**A case-based reasoning method for discriminating damage levels in ancient wood components based on fuzzy similarity priority**,"

*BioResources*16(3), 4814-4830.

#### Abstract

In order to rapidly identify internal damage levels accurately in ancient wood components, stress wave detection technology was used to perform simulated damage tests on pine specimens. Based on the detected wave velocity data, the diameter of the specimen, the attenuation coefficient, and the ratio of the wave velocities on the four paths were selected as the discriminant factors for identifying the level of internal damage in the specimens. A case-based reasoning method for discriminating internal damage levels in ancient wood components based on fuzzy similarity priority was proposed. A fuzzy similarity priority relationship between the target case and the source case was established. By introducing the idea of variable weights, the weight of each discriminant factor was determined via the “penalize-excitation” variable weight function. The comprehensive similarity sequences between the target case and the source case were obtained. The source case that was most similar to the target case was used to determine the damage level of the target case. The results showed that this method can quickly and accurately identify the damage levels in ancient wood components, which provides a new method for the safe evaluation of ancient wood buildings.

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#### Full Article

**A Case-Based Reasoning Method for Discriminating Damage Levels in Ancient Wood Components Based on Fuzzy Similarity Priority**

Ziyi Wang,^{a,b} Donghui Ma,^{a,b,c} Wei Wang,^{a,b,}* Wei Qian,^{a,c} Xiaodong Guo,^{a,b,c }Junhong Huan,^{d} and Zhongwei Gao ^{a,b}

In order to rapidly identify internal damage levels accurately in ancient wood components, stress wave detection technology was used to perform simulated damage tests on pine specimens. Based on the detected wave velocity data, the diameter of the specimen, the attenuation coefficient, and the ratio of the wave velocities on the four paths were selected as the discriminant factors for identifying the level of internal damage in the specimens. A case-based reasoning method for discriminating internal damage levels in ancient wood components based on fuzzy similarity priority was proposed. A fuzzy similarity priority relationship between the target case and the source case was established. By introducing the idea of variable weights, the weight of each discriminant factor was determined *via* the “penalize-excitation” variable weight function. The comprehensive similarity sequences between the target case and the source case were obtained. The source case that was most similar to the target case was used to determine the damage level of the target case. The results showed that this method can quickly and accurately identify the damage levels in ancient wood components, which provides a new method for the safe evaluation of ancient wood buildings.

*Keywords: Ancient wood components; Stress wave; Damage level; Case-based reasoning; Fuzzy similarity priority*

*Contact information: a: Faculty of Architecture, Civil and Transportation Engineering, Beijing University of Technology, Beijing 100124, China; b: Key Scientific Research Base of Safety Assessment and Disaster Mitigation for Traditional Timber Structure (Beijing University of Technology), State Administration for Cultural Heritage, Beijing 100124, China; c: Beijing Engineering Research Center of Historic Buildings Protection, Beijing University of Technology, Beijing 100124, China; d: School of Civil Engineering, Shijiazhuang Tiedao University, Shijiazhuang 050043, China; *Corresponding author: ieeww@bjut.edu.cn*

**GRAPHICAL ABSTRACT**

**INTRODUCTION**

As an important part of ancient Chinese architecture, wood structures are the crystallization of the sweat and wisdom of ancient people and have high historical, artistic, and scientific value. Since most ancient wooden structures are exposed, long-term erosion from natural sources, *e.g.*, wind, rain, and temperature difference, leads to various sources of decay, *e.g.*, moths, damages, and cracks. This not only brings hidden dangers to the structural safety of the whole ancient building, but it also causes the original and true historical information of ancient buildings to gradually disappear (Li 2015). Therefore, it is necessary to obtain accurate information detailing the damage through appropriate and reasonable detection technology, which is of great importance in evaluating the safety and health status of ancient wood structures.

In recent years, great progress has been made in terms of the detection and protection of ancient wood structures. Among them, as a modern nondestructive testing technology, stress wave testing is widely used in the exploration, analysis, and evaluation of ancient wooden structures. Lee (1965) first used stress wave detection technology to determine the propagation time of lateral stress waves and longitudinal stress waves in wood components, and finally obtained the propagation velocity of the stress waves. Lin and Wu (2013) used the stress wave method to test elm specimens. The results showed that the moisture content of the wood, cracks, size of the holes, and number of holes had significant effects on the expansion parameters and the dynamic elastic modulus. Dackermann *et al*. (2014) used stress waves to measure structural wood properties and proposed how to evaluate the test results and the health status of the analyzed wood components. Li *et al*. (2015) selected six variation factors as the discriminant basis to classify the defect grades of wood components based on the stress wave detection data. Morales-Conde and Machado (2017) predicted the elastic modulus of wood *via* stress wave detection technology considering the spatial variability of wood cross sections. Du *et al*. (2018) proposed a three-dimensional stress wave imaging method based on TKriging to reconstruct internal defect images of wood. Huan *et al*. (2018) proposed a stress wave tomography algorithm with a velocity error correction mechanism based on the wave velocity data set measured *via* stress waves. Finally, a sectional image of the test sample image was generated. Wang *et al*. (2019) used a stress wave and impedance meter to detect the internal damage of wood components, and induced ordered weighted averaging (IOWA) operators, induced ordered weighted geometric averaging (IOWGA) operators, and induced ordered weighted harmonic averaging (IOWHA) operators were introduced to establish a combined prediction model to predict the internal damage of ancient wood structures. Yue *et al*. (2019) compared electric resistance tomography and stress wave tomography for decay detection in trees. The results showed that electric resistance tomography was better than stress wave tomography for testing the early stages of decay, while stress wave tomography was more effective for the late stages of decay. Wang *et al*. (2020) proposed a coupling model of a fuzzy soft set and the Bayesian method to forecast internal defects in wooden structures based on stress waves and an impedance test. Bandara *et al*. (2021) used time-frequency analysis techniques and stress waves to evaluate health monitoring of timber poles.

According to the existing research, when stress waves are used to detect the damage of ancient wood components, the internal damage is generally judged *via* analyzing the 2D image. Although the damage can be qualitatively judged, the analysis process is cumbersome and there is not a high degree of accuracy (Xu *et al*. 2011). There are few scholars who have studied the relationship between the grade of the internal wood damage and the wave velocity of the stress wave. In view of this, the diameter of the specimen, the attenuation coefficient, and the ratio of the wave velocities on the four paths were selected as the discriminant factors for identifying the internal damage levels of specimens. Meanwhile, the concept of fuzzy similarity priority has been introduced to propose case-based reasoning (CBR) for the identification of internal damage in ancient wood components (Zhao and He 2008; He *et al*. 2009; Zhang *et al*. 2015). Concretely speaking, the fuzzy similarity priority relationship between the target case and source case is first established for each discriminant factor. Secondly, the weight of each discriminant factor is determined *via* the “penalize-excitation” variable weight model. Finally, a comprehensive similarity sequence between the target base and source base was obtained to find the source base most similar to the target base. This method can quickly and accurately identify the level of internal damage in ancient wood components.

**EXPERIMENTAL**

**Materials**

Pine, which commonly has been used in ancient timber structures, was selected as the test specimen. After visual examination and compression examination, no obvious defects, *e.g.*, joints and cracks, were found in these specimens. The measured average moisture content of the specimens met the requirements of GB/T standard 50005 (2017) and GB/T standard 50329 (2012). Reverse simulation tests were conducted on six specimens, and the simulated type was hollow. The simulated damage ratios were 0, 1/32, 1/16, 1/8, 1/4, and 1/2, respectively. It was assumed that the specimen and the damage shape were a standard complete circle. Table 1 shows the different parameters of the specimens.

**Stress Wave Detection**

A stress wave detector produced by FAKOOP (Ágfalva, Hungary) was used to test these specimens.

**Fig. 1. **(a) Stress Wave Detection; (b) Propagation path; and (c) Two-dimensional image

First, eight sensors were selected to place around the specimen using eight steel nails (Fig. 1a). The sensors were connected with a signal amplifier, which could realize wireless connections with a computer *via* Bluetooth. After tapping each sensor with a hammer three times, each sensor transmitted the wave velocity information to the computer through a signal amplifier. Finally, the 2D image of internal damage of the wood was calculated using the ArborSonic 3D software (Version 5.2.107, FAKOPP, Ágfalva, Hungary) (Fig. 1c). Figure 1b shows the propagation path and the 2D image.

**Selection of Discriminant Factors and Damage Level**

In this experiment, six intact specimens were first tested to obtain the wave velocity value of each path, which was denoted as the initial comparison value *V _{m0}*

*(m = a, b, c, d)*. Then, the wave velocities of the specimens were measured using different damage ratios. Wave velocities can be divided into four categories, according to the propagation path (Fig. 1b). Concretely speaking, the wave velocity values between two adjacent points included

*V*,

_{a12}*V*,

_{a23}*V*,

_{a34}*V*,

_{a45}*V*,

_{a56}*V*,

_{a67}*V*, and

_{a78}*V*, of which the mean values were uniformly denoted as

_{a81}*V̅*. The wave velocity values of two points with one interval included

_{a}*V*,

_{b13}*V*,

_{b24}*V*,

_{b35}*V*,

_{b46}*V*,

_{b57}*V*,

_{b68}*V*, and

_{b71}*V*, of which the mean values were uniformly denoted as

_{b82}*V̅*. The wave velocity values of two points with two intervals included

_{b}*V*,

_{c14}*V*,

_{c25}*V*,

_{c36}*V*,

_{c47}*V*,

_{c58}*V*,

_{c61}*V*, and

_{c72}*V*, of which the mean values were uniformly denoted as

_{c83}*V̅*. The wave velocity values of two points with three intervals included

_{c}*V*,

_{d15}*V*,

_{d26}*V*, and

_{d37}*V*, of which the mean values were uniformly denoted as

_{d48}*V̅*. The attenuation coefficient of the wave velocity (

_{d}*δ*) can be calculated according to Eq. 1, (Li 2015).

_{m}(1)

The ratio of the wave velocity mean values under the four paths were selected as the discriminant factor, which included *V̅ _{a}*/

*V̅*,

_{b}*V̅*/

_{a}*V̅*,

_{c}*V̅*/

_{a}*V̅*,

_{d}*V̅*/

_{b}*V̅*,

_{c}*V̅*/

_{b}*V̅*, and

_{d}*V̅*/

_{c}*V̅*. Meanwhile, the diameter (

_{d}*D*) of the specimen was also taken as a discriminant factor. Therefore, a total of 11 discriminant factors were selected in this model. The classification of the damage level was divided into 6 levels, as shown in Table 2 (Li 2015).

**Data Statistics**

In this paper, *C _{1}* through

*C*with known damage levels were taken as training samples to form the source case base. Meanwhile,

_{36}*C*through

_{01}*C*with known damage levels were randomly selected as testing samples to form the target case base. The data statistics of the source and target examples are shown in Table 3.

_{06}**Model Construction**

It can be seen in Table 3 that there are many discriminant factors and there is a high degree of nonlinearity between these discriminant factors. It is debatable whether simply relying on a certain discriminant factor to determine the damage level of the target case is suitable. In response to this, this section tentatively proposes a case-based reasoning method for determining the internal damage level of ancient wood components based on a fuzzy similarity priority ratio.

*Fuzzy analogy preferred ratio*

Suppose *A* is a set with *K* objects in the domain *U*, where *A *= {*a*_{1}, *a*_{2},···, *a _{K}*}, ∀

*a*,

_{i}*a*∈

_{j}*A*(

*i*,

*j*=1, 2,···,

*K*). Let

*a*and

_{i}*a*compare with object

_{j}*a*, then the fuzzy similarity priority relationship

_{0}*R*is the following mapping shown in Eq. 2,

(2)

where, *γ _{ij}*+

*γ*= 0 (

_{ji }*i*≠

*j*,

*i*,

*j*= 1, 2,···,

*K*),

*γ*= 0 (

_{ij }*i*= 1,2,···,

*K*) (Liu and Zhu 2002).

The above conditions indicate that if *a _{i}* is compared with itself, there is no so-called priority, and

*γ*= 0. If

_{ii }*a*has a priority of

_{i}*γ*when compared with

_{ij}*a*, then

_{j}*a*has a priority of

_{j}*γ*= 1-

_{ji }*γ*when compared with

_{ij}*a*. If

_{i}*γ*= 1, it means that

_{ij }*a*is much more similar to

_{i}*a*than

_{0}*a*. If

_{j}*γ*= 0.5, it means that

_{ij }*a*and

_{i}*a*are the same degree of similarity to

_{j}*a*. Therefore,

_{0}*γ*is called the fuzzy similarity priority ratio of

_{ij}*a*similar to

_{i}*a*when compared with

_{0}*a*, and

_{j}*R*is called the fuzzy similarity priority relationship.

*Representation of the source case and target case*

Suppose *B *= *B _{1 }*×

*B*×···×

_{2 }*B*×···×

_{j }*B*is a

_{n}*n*-dimensional factor discrete space and

*B*(j = 1,2,···,

_{j}*n*) is a finite real number set. The case can be defined as

*C*= (

*b*,

_{1}*b*,···,

_{2}*b*,···,

_{j}*b*), where,

_{n}*b*∈

_{j }*B*(

_{j}*j*= 1, 2,···,

*n*) and

*b*is the discriminant factor of the source case. Correspondingly, the source case can be expressed as

_{j}*BC*= {

*C*,

_{1}*C*,···,

_{2}*C*,···,

_{k}*C*}, where

_{K}*C*∈

_{k}*BC*(

*k*= 1, 2,···,

*K*) and

*C*is the source case. Then, the target case can be expressed as

_{k}*C*= (

_{0 }*b*,

_{01}*b*,···,

_{02}*b*,···,

_{0j}*b*), where

_{0n}*b*(

_{0j}*j*= 1, 2,···,

*n*) is the discriminant factor of the target case.

*Similarity measure between discriminant factors*

Suppose *C _{p}* and

*C*are the source cases, where,

_{q}*C*,

_{p}*C*∈

_{q }*BC*,

*C*≠

_{p }*C*.

_{q}*C*is the target case, leading to Eq. 3, Eq. 4, and Eq. 5,

_{0}*C _{p} = (b_{p1}*,

*b*,

_{p2}*···*,

*b*,

_{pj}*···*,

*b*(3)

_{pn})*C _{q}= (b_{q1}*,

*bq2*,

*···*,

*b*,

_{qj},···*b*(4)

_{qn})*C _{0} = (b_{01}*,

*b*,

_{02}*···*,

*b*,

_{0j}*···*,

*b*(5)

_{0n})The similarity measurement between the discriminant factors can be expressed by the semantic distance between the discriminant factors, which can be solved using the Hamming distance formula. The semantic distance between the *j*-th discriminant factor of *C _{p}* and the

*j*-th discriminant factor of

*C*is expressed as Eq. 6,

_{0}*D*(*C _{pj}*,

*C*) = |

_{0j}*b*–

_{pj}*b*| (6)

_{0j}and the semantic distance between the *j*-th discriminant factor of *C _{q}* and the

*j*-th discriminant factor of

*C*is expressed as Eq. 7,

_{0}*D*(*C _{qj}*,

*C*) = |

_{0j}*b*–

_{qj}*b*| (7)

_{0j}where, *D*(*C _{pj}*,

*C*) is the semantic distance between the

_{0j}*j*-th discriminant factor

*b*of

_{pj}*C*and the

_{p}*j*-th discriminant factor

*b*of

_{0j}*C*.

_{0}*D*(

*C*,

_{qj}*C*) is the semantic distance between the

_{0j}*j*-th discriminant factor

*b*of

_{qj}*C*and the

_{q}*j*-th discriminant factor

*b*of

_{0j}*C*.

_{0}When the semantic distance between the two cases is used to indicate the similarity degree, it can be considered that the smaller the semantic distance is, the more similar the two discriminant factors are.

*Constructing the fuzzy similarity priority relationship*

The fuzzy similarity priority ratio of *C _{pj}* similar to

*C*when compared with

_{0j}*C*can be defined as Eq. 8,

_{qj}(8)

where *D ^{j}_{pq }*∈ [0, 1],

*D*=1-

^{j}_{qp}*D*∈ [0, 1]. The bigger

^{j}_{pq }*D*is, the bigger the similarity degree of

^{j}_{pq}*C*similar to

_{pj}*C*when compared with

_{0j}*C*is.

_{qj}The fuzzy similarity priority relationship *D*(*j*) corresponding to the *j*-th discriminant factor can be constructed by the following steps:

Taking *p* and *q* equal to 1, 2,···, *K *in turn, if *p *= 1, and *q *= 2, 3,···, *K*, *D ^{j}_{12}*,

*D*,···,

^{j}_{13}*D*is solved. If

^{j}_{1K}*p*= 2 and

*q*= 1, 3,···,

*K*,

*D*,

^{j}_{22}*D*,···,

^{j}_{23}*D*is solved. Meanwhile, if

^{j}_{2K}*p*is equal to

*q*, then

*D*= 1. Finally, the matrix

^{j}_{pq }*D(j)*can be expressed as Eq. 9,

* *(9)

where the matrix is called the fuzzy similarity priority relationship of the *j*-th discriminant factor. By taking *j* equal to 1, 2,···, *n*, in turn, the *n* fuzzy similarity precedence relationships corresponding to the *n* discriminant factors can be obtained.

*Discriminating the level of internal damage in ancient wood components based on fuzzy similarity priority*

Taking a* λ*-cut set for *D(j)*, the *K* similarity degree sequences between the *j*-th discriminant factor of the source cases and the target case *C _{0}* is obtained. Let

*λ*be from big to small to check

*C*, respectively. If the elements on the

_{0}*p*-th row are all 1 (except for the elements on the diagonal, which are 0), it can be considered that

*C*and

_{p}*C*are most similar. At this point, the row and column where

_{0}*C*is located are deleted. By that analogy, the similarity degree sequences between the

_{p}*K*source cases and

*C*can be obtained.

_{0}Suppose that the source case that is most similar to the target case *C _{0}* is listed at the top of the sequence and its sequence number is 1 and the source sample that is least similar to the target case

*C*is listed at the end of the sequence and its sequence number

_{0}*K*. Then, the sequence numbers of the

*K*source cases can constitute the following sequence number set, shown in Eq. 10,

*T _{j}*=(

*t*,

_{1j}*t*,···,

_{2j}*t*,···,

_{kj}*t*) (10)

_{Kj}There are *n* sequence number sets corresponding to *n* discriminant factors, as shown in Eq. 11,

(11)

The sequence number of the similarity degree between the *k-*th source case and the target case *C _{0}* can be expressed as Eq. 12,

(12)

where *ω _{j}* is the weight of

*n*discriminant factors. Taking

*k*equal to 1, 2,···,

*K*, in turn, the sequence number of

*K*source cases can be obtained by using Eq. 12. The smaller

*F*is, the more similar

_{k}*C*and

_{k}*C*is.

_{0}*Establishment of the variable weight model*

Weights are used to measure the relative importance of influencing factors. Considering the sensitivity of the weights, the same factor will have different influences on the decision output in different decision-making environments (Zhou *et al.* 2010). During the determination of the weight (*ω _{j}*) of the discriminant factor, whether the weight selection is reasonable or not directly determines the accuracy of the discriminant result. However, the evaluation method adopted by the current evaluation standard is a constant weight evaluation method,

*i.e.*, regardless of how the value of the discriminant factors change, the weights of the discriminant factors are constant. If one or two discriminant factors are particularly dangerous, it may be neutralized by other discriminant factor regardless of the utilized evaluation method. This may reduce the accuracy of the evaluation system and decrease the objective impartiality of the evaluation.

In addition, the more discriminant factors there are, the more average the weights are. Therefore, the greater the possibility of misjudgment (Liu 2010). To verify whether the evaluation results were consistent, this paper introduced entropy weight (Zhou *et al.* 2010), fractal theory (Liu *et al.* 2005), and projection pursuit (Wang 2019) to determine the weights (*ω’ _{j}*) of the discriminant factors, respectively. In the terms of the limitations of the constant weight method, three variable weight evaluation models integrating “penalize” and “excitation” were established to improve these three constant weight methods, respectively. The variable weight model can effectively solve the unreasonable evaluation results caused by multiple discriminant factors. According to the principle of state variable weight, the state variable weight vector (

*S*) can be expressed as Eq. 13,

*S *= (*s _{1}*,

*s*,···,

_{2}*s*,···,

_{j}*s*) (

_{n}*j*= 1,2, ···,

*n*) (13)

where *s _{j}* is the function of

*x*, which can be determined by Eq. 14,

_{j}(14)

where *s _{j}* is a state variable weight function of the penalize-excitation,

*x*is the state value of each discriminant factor of the target case, which can be obtained by normalizing 11 discriminant factors,

_{j}*α*is the level of “penalize”,

*β*is the level of “excitation”,

*C*,

*c*, and

_{1}*c*are the evaluation strategies, and

_{2}*R*is the adjustment factor (Duan 2003; Fan and Chen (2008). These parameters need to meet two conditions: 0 <

*μ*<

*λ*<

*α*<

*β*< 1 and 0 <

*C*<

*c*<

_{1 }*c*< 1.

_{2 }The characteristics of the variable weight evaluation are specifically reflected in the following three aspects:

(1) For the evaluation value of each discriminant factor, there is *x _{j }*∈ (0,1). The closer the level of “excitation”

*β*is to 1,

*i.e.*, the narrower the interval (

*β*, 1), the faster the strong “excitation” that the state in this interval receives is.

(2) If the level of “penalize” *α *is very close to the level of “excitation” *β*, *i.e.*, the qualified interval [*α*,* β*] is narrow, then there is neither “penalize” nor “excitation” for the state in this interval.

(3) The interval (0, *α*) of “penalize” is wide and is divided into three stages, including the initial “penalize” stage (*λ*, *α*), the strong “penalize” stage (*μ*, *λ*], and the veto stage (0, *μ*]. The level of “penalize” that the state receives is low during the initial “penalize” stage, while the level of “penalize” that the state receives is large during the strong “penalize” stage. In the veto stage, the evaluation value of the evaluation project is too low, but it has a large weight; therefore, the overall evaluation value of the overall target sharply drops. When the evaluation value is lower than or equal to the specified veto value, the entire target will be rejected.

If 0 < *C *< 1, 0 < 1-*β *< *C*, , and 1 < R <c then *s _{j}* is a state variable weight function of the strong local penalize-excitation (Duan 2003). According to

*s*, the variable weight model can be obtained according to Eq. 15,

_{j}(15)

Combining Eq. 15 into Eq. 12 to obtain the weighted sum *F _{k} (k = 1,2,···, K)* of the sequence number of all source cases in

*K*sequences yields Eq. 16,

(16)

**RESULTS AND DISCUSSION**

**Determining the Similarity Degree Sequence**

Taking the target case *C _{01}* as an example to illustrate the discriminating process, the fuzzy similarity relations

*D*(1) corresponding to 11 discriminant factors can be established according to Eqs. 6 through 9. Simultaneously, the similarity priority matrix of each discriminant factor was established, which was represented by

*D*(

*j*) (

*j*= 1,2,···, 11). Secondly, the similarity degree sequence

*T*(

_{j}*j*= 1,2,···, 11) between each discriminant factor of the source cases and each discriminant factor of the target case

*C*could be obtained by using the obtained fuzzy similarity priority relationship

_{01}*D*(

*j*), which is shown in Table 4.