Factors leading to buildings being demolished and probability of remainder

Transcription

1 Factors leading to buildings being demolished and probability of remainder T. Osaragi Department of Mechanical and Environmental Informatics, Tokyo Institute of Technology, Japan Abstract Land use changes generally arise when older style buildings are demolished and replaced with contemporary buildings. The decision making of landowners can be observed through such phenomena. Hence the direction and the speed of land use change are dependent on the possibility and probability that buildings will be demolished or will remain. Various research studies about the life span of buildings have been carried out, since the life span of buildings is one of the basic concerns for land use forecast. The reliability theory has been applied in many studies, and they, in turn, proposed methods for assessing the probability that buildings will be demolished or remain in the future. However, it has been generally considered that the life span of buildings is a simple function only involved with the age of buildings. In this paper we discuss factors leading to buildings being demolished and probability that buildings will be demolished or remain in the process of land use transition. We examine the characteristics of buildings (age of building, construction materials, building type, etc.) and the characteristics of place (land use zoning, building area to plot ratio, accessibility to railway station, etc.), and propose a statistical model that can evaluate how these factors affect the life span of buildings. Also, some numerical examples are shown using actual data compiled by local government in Tokyo, Japan. Keywords: land use, life span, buildings being demolished, probability of remainder, reliability theory, AIC, moment-generating function. 1 Introduction Analysing land use transition process is one of the basic concerns for land use planners. Since the 1960s many theoretical and numerical studies have been

2 326 The Sustainable City III carried out for forecasting land use change. Much of this work can be found in Klosterman et al. [1] and Wegener [2], [3] and there has also been interest among urban researchers. The latter analyse land use transition across coarse-grained urban lattices (Osaragi and Kurisaki [4]). Such work has recently been enlivened by the increasing availability of computer facilities, associated software and detailed geographical databases (Batty [5], Landis and Zhang [6]). In respect of land use change, household or business proprietors correspond to the decision maker. That is especially true in Japan. Land use is now deeply dependent on their views, since their rights as landowners have become very strong post the Pacific War. This single factor alone often makes it difficult to develop cities according to general plans in Japan. Even if we would have a proposed plan for important new facilities such as an airport or high ways, for example, the rights of the landowner could eopardize the policy. In the field of city planning in Japan, it is therefore said that difficulties of actual planning are proportional to the number of landowners. In such circumstances it should be noted that a land use model, which can describe decision making of each landowner, is necessary for a micro-simulation model of land use. Note also that a basic unit of land use change is a lot owned by a landowner or business proprietor. Land use changes generally arise when older style buildings are demolished and replaced with contemporary buildings. Such phenomena are often observed in established high-density areas. Hence the direction and the speed of land use change is dependent on the possibility and probability that buildings will be demolished or will remain. Additionally, we often observe the phenomenon that a part of a lot is divided into some smaller lots and used for new land use. Hence, we need to discuss not only change of land-use-classification but also this sub-division of lots. Thus, a more detailed land use model, which identifies "buildings being demolished", "change of building type", and "division of lot", is necessary for forecasting land use change in established city areas. This is quite different from land use change in suburban areas where a maor concern is "change of land-useclassification, for example, change from field/forest to residential area". In this paper we discuss "factors leading to buildings being demolished" and probability that buildings will be demolished or remain" in the process of land use transition. Various research studies about the life span of buildings have been carried out (Komatsu [7]), since the life span of buildings is one of the basic concerns for land use forecast as described above. The reliability theory has been applied in many studies, and they, in turn, proposed methods for assessing the probability that buildings will be demolished or remain in the future. It has been generally considered that the life span of buildings is a simple function only involved with the age of buildings. However, a mathematical method to evaluate statistical significance has not been developed yet. Also such a microscopic phenomena that buildings will be demolished or remain, has not yet been considered in the previous land use models. This paper examines characteristics of buildings and place, and proposes a model that can evaluate how these characteristics affect the life span

3 The Sustainable City III 327 of buildings. Also some numerical examples are shown using actual data compiled by local government in Tokyo, Japan. 2 Probability of remainder combined with factors leading to buildings being demolished Various factors, which include characteristics of place (zone regulations, regulation of floor area to plot area ratio, accessibility to railway station, etc.) and the characteristics of building (construction materials, building type, etc.), affect the life span of buildings. It is, however, difficult to know directly which factor has conspicuous influence on the life span of buildings. Hence, we need a statistical method to examine which factor significantly influences the life span of buildings. The variable t denotes age of building. Age t is assumed to be a discrete variable, in order to correspond with actual data obtained. Namely t will be expressed as t=1,2,...,n. Probability that a building, which has remained to age (t-1), continues to remain to age t is expressed by P (t t-1), where an index (= 1,...,m) denotes a classification of a factor. This probability P (t t-1) is hereafter called the interval probability of remainder. Since P (t t-1) is a conditional probability, it has the following relationship with the probability of remainder, denoted by P (t): P () t P ( t t 1) =, (1) P ( t 1) where P (t) represents probability that a building of age t is remaining. That is, the probability of remainder P (t) can be described using the interval probability of remainder P (t t-1) as follows: t P() t = P( k k 1). (2) k = 1 As corollary, probability that a building of classification remains to age (t-1) and will be demolished at age t can be written as 1-P (t t-1). The number of buildings of age (t-1) is expressed as N (t), and the number of buildings demolished at age t is expressed as d (t). Simultaneous probability, denoted by P, that the data is observed under the above probability distributions can be described as follows: P = m n N ()! t ()! () ()! = 1 t= 1 d t { N t d t } { P t t } d () t N() t d() t 1 ( 1) P ( t t 1), (3) where the maximum number of age is assumed to be n. The maximum likelihood estimator of P (t t-1) is obtained using Lagrange's undecided multiplier method as follows:

4 328 The Sustainable City III N () () ˆ t d t P ( t t 1) =. (4) N () t As for a case of life span examination on industrial products, start-time of observation can be set to the same for all samples. However, in the case of buildings, we cannot start observation at the same time for all buildings. Hence, in case of t=1, we have to estimate P(1 0) from incomplete observations using the following equation: N (1) 2 (1) ˆ d P (1 0) =. (5) N (1) The details are described in Kaplan and Meier [8]. The maximum log-likelihood of a model based on a classification (=1,...,m) of a factor can be described as follows: m L( P) max = d ()ln t d () t N ()ln t N () t = 1 t= 1,(6) + { N ( k) d ( k) } ln { N ( k) d ( k) } where the constant term of eqn (3) is omitted. On the other hand, the number of free parameters is m by n, since free parameters of a model are only P (t t-1). From the above discussion, AIC (Akaike's Information Criterion) of this model can be expressed by the following equation: AIC = 2 LP ( ) max + 2mn. (7) AIC is an index for synthetically evaluating mathematical models from a viewpoint of fitness and simplicity (Akaike [9], [10]). Thus, a classification of a factor, which produces small value of AIC, is statistically considered a predominance factor, which significantly influences the life span of buildings. n 3 Estimation of probability of remainder using actual data of buildings In this paper, the data (1994 to 1998) compiled by local government in Tokyo are analysed. Total number of buildings analysed in this research is 53,918. Area of building, construction year, construction materials, building type, and building area, etc. are included as information about buildings. On the other hand, width of road, land use zoning, regulation of building area to plot ratio, land use zoning, etc. are also included as information about places in which buildings are located. Using geographic information systems, integration of this database is achieved so that all information can be used seamlessly. Moreover, secondary data such as distance to railway station and distance to bus stop are also created and built into an integrated database. Eighteen factors, which could be concerned with the life span of buildings, are elected as candidates of factors leading to the probability that a building will be demolished, and are shown in table 1 below.

5 The Sustainable City III 329 Table 1: Candidates of factors leading to buildings being demolished. Building Regulations Place No. Factors Categories 1 Building area (m2) less than 50/ / / / / / / 1000 or more 2 Construction material wood/ reinforced concrete/ steel/ others 3 Building type residence/ apartment/ complexed house/ office+bank/ shop/ factory+storehouse/ attached+garage/ others 4 Number of stories 1/ 2/ 3/ 4/ 5/ 6-11/ 12 or more 5 Basement yes/ no 6 Area of plot (m2) less than 100/ / / / / 600 or more 7 Land use zoning first+second-low rise residence/ first+second-high-rise residence/ first+second residence/ neighborhood residence/ commercial/ semi-industrial 8 Building area to plot ratio (%) / / 200/ 300/ 400 or more 9 Building to plot ratio (%) 30+40/ 50/ 60/ Fire prevention zoning fire prevention/ semi-fire prevention/ none 11 Height control zoning first category/ second category/ third category/ none 12 Height regulations (m) 10/ none 13 Distance to bus stop (m) less than 100/ / / / / 600 or more 14 Distance to railway station (m) less than 500/ / / / 2000 or more 15 Time-distance to railway station (min) less than 5/ 5-10/ 10-15/ 15-20/ 20 or more 16 Time-distance to Yamate-line (min) less than 15/ 15-17/ 17-19/ 19 or more 17 Width of adacent road (m) less than 2/ 2-4/ 4-6/ 6-8/ 8-10/ 10 or more 18 Land use district commercial/ industrial/ copmlexed house/ residential/ others In order to evaluate the influence of each factor, we have to check whether the life span characteristic of buildings obtained under a certain classification of a factor is statistically significant. In this research the best classification is obtained by comparing values of AIC obtained under a classification of buildings according to various factors. The following procedure is used. (1) Compute AIC under the most detailed classification of each factor shown in table 1 (Model I). (2) Reclassify categories of each factor and search a new classification by which a value of AIC becomes the smallest (Model II). (3) Constitute a new classification mutually by combining two or more factors, which produce the smallest value of AIC (Model III). In order to exclude special cases, for example, many buildings were demolished during the Pacific War, only buildings 50 years old or less are analysed. (1) Model I: Values of AIC calculated under the detailed classification of each factor are shown in table 2. Table 2: Results of Model I based on the most detailed classification. No. Factors NC AIC No. Factors NC AIC 1 4 Number of stories 8 25, Time-distance to railway station 5 25, Construction material 4 25, Distance to railway station 5 25, Area of plot 6 25, Basement 2 25, Landuse zoning 6 25, Height regurations 2 25, Width of adacent road 6 25, Time-distance to Yamate-line 4 25, Building area to plot ratio 5 25, Building area 8 26, Landuse district 5 25, Building type 8 26, Height control zoning 4 25, Basic Model 1 26, Building to plot ratio 4 25, Distance to bus stop 6 26, Fire prevention zoning 3 25,911 NC: the number of categories

6 330 The Sustainable City III The result of a model, which includes only one category, is also shown. This model is called "the basic model", and assumes that any factors which do not influence the life span of buildings. Table 2 shows that most of all the factors excel the basic model. That is, the detailed classifications are statistically significant to explain the life span of buildings. (2) Model II: A new model is estimated by reclassifying categories within each factor. The results are shown in table 3. Table 3: Results of Model II based on reclassified categories. No. Factors Categories NC AIC 1 4 Number of stories 1/ 2/ 3/ 4 or more 4 25, Construction material wood/ reinforced concrete/ steel+others 3 25, Area of plot less than 100/ / / 600 or more 4 25, Landuse zoning first+second-low rise residence/ first+second-high-rise residence/ first+second residence/ neighborhood residence/ commercial/ semi-industrial 6 25, With of adacent road less than 2/ 2-6/ 6-10/ 10 or more 4 25, Landuse district commercial/ industrial+copmlexed house/ residential/ others 4 25, Building area less than 50/ / / 300 or more 4 25, Building area to plot ratio / / 200/ 300/ 400 or more 5 25, Height control zoning first category/ second category/ third category/ none 4 25, Building to plot ratio 30+40/ 50/ 60/ , Building type residence/ apartment/ complexed house/ office+bank+shop+factory+storehouse/ attached+garage+others 5 25, Fire prevention zoning fire prevention/ semi-fire prevention/ none 3 25, Time-distance to railway station less than 5/ 5-10/ 10-15/ 15-20/ 20 or more 5 25, Distance to railway station less than 500/ / / 1500 or more 4 25, Basement yes/ no 2 25, Height regurations 10/ none 2 25, Time-distance to Yamate-line less than 15/ 15-17/ 17 or more 3 25, Distance to bus stop less than 200/ 200 or more 2 26, Basic Model (none) 1 26,055 NC: the number of categories When reclassification is achieved, we can obtain smaller value of AIC than the detailed classification (Model I). Especially, models of the 4-categoryclassification based on Number of stories and the 3-category-classification based on Construction materials are excellent. That is, these classifications have strong relationships with whether buildings will remain or be demolished. Also, it is shown that factors of Area of plot, Land use zoning and Width of adacent road, which indicate the characteristics of places, influence on the life span of buildings. (3) Model III: A third model is estimated by combining the above factors. If three or more kinds of factors are combined, a model has a tendency to become redundant. Hence, reclassification of categories is achieved using only two factors. All of the combination of two factors is examined by changing classification of categories. As a result, it can be shown that the classification based on combination of Number of stories and Building area is the best, although Building area is not so significant itself to explain the life span of buildings (see table 3). The probability of remainder estimated under this classification is shown in figure 1. The classification categories C1-C6 in figure 1 is shown in table 4.

7 The Sustainable City III 331 Probability of remainder Age of building C1 C2 C3 C4 C5 C6 Figure 1: Probability of remainder (categories C1-C6 are shown in table 4). Table 4: Results of Model III based on combination of factors. Building Area (m2) less than or more 1 C1 2 C2 C3 C4 3 4 or more C5 C6 AIC=24818 Number of categories=6 Number of Stories Figure 1 shows that, the probability of remainder is different according to Number of stories of buildings. Furthermore, the probability of remainder is different according to Building area, even if Number of stories is the same. This figure expressing the characteristics of life span can be used for a microsimulation of land use forecasts. 4 Confidence interval of probability of remainder The probability of remainder of buildings can be estimated using the above method. However a way of placing confidence margins around predicted values of the probability is required, since statistical errors are included in the estimated values. Some studies have attempted to obtain confidence intervals using the Monte-Carlo simulation. In this research, however, a method for estimating the variance of expected errors is developed using the theory of the moment generating functions. From eqn (1), the probability of remainder P(t) can be written as follows: P ( t) = P( t 1) P( t t 1). (8) In practice, however, the predicted numbers of buildings being demolished can often be well wide of the mark for a variety of reasons. In this paper we will look at one type of error δp(t) in the probability of remainder. Namely it is assumed that statistics error δp(t t-1) is included in P(t t-1), since P(t t-1) is estimated from observation data.

8 332 The Sustainable City III Substituting P(t)+δP(t) for P(t) and P(t t-1)+δ P(t t-1) for P(t t-1), as shown in eqn (8) above, we can rewrite eqn (8) as follows: Pt () + δpt () = { Pt ( 1) + δpt ( 1)}. (9) { Pt ( t 1) + δ Pt ( t 1)} Then, if we assume that the second-order moment in eqn (9) is small and can be ignored, the total error in the system is given by, δpt ( ) = Pt ( 1) δpt ( t 1) + δpt ( 1) Pt ( t 1). (10) The maximum likelihood estimator for P(t t-1) can be estimated by eqn (4) using the observation data on the number of buildings being demolished within a certain time period. The standard theory of multinomial sampling gives: E[ δ P( t t 1) ] = 0, (11) 2 Pt ( t 1){1 Pt ( t 1)} σ Ptt ( 1) =. (12) Nt () Whenever the sample size, N(t), is sufficiently large this multinomial distribution will approach the normal distribution. Hence we can regard the distribution of δp(t t-1) as a normal distribution, N(0, σ 2 P(t t-1)). It therefore follows that δp(t) in eqn (10) is also a normal random variable with a mean of zero and a standard deviation of σ 2 P(t) --- provided we know the variable of P(1 0) and provided σ 2 P(1 0) is assumed to be zero, because δp(t) is then simply a linear function of δp(t t-1). The variance of prediction, σ 2 P(t), can be obtained by utilization of the concept of the moment-generating function. Namely, if we let the moment-generating functions of the left-hand and right-hand sides of eqn (10) be φ L (θ) and φ R (θ), respectively, we obtain: 2 θ 2 φl( θ) = E[exp{ θδp( t)}] = exp[ σ P() t ], (13) 2 2 θ φr( θ) = exp[ { P ( t 1) σ P( t t 1) + P ( t t 1) σ p( t 1) }]. (14) 2 Hence we obtain a formula for expressing the variance of δp(t) by comparing eqn (13) with eqn (14): σpt () = P ( t 1) σptt ( 1) + P ( t t 1) σpt ( 1). (15) where σ 2 P(0)=0 because the value of δp(0) is assumed to be zero. Using the actual data on buildings being demolished, the confidence interval of probability of remainder is estimated, and a part of the results are shown in figure 2 and figure 3, where confidence intervals, 2σ, have been placed around the predicted values. Thin lines indicate the expected errors. Since many samples are observed in the category C3 (Number of stories=2, Building area<50m2), the confidence interval is narrow. Conversely, since fewer samples are observed in the category C2 (Number of stories=2, 50m2<=Building area<200m2), the confidence interval is large. Therefore, it is necessary to consider confidence intervals, when we use predicted values of the probability of remainder.

9 The Sustainable City III 333 Probability of remainder P (t )-2σ C3 (Number of stories=2, Building area<50m 2 ) P (t )+2σ Age of building Figure 2: Probability of remainder and its confidence intervals, C3. Probability of remainder P (t )-2σ C2 (Number of stories=2, 50m 2 <=Building area<200m 2 ) P (t )+2σ Age of building Figure 3: Probability of remainder and its confidence intervals, C2. 5 Conclusion A statistical method to examine the influence of factors leading to buildings being demolished is proposed in this paper. This method can evaluate factors relating to the characteristics of buildings and place, which influence the life span of buildings. The value of AIC is used for an index for evaluating factors and their classifications. The small value of AIC shows that the corresponding factor significantly influences whether buildings will remain standing or be demolished. In addition, when analysing the database on actual buildings being demolished, it can be shown that the number of stories and its construction materials are important considerations in regard to the life span of buildings. When combining factors, number of stories and building area, it is possible to grasp more detailed characteristics of buildings being demolished. Moreover, a method for estimating confidence intervals of predicted values of the probability of remainder is presented.

10 334 The Sustainable City III Acknowledgements The author would like to acknowledge very valuable discussions with Associate Professor Tohru Yoshikawa of the Tokyo Metropolitan University and Mr. Alexander Macgregor. References [1] Klosterman, E.R., Batty, M., Wegener, M., Harris, B., Lee, B.D., Large- Scale Urban Models: Twenty Years Later. Journal of American Planning Association, 60, pp.3-44, [2] Wegener, M., Operational Urban Models. Journal of American Planning Association, 60, pp.17-29, [3] Wegener, M., Current and Future Land use models. Proceedings of the Travel Model Improvements Program, Land use Modeling Conference, February, US Departments of Transportation and Energy, and the Environmental Protection Agency: Washington DC, pp.13-40, [4] Osaragi, T. and Kurisaki, N., Modeling of Land use Transition and Its Application. Geographical and Environmental Modelling, 4(2), pp , [5] Batty, M., Urban modeling in computer-graphic and geographic information system environments. Environment and Planning B, 19, pp , [6] Landis, J. and Zhang, M., The second generation of the California urban futures model. Environment and Planning B, 25, pp , [7] Komatsu, Y., Some theoretical studies on making a life table of buildings. Journal of Architecture, Planning and Environmental Engineering, 439, pp.91-99, 1992 (in Japanese). [8] Kaplan, E.L. and Meier, P., Nonparametric Estimation from Incomplete Observations. Journal of American Statistic Association, 53, pp , [9] Akaike, H., Information theory and an extension of the maximum likelihood principle. Proceeding of 2nd International Symposium on Information Theory, eds. B.N Petron and F. Csak: Budapest, Akadeniai kaido, pp , [10] Akaike, H., A new look at the statistical model identification. IEEE Transactions on Automatic Control, AC-19, pp , 1974.