CN103440368B - Multi-model dynamic soft measurement modeling method - Google Patents

Abstract

A multi-model dynamic soft measurement modeling method is characterized in that a plurality of sub-models are established by utilizing a self-adaptive fuzzy kernel clustering method and a least square support vector machine; then, a probability distribution function constructed by an evidence synthesis rule is used as a weight factor to fuse the sub-model outputs to obtain multi-model outputs; and finally, dynamically estimating the prediction error of the multiple models by combining an autoregressive moving average model.

Description

A multi-model dynamic soft sensor modeling method

technical field

The invention relates to a soft-sensing method for esterification rate in the polyester industrial production process, in particular to a multi-model dynamic soft-sensing method based on evidence theory synthesis rules and an autoregressive sliding average model.

Background technique

Figure 1 shows the basic process of esterification reaction. As a key link in the entire polyester production process, esterification plays a decisive role in stabilizing polyester production. The key quality index of the outlet of the first esterification tank in the reaction device - the esterification rate directly affects the progress of subsequent reactions and the crystallization performance of polyester products, so the entire production process is often controlled by controlling the esterification rate. However, different polycondensation processes have different requirements on the esterification rate, so operating conditions such as reaction pressure and raw material ratio must be adjusted in the production process to achieve the required esterification rate. However, sudden changes in operating conditions will cause quality fluctuations in the esterification rate, which is not conducive to real-time control of the entire production process. On the other hand, the esterification process mostly uses two esterification reactors to achieve the esterification rate required by the process, but the highly nonlinear reaction system, time-varying and uncertainties increase the difficulty of online measurement of the esterification rate.

On-site analytical instruments are not only expensive and complicated to maintain, but also there is usually a long time lag when using analytical instruments to measure the esterification rate, which will lead to a decline in the performance of control quality and it is difficult to meet production requirements. The basic method of soft measurement is to organically combine the automatic control theory with the knowledge of the production process, apply computer technology, and select some auxiliary variables that are easy to measure for the leading variables that are difficult to measure or cannot be measured temporarily, and infer by forming a certain mathematical relationship. And estimate, replace the function of on-site analysis instrument with software. The soft sensor method has been widely studied and applied in various fields because of its rapid response, continuous information on leading variables, low investment, and simple maintenance. However, in recent decades, with the advancement of science and technology, modern industrial production has higher and higher requirements for the production process, the amount of data has increased sharply, the data types have become more and more complex, and the working conditions are complex and changeable. On the other hand, industrial processes are generally dynamic, and static soft-sensing methods usually cannot reflect the dynamic information and global characteristics of industrial processes, resulting in poor adaptability of the model and long-term use. Therefore, the simple and conventional soft-sensing methods in the past can no longer meet the needs of modern production processes, and are prone to problems such as poor matching of process characteristics, low prediction accuracy, and poor adaptability.

In order to obtain a soft-sensing method suitable for predicting and analyzing esterification rate data in a more general sense, many improved methods have been proposed, and fruitful research results have been formed, mainly in the following aspects: using various modeling methods, Such as mechanism analysis, artificial neural network, least square support vector machine and Gaussian process to build a model for the sample set to predict the output of the leading variable; use various intelligent optimization methods, such as: particle swarm algorithm, genetic algorithm and evolutionary algorithm, etc. to model the model Optimize parameters; use various clustering methods such as: K-means clustering, fuzzy C-means clustering, manifold clustering and affine propagation clustering methods to cluster samples into several sub-categories, and build several sub-models to improve Model predictive performance and more.

Contents of the invention

Aiming at the above-mentioned deficiencies in the prior art, the present invention provides a multi-model dynamic soft-sensing modeling method, which is based on the synthesis rule of evidence theory (D-S rule) and the autoregressive moving average model (ARMA), which has better performance than the prior art. The adaptability is higher in the soft measurement of the esterification rate.

The present invention is realized through the following technical solutions:

A multi-model dynamic soft sensor modeling method, comprising the following steps:

S1. Data preprocessing: select the training sample data set X _m*n , m is the sample dimension, n is the number of samples, remove abnormal data and normalize the data;

S2. Adaptive fuzzy kernel clustering analysis: use the adaptive fuzzy kernel clustering method to cluster the training sample data set X _m*n , obtain the fuzzy class membership degree and each cluster center of each sample, and automatically determine the The optimal number of clusters c;

S3. Establish sub-models: use the least squares support vector machine to train and learn the training sample sets of c clusters, select the Gaussian kernel function as the kernel function of the least squares support vector machine, and establish and determine c sub-models by cross-validation method Model parameters: penalty factor C and kernel parameter σ, and get the output of each sub-model

S4. Sub-model output fusion based on evidence theory synthesis rules: Calculate the evidence probability distribution function value of each sub-model, use it as the weight factor of the sub-model, and then perform evidence fusion on the output of each sub-model to obtain static multi-model output

S5. Dynamic model output: use the autoregressive moving average model to output multiple models at the current moment t, that is, to For dynamic adjustment, first judge Whether it is a stationary sequence, if not, will converted to a stationary sequence; otherwise, directly Subtract the real measurement value y to get a time series about the output value error Δy, and then use the autoregressive moving average model (p, q) to model the time series to get the autoregressive moving average model about the forecast error, Finally, combining the above two models for model prediction, the final dynamic multi-model output is

Preferably, in step S2, the steps of the adaptive fuzzy kernel clustering method include:

S21: Clustering objective function: for the training sample set X={ _xi |i=1,2...n}, the objective function of the adaptive fuzzy kernel clustering method is defined as

In the formula, m is the fuzzy control index, μ _ij is the membership value of the i-th sample corresponding to the j-th cluster, v _j is the j-th cluster center, and K( _xi , v _j ) is a Gaussian kernel function;

S22: Membership update:

S23: Cluster center update:

S23: Evaluation of clustering results: after the clustering is completed, evaluate the clustering results of the effectiveness indicators

Preferably, step S4 includes:

S41: Evidence probability distribution function of the first sub-model: use all the c sub-models obtained by clustering as the identification framework in the evidence theory, and regard any sub-model as the focal element C _j (j=1,2.. .c), for the sample x ₁ , calculate its fuzzy class membership degree for the first sub-model, that is, the first focal element C ₁ , and use it as a piece of evidence according to the evidence theory, and record the probability distribution function of the evidence is m({C ₁ }|x ₁ )=μ ₁₁ ;

And for all n test sample data X={ _xi |i=1,2...n}, also get n pieces of evidence, and its probability distribution function is recorded as m({C ₁ }| _xi )=μ _i1 , i=1,2...n;

Then, these probability distribution functions are fused using evidence-theoretic composition rules, and the fused probability distribution function is used as the probability distribution function of the first sub-model:

Among them, the contradiction factor used to reflect the degree of conflict in the evidence;

S44: Evidence probability distribution functions of all sub-models: by analogy, for all c sub-models, follow step S43 to obtain c evidence probability distribution functions m({C ₁ }|X)...m({C _c }|X);

S45: Multi-model output: separately calculate the sub-output of X for each sub-model The c probability distribution functions in S44 are used as the weight factors of each sub-model, and the output results of the obtained sub-models are weighted and fused, then the multi-model output of the training sample data set is expressed as:

Preferably, step S5 includes:

S51: Using autoregressive moving average model to static multi-model output For dynamic correction, the autoregressive moving average model describes the response of the system at the current moment t which is It is not only related to its previous observations in time, but also has a certain dependence relationship with the present value and lag value of the system disturbance. The autoregressive moving average model (p,q) can be expressed as

Among them, p is the autoregressive item; q is the number of moving average items;

S52: Introducing the linear transition operator B, then there is Therefore, the formula in S51 can be transformed into

In the formula, ε _t is a white noise sequence satisfying N(0,σ ² ), Φ(B) and θ(B) are m-order and n-order polynomials of shift operator B,

According to the basic theory of linear operators on Hilbert space, for random time series that satisfy stationary, normal, zero mean It can be approximated with arbitrary precision by an autoregressive moving average model (p,q).

This method first utilizes the focusing advantage of evidence synthesis rules to deal with uncertain information, establishes multiple evidence probability distribution functions for each sub-model obtained by the affine propagation clustering method, and uses it as the weight factor of each sub-model, for each sub-model The output of the model is weighted and fused to obtain the multi-model output of the test sample, which avoids the shock caused by the switching mode, eliminates the influence of the wrong division of samples on the output accuracy of the model, and effectively improves the predictive ability of the model; The obtained static multi-model output error information is dynamically corrected, which significantly improves the dynamic response characteristics of the system.

Description of drawings

What Fig. 1 shows is the process schematic diagram of esterification;

What Fig. 2 shows is the flow chart of multi-model dynamic soft sensor modeling method of the present invention;

What Fig. 3 shows is the parameter table of submodel C and σ;

What Fig. 4 shows is the comparison result schematic diagram of LSSVM measurement method to the predicted value of esterification rate test sample and artificial value;

Shown in Fig. 5 is the contrast result schematic diagram of SFKCM-LSSVM measurement method to the predicted value of esterification rate test sample and artificial value;

Shown in Fig. 6 is the comparison result schematic diagram of AP-LS-SVM measurement method to the predicted value of esterification rate test sample and artificial value;

What Fig. 7 shows is the comparison result schematic diagram of the present invention to the predicted value of esterification rate test sample and artificial value;

Fig. 8 is a schematic diagram showing the performance comparison between the present invention and the existing measuring method.

detailed description

The technical solutions in the embodiments of the present invention will be clearly and completely described and discussed below in conjunction with the accompanying drawings of the present invention. Obviously, what is described here is only a part of the examples of the present invention, not all examples. Based on the present invention All other embodiments obtained by persons of ordinary skill in the art without creative efforts fall within the protection scope of the present invention.

In order to facilitate the understanding of the embodiments of the present invention, specific embodiments will be taken as examples for further explanation below in conjunction with the accompanying drawings, and each embodiment does not constitute a limitation to the embodiments of the present invention.

First, using the focusing advantage of evidence synthesis rules to deal with uncertain information, multiple evidence probability distribution functions are established for each sub-model obtained by the affine propagation clustering method, which are used as the weight factors of each sub-model, and the weight factors of each sub-model The output is weighted and fused to obtain the multi-model output of the test sample, which avoids the shock caused by the switching mode, eliminates the impact of sample error division on the model output accuracy, and effectively improves the prediction ability of the model; then combined with the autoregressive moving average model (ARMA, Auto-Regressive Moving Average Model) dynamically corrects the obtained static multi-model output error information, which significantly improves the dynamic response characteristics of the system.

The technical scheme that this method solves technical problem takes is:

Please refer to Figure 2, a multi-model dynamic soft sensor modeling method based on evidence theory synthesis rules and autoregressive moving average model, including the following steps:

S1: Data preprocessing: select the training sample data set X _m*n , m is the sample dimension, n is the number of samples, remove abnormal data and normalize the data;

S2: Adaptive fuzzy kernel clustering analysis: use the adaptive fuzzy kernel clustering method to cluster the sample set X _m*n , get the fuzzy class membership and each cluster center of each sample, and automatically determine the best The number of clusters c;

S3: Building sub-models: For each sub-training sample set, use least squares support vector machine (LS-SVM, hereinafter replaced by LS-SVM) to train and learn it, and determine the parameters of each sub-model. Select the Gaussian kernel function as the kernel function of LS-SVM, and determine the parameters of each sub-model through the cross-validation method: the penalty factor C and the kernel parameter σ, as shown in Figure 3;

S4: DS-based sub-model output fusion: According to the method of formula (6), the evidence probability distribution function value of each sub-model is obtained, and it is used as the weight factor of the sub-model, and then the output of each sub-model is calculated by formula (7). Perform evidence fusion to obtain multi-model output

S5: Dynamic model output: use the above static model to get the multi-model output of the sample After that, use the ARMA model to output the multi-model at the current time t right Make dynamic adjustments. judge first Whether it is a stationary sequence, if not, will converted to a stationary sequence; otherwise, directly Subtract it from the real measurement value y to get a time series about the output value error Δy, and then use the ARMA model (p, q) to model the time series to get the ARMA model about the forecast error. Finally, combining the above two models for model prediction, the final output of the sample is

In step S2, the steps of the "adaptive fuzzy kernel clustering method" are as follows:

S21: Clustering objective function: for the training sample set X={ _xi |i=1,2...n}, the objective function of the adaptive fuzzy kernel clustering method is defined as

In the formula, m is the fuzzy control index, μ _ij is the membership value of the i-th sample corresponding to the j-th cluster, v _j is the j-th cluster center, and K(x,y) is a Gaussian kernel function.

S22: Membership update:

S23: Cluster center update:

S23: Evaluation of clustering results: after the clustering is completed, the following validity indicators are used to evaluate the clustering results

In step S4, the specific steps of "model prediction output based on evidence theory synthesis rules" are as follows:

S41: Evidence probability distribution function of the first sub-model: use all the c sub-models obtained by clustering as the identification framework in the evidence theory, and regard any sub-model as the focal element C _j (j=1,2.. .c). Then, for the sample x ₁ , first calculate its fuzzy class membership degree for the first sub-model, that is, the first focal element C ₁ , according to formula (3). And according to the evidence theory, take it as a piece of evidence, record the probability distribution function of this evidence as m({C ₁ }| _xi )=μ ₁₁ .

And for all n test sample data X={ _xi |i=1,2...n}, similarly, n pieces of evidence can be obtained, and its probability distribution function is recorded as m({C ₁ }| _xi )=μ _i1 , i=1,2...n.

Then, these probability distribution functions are fused using evidence theory synthesis rules, and the fused probability distribution function is used as the probability distribution function of the first sub-model, as shown in formula (6):

Among them, the contradiction factor Its size reflects how conflicting the evidence is.

S44: Evidence probability distribution functions of all sub-models: by analogy, for all c sub-models, according to step S43, c evidence probability distribution functions m({C ₁ }|X)...m({C _c }|X).

S45: Multi-model output: Calculate the sub-output of X for each sub-model LS-SVM1,...LS-SVMc respectively Using the c probability distribution functions obtained above as the weight factors of each sub-model, and performing weighted fusion on the obtained sub-model output results, the multi-model output of the test sample set can be expressed as

In step S5, the specific steps of "dynamization of model output" are:

S51: Use the autoregressive moving average model (ARMA) to output the static multi-model obtained in the previous section Make dynamic corrections. The ARMA model describes the response of the system at the current time t which is It is not only related to its previous observation value in time, but also has a certain dependence relationship with the present value and lag value of the system disturbance. The ARMA model (p,q) can be expressed as

Among them, AR is autoregressive, p is autoregressive item; MA is moving average, and q is the number of moving average items.

S52: Introducing the linear transition operator B, then there is So formula (8) can be transformed into

In the formula, ε _t is a white noise sequence satisfying N(0,σ ² ), Φ(B) and θ(B) are m-order and n-order polynomials of shift operator B,

According to the basic theory of linear operators on Hilbert space, for random time series that satisfy stationary, normal, zero mean It can be approximated with arbitrary precision by an ARMA model (p,q).

Give an embodiment according to actual data below:

Step 1: Process the data collected on site to obtain 1000 sets of standard data. Among them, 900 sets of data are used as the training data set X for the establishment of the model; the remaining 100 sets are used as the test data set to test the predictive ability of the model.

The second step: use the affine propagation clustering method to cluster the training data set, and obtain the optimal number of clusters c=4, and the corresponding cluster center v.

Step 3: For the four sub-training sample sets obtained by clustering, use the LS-SVM method to establish four sub-models, and train and learn, and determine the parameters of the LS-SVM through the cross-validation method, as shown in Figure 3.

Step 4: Calculate the probability distribution function of the test sample set to each sub-model according to formula (6), use it as the weight factor of each sub-model, and then calculate the output of the test sample X ^test relative to each sub-model Then use formula (7) to fuse the output of each sub-model to get the output of the test sample

Step 5: The predicted value of the test sample at the current moment t Subtract it from the manual analysis value y, and perform ARMA modeling on the time series of the output error Δy. It is concluded that when the optimal order p=4, the prediction performance of the algorithm is the best.

Figures 4-7 are plots of predicted performance for three different measurement methods and the measurement method of the present invention. As can be seen from the simulation results, the present invention's multi-model dynamic soft sensor modeling method based on evidence theory synthesis rules and autoregressive moving average model has a significant effect on the predictive performance of esterification rate compared with single model and traditional multi-model method. greatly improved. This is because the esterification reaction has the characteristics of high nonlinearity and multiple working conditions, and a model needs to consider all training samples when modeling a single model, which limits the accuracy of the model; The training data set is clustered and divided, and different sub-models are established respectively, but the difference between the test sample and the training sample, the division of the test sample and the impact of the dynamic change of the process on the output results of the multi-model are not considered in depth in the stage of predicting the output of the test sample , so there is no significant improvement in predictive performance. The method of the present invention uses the affine propagation clustering method to first cluster and divide the samples with the same working conditions and similar characteristics, and then fully considers the influence of the output of each sub-model on the final output of the sample in the output stage of the prediction test sample, and uses evidence Theoretical synthesis rules construct weight factors to perform multi-model fusion on the output of the sub-models to obtain the final output of the test sample, avoiding the oscillation caused by the switching mode and the prediction deviation caused by the misclassification of the sample; in addition, considering the dynamic characteristics of the actual industrial process , the autoregressive moving average model is used to dynamically correct the output of multiple models, which improves the dynamic response characteristics of the system, so it has better adaptability, and has achieved better fitting results in the soft measurement of the esterification rate.

Figure 8 lists the performance parameters of different soft sensing methods. It can be seen from FIG. 8 that, compared with the existing modeling method, the root mean square error and the maximum relative error have been improved by adopting the multi-model dynamic soft sensor modeling method provided by the present invention.

The above is only a preferred embodiment of the present invention, but the scope of protection of the present invention is not limited thereto. Any person skilled in the art within the technical scope disclosed in the present invention can easily think of changes or Replacement should be covered within the protection scope of the present invention. Therefore, the protection scope of the present invention should be determined by the protection scope of the claims.

Claims (1)

1. a kind of multi-model dynamic soft measuring modeling method, it is characterised in that comprise the following steps：

S1, data prediction：Selection training sample data collection X_m*n, m is sample dimension, and n is number of samples, rejecting abnormalities data And data are normalized；

S2, the analysis of adaptive fuzzy kernel clustering：Using adaptive fuzzy kernel clustering method to training sample data collection X_m*nGathered Class, obtains the fuzzy class degree of membership and each cluster centre of each sample, and automatically determines out preferable clustering number mesh c；

S3, set up submodel：Study is trained to the training sample set of c cluster using least square method supporting vector machine, is selected Kernel function of the gaussian kernel function as least square method supporting vector machine is selected, c submodel is set up and determined by cross-validation method Parameter：Penalty factor and nuclear parameter σ, and obtain the output of each submodel

S4, the submodel output fusion based on Combination Rules of Evidence Theory：The evidential probability partition function value of each submodel is calculated, As the weight of submodel, the then output to each submodel carries out evidence fusion, obtains static multiple mode output

S5, the mobilism of model output：The multi-model of current time t is exported using autoregressive moving-average model, i.e., it is rightEnter Mobile state is adjusted, and is first determined whetherWhether it is stationary sequence, if it is not, willBe converted to stationary sequence；Otherwise directly willWith it is true Actual measurement value y subtracts each other, and a time series on output valve error delta y is obtained, then using autoregressive moving-average model (p, q) is modeled to the time series, obtains the autoregressive moving-average model on predicated error, finally, by the step The output of static multiple mode in rapid S4Being combined with the output of autoregressive moving-average model in S5 carries out model prediction, then most Whole dynamic multi-model is output as

Include in step S2, the step of the adaptive fuzzy kernel clustering method：

S21：Cluster object function：To training sample set X={ x_i| i=1,2...n }, the target of adaptive fuzzy kernel clustering method Function is defined as

$\left\{\begin{array}{c}{J}_{\mathrm{\Φ}}(U,c)=\underset{i=1}{\overset{n}{\Σ}}\underset{j=1}{\overset{c}{\Σ}}{\mathrm{\μ}}_{ij}^m\left|\right|\[1-K(x_i,v_j)\]|{|}^2\\ \begin{array}{cc}s.t.& U\∈M_{fc}\end{array}\end{array}\right.$

In formula,M is fuzzy control index, μ_ijIt is What i sample corresponded to j-th cluster is subordinate to angle value, v_jIt is j-th cluster centre, K (x_i, v_j) it is gaussian kernel function；

S22：Degree of membership updates：

${\mathrm{\μ}}_{ij}={(1-K(x_i,v_j\left)\right)}^{-1/(m-1)}/\underset{j=1}{\overset{c}{\Σ}}{(1-K(x_i,v_j\left)\right)}^{-1/(m-1)};$

S23：Cluster centre updates：

$v_i=\underset{i=1}{\overset{n}{\Σ}}{{\mathrm{\μ}}_{ij}}^{m}K(x_i,v_j)x_i/\underset{i=1}{\overset{n}{\Σ}}{{\mathrm{\μ}}_{ij}}^{m}K(x_i,v_j);$

S24：Cluster result is evaluated：After cluster terminates, Validity Index is evaluated the result for clustering

$V_{GX}\left(c\right)=\frac{\underset{i=1}{\overset{c}{\Σ}}\underset{k=1}{\overset{n}{\Σ}}{\mathrm{\μ}}_{ik}^m(1-K(v_k,x_i\left)\right)+\frac{1}{c}\underset{i=1}{\overset{c}{\Σ}}(1-K(v_i,\stackrel{\‾}{v}\left)\right)}{\underset{i\≠k}{min}(1-K(v_k,v_i\left)\right)};$

Step S4 includes：

S41：First evidential probability partition function of submodel：Using all c submodels obtained by cluster as evidence theory In framework of identification, and any submodel is considered as Jiao unit C_j, wherein, j=1,2...c, for sample x₁, it is calculated for One submodel, i.e., first Jiao unit C₁Fuzzy class degree of membership, and according to evidence theory, as an evidence, note should The probability distribution function of evidence is m ({ C₁}|x₁)=μ₁₁；

And for all of n test sample data X={ x_i| i=1,2...n }, n bar evidences are similarly obtained, its probability assignments letter Number scale is m ({ C₁}|x_i)=μ_i1, i=1,2...n；

Then, these probability distribution functions are merged using Combination Rules of Evidence Theory, by the probability assignments letter after fusion Number is used as first probability distribution function of submodel：

$\left\{\begin{array}{c}m\left(\right\{C_1\left\}\right|X)=\frac{\underset{\left\{C_1\right\}|x_1\∩\mathrm{...}\∩\{C_1\left\}\right|x_n=\left\{C_1\right\}|X}{\Σ}{m}_1\left(\right\{C_1\left\}\right|x_1)...m_n\left(\right\{C_1\left\}\right|x_n)}{1-k}\\ m\left(\mathrm{\Φ}\right)=0\end{array}\right.$

Wherein, the contradiction factorIt is used to reflect the conflict journey of evidence Degree；

S44：The evidential probability partition function of all submodels：The rest may be inferred, for all of c submodel, obtains c evidence Probability distribution function m ({ C₁}|X)...m({C_c}|X)；

S45：Multi-model is exported：Son outputs of the X for each submodel is calculated respectivelyBy c probability in S44 point With function as each submodel weight, the submodel output result to gained is weighted fusion, then number of training It is expressed as according to the multi-model output of collection：

$\hat{y}=m\left(\right\{C_1\left\}\right|X){\hat{y}}_1+m\left(\right\{C_2\left\}\right|X){\hat{y}}_2+...m\left(\right\{C_c\left\}\right|X){\hat{y}}_c;$

Step S5 includes：

S51：The static multi-model is exported using autoregressive moving-average modelDynamic calibration is carried out, autoregression is slided The response of averaging model descriptive system current time tI.e.It is not only relevant with the observation before it in time, also be There is certain dependence in the present worth and lagged value of system disturbance, autoregressive moving-average model (p, q) is represented by

Wherein, p is autoregression；Q is rolling average item number；

S52：Linear shift operator B is introduced, is then hadTherefore the formula in S51 can transform to

$\mathrm{\Φ}\left(B\right){\hat{y}}_t=\mathrm{\θ}\left(B\right){\mathrm{\ϵ}}_t$

In formula, ε_tTo meet N (0, σ²) white noise sequence, Φ (B) and θ (B) for Shift operators B m ranks and n-order polynomial,

$\left\{\begin{array}{c}\mathrm{\Φ}\left(B\right)=(1-{\mathrm{\φ}}_{1}B-...{\mathrm{\φ}}_m{B}^m)\\ \mathrm{\θ}\left(B\right)=(1-{\mathrm{\θ}}_{1}B-...{\mathrm{\θ}}_m{B}^m)\end{array}\right.$

According to the basic theories of Hilbert space Linear Operators, the Random time sequence to meeting steady, normal state, zero-meanCan be approached with arbitrary accuracy with an autoregressive moving-average model (p, q).