Behavior of Carbon Adhesion on Aged Coking-chamber Walls to Pushing Load

Abstract

A common problem observed in aged coke plants is the increase in pushing load during the discharge of coke mass, arising from irregularities on the damaged coking-chamber walls. Usually, the chamber wall is partially covered with adhered carbon. Because carbon growth is influenced by a number of factors, the chamber wall has a complicated carbon adhesion distribution, and the amount of carbon-covered area differs from chamber to chamber. Carbon adhesion locally affects the pushing load both positively and negatively. Small carbon deposits filling the surface depressions lower the pushing load. In contrast, excess carbon growth creating a protrusion shape occasionally behaves as a resisting force during pushing. This study is performed to elucidate the influence of the carbon-covered area on the pushing load. Chamber wall images were gathered at operating coke plants by means of an inspection apparatus, which was inserted into the high-temperature chambers. An image processing technique was devised for classifying the wall surface into three states: bare brick, dense carbon, and patchy carbon. It was confirmed that the dense carbon has the optimum amount for suppressing high pushing loads. Statistical analysis using a probability model demonstrated that stable pushing can be obtained in specific dense and patchy carbon amounts.

1. Introduction

Metallurgical coke, which is necessary for the iron-making process by means of blast furnaces, is produced in a coke plant having a number of coking chambers. Many coke plants were built in the 1970s in Japan. When the usage period of coke oven batteries exceeds 40 years, the production capacity and energy efficiency gradually decrease due to the deteriorated ovens. One of the problems commonly observed in aged coke plants is the inevitable increase in pushing load when extracting coke cake. Irregularities on the coking-chamber walls arising from wear damage and partial defects of brick surfaces worsen the pushing load. Experimental studies on this phenomenon have been reported.^1,2,3,4) The relationship between the deformed wall shape and the pushing load was examined in detail using irregularity data of coking chamber walls obtained by a diagnosis apparatus equipped with a 3D profile measurement system.⁵⁾

Usually, the wall of the coking chamber is partially covered with adhered carbon that is produced during the coking process. Carbon growth is influenced by a number of factors, such as wall temperature, brick surface roughness, chemical nature of coal, and air flow condition of a carbon incineration lance.^6,7,8) Because these factors are uneven in the coking chamber, the chamber wall has a complicated carbon adhesion distribution. In addition, the amount of carbon-covered area differs from chamber to chamber. Carbon adhesion locally affects the pushing load both positively and negatively depending on the circumstance. A favorable thin carbon layer flattens the relatively small surface depressions and lowers the pushing load. In contrast, excess carbon growth creating a protrusion shape on the chamber wall occasionally blocks extracting coke cake. Effective carbon control for stable production, which is particularly desirable at a coke plant in operation for a few decades, counts on the experience of factory workers.

In our previous study, an inspection apparatus that obtained images of huge walls with a high temperature exceeding 1000°C inside a coking chamber was developed and applied.⁹⁾ We then developed an image processing technique that enabled quantification of the carbon distribution.¹⁰⁾ The effects of unevenness on the coking-chamber wall on pushing load has not been separated from the effects of carbon on the coking-chamber wall. Since concavo-convex shape data of carbon depositions had not been obtained, we conducted statistical analyses on the influence of the carbon-covered area size on the coking-chamber walls on the pushing load in the present study. First, in Chapter 2, an image processing technique for recognizing the carbon state on a high-temperature coking chamber wall is explained. A texture analysis in conjunction with machine learning was applied to classify the wall surface into three states: brick, dense carbon, and patchy carbon. In Chapter 3, the actual situation of carbon deposition at operating coke plants and its relationship to the pushing load are statistically analyzed. In addition to a simple scatter graph method, a probability model using a mixture of Gaussians is proposed because the pushing load data have a large dispersion. The results of the analyses are discussed, and some findings on the characteristics of carbon adhesion are summarized in Chapter 4.

2. Recognition of Carbon Adhesion on Coking-chamber Walls

Coking chamber wall images are obtained using a heat-resistant diagnostic apparatus.⁹⁾ The diagnosis is carried out in an empty chamber after coke cake is discharged. The heat-resistant imaging apparatus is inserted from the pusher side and promptly goes forward and backward in the chamber within approximately four minutes while taking the left and right wall images, respectively. Since carbon deposited on the chamber wall combusts due to the air entering from the pusher side opening, a carbon region is brighter than the surrounding brick region. Line scan cameras moving inside the chamber capture a thermal image of the chamber wall, which emits visible light radiation. Figure 1 presents an example of a portion of the thermal image. The bare brick region has a vertical and horizontal line pattern composed of mortar joints and vertical cracks filled with bright carbon. Broadly, carbon can be recognized as a region with less brightness distribution. The carbon surface tends to be brighter than the brick surface because the high-temperature carbon is supposed to be in a state of combustion.

Fig. 1.

Appearance of coking-chamber wall.

The surface of the chamber wall was classified into three states, “brick,” “dense carbon,” and “patchy carbon,”¹⁰⁾ as shown in Fig. 1. A lattice pattern of brick structure can be distinctly observed in the “brick” area. The “dense carbon” area is continuously covered with a carbon layer. The authors use “dense” to denote the spatially continuous situation. In the “patchy carbon” area, each brick is partially covered with carbon that spreads from the joints and cracks. It is difficult to define the objective boundary between the brick region and patchy carbon region in the case of the intermediate amount of carbon growth. Accordingly, sample data of the patchy carbon area were subjectively selected and utilized as training data for after-mentioned analyses. In the patchy carbon region, small-scale dips on the wall arising from brick edge defects are smoothed by carbon deposition, and the frictional resistance of the wall during coke cake pushing is lower than one of the bare brick region.

Although it is easy for the human eye to distinguish the carbon region on the wall, simple image processing, such as binarization focusing on the brightness difference between the brick and carbon, does not work. This is because the gray level in the thermal image fluctuates depending on its temperature. In this study, we applied texture analysis, which is a common technique for locating specific objects on a given image using texture features defined in advance. The details of the texture analysis and related image processing are described as follows.

The data processing flow consisting of image processing and classification is presented in Fig. 2. In the first half of the data processing, images are created to derive texture features. Figure 3(a) shows an example of an entered original image of the entire wall of one side of a coking chamber (16 m in length and 6 m in height). It has a resolution of approximately 16000 pixels in the lateral direction and 4096 pixels in the vertical direction. This resolution is halved after the shrinking process. The vertical and horizontal line patterns observed in the brick region are considered to be distinctive characteristics for recognizing the wall surface state. As the first step of the image analysis, preprocessing by a top-hat transform is performed to enhance the line pattern. The top-hat transform is defined as the difference between the input image and its opening. The line-enhanced image is shown in Fig 3(b). The line-enhanced image is used for calculating gray level co-occurrence matrix features and histogram features mentioned later. By adding grayscale dilation before the top-hat transform to remove fine gray-level unevenness of the carbon areas, a binary image composed only of brick joints and cracks can be obtained, as shown in Fig. 3(c). Then, consider an image in which each gray level is determined as the distance from a reference pixel to the nearest bright pixel in the binary image. Such a “distance image” generated from image (c) is (d) in Fig. 3. Here, the shorter the distance, the brighter the gray level. This property is considered to be another useful texture feature for detecting carbon.

Fig. 2.

Data processing flow for classification of wall carbon states.

Fig. 3.

Images in processing (a) input original image, (b) line-enhanced image, (c) line-extracted image, and (d) distance image.

In the latter half of the data processing flow in Fig. 2, the line-enhanced image and the distance image are divided into small areas called the region of interest (ROI). The ROI is a unit for calculating the texture features. We set the ROI size to 128 pixels × 128 pixels. Generally, suitable features for a texture analysis are selected by trial and error. The features used in the current study are derived from three different methods,¹⁰⁾ as mentioned below.

2.1. Gray Level Co-occurrence Matrix Features

Gray-level co-occurrence matrices (GLCMs)¹¹⁾ are widely used in texture analysis because they can extract the spatial dependence of gray-level values in an image. The GLCM is a matrix that is defined from the distribution of co-occurrence grayscale values between two pixels at a constant offset. Several statistics that can be calculated from the co-occurrence matrix are used as features for distinguishing the texture of an image. The features selected in this study include angular second moment, contrast, entropy, and correlation.

2.2. Histogram Features

A histogram of gray levels provides a concise summary of the statistical information contained in an image. The shape of the histogram varies, reflecting the character of the image. For example, histograms of “brick,” “dense carbon,” and “patchy carbon.” ROIs have distinctly different shapes, as illustrated in Fig. 4. Features describing the shape of the histogram, such as skewness and coefficient of variation, can be used for texture analysis.

Fig. 4.

Schematic illustration of histogram shapes.

2.3. Distance Image Feature

It is obvious that dense carbon areas in the original image have high brightness on the distance image as can be seen in Fig. 3(d). The average gray level of a ROI on the distance image is expected to be a significant feature.

A support vector machine (SVM) is utilized for the classification of “brick,” “dense carbon,” and “patchy carbon” regions. Training data provided for the learning were collected from a number of chamber-wall images. Patchy carbon data were subjectively selected. The results of the analysis based on the GLCM and histogram and distance image features are depicted in Fig. 5. “Brick,” “dense carbon,” and “patchy carbon” are successfully recognized at the most ROIs. It was confirmed that GLCM and distance image features were more useful than histogram features.

Fig. 5.

Example of raw classification result.

In the final step of the data processing flow, a majority vote algorithm, which was devised to deal with false classification, is performed. Some scattered ROIs in the dotted ovals in Fig. 5 are the concrete target of the majority vote algorithm. These ROIs in brick or patchy carbon regions were mistakenly recognized as carbon. Consider that the length of the horizontal and vertical sides of a ROI (128 × 128 pixels) is divided evenly into n parts. Here, n is an integer value greater than 1. Then, n × n origins are determined by shifting the starting points at intervals of 128/n in both the horizontal and vertical directions. Texture features are calculated for a set of ROIs that are lined up from each origin. The classification of a small area with 128/n × 128/n pixels is decided on a majority vote as it is concerned with n² candidates of analysis calculation results. Figure 6 presents a simplified illustrative example for n = 2. Here, 2² = 4 origins (O₁, O₂, O_3, and O₄) are designated. All gray small areas with 64 × 64 pixels in the figure are located at the same position. The four ROIs indicated by heavy line squares contain the gray area. The most common wall surface state among 4 feature analyses is selected. Setting the division number of the ROI to 16 (n = 4) was appropriate in practice. The classification result presented in Fig. 7 includes the majority vote algorithm. A glance at the small regions indicated by the dotted line ovals in Fig. 5 indicates that the separation accuracy explicitly improves in Fig. 7. The majority vote algorithm also contributes to drawing smooth boundary lines.

Fig. 6.

Simplified depiction of majority vote algorithm.

Fig. 7.

Example of final classification result.

3. Data Analysis Results

At a coke plant that is equipped with a diagnostic apparatus, high-temperature brick walls inside coking chambers are inspected during the interval of the manufacturing operation. A hundred coking chambers are observed in sequence to reduce the impact on production. The inspection frequency differs from plant to plant. The obtained wall image data is stored at each plant. Pushing load data, which is monitored by means of an ammeter or a torque meter placed at the driving motor of a pusher, is recorded as a crucial indicator of daily plant operation. In this study, we attempted to clarify the influence of carbon adhesion on the pushing load based on statistical analyses.

We selected two coke-making plants presented in Table 1 for the analyses. Coke plant A has been in operation for more than 50 years, and its coking chamber walls may have relatively large roughness. Coke plant B was expected to have a wide range of adhered carbon conditions because chamber wall diagnosis is frequently carried out and a large amount of inspection data is accumulated. Our investigation on the influence of carbon deposits was intended not to be limited to specific operating conditions. For this reason, six and eight years of analysis data were gathered from coke plant A and coke plant B, respectively. The operating conditions, such as working rate, coking temperature, carbon blend, and carbon moisture, varied many times over several years.

Table 1. Specification of the coke plants.

	Coke plant A	Coke plant B
Start year of operation	1964, 1967	1970
Coking chamber dimension (height, width, length)	5.0 × 0.42 × 13.6 m	6.0 × 0.45 × 15.7 m
Data gathering year	2010 to 2015	2008 to 2015
Number of inspection data	97	396

The wall image data is combined with the pushing load data immediately before the inspection were prepared. The pushing load is calculated from the maximum torque of the pusher’s driving motor at coke plant A. The maximum torque is observed when an entire coke cake begins to move. Regarding coke plant B, the maximum current of the motor is used as a substitute for the pushing force. The analyses were conducted by the two approaches described below.

3.1. Scatter Graph Method

The carbon deposit on a chamber wall can be quantified by means of the image processing software mentioned in the previous chapter. When the wall surface state is classified as shown in Fig. 7, the pixel numbers of the brick regions, dense carbon regions, and patchy carbon regions obtained are normalized as area ratios ranging from 0 to 1. If the whole wall is covered with one state, its area ratio becomes 1 and the area ratios of the other states are 0. Here, the total wall pixel includes unclassified portions such as the dark region and saturated region. The image regions adjacent to the chamber doors are too dark for the classification because of the decrease in temperature. Such unclassified portions account for 10% to 20% of the entire area in many cases. Wall image data that have more than 20% of the unclassified portions were not utilized for the analyses.

Figure 8 presents histograms of dense carbon area ratios of coking chamber walls. Here the dense carbon area is the average of left and right walls. The amount of carbon deposit at coke plant A is small compared to coke plant B. The difference of the dense carbon area ratios between two plants implies that the empirical carbon control policy differs from plant to plant. As illustrated in Fig. 9, dense carbon amounts of left and right walls facing each other are not equal but are positively correlated. The bilateral difference tends to be large when the carbon area ratio is in the middle range at coke plant B.

Fig. 8.

Histograms of dense carbon area ratio.

Fig. 9.

Relationship of dense carbon area ratios between right and left walls.

Figure 10 depicts the relationship between the patchy carbon area ratios of the left and right walls of coke plant A. The patchy carbon area ratios of facing walls in a coking chamber increase and decrease relative to one another in the same way as the dense carbon area ratio. On the grounds of the left-right symmetry of adhered carbon, as can be seen in Figs. 9 and 10, the area ratios of facing walls in a coking chamber are averaged for both the dense carbon and patchy carbon in the analyses described later.

Fig. 10.

Patchy carbon area ratios at plant A.

The relationship between dense carbon area ratio and patchy carbon area ratio is presented in Fig. 11. As for coke plant A, in which carbon adhered to chamber walls is holistically small, the patchy carbon area ratio fluctuates along with the dense carbon in the range of less than 0.2. In coke plant B, patchy carbon area ratio varies widely in the dense carbon area ratio range of 0.2–0.4.

Fig. 11.

Relationship between dense carbon area ratio and patchy carbon area ratio.

Figure 12 shows the relationship between dense carbon area ratio and pushing load. The left graph of coke plant A denotes that the pushing load is consistently low at the dense carbon area ratio of 0.2 to 0.3. On the other hand, regarding coke plant B, a stable zone of the pushing load exists at the dense carbon area ratio of 0.2 to 0.3.

Fig. 12.

Distribution of dense carbon area ratio and pushing load.

3.2. Probability Model

In this section, the pushing load variation with two variables of dense carbon area ratio and patchy carbon area ratio is analyzed. As can be seen in Fig. 12, the pushing load characteristically has a large dispersion because it is influenced by not only the chamber wall appearance but also by a number of factors, such as the material properties of coal and coking conditions. Therefore, a simple analysis approach using the scatter graph method is not available. To solve this challenge, we used mixtures of Gaussians¹²⁾ to generate the probability model from the observation date. Mixtures of Gaussians are linear combinations of multiple Gaussian distributions, and any continuous density can be approximated by tuning the parameters of mixtures of Gaussians. The model by mixtures of Gaussians is fitted to the actual data of the pushing load and dense/patchy carbon area ratio, and the relationship between the pushing load and dense/patchy carbon area ratio is derived from the fitting model.

Figure 13 shows the procedure to generate probability density based on mixtures of Gaussians in the case of a simple two-dimensional description. Two-dimensional data x y is obtained as shown in (1) of Fig. 13. Mixtures of Gaussians are linear combinations of multiple Gaussian distributions. The probability density p(x,y) is modeled by   

$$p\left(x,y\right)={\sum }_{k=1}^k{\pi }_k\mathrm{\hspace{0.17em} }N\text{(}x,y\text{|}\mathrm{\hspace{0.17em} }{\mu }_k,{\mathrm{\Sigma }}_k)$$(1)

  

$${\sum }_{k=1}^K{\pi }_k=1\mathrm{\hspace{0.17em} }\mathrm{\hspace{0.17em} }\mathrm{\hspace{0.17em} }\mathrm{\hspace{0.17em} }0\le {\pi }_k\le 1$$(2)

where, k is the index of Gaussian distributions, K is the number of Gaussian distributions, $N\text{(}x,y\text{|}\mathrm{\hspace{0.17em} }{\mu }_k,{\mathrm{\Sigma }}_k)$ is a Gaussian distribution of the mixture, has its own mean μ_k and covariance Σ_k, and π_k is the coefficient in the linear combination.

Fig. 13.

Schematic explanation for deriving probabilistic density based on mixture of Gaussians.

Sufficiency Gaussian distributions ((2) of Fig. 13: 3 component of Gaussians) are scattered so as to match the data distribution. Then, Gaussian distributions are linearly combined after adjusting the coefficients π_k, so that the probability density of the given data can be approximated as shown in (3) of Fig. 13.

One way to set the values of these parameters is to use maximum likelihood (the evaluation value that quantitatively evaluates how likely the data is for the model) to adjust the means μ_k and covariances Σ_k as well as the coefficients π_k in the linear combination. A well-known adjusting method is the EM algorithm. Regarding the optimization number K of Gaussian distributions, it is possible to search for the best number by using a cross-validation or Bayesian information criterion.

The relationship between x-and y is estimated by using the approximated probability density p(x,y). Considering a particular probability of y for any given value of x, by calculating y at an arbitrary cumulative probability from the approximated probability density p(x,y), the relationship between x-and y can be described in (4) of Fig. 13. For example, when the cumulative probability is set to 50%, at the cumulative probability of 50%, y corresponds to the median. The median of y for any given value of x can be visualized (denoted by the broken line in (4) of Fig. 13).

In this study, the cumulative probability of y is set to 80%, and the evaluation is performed with a value larger than the median (represented by the solid line in (4) of Fig. 13). It should be noted that the description has been detailed for the case of a simple two-dimensional situation, but it is also applicable to the cases having three or more dimensions.

The pushing load analysis, based on the probability model described above, was applied to each coke plant data. In Fig. 14, the dense carbon area ratios and patchy carbon area ratios are plotted on the horizontal axis and vertical axis, respectively. Colors ranging from blue to red estimate the pushing load derived from the probability model. The pushing load is indicated by force (N) calculated from a torque meter at Plant A and is monitored as electric current (A) of the pushing motor at Plant B. The results for coke plant A reveal that excess carbon deposition (region B1 in the figure) and deficiency in carbon deposit (region B2) explicitly cause the rise of pushing load. The color-coded graph for coke plant B differs slightly from that of coke plant A. However, both coke plants A and B exhibit a similar tendency wherein a stable pushing load zone exists against the dense and patchy carbon (regions A).

Fig. 14.

Pushing load distribution in the plane of dense and patchy carbon area ratios.

4. Discussion

The characteristics of dense carbon adhesion derived from Figs. 8 and 9 have been explained here. The dense carbon area ratio has an upper limit of around 0.7. The difference in dense carbon area ratios between the right and left walls in a coking chamber increases from zero to approximately 0.3, and then gradually decreases at 0.4 or above. In the range of high dense carbon area ratios exceeding 0.6, which is deemed to be excessive carbon deposits, facing chamber walls exhibit similar dense carbon area ratios.

Regarding the relationship between the two carbon adhesion states, the dense carbon area ratio is not proportional to the patchy carbon area ratio, as visible in Fig. 11. In particular, the two area ratios are negatively correlated in the range of dense carbon area ratios above 0.3. It is surmised that the patchy carbon is the intermediate state of the creation process, where it is transforming from a bare brick surface to a carbon layer. For example, in Fig. 11, the wall state located at a dense carbon area ratio of 0.3, and a patchy carbon area ratio of 0.3 might move to the highly adhered carbon state with a dense carbon area ratio of 0.6 during the carbon growing phase, due to the change of patchy carbon regions to dense carbon regions. Such a temporal wall state change in one coking chamber is a task to be studied in the future.

The material form of dense carbon is a carbon layer with uneven thickness, and it is a factor that raises the pushing load. In contrast, patchy carbon decreases the pushing load because it smoothens small-scale depressions arising from brick edge defects along vertical cracks. From this point of view, it is sometimes stated that an optimally controlled chamber wall is wholly covered with patch carbon. However, the ideal carbon distribution cannot be confirmed at the actual working aged coke plants.

The optimal amount of carbon deposition was investigated in previous studies by Niinou and Edano.^13,14) They proposed a carbon control technique by means of a carbon incineration lance, in which the discharged gas CO₂ concentration related to the total area of adhered carbon was monitored. The present study directly quantified the amount of adhered carbon. Figure 12 explicitly indicates that the dense carbon area ratio has the optimum range for suppressing high pushing loads. Deteriorated brick walls have small-sized depressions caused by brick edge defects. A small amount of carbon depositions, which appears as the patchy carbon, flattens the surface depressions and lowers the pushing load. Conversely, excess carbon growth creates a protrusion shape, which increase the pushing load. The dense carbon area ratio can be a simple indicator for managing the pushing load in daily plant operation.

From the analysis results of the probability model, it was confirmed that stable pushing can be obtained in the specific range of the two variables: dense and patchy carbon area ratios. Figure 14 indicates that excessively small carbon deposition significantly worsens the pushing load, and the maximal level of dense carbon area ratio should also be avoided. It can be explained by the different effects between dense carbon and patchy carbon on the pushing load, as mentioned before. Because the two favorable area ratios are different for each plant, as shown in Fig. 14, the probability model analysis is helpful for finding a carbon adhesion state targeted.

5. Conclusion

This study presents a new approach to investigate the influence of carbon deposition on the pushing load targeting long-used coking chambers. Thermal images of brick walls inside high-temperature coking chambers played a pivotal role in our study. These images were obtained by means of a heat-resistant diagnosis apparatus. The carbon that was adhered to the chamber wall was classified into two states: “dense carbon” and “patchy carbon”. The actual situations of carbon deposition at two operating coke plants were quantified. The relationship between the dense and patchy carbon areas changes characteristically depending on the total carbon amount. In addition to a simple scatter graph method, a probability model was applied to investigate the relationship between the carbon area and pushing load. The analysis results revealed that an appropriate carbon control for suppressing high pushing loads can be realized for a specific amount of adhered carbon.

References

• 1)   K.  Nishioka,  H.  Ueda,  M.  Ogawa and  S.  Yoshida: Coke Circular, 35 (1986), No. 1, 21 (in Japanese).
• 2)   S.  Watakabe,  K.  Takeda,  H.  Suginobe and  H.  Itaya: Tetsu-to-Hagané, 84 (1998), No. 3, 165 (in Japanese).
• 3)   T.  Nakagawa,  T.  Arima,  K.  Kato and  M.  Ando: CAMP-ISIJ, 15 (2002), 766 (in Japanese).
• 4)   S.  Aizawa,  K.  Uebo and  S.  Yoshida: CAMP-ISIJ, 20 (2007), 877 (in Japanese).
• 5)   M.  Sugiura,  T.  Nakagawa,  T.  Arima,  K.  Kato,  M.  Sakaida,  Y.  Morizane,  A.  Sano and  K.  Irie: ISIJ Int., 53 (2013), No. 4, 583. https://doi.org/10.2355/isijinternational.53.583
• 6)   Y.  Jōmoto,  H.  Matsuoka and  S.  Ōta: J. Fuel Soc. Jpn., 48 (1969), No. 510, 732 (in Japanese).
• 7)   K.  Uebo,  H.  Kunimasa and  S.  Suyama: Tetsu-to-Hagané, 90 (2004), No. 9, 721 (in Japanese).
• 8)   T.  Nakagawa,  K.  Kato and  M.  Naito: Nippon Steel Tech. Rep., (2006), No. 94, 64. https://www.nipponsteel.com/en/tech/report/nsc/pdf/n9411.pdf
• 9)   M.  Sugiura,  M.  Sakaida,  Y.  Fujikake and  K.  Irie: Trans. Soc. Instrum. Control Eng., 47 (2011), No. 10, 435 (in Japanese).
• 10)   M.  Sugiura and  Y.  Abe: Proc. SICE Annual Conf. 2017, (Kanazawa), SICE, Tokyo, (2017), 429, USB memory.
• 11)   R. M.  Haralick,  K.  Shanmugam and  I.  Dinstein: IEEE Trans. Syst. Man Cybern., SMC-3 (1973), No. 6, 610.
• 12)   C. M.  Bishop: Pattern Recognition and Machine Learning, Springer, New York, (2006), 110.
• 13)   T.  Niinou,  Y.  Fukasawa,  M.  Koga,  H.  Oshima,  Y.  Watanabe and  T.  Tsutsumi: CAMP-ISIJ, 24 (2011), 147, CD-ROM (in Japanese).
• 14)   T.  Edano and  Y.  Fukasawa: CAMP-ISIJ, 31 (2018), 618, CD-ROM (in Japanese).