On the modeling of tensile index from larger data sets

Detection of outliers in laboratory samples

To show the outlier detection principals, samples for analyzing tensile index are used in this section. Assume that the dynamic variations in the process, during each pulp sampling, can be considered as small and that the obtained average of each pulp sample (fourteen samples in our case) is representative for the process conditions during each sampling interval. By preparing handsheets from each pulp sample this means that possible outliers are related to the handsheets and not necessarily to the variations in the process. Suppose that seven handsheets are prepared. From each handsheet, three strips are provided for analysis according to Figure 1.

Figure 1

A schematic drawing of three strips obtained from each of the seven handsheets.

This means that mainly three different approaches can be formulated when analysing the tensile index

where θ is the tensile strength while the denominator in Case A is the average basis weight for handsheets, i. e. one measure for the complete batch of handsheets. For Case B the denominator can be seen as the most logical average basis weight to use for each handsheet. Case C covers each tensile index sampled from each handsheet.

Thus, for the samples j=1,…,m we have to consider i=1,…,3l tensile strength (θ) measurements.^[1] Introduce n=3l, i. e. in our case nm=294 elements to analyze in the vector. In our example x=[x1,…,xnm,] and when the generalized ESD is applied on the data series the multiple discordant outliers can be recursively identified if the dynamic process variations between the pulp sampling intervals are handled carefully. This statement will be further penetrated in the next section but we need to describe the basic outlier detection procedure first.

Sort the nm observed values in the vector x from the mean x¯ in ascending order and calculate statistics for up to nm−1 outliers, i. e. i=1:nm−1.

where the denominator s represents the standard deviation. The recursive process is continued until R1,R2,…,Rnm have been computed.

Compare each Ri with the critical value λi for a pre-specified significance level α, defined as

where tnm−i−1,p is the inverse of Student’s t-cumulative distribution function with nm−i−1 degrees of freedom and the percentile values of the t-distribution

where α is the significance level, see further EPA (2006). By this definition, the critical value λi represents decision cut-point to label whether an observation is a potential outlier, see Barnett and Lewis (1994).

The null hypothesis (i. e. no outliers) can be rejected if |Ri>λi| which results in i extreme values classified as outliers, EPA (2006). This process is continued until i=nm−1 and we can conclude that there are a certain number of outliers, or until all the tests have been performed and none were found to be significant. In other words, if none of the tests are significant, i. e. Ri<λi, then there are no outliers in x and the null hypothesis holds. Note, this procedure is not linked to possible outliers in a dynamic perspective, i. e. laboratory samples related to process variations. Therefore, it is necessary to develop it further to be useful in a broader perspective which will be discussed later in this paper.

Selection procedure for reliable pulp and handsheet property candidates for process modeling

As the process data are oversampled (every second) with respect to pulp sampling (equidistantly for 3 minutes), it is wise to maintain the high-frequency information to get a possibility to illustrate the process noise and its impact on the pulp property variations. However, it is unknown whether the measured laboratory samples of the pulp and handsheet properties are reliable in a dynamic perspective as rapid process changes or natural fluctuations are not available. Therefore, it is necessary to strengthen the hypothesis that the dependent variables that are selected can be predicted.

Consider that there are m different test series to study and that each test series comprises i different pulp samples. As the process data are recorded every second during the time interval for each batch of the pulp sample, both vector xmi including internal and/or external variables and the sampling rate are known and thereby also the number of samples, N. By estimating the mean values of the internal and external variables for each time interval, a common timeframe for the analysis of the pulp and handsheet property fmi is obtained.

As seen in Equation 5, the pulp and handsheet property is only defined as an averaged measure during the sampling interval and most likely is non-linearly dependent on the process conditions. This means that it is natural to model the selected pulp and handsheet property as a collection of piece-wise linear functions of the form

where {θm1,…,θmk} represents the parameter vector and k the number of predictors (Lowe and Zohdy (2010)).

The number of linear regions into which the non-linear function is broken up is represented by q. By using the parameters in Equation 6, the pulp property is assumed to be predictable for sampling rates related to the internal and external variables i. e. fˆm(x¯m)→fˆm(t) and this extension of course requires minimized process fluctuations during the sampling interval.

In the work reported here, the test was performed over a limited period (three days), see Figure 18, and it is assumed that the test series can be modelled by one piece-wise linear function, i. e. q=1.

A refinement using the adjusted R2 will be included to set a penalty for the number of predictors in the model, i. e.

where ∑(f−fˆ)2 represents the sum of the squared residuals from the regression^[2] and ∑(f−f¯)2 the sum of the squared differences from the mean of the dependent variable, while n is the number of observations and d is the degree of the polynomial, see further Draper and Smith (1998).

Compare two test series (1) and (2) of the same dependent variable using the Kolmogorov-Smirnov test (Stephens (1974)) to determine if the test series differ significantly^[3] as each element i in the test series is related to the same pulp sample in the blow-line.

Real non-zero values are considered, and the comparison is made to find all pulp or handsheet properties fi(1) and fi(2) that fulfill the relation

Constraint c is related to the accepted variation in the laboratory equipment and the number of samples required for further analysis.

The procedure results in a reduced vector fj where j={1,…,2ψ} and ψ is the length of fj(1) and fj(2). All other pulp or handsheet properties are saved separately for later use. Select different combinations of predictors according to Equation 6 and perform a polynomial fit of the models m using the 2ψ accepted samples.

To provide the initial models, a low constraint is introduced on the adj.R2≥0.3. This constraint is normally not acceptable in modeling procedures but in this step we want to find enough data for further analysis.

Note, when selecting the permitted residuals between two sets the elements to be compared can be out of range and still be selected as candidates for analysis if they are both outliers. To reduce that risk, each sample can be tested iteratively by estimating the adj.R2 for the remaining 2ψ−k samples. If the model is improved, the rejected k samples are left for further analysis.

The best model fulfilling the constraint will be used by estimating the dependent variables fˆjm. A vector of the differences is created between measured and estimated variables of all 2ψ−k samples. Find the smallest and largest elements multiplied by r, i. e. the coefficient of multiple correlation, and use these scalars as constraints.^[4] Estimate the dependent variables fˆlm based on the data rejected and define the differences ξl=fl−fˆlm. If the elements of ξl are within the constraints, i. e.

and l={1,…,2(Ψ−ψ)+k}, the corresponding measures fl are accepted for further analysis.

If the assumptions introduced above are incorrect for the given data set, the methods will likely give erroneous results. This means that more laboratory measurements should be performed and analyzed before changing the data in the test series studied.

However, if it is not accepted to re-run the tedious laboratory testing it is still possible to improve the models by ranking the absolute differences between the measured and estimated properties in ascending order. Thereby, new models of the pulp and handsheet properties can be derived and validated. In this step the samples rejected are also tested to find other acceptable measures.^[5]

The selection of predictors has been discussed in several articles and the reader is referred to Karlström et al. (2015, 2016a, 2016b) for more details. In this paper, only internal variables like consistency and fiber residence time will be considered as they outperform the external variables as independent variables (predictors) when making polynomial fits of pulp and handsheet properties, see Karlström and Hill (2017a, 2017b, 2017c).

Detection of outliers in laboratory samples

In our example 3nm=291 elements in the sample vector are acceptable for further analysis.^[6]

To find out if the selected variables are acceptable for further analysis a second iteration based on the modified generalized ESD procedure, Equation 1–Equation 4 can be applied.

The assumption that the pulp property measurements, excluding the suspected outliers, are approximately normal distributed is appropriate when the vector size is greater than or equal to 25 Rosner (1983). In the pulp and paper industry, this required vector size can be a problem due to tedious laboratory tests.^[7] Therefore, other complementary methods where the sampling vectors are extended must be added to get a reliable set of data. In our example the condition is thereby fulfilled if we can handle the process dynamics when extending the vector size by adding new measurements from other sampling intervals. However, most often such procedures result in situations where the mean values obtained from each sampling interval can differ considerably see Figure 4. If these deviations are caused by uncertain test procedures it is important to handle the data set with care.

Figure 4

Tensile index for Case A for the entire data series when all outliers caused by measurement errors have been rejected (1^st iteration).

Obviously as seen in Figure 4, the standard deviation differs quite much for each sample as well as for Case A–Case C in Table 1. Which case to choose as a standard when analyzing tensile index is hard to pre-specify as the means for all samples are almost equal.

We can conclude that the number of samples in each sample is ≤21 which means that the generalized ESD most likely does not approach a normal distribution. Moreover, as seen in Equation 2 and Equation 3 the outlier criterion Rj≥λi differs from the discretized Rj≥λj. To overcome such problems we introduce a procedure where the mean of each sample is extracted from each measurement. This is shown in Figure 5 for Case A and Case C. The discrepancies between the two cases are small and we can expect that Case B, which is in between the two cases in Table 1, has similar characteristics.

In Figure 5, it is also shown that several potential outliers can exist depending on the chosen significance level α.

Following the procedure outlined in Appendix A, outliers can be detected in samples of pulp and handsheet properties. When the deviations in the mean values are caused by variable process conditions it is even more relevant to introduce the procedure above. A good checkpoint is to see if the distributions tend to be skew. When the skewness is far from zero this indicates that some of the tensile index measurements are in the lower region of acceptable measures. Nevertheless, the key questions are whether or not it is acceptable with a deviation in tensile index of ±3 Nm/g and if it is acceptable to use measurements with such spread for modeling purposes, i. e. link the results to different types of process dynamic evaluations.

Figure 5

Detrended tensile index for Case A and Case C after the 1^st iteration. Each sample is detrended individually.

Selection procedure for reliable pulp and handsheet property candidates for process modeling

Even though, an appropriate generalized ESD-procedure is used to detect outliers it is not guaranteed that the laboratory data will be useful in a dynamic perspective. This is best illustrated by studying the time plot for specific energy where the time for pulp samples are included. As seen in Figure 6 the specific energy and most likely also the pulp and handsheet properties can vary considerably during the sampling. This is of course a challenge when it comes to validation of pulp and handsheet properties in a dynamic perspective.

Harrell et al. (1985) and Freedman and Pee (1989), who presented a general guideline for the minimum number of events per variable (EPV) in multivariate analysis, demonstrated that overfitting was inflated when the ratio of the number of variables to the number of observations was greater than 1/4, which corresponds to an EPV ≥ 4. Peduzzi et al. (1996) suggested increasing that number to at least ten events per variable analyzed to maintain the validity of the final model. This analysis was based on data from a cardiac trial with good quality data, and, in our situation, this recommendation would result in at least 50 samples if five predictors are used.

Draper and Smith (1998) suggested the use of an EPV of 10 as a good choice but in industrial (especially in pulp and paper industry) applications, this is usually not possible due to tedious laboratory analysis and uncertainties in the measurements which limits the number of reliable samples.

Figure 6

Specific energy and pulp sampling occasions versus time.

Vittinghoff and McCulloch (2007) conducted a large simulation study of other influences on confidence interval coverage relative bias and other model performance measures and found a range of circumstances in which coverage and bias were within acceptable levels despite an EPV less than 10. In short, they concluded that the “one in ten” rule can be relaxed and, in this paper, it is assumed that reliable models can be derived using an EPV ≈ 4 if it is possible to confirm that some of the measurements obtained during the major step changes in production, plate gap and dilution water feed rate in Figure 18 (Appendix B) are covered.

In summary, the methodology for large data sets is based on three different steps using double tests of each sample and it is always appropriate to be critical of the results if too few laboratory samples are available relative to the number of predictors analyzed in the model. This is central in this section and the idea is to extract the laboratory measurements which are possible to link to process data. Thereafter the data selected is ranked before training and verifying the models according to the procedure outlined by Karlström and Hill (2017a, 2017b, 2017c).

The idea so far, has been to extract the laboratory measurements which are possible to link to process data. Originally, 160 pulp samples^[9] were included in the proposed test series, although only about 100 tensile index measurements were analyzed.

In this paper, the measurements will be ranked by using the absolute difference between the measured property and the estimated property. This is illustrated in Table 2, which gives the accepted measurements of tensile index. As seen in Table 2, about 60 % of the pulp samples are accepted if selecting a constraint of ±2 Nm/g according to EPV = 4. The original data selection is shown in the right column, i. e. the modeling and hold-out sets. The original data selection is shown in the right column, i. e. the modeling and hold-out sets together with some of the rejected samples obtained as a result of the asymmetry in the upper and lower constraints in Equation 9. If such rearranged measurements are used, it is possible to analyze a new set of “accepted” data at different adj. R² according to the procedure outlined above, see Table 3.

As stated by Karlström and Hill (2017a, 2017b, 2017c), only internal variables like consistency and fiber residence time are necessary to consider as independent variables when making polynomial fits of pulp and handsheet properties. Thereby, Equation 6 can be expressed as

where τ corresponds to the tensile index estimation in a CD-82 refiner. The independent variables C and η represents the consistencies and fiber residence times in the flat zone and conical zones {FZ,CD} respectively. For details, see Karlström and Hill (2017a, 2017b, 2017c).

Figure 7

Measured tensile index versus estimates tensile index.

Figure 8

Normalized properties to show where samples for tensile index are taken.

Figure 9

Estimated and measured (TestA and TestB) tensile index.

In this paper, we choose to focus on three figures summarizing findings for estimated and measured values for tensile indices.

In Figure 7, it is obvious that the model can estimate the tensile index within ±2 Nm/g. It is also important to confirm that the major dynamics in the predictors are covered by the step changes, see Figure 8. Finally, to get an understanding of the process fluctuations in the estimation of the tensile index it is wise to also include a time plot, see Figure 9.

Thus, to use the tensile index model in on-line applications it is important to verify it over long periods and different process conditions. The reason is of course that the model parameters derived can change when using e. g. other refining segments, production levels etc.

It is finally, interesting to note that the models can be used in other processes based on CD-82 refiners as well. This statement was confirmed by implementing the model (with another intercept of course) in a TMP process, see Karlström et al. (2018).