi>x</mi>
</mrow>
<mrow>
<mo>-</mo>
<mn>1</mn>
<mo>/</mo>
<mn>2</mn>
</mrow>
</msubsup>
<mi>x</mi>
<mo>-</mo>
<mo>-</mo>
<mo>-</mo>
<mrow>
<mo>(</mo>
<mn>7</mn>
<mo>)</mo>
</mrow>
</mrow>
And then obtain estimating residual errorEstimate residual error exCorresponding data matrix is：
<mrow>
<msub>
<mi>E</mi>
<mi>x</mi>
</msub>
<mo>=</mo>
<mi>X</mi>
<mo>-</mo>
<mover>
<mi>X</mi>
<mo>^</mo>
</mover>
<mo>-</mo>
<mo>-</mo>
<mo>-</mo>
<mrow>
<mo>(</mo>
<mn>8</mn>
<mo>)</mo>
</mrow>
</mrow>
By estimation residual error exThe subspace opened is non-related processing subspace Φex；
According to formula (9) to non-related processing subspace ΦexPivot analysis is carried out, the expression formula of formula (9) is as follows：
<mrow>
<msub>
<mi>E</mi>
<mi>x</mi>
</msub>
<mo>=</mo>
<msub>
<mover>
<mi>E</mi>
<mo>^</mo>
</mover>
<mi>x</mi>
</msub>
<mo>+</mo>
<msubsup>
<mi>E</mi>
<mi>x</mi>
<mi>R</mi>
</msubsup>
<mo>=</mo>
<msub>
<mi>T</mi>
<mrow>
<mi>e</mi>
<mi>x</mi>
</mrow>
</msub>
<msubsup>
<mi>P</mi>
<mrow>
<mi>e</mi>
<mi>x</mi>
</mrow>
<mi>T</mi>
</msubsup>
<mo>+</mo>
<msubsup>
<mi>E</mi>
<mi>x</mi>
<mi>R</mi>
</msubsup>
<mo>-</mo>
<mo>-</mo>
<mo>-</mo>
<mrow>
<mo>(</mo>
<mn>9</mn>
<mo>)</mo>
</mrow>
</mrow>
In formula,Represent TexReturn to non-related processing subspace ΦexProjection matrix,Represent pivot residual matrix, Tex
It is k to represent column vector numberexPivot sub-matrix, kexRepresent pivot number, PexRepresent loading matrix；
ByThe subspace opened is non-related processing principal component subspaceByThe subspace opened is residual for non-related processing
Poor subspace
According to non-related processing principal component subspaceScore vectorAnd residual vectorObtain non-related processing principal component subspaceStatisticAnd non-related processing residual error
SpaceStatistic SPEexFor：
<mrow>
<msubsup>
<mi>T</mi>
<mrow>
<mi>e</mi>
<mi>x</mi>
</mrow>
<mn>2</mn>
</msubsup>
<mo>=</mo>
<msubsup>
<mi>t</mi>
<mrow>
<mi>e</mi>
<mi>x</mi>
</mrow>
<mi>T</mi>
</msubsup>
<msubsup>
<mi>&Lambda;</mi>
<mrow>
<mi>e</mi>
<mi>x</mi>
</mrow>
<mrow>
<mo>-</mo>
<mn>1</mn>
</mrow>
</msubsup>
<msub>
<mi>t</mi>
<mrow>
<mi>e</mi>
<mi>x</mi>
</mrow>
</msub>
<mo>-</mo>
<mo>-</mo>
<mo>-</mo>
<mrow>
<mo>(</mo>
<mn>10</mn>
<mo>)</mo>
</mrow>
</mrow>
<mrow>
<msub>
<mi>SPE</mi>
<mrow>
<mi>e</mi>
<mi>x</mi>
</mrow>
</msub>
<mo>=</mo>
<msup>
<mrow>
<mo>(</mo>
<msub>
<mi>e</mi>
<mi>x</mi>
</msub>
<mo>-</mo>
<msub>
<mover>
<mi>e</mi>
<mo>^</mo>
</mover>
<mi>x</mi>
</msub>
<mo>)</mo>
</mrow>
<mi>T</mi>
</msup>
<mrow>
<mo>(</mo>
<msub>
<mi>e</mi>
<mi>x</mi>
</msub>
<mo>-</mo>
<msub>
<mover>
<mi>e</mi>
<mo>^</mo>
</mover>
<mi>x</mi>
</msub>
<mo>)</mo>
</mrow>
<mo>-</mo>
<mo>-</mo>
<mo>-</mo>
<mrow>
<mo>(</mo>
<mn>11</mn>
<mo>)</mo>
</mrow>
</mrow>
In formula, ΛexRepresent kexThe eigenvalue cluster of individual pivot into diagonal matrix；
Vector v is projected, obtained
<mrow>
<mover>
<mi>y</mi>
<mo>^</mo>
</mover>
<mo>=</mo>
<msubsup>
<mi>&Sigma;</mi>
<mrow>
<mi>y</mi>
<mi>y</mi>
</mrow>
<mrow>
<mn>1</mn>
<mo>/</mo>
<mn>2</mn>
</mrow>
</msubsup>
<msub>
<mi>V</mi>
<mi>k</mi>
</msub>
<msubsup>
<mi>V</mi>
<mi>k</mi>
<mi>T</mi>
</msubsup>
<msubsup>
<mi>&Sigma;</mi>
<mrow>
<mi>y</mi>
<mi>y</mi>
</mrow>
<mrow>
<mo>-</mo>
<mn>1</mn>
<mo>/</mo>
<mn>2</mn>
</mrow>
</msubsup>
<mi>y</mi>
<mo>-</mo>
<mo>-</mo>
<mo>-</mo>
<mrow>
<mo>(</mo>
<mn>12</mn>
<mo>)</mo>
</mrow>
</mrow>
And then obtain estimating residual errorEstimate residual error eyCorresponding data matrix is：
<mrow>
<msub>
<mi>E</mi>
<mi>y</mi>
</msub>
<mo>=</mo>
<mi>Y</mi>
<mo>-</mo>
<mover>
<mi>Y</mi>
<mo>^</mo>
</mover>
<mo>-</mo>
<mo>-</mo>
<mo>-</mo>
<mrow>
<mo>(</mo>
<mn>13</mn>
<mo>)</mo>
</mrow>
</mrow>
By estimation residual error eyThe subspace opened is uncorrelated quality subspace Φey；
According to formula (14) to non-related processing subspace ΦeyPivot analysis is carried out, the expression formula of formula (14) is as follows：
<mrow>
<msub>
<mi>E</mi>
<mi>y</mi>
</msub>
<mo>=</mo>
<msub>
<mover>
<mi>E</mi>
<mo>^</mo>
</mover>
<mi>y</mi>
</msub>
<mo>+</mo>
<msubsup>
<mi>E</mi>
<mi>y</mi>
<mi>R</mi>
</msubsup>
<mo>=</mo>
<msub>
<mi>T</mi>
<mrow>
<mi>e</mi>
<mi>y</mi>
</mrow>
</msub>
<msubsup>
<mi>P</mi>
<mrow>
<mi>e</mi>
<mi>y</mi>
</mrow>
<mi>T</mi>
</msubsup>
<mo>+</mo>
<msubsup>
<mi>E</mi>
<mi>y</mi>
<mi>R</mi>
</msubsup>
<mo>-</mo>
<mo>-</mo>
<mo>-</mo>
<mrow>
<mo>(</mo>
<mn>14</mn>
<mo>)</mo>
</mrow>
</mrow>
In formula,Represent TeyReturn to uncorrelated quality subspace ΦeyProjection matrix,Represent pivot residual matrix, Tey
It is k to represent column vector numbereyPivot sub-matrix, keyRepresent pivot number, PeyRepresent loading matrix；
ByThe subspace opened is uncorrelated quality principal component subspaceByThe subspace opened is residual for uncorrelated quality
Poor subspace
According to non-related processing principal component subspaceScore vectorAnd residual vector
Obtain uncorrelated quality principal component subspaceStatisticAnd uncorrelated quality residual subspaceStatistic SPEey
For：
<mrow>
<msubsup>
<mi>T</mi>
<mrow>
<mi>e</mi>
<mi>y</mi>
</mrow>
<mn>2</mn>
</msubsup>
<mo>=</mo>
<msubsup>
<mi>t</mi>
<mrow>
<mi>e</mi>
<mi>y</mi>
</mrow>
<mi>T</mi>
</msubsup>
<msubsup>
<mi>&Lambda;</mi>
<mrow>
<mi>e</mi>
<mi>y</mi>
</mrow>
<mrow>
<mo>-</mo>
<mn>1</mn>
</mrow>
</msubsup>
<msub>
<mi>t</mi>
<mrow>
<mi>e</mi>
<mi>y</mi>
</mrow>
</msub>
<mo>-</mo>
<mo>-</mo>
<mo>-</mo>
<mrow>
<mo>(</mo>
<mn>15</mn>
<mo>)</mo>
</mrow>
</mrow>
<mrow>
<msub>
<mi>SPE</mi>
<mrow>
<mi>e</mi>
<mi>y</mi>
</mrow>
</msub>
<mo>=</mo>
<msup>
<mrow>
<mo>(</mo>
<msub>
<mi>e</mi>
<mi>y</mi>
</msub>
<mo>-</mo>
<msub>
<mover>
<mi>e</mi>
<mo>^</mo>
</mover>
<mi>y</mi>
</msub>
<mo>)</mo>
</mrow>
<mi>T</mi>
</msup>
<mrow>
<mo>(</mo>
<msub>
<mi>e</mi>
<mi>y</mi>
</msub>
<mo>-</mo>
<msub>
<mover>
<mi>e</mi>
<mo>^</mo>
</mover>
<mi>y</mi>
</msub>
<mo>)</mo>
</mrow>
<mo>-</mo>
<mo>-</mo>
<mo>-</mo>
<mrow>
<mo>(</mo>
<mn>16</mn>
<mo>)</mo>
</mrow>
</mrow>
In formula, ΛeyRepresent keyThe eigenvalue cluster of individual pivot into diagonal matrix.
4. penicillin fermentation process quality dependent failure detection method as claimed in claim 3, it is characterised in that step (3)
In, according to formula (17), (18) to the process data collection of real time executionWith qualitative data collectionIt is standardized, it is public
Formula (17), the expression formula of (18) are as follows：
<mrow>
<msup>
<mi>X</mi>
<mo>*</mo>
</msup>
<mo>=</mo>
<mrow>
<mo>(</mo>
<msubsup>
<mi>X</mi>
<mn>0</mn>
<mo>*</mo>
</msubsup>
<mo>-</mo>
<mi>m</mi>
<mi>e</mi>
<mi>a</mi>
<mi>n</mi>
<mo>(</mo>
<msub>
<mi>X</mi>
<mn>1</mn>
</msub>
<mo>)</mo>
<mo>)</mo>
</mrow>
<mo>/</mo>
<mi>s</mi>
<mi>t</mi>
<mi>d</mi>
<mrow>
<mo>(</mo>
<msub>
<mi>X</mi>
<mn>1</mn>
</msub>
<mo>)</mo>
</mrow>
<mo>-</mo>
<mo>-</mo>
<mo>-</mo>
<mrow>
<mo>(</mo>
<mn>17</mn>
<mo>)</mo>
</mrow>
</mrow>
<mrow>
<msup>
<mi>Y</mi>
<mo>*</mo>
</msup>
<mo>=</mo>
<mrow>
<mo>(</mo>
<msubsup>
<mi>Y</mi>
<mn>0</mn>
<mo>*</mo>
</msubsup>
<mo>-</mo>
<mi>m</mi>
<mi>e</mi>
<mi>a</mi>
<mi>n</mi>
<mo>(</mo>
<msub>
<mi>Y</mi>
<mn>1</mn>
</msub>
<mo>)</mo>
<mo>)</mo>
</mrow>
<mo>/</mo>
<mi>s</mi>
<mi>t</mi>
<mi>d</mi>
<mrow>
<mo>(</mo>
<msub>
<mi>Y</mi>
<mn>1</mn>
</msub>
<mo>)</mo>
</mrow>
<mo>-</mo>
<mo>-</mo>
<mo>-</mo>
<mrow>
<mo>(</mo>
<mn>18</mn>
<mo>)</mo>
</mrow>
<mo>.</mo>
</mrow>
5. penicillin fermentation process quality dependent failure detection method as claimed in claim 3, it is characterised in that step (4)
In, during by analyzing the monitoring statisticss amount of five sub-spaces beyond threshold condition acquisition failure detection result, if when continuous multiple
The procedure quality correlation subspaces Φ at quarterxyStatisticMore than threshold value, then show related to quality in penicillin fermentation process
Process variable break down；If non-related processing principal component subspaceStatisticOr non-related processing residual error is empty
BetweenStatistic SPEexBeyond threshold value, then show to pertain only to the process variable hair unrelated with quality in penicillin fermentation process
Raw failure；If uncorrelated quality principal component subspaceStatisticOr uncorrelated quality residual subspaceStatistic
SPEeyBeyond threshold value, then show the fault impact quality variable occurred in penicillin fermentation process, it is off quality.
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| CN108664002A (en) * | 2018-04-27 | 2018-10-16 | 中国石油大学(华东) | A kind of nonlinear dynamic process monitoring method towards quality |
| CN109471420A (en) * | 2018-09-21 | 2019-03-15 | 浙江大学 | Intelligent power plant's large size Thermal generation unit air preheater control performance monitoring method based on CVA-SFA |
| WO2019080489A1 (en) * | 2017-10-26 | 2019-05-02 | 东北大学 | Process fault detection method based on concurrent partial least squares |
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