Proceedings of the Institute of Statistical Mathematics Vol. 51, No. 1, 3-9(2003)

and Management Research

Kenji Naito

(Department of Forest Science, Faculty of Agriculture, Utsunomiya University)

This paper overviews the application and development of statistical methods in forest research in Japan from the end of World War II to the 1980s. Just after World War II, the sampling theory and ANOVA were often applied to the forest inventory. In the late 1950s, the first nation-wide forest resource inventory was compiled by the research staff of the Institute of Statistical Mathematics. Since Bitterlich W. proposed a very simple, fast, easy method for estimating forest stand volume in 1947, many Japanese researchers in forest biometrics modified it from 1955 to 1979. The theory of Bitterlich’s method follows statistical theory, although it was developed through practical forestry works. The Gentan probability theory proposed by Dr. Suzuki T. around 1960 is also an important aspect in forest planning, because it generalized the concept of the normal forest and has been used in Japan for the nation-wide prediction of timber supply from private forests. In August 1965, the Japan Association for Forestry Statistics was established to develop and provide statistical methods for Japanese forestry research and practice. Many topics on the sampling method, Bitterlich’s method and the application of computers to forestry research were discussed during the meeting of the association with forestry practitioners. A new statistical method and concepts requested in the 21st century may deal with an evaluation method for non-market products related to forests, and a system for a wide range of information needed for public involvement or participation into forest resource management.

**Key words: Forest inventory, quantification method, point sampling, Gentan probability.**

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Proceedings of the Institute of Statistical Mathematics Vol. 51, No. 1, 11-18(2003)

from the Theory of Column Buckling

Naoto Matsumura

(Faculty of Bioresources, Mie University)

A model selection of suitable height curves is used as a basis for estimating forest stand statistics. Using selected empirical and theoretical height curves, its characteristics, goodness of fit and response to data types are discussed. The height curve derived from buckling theory was tested as a theoretical equation, and Näslund, Henricksen, Michailoff, Allometry equations were selected for empirical use and proved.

The mean square of errors showed the worst performance in case of buckling height curve and almost the same by other equations. Because of the number of free parameters, the buckling curve with only one parameter showed less flexibility. However, this buckling height curve provides theoretically a natural growth limit and it is also be expected to be useful for an iteration of measurements, multi-species and regional inventory.

**Key words: Forest measurements, buckling, height curve, theoretical equation, allometry.**

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Proceedings of the Institute of Statistical Mathematics Vol. 51, No. 1, 19-35(2003)

to Tree Diameter Growth Analysis

Hirokazu Yanagihara and Atsushi Yoshimoto

(The Institute of Statistical Mathematics)

A generalized multivariate analysis of variance model (GMANOVA model) has been widely accepted for analyses of repeatedly measured data. Because of the frequent use of the GMANOVA model for growth data analysis, it is generally called a growth curve model. In the field of forest planning, there exist many analyses on forest growth data. However, the GMANOVA model has not yet been used for the analysis. This paper investigates the applicability of the GMANOVA model to forest growth analysis in forest planning. We conduct curve fitting, hypothesis testing and variable selection to see its applicability to the analysis. Since thinning effects on forest growth are one of the main concerns of forest managers and growth analysts, we apply the model to the analysis of thinning effects on forest stand growth.

**Key words: Forest planning, growth curve model, polynomial curve, test for equality, variable selection.**

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Proceedings of the Institute of Statistical Mathematics Vol. 51, No. 1, 37-58(2003)

Structure of Tree Species Using Genetic Markers

Makoto Takahashi

(Breeding Department, Forest Tree Breeding Center)

Kenichiro Shimatani

(The Institute of Statistical Mathematics)

Recent development of molecular genetic techniques have involved intensive studies of within-population spatial genetic structures for forest tree species. Genetic variation data have stratified structures of three levels; allele, genotype, and multilocus genotype. Studies of spatial genetic structures should simultaneously deal with the stratified information together with the locations of individuals. This article reviews spatial statistical methods for analyzing spatial genetic datasets and discusses their properties, problems, biological background, and the relationships among the statistics.

**Key words: Allele, ecology, genetic markers, genotype, multilocus genotype, natural forest, population genetics, spatial statistics.**

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Proceedings of the Institute of Statistical Mathematics Vol. 51, No. 1, 59-72(2003)

Forests in Toyama Prefecture

Megumi Ishida

(Toyama Forestry and Forest Products Research Center)

The distribution and dynamics of beech ({\it Fagus crenata}) forests have been surveyed in Toyama Prefecture located on the Japan Sea side of central Japan. The relationships among the distribution of beech forests and 7 environmental factors (altitude, warmth index, annual precipitation, annual maximum snow depth, distance from village, slope position, slope direction) were examined on the basis of stand data of 295 plots by canonical correspondence analysis. The distribution of beech forests was principally determined by environmental factors that have high correlation with altitudes, such as warmth indices and maximum snow depths. The upper limit of the vertical forest zone was higher on the south face slope. The relative basal areas of beech were increasing with distance from the nearest villages. For about 10 years, the densities of trees decreased in all 13 plots observed. The DBH (diameter at breast height of tree) growths and survival ratios of the trees were larger and lower, respectively, in trees with larger DBH. The relative basal areas of beech increased with time.

**Key words: Beech forest, canonical correspondence analysis, vertical distribution, forest dynamics.**

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Proceedings of the Institute of Statistical Mathematics Vol. 51, No. 1, 73-94(2003)

Management Using MSPATH

Algorithm

Atsushi Yoshimoto

(The Institute of Statistical Mathematics)

A dynamic programming model is constructed to search for an optimal thinning regime and rotation age in the Kyushu region. This model incorporates the MSPATH (Multi-Stage Projection Alternative Technique) algorithm into a growth simulator derived from a stand density control diagram. The MSPATH algorithm becomes effective where there is a long-term effect of thinning on the objective function, e.g., where there is a price premium by DBH (diameter at breast height). The proposed dynamic programming model with MSPATH is classified as a one-stage and one-state dynamic programming model, where the forest stand age is regarded as the stage and the thinning intensity as the state as well as a control variable. We elaborate how the MSPATH algorithm works and differs from the traditional dynamic programming algorithm, then conduct an experimental analysis of the proposed dynamic programming model for forest stand management.

**Key words: Forest economics, operations research, dynamic programming, stand density control diagram.**

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Proceedings of the Institute of Statistical Mathematics Vol. 51, No. 1, 95-109(2003)

Based on the Forest Resource Table

Ichiro Fujikake

(Faculty of Agriculture, Miyazaki University)

The concept of Gentan Probability has been introduced and used in forest planning in Japan to describe the age class in which a forest stand is cut. Gentan Probability is usually estimated on the basis of data from the forest resource table. However, the characteristics of the data obtained from such a source have not been explored thoroughly. This paper first discusses the censoring and truncation in such data in estimating the Gentan Probability. Based on the discussion, a method of maximum likelihood is developed in estimating the Gentan Probability, and the properties of the procedure developed by Suzuki based on the method of moments and the method introduced by Blandon are reconsidered. Both Blandon’s method and the newly developed method are considered to be suitable for data with the censoring and truncation and performed well in a simulation. However, the application of the method of moments appeared to have problems and performed worse. Although it has been said that the estimation of the Gentan Probability shouldn’t depend on the age structure at the beginning of the observation period, it is concluded that the dependence itself is not a problem if censoring and truncation are adequately taken into account.

**Key words: Cutting age, Gentan Probability, maximum likelihood method, simulation.**

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Proceedings of the Institute of Statistical Mathematics Vol. 51, No. 1, 111-120(2003)

Based on Price of Logs

Takuya Hiroshima

(Graduate School of Agricultural and Life Science, The University of Tokyo)

Gentan probability is defined as the probability that a newly planted forest stand will be harvested after a certain period. This probability is derived from the Poisson process and the waiting time theory, and has been applied to yield prediction in Japan. For estimating the probability, the mean and variance of cutting age are required, although it is hard to obtain such a data in practice. In this paper, the analysis of Yoshimoto’s Gentan probability is conducted using the price function of logs. First, two types of time-dependent log price curves are estimated by the OLS method for the growth function of Yoshimoto’s Gentan probability. Next, parameters of the Gentan probability density function are determined on the basis of three hypothetical cases. Finally, six cases of the Gentan probability distribution are derived and are compared with the actual distribution derived from the observed mean and variance of the cutting age. The result is that one case of the distributions of Yoshimoto’s Gentan probability is almost consistent with the actual one. This implies that the Gentan probability based on the price function of logs may be useful for yield prediction.

**Key words: Gentan probability, log price, Poisson process, waiting time, yield estimation.**

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Proceedings of the Institute of Statistical Mathematics Vol. 51, No. 1, 121-133(2003)

under Uncertain Environments

—Toward Soft Zoning —

Atsushi Yoshimoto

(The Institute of Statistical Mathematics)

The physical characteristics of forest areas are often utilized for the classification of forest patches for management (called “Hard Zoning”). This is because all costs and benefits are derived on the basis of physical characteristics with certainty. Since log price and working costs change over time, their dynamics under uncertain environments have to be considered, and their classification has to be implemented on the basis of managerial or economic conditions. This can be done by searching for a minimum threshold price line for sustaining forest stands under management. This is called “Soft Zoning.” In this paper, a stochastic control model is constructed for seeking a minimum threshold price line to maintain forest stand management. Only log price is regarded as an uncertain phenomenon. Other economic factors, e.g., harvesting costs and planting costs, are assumed to be constant over time. Our analysis shows that a minimum threshold price level is affected by the target forest stand growth. For a slow-growing forest stand, the level becomes higher than that for a fast-growing forest stand. With a downward price expectation, the transition of sustainable forest management into abandonment could occur with little change in optimal rotation age, while without such an expectation, the transition causes lengthening of the rotation age first.

**Key words: Forest economics, forest management, stochastic differential equation, stochastic dynamic programming.**

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Proceedings of the Institute of Statistical Mathematics Vol. 51, No. 1, 135-146(2003)

Demand and Supply : Towards Discussing Measures

for Japanese Forestry Promotion

Satoshi Tachibana

(Forestry and Forest Products Research Institute)

Japanese softwood is the most important player in the Japanese timber market. This study aimed to reveal the demand and supply structures of Japanese softwood logs and to propose important measures toward promoting domestic softwood log supply. The two-stage least-squares (2SLS) method was adopted to estimate the log market for three Japanese softwoods: Japanese cedar (*sugi*), Japanese cypress (*hinoki*) and all Japanese softwood logs. The major results are as follows; (1) log suppliers (mainly forest owners) decide to provide a volume of softwood logs based on price changes of at least two years, (2) Japanese cedar logs are supplied and demanded with higher elasticity than other softwood logs, (3) the growing stock of softwood has a significant effect on log supply and may derive a larger supply, being followed by its aging, (4) the US softwood log price has a significant impact on Japanese softwood demand with higher elasticity, and (5) housing starts (house construction starts) in Japan significantly influence log demand. These results suggest mainly that measures in price and housing construction aspects can play an important role in developing the market for Japanese softwood logs.

**Key words: Japanese cedar, Japanese cypress, round log demand and supply model, two-stage least-squares, price elasticity.**

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Proceedings of the Institute of Statistical Mathematics Vol. 51, No. 1, 147-165(2003)

for Forest Products Trade

Using FAO Data

Kiyoshi Yukutake

(Department of Agriculture, Miyazaki University)

Atsushi Yoshimoto

(The Institute of Statistical Mathematics)

Hiroe Hamada

(Department of Agriculture, Miyazaki University)

The problem of environmental conservation and timber products trade has been one of the major issues in international meetings, e.g., WTO, where quantitative analysis using trade models has been playing an important role in negotiations. In constructing the trade models, it is necessary to have the export/import functions of the target products in the target regions. These functions are derived by applying the price elasticity of the corresponding export and import of the products. In this study, we estimate the price elasticity of the export and import of timber products, where the sign test of the derived elasticity is mainly implemented. There are ten target regions: Japan, developing countries in Asia, North America, Central America, South America, East Europe, West Europe, Oceania, Russia, and Africa. The target products are industrial logs, lumber, wooden panels, chips and particles. The data source is the FAO statistics data on a monthly basis from 1970 to 1999. We apply OLS, 2SLS, and 3SLS estimation methods to the log-transformed linear functions. Furthermore, considering the effect of time-lagged variables, the Almon lag model is also estimated. Our experiments show that if the results from OLS are unsatisfactory for the sign test, none of the others, i.e., 2SLS, 3SLS and Almon lag could outperform it. The supply side estimation would not be satisfactory in terms of the sign test in most cases.

**Key words: Trade structure of forest products, function of exports & imports, price elasticity, econometric analysis.**

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Proceedings of the Institute of Statistical Mathematics Vol. 51, No. 1, 167-172(2003)

Game and Related Properties

Katuomi Hirano

(The Institute of Statistical Mathematics;

Department of Statistical Science, Graduate University for Advanced Studies)

Department of Statistical Science, Graduate University for Advanced Studies)

Sigeo Aki

(Division of Mathematical Science, Graduate School of

Engineering Science, Osaka University)

Engineering Science, Osaka University)

Maehara and Ueda (2000, *J. Appl. Probab.*, **37**, 601--605) gave a lucid explanation of a janken game as follows: In the janken game, each player can choose one of the three strategies, S (scissors), P (paper) and R (rock) independently from other players; S wins against P, P wins against R, and R wins against S. The janken game started by *n* ( __>__2) players is a kind of survival game. If *n* is large, it probably takes many rounds. In a round, if just two strategies (say, S and P) are chosen by the participants of that round, then those who chose the weaker strategy (P) are losers, and have to retire from the rest of the game. If either just one strategy or all three strategies are chosen, then there are no losers, and all participants of that round participate in the next round. Started by *n* players in the first round, the game continues in this way until a sole survivor (winner) is left. Let *W _{n}* denote the number of rounds played through the game. We call

Let \phi_{n}(*t*) be the probability generating function of *W _{n}*, that is \phi

**Key words: Janken game, waiting time distribution, probability generating function.**

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