Multivariate lognormal mixture for pulp particle characterization

First, we review the multivariate normal distribution with \({\varvec{\mu }}\in {\mathbb {R}}^d\) the vector of mean values, and \({\varvec{\Sigma }}\in {\mathbb {R}}^{d \times d}\) the covariance matrix, which is symmetric, \({\varvec{\Sigma }}^{\top }= {\varvec{\Sigma }}\), and positive definite, \({\varvec{\Sigma }} \succ {\varvec{0}}\). Its probability density function (PDF) is,where \({\varvec{\xi }}\in {\mathbb {R}}^d\) is a random-variable vector of dimension d. Its multivariate, cumulative distribution function (CDF) evaluated on a rectangular domain is,with \({\varvec{\lambda }}\in {\mathbb {R}}^d\) the lower limits and \({\varvec{\upsilon }}\in {\mathbb {R}}^d\) the upper limits. The CDF is not available in closed form but can be evaluated numerically using mvncdf in Matlab (MathWorks 2023). The PDF of the truncated, multivariate normal distribution is thenHere, if \({\varvec{a}}\in {\mathbb {R}}^d\) and \({\varvec{b}}\in {\mathbb {R}}^d\), then \({\varvec{a}}<{\varvec{b}} = \bigwedge _{i=1}^d (a_i < b_i)\), with analogous definitions for other vector inequalities. Henceforth, a superposed circumflex indicates a truncated distribution.

The truncated, multivariate lognormal distribution is obtained through a change of variables \({\varvec{x}} = \exp ({\varvec{\xi }})\), so that \({\varvec{\xi }}({\varvec{x}}) = \ln ({\varvec{x}})\). The Multivariate change of variables theorem gives the multivariate lognormal PDF,where \({\varvec{\ell }} = \exp ({\varvec{\lambda }})\), \({\varvec{u}} = \exp ({\varvec{\upsilon }})\), and where we used that \({\varvec{x}} > {\varvec{0}}\).

The likelihood of a set of n lognormally distributed samples \({\varvec{x}}^*_1, {\varvec{x}}^*_2,..., {\varvec{x}}^*_n\), truncated so that \({\varvec{\ell }}<{\varvec{x}}^*_i<{\varvec{u}}\), is given byFor maximum likelihood estimation (MLE), one seekswhere Eq. (4) yields,

The pulp particle geometry follows a multivariate distribution \(\psi ({\varvec{x}})\), where \(x_1=L\) is the fiber contour length and \(x_2 = W\) is the fiber width, and so on. Due to the window of observation, however, only a truncated distribution \({\hat{\psi }}({\varvec{x}}; {\varvec{\ell }}, {\varvec{u}})\) is experimentally accessible. The pulp may contain fractions of different character, such as shives, fibers, and fines, which implies that \({\hat{\psi }}\) may be multimodal with N modes,where \({\varvec{c}} \ge {\varvec{0}}\) is a vector such that \(\sum _{p=1}^N c_p=1\). We call this \({\hat{\psi }}\) a truncated lognormal mixture (TLM), inspired by the Gaussian-mixture concept. Each choice of N represents a different pulp particle model withfree parameters, including \(N-1\) elements of \({\varvec{c}}\), the vectors \({\varvec{\mu }}^{1},...,{\varvec{\mu }}^{N}\), and the lower triangles of \({\varvec{\Sigma }}^{1},...,{\varvec{\Sigma }}^{N}\).

The logarithm of the likelihood of a set of particle geometry samples \({\varvec{x}}^*_i\), \(i=1,2,...,n\), such that \({\varvec{\ell }}< {\varvec{x}}^*_i < {\varvec{u}}\), drawn from the TLM iswhere the logarithm of the sum is evaluated using the method in Appendix A to avoid numerical overflow, with \(\ln ({\hat{\phi }})\) given by Eq. (7). The maximum likelihood for the observed samples is found by solving the optimization problemIn this work, problem (11) is solved using the Nelder–Mead simplex algorithm, i.e. fminsearch in Matlab (Lagarias et al. 1998; MathWorks 2023), with a penalty step function to enforce \({\varvec{c}} \ge {\varvec{0}}\) and \({\varvec{\Sigma }}^{p} \succ {\varvec{0}}\).

Within this modeling framework, the parameter values \({\varvec{\mu }}^{1}, \ldots , {\varvec{\mu }}^{N}\) and \({\varvec{\Sigma }}^{1}, \ldots , {\varvec{\Sigma }}^{N}\) are independent of the truncation limits, \({\varvec{\ell }}\) and \({\varvec{u}}\). Therefore, when fitting these parameters using maximum likelihood estimation, the expected values remain consistent, irrespective of the chosen truncation limits. This consistency does not extend to \({\varvec{c}}\), which describes the number fractions for truncated distributions.

It is often of interest to calculate the marginal distributions of multivariate distributions; the qth marginal distribution of \({\hat{\psi }}\) is defined as,where \({\hat{\psi }}_q(x_q) = {\hat{\psi }}_q(x_q; {\varvec{c}},{\varvec{\mu }}^{1},...,{\varvec{\mu }}^{N},{\varvec{\Sigma }}^{1},...,{\varvec{\Sigma }}^{N}, {\varvec{\ell }}, {\varvec{u}})\). It is easy to show by direct integration that, if \({\varvec{\Sigma }}^1,...,{\varvec{\Sigma }}^N\) are all diagonal, then the marginal distributions of \({\hat{\psi }}\) are univariate TLMs, \({\hat{\psi }}_q(x_q) = {\hat{\psi }}_q(x_q; {\varvec{c}},\mu _q^{1},...,\mu _q^{N},\Sigma ^{1}_{qq},...,\Sigma ^{N}_{qq}, \ell _q, u_q)\). However, if the random variables are correlated in any way, then the marginal distributions are not univariate TLMs.

One use of the marginal distribution is ranking of the goodness-of-fit using the one-sample Kolmogorov–Smirnov (KS) statistic (Kolmogorov 1933; Smirnov 1944). This requires the marginal CDFThe KS statistic of the qth random variable is defined as,where \({\hat{\Psi }}_q^*(x_q)\) is the empirical marginal CDF.

The marginal distributions and corresponding CDFs generally depend on all elements of the vectors \({\varvec{\ell }}\) and \({\varvec{u}}\), and the analytical form of those marginal distributions is difficult to access. It is, however, possible to draw samples from the TLM by applying the exponential function to samples drawn from multivariate Gaussian distributions, i.e. by using mvnrnd in Matlab (MathWorks 2023). Hence, it is straight-forward to construct the marginal distributions and CDFs using a Monte-Carlo approach with a posteriori truncation (Fig. 1).

We consider data from previously published mill trials with different assortments of spruce wood at the CTMP plant in Skoghall papermill (Stora Enso) producing board grades. (Ferritsius et al. 2018, 2022) Briefly, the wood raw materials are three assortments of Norway spruce (Picea abies L. Karst.): roundwood chips, sawmill chips, or a 50/50 mixtures of roundwood and sawmill chips. The chip refiner is a Valmet CD 82 single disc. Each one of the chip assortments are run at three different levels of specific energy input by adjusting the flow rate of dilution water to the flat zone of the refiner.

CTMP from each chip assortment are produced for 4 h. Composite samples of screened chips going to the presteaming bin are collected before CTMP samples are taken. The refiner runs for 1 h with fixed process settings. Then, a composite CTMP sample is collected from the latency chest. Subsequently, the process settings are adjusted for the next specific energy input, and so on.

The chip samples from the mill trials are digested in a laboratory kraft cook to a yield of 49 %. In addition, a fraction of the CTMP samples is post-processed in the laboratory kraft cook. The CTMP, cooked CTMP, and cooked chip samples from different material sources are compiled in first three columns of Table 1.

Importantly, it should be noted that the scope of our study is limited to selected samples. These serve as distinct yet illustrative examples to demonstrate the potential utility of our novel characterization method. Therefore, while our findings offer insights into the method’s capabilities, they should not be interpreted as universally applicable.

After hot disintegration according to ISO 5263-3, the pulp samples are analyzed according to ISO 16,065-2 using the FiberLab™ optical analyzer (Valmet Automation Oy, Kajaani, Finland). Fiber length is measured with 10 \(\mu\)m resolution, and fiber width is measured with 1.5 \(\mu\)m resolution (FiberLab 2008). It is important to note that the FiberLab analyzer imposes inherent truncation limits on particle length between \(\ell _1 = 200\) \(\mu\)m and \(u_1 = 8900\) \(\mu\)m. Measurements falling outside these length parameters are automatically discarded by the analyzer. In our analysis, a lower width truncation limit of \(\ell _2 = 10\) \(\mu\)m set, below which measurements are discarded. A justification for selecting this particular truncation limit can be found in Appendix B. We also introduce the variable \(\gamma\) to quantify the fraction of excluded particles due to this width constraint (Table 1).

Laboratory sheets are prepared in compliance with ISO 5269-1 to facilitate mechanical testing. The apparent density of the sheets is assessed following the procedures outlined in ISO 534 (10 samples). Tensile attributes, including tensile index (TI), tensile stiffness index (TSI), strain-at-break, and tensile energy absorption (TEA), are evaluated using the methods prescribed in ISO 1924-3 (20 samples). The Scott-Bond energy is determined in accordance with the Tappi T 569 standard (6 samples), and the short span compression test is executed following the guidelines of ISO 9895 (10 samples).

We use AI language support provided by OpenAI’s ChatGPT to assist in the writing and editing of this article. We declare that the use of AI language support was for purpose of improving the clarity and accuracy of the language used in the article.

The probability that an observation \({\varvec{x}}^*_i\) originates from fraction p isParticularly, the probability that \({\varvec{x}}^*_i\) originates from the large-particle fraction is, \(P_{\textrm{large}}({\varvec{x}}^*_i) = P_2({\varvec{x}}^*_i)\) in the bimodal case. This probability can be used to categorize individual particles, and particles with \(P_{\textrm{large}} \ge 0.5\) are indicated by red markers in Fig. 2. We also define a third category of outliers, taken as the \(\alpha = 0.1\) % percentile of the likelihood. These objects are the most unlikely to be observed under the assumption of a bimodal TLM, and are indicated by ‘\(\times\)’ symbols in Fig. 2. Examples of possible outliers are shives or aggregates.

We exclude outliers and construct the marginal distributions of the observed values and the TLM for the roundwood CTMP in Fig. 3a, b. The agreement of the TLM marginal distribution with the empirical marginal distribution is excellent for the particle length and width in this particular example.

We proceed to consider bimodal TLMs fit for each of the CTMP samples, and to each one of the kraft post-processed samples. For convenience, we introduce the logspace-average length \({\bar{L}}^p = \exp (\mu ^p_1)\) and width \({\bar{W}}^p = \exp (\mu ^p_2)\) for fractions \(p=1,2\). The fit values of \({\bar{L}}^p\) and \({\bar{W}}^p\) are consistently reproducible for different pulps produced with the same raw material and processing conditions within our dataset (Fig. 4a). Pulps made from sawmill chips have wider large particles than roundwood-based pulp, and the width is reduced by kraft post-processing due to loss of lignin. For the CTMP samples, neither \({\bar{L}}^p\) nor \({\bar{W}}^p\) shows significant dependence on specific energy (not shown). The number fraction \(c_2\) of large particles is averaged across pulp samples with the same raw material and processing conditions, and plotted in Fig. 4b. We observe that there is a relatively larger fraction of large particles in cooked wood chips, which suggests fiber length reduction in the CTMP process. Figure 4b also shows that wood-chip samples are roughly equally divided between small and large particle fractions. This could be seen as conflicting with the initial assumption that the unprocessed fiber in the raw material is lognormal distributed. However, it has been previously established that the native fiber length–width distribution of Norway spruce does not include the small-particle component observed in this study (Havimo et al. 2008). This implies that even the cooked chips contain a substantial number fraction of debris.

We study the large-particle fraction, \(p=2\), of CTMP and kraft post-processed CTMP in more detail. The logspace-average length and width are plotted in Fig. 5, together with the elliptical region in logspace, which represents one standard deviation. These ellipses give a visual representation of the covariance matrix \({\varvec{\Sigma }}^2\). The covariance matrices are also robustly reproduced for pulps produced with the same raw material and processing conditions.

Let \(D_L\) be the KS statistic of the truncated length distributions, while \(D_W\) is the KS statistic of the truncated width distribution. For a large sample size n, a good fit is characterized by small values \(\sqrt{n}D_L\) and \(\sqrt{n}D_W\). We plot \((\sqrt{n}D_L,\sqrt{n}D_W)\) for all samples in Fig. 6a. We observe that the four cooked chip distributions exhibit the worst fits to the marginal length distribution. Instead of a lognormal small-particle fraction, the particles of chips exhibit an essentially uniform length distribution in the small-particle range, as exemplified by the sawmill chips (Fig. 6b).

The lognormality of the small-particle fraction appears to be contingent on mechanical treatment, as evidenced by a better fit to CTMP compared to cooked chips (Fig. 6a). Moreover, a possible explanation for the particularly poor fit of sawmill chip pulp is that the sawmill raw material represents conditional sampling of wood fibers; the sample is not drawn from all natural fibers of Norway spruce, but only those from mature trees, and only the most distal fibers from the pith.

The laboratory sheet density, TI, TSI, strain-at-break, TEA, Scott-Bond, and SCT index for uncooked CTMP are compiled in Table 2. It is of interest to predict these properties using particle size data. Therefore, we compare the predictive capability of the TLM parameters with those of conventional averages of the length and width marginal distributions.

We first define different types of weighted averages often encountered in the literature (Ferritsius et al. 2018). Let t be an element of the vector \({\varvec{x}}\) of particle properties to be averaged. For each observed particle \(i=1,...,n\), with properties \({\varvec{x}}^*_i\), \(t^*_i\) is the element of interest, and \(L^*_i\) is the particle length. We define the jth-order length-weighted average of t asThat is, \(\left\langle t \right\rangle _0\) is the arithmetic average, \(\left\langle t \right\rangle _1\) is the length-weighted average, and \(\left\langle t \right\rangle _2\) is the length-length-weighted average.

To formalize the question of whether TLM parameters, in some sense, are better predictors than weighted averages, we define the predictor variable vector of weighted averages as \({\varvec{X}}_{\textrm{avg}} = [\left\langle L \right\rangle _0,\left\langle W \right\rangle _0,\left\langle L \right\rangle _1,\left\langle W \right\rangle _1,\left\langle L \right\rangle _2,\left\langle W \right\rangle _2]\), and the TLM predictor variable vector as \({\varvec{X}}_{\textrm{TLM}} = [c_2,\mu ^1_1,\mu ^1_2,\mu ^2_1,\mu ^2_2]\). The response variable \({\varvec{Y}}\) comprises laboratory sheet density, TI, TSI, strain-at-break, TEA, Scott-Bond, and SCT index. The corresponding standardized variable vectors, \({\varvec{{\tilde{X}}}}_{\textrm{avg}}\), \({\varvec{{\tilde{X}}}}_{\textrm{TLM}}\) and \({\varvec{{\tilde{Y}}}}\), are obtained by scaling each element so that it has zero mean and unit variance. Finally, we fit each one of the standardized predictor variables to the standardized response variables in a least-squares sense, using fitlm in Matlab (MathWorks 2023). The predictive capability of weighted averages, \({\varvec{{\tilde{X}}}}_{\textrm{avg}}\) (Fig. 7a), is outperformed by that of the TLM, \({\varvec{{\tilde{X}}}}_{\textrm{TLM}}\) (Fig. 7b), for all responses, as demonstrated by comparing the \(R^2\) values (Fig. 7c) of the respective fits. This shows that the TLM characterization captures the details of the particle distribution better than any combination of conventional averages of the length–width distribution, when judged as predictors of sheet properties. We believe that the TLM description of the length–width distribution is more precise than averages formed using marginal distributions of width and length, as evidenced by better correlation with laboratory sheet properties even though \({\varvec{{\tilde{X}}}}_{\textrm{TLM}}\) has a lower dimensionality than \({\varvec{{\tilde{X}}}}_{\textrm{avg}}\).