he United Nations Conference on Environment and Development (UNCED) in 1992 recommended compact urban patterns with high density and mixed land use as ways to control urban sprawl and save energy (Mindali et al., 2004). The concept of compact city has been practiced to single-core urban area to encourage the aggregation tendency from periphery to downtown area (Breheny, 1995). The concept of compact city has evolved from the beginning the protection of environment and agricultural land to contemporary livability and diversity. With the challenges of global climate change and energy crisis, compact city become paradigm to integrate economic development, urban reconstruction and growth adaptation. Previous studies emphasized comprehensive analysis comparing city compactness (Burton, 2002;Thinh et al., 2002;Kasanko et al., 2006;Schneider and Woodcock, 2008). However, the results might able to cluster the cities but unable to sketch out the interaction within cities accurately.

Many measurements have been proposed to analyze the physical environment and urban function of compact city. Li and Yeh (2004) used landscape fragmental index to analyze the physical pattern of compact city. Burton (2002) constructed three dimensional indicators including density, mixed, and intensity to analyze urban function. The application of compact city measurement can help to categorize cities according to the compact degree but ignore other urban development features. In fact, urban development might be varied for different location, terrain, scale or industry (Catalan et al., 2008). The combination of natural resource, industrial type, technical progress might construct various compact city types. Therefore, the single measurement of city compactness should be the first step to detect the compact city, and there is necessary to apply other measurement to explore the relationship between compact city and urban feature.

With the completeness of compact city concept, there are four aspects altitude, density, efficiency, and flexibility (Dantzig andSaaty, 1973?Burton, 2002).High density urban space, clustered economic effect, the decrease of travel distance and high efficiency urban development might help to practice sustainable development. However, high population density and intensified activities have already impacted livability seriously such as congestion (Breheny, 1995;Balcombe and York, 1993) and the increment of crime ratio and housing price (Lin and Yang, 2006). According to the statistic of Ministry of Interior, the housing price has increased 20% in the past 10 years while the household income has only increased 5%, such housing price to income ratio (PIR) has indicated the decrease of livability in urban space.

Compact city pattern has way beyond singlecore aggregation to disperse multi-core connection. Therefore, this study attempts to categorize urban space according to the urban features. Furthermore, the limited urban space with increasing population emigration might impact urban livability while such impact might be varied due to the compact city pattern. Firstly, this study applies principle component analysis (PCA) to analyze compact city pattern and the change tendency in two different time periods. Next, the impact of housing price to income ratio (PIR) is then discussed by using geographical weighted regression (GWR). Section 2 presents the evolution process of compact city. Sections II.

The concept of compact city has evolved. The original concept of compact city is the protection of natural environment and agricultural land from urban expansion. Recently, compact city has become a measurement to fight against global climate change and energy crisis. The followings are the evolution of compact city (OECD, 2012).

The ancient compact city emerged in the Middle Ages. Residents got well protected inside the wall which become an ancient compact city pattern. However, the eighteen-century Industrial Revolution and large amount of people moved into cities had radical impacts on the wall.

In eighteen-and nineteen-century, large-scale urbanization has cut down open spaces. In addition, insufficient public facilities were unable to process sewage water and garbage and resulted in serious public health issue. During that time, garden city proposed by Ebenezer Howard and radiant city proposed by Le Corbusier had become the transforming compact city. Such buffer zone of urban environment and natural environment has contentedly become the core of urban planning in England, Japan, Hong Kong, and other countries (UK Department of the Environment, 1995; Kuhn, 2003;Tang et al., 2007;Kim, 2010).

After 1960, livability became an important issue in urban planning field. The green buffer zone is not only a segregation of urban space and natural environment but open space and leisure. In addition, the vitality of urban activities and mixed land use might improve livability in urban space (Jacobs, 1962). Until Dantzig and Saaty (1973), compact city has finally addressed with high density development and avoiding excessive urban sprawl.

Green Paper on the Urban Environment (Commission of the European Communities, 1990) indicated that compact city is one of the planning measures to achieve sustainable development. In fact, the compact city not only achieve sustainability but satisfy multiple purposes such as the clustered economic effect, the decrease in travel distance and urban efficiency (Thomas and Cousins, 1996;Churchman, 1999).

This study subjects to Taipei metropolitan area as the study area where has been regarded as the 46th metropolitan area in the world, 40% entire population clustered in 3,700 km2 (see Fig. 1). Large amount population and industries aggregation extends the development area and become multi-core metropolitan area. (See Table 1) Shi, 2013). Through the process, the first principle component (PC) is designed to have the largest variance, and the ranking of PCs is according to eigenvalues. The following is the PCA formula:

where n denotes to spatial units, p denotes the number of variables, ?? ?? denotes the original variables, and ?? ?? denotes principle components. ?? 1 , ?? 2 , ?,?? ?? (m?p) are linear combinations of ?? ?? .

Ordinary Least Squares (OLS) is one of the conventional global regression models to analyze the pattern of the data by fitting a model to the observed data (Hutcheson and Moutinho, 2008;Hutcheson, 2011). However, conventional global regression models ignore spatial heterogeneity and summarize across the entire area. In fact, many processes are spatial heterogeneity and might produce various responses (Fotheringham et al., 2002). However, geographically weighted regression is an increasingly popular method of analyzing spatial heterogeneity in urban geographic analyses (Lafary et al., 2008).

In order to identify the spatial relationships between urban compactness and housing price to income ratio (PIR), this study applies GWR model. The following is the GWR model:

where (?? ?? , ?? ?? ) refers to the coordinate location of each observationiin a space, ?? 0 and ?? ?? are estimated parameters, and ?? ?? is the random error at i.

Bandwidth selection is important in GWR model, and there are two measures: a fixed-distance kernel and an adaptive kernel. A fixed-distance kernel indicates a constant radius while an adaptive kernel indicates a constant number of neighbors. Due to the wide range of spatial units, the application of adaptive bandwidth might be more appropriate. In addition, there are two ways to measure the number of neighbors: cross-validation (CV) and the Akaike information criterion (AIC). Both measures will be applied and compared to determine the appropriate bandwidth.

The following is the adaptive Gaussian kernel:

where ?? ???? refers to the spatial distance between observations, and ?? refers to the bandwidth of variables.

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According to the compact measurement, both the high density and intensity are extended outward and they are increasing in both periphery townships, and the mixed-use degree remains high and increasing. The cross analysis of compact degree in 1995 and 2006 shows that there are four types of compact city pattern in Taipei metropolitan area including high compact, medium compact, low compact, and special urban development. (See Fig. 2) In medium compact cluster, population aggregation seems to be the reason of the compact development for "employment in tertiary industrial sector", "population density", "household amounts per hectare", "residential area", and "industrial area." In 2006, industrial development is still the main driving force of such compact pattern but attracts commercial activities and public infrastructure. In low compact cluster, both urban development features in 1995 and 2006 are similar. The comprehensive development for "residential area", "commercial area", "industrial area", "public infrastructure area", "employment in manufacturing sector", "household amounts per hectare" and "the ratio of population increment" are positively significant in both years. V.

This study compares traditional ordinary least square (OLS) and geographically weighted regression (GWR) to see if there is any spatial heterogeneity. The result shows that R square is higher and AICc value is lower in GWR suggested that GWR has better explanation for considering spatial heterogeneity. (See Table 6) The significant spatial scale dependence occurs in the relationships between urban compactness and PIR. At a spatial scale of 25 neighbors, the AICc value has the lowest value. Therefore, 25 neighbors has become the acceptable bandwidth to model the relationship between compact city and PIR. shows that all variables except "facilities", "the increment of population density in sub-core", and "population increment" are significant. (See Table 7 and Fig. 4) a) The impact of population density to PIR The average coefficient value in population density is -3.066 indicating a negative relationship between population density and PIR. However, in the city center and the eastern area have relatively positive effect suggesting a relative high housing price.

The average coefficient value in sub-core population density is -0.02 indicating a negative relationship between sub-core population density and PIR. Only western districts and some southern districts have positive effect while those districts are sub-core area in practice. Therefore, the increment of population might have relatively impact on the housing price and further increase the pressure on house affordability.

The average coefficient value in building density is 2.187 indicating a positive relationship between building density and PIR. Only partial districts show negative effect for relatively lower built environment in the southern districts and rapid developing in city center. In rapid development districts, the increment of building density is able to mitigate the housing affordability.

The average coefficient value in residential density is 2.184. The study area except the eastern districts show positive effect indicating the higher residential density equals to the higher housing demand and might result in an increasing pressure in housing affordability.

The average coefficient value in mixed land use is -0.776and all the study area show negative relationship between mixed land use and PIR. The increment of mixed land use is not only improving livability but increasing local employment. The satellite town is able to stabilize housing affordability.

The average coefficient value in employment is 0.106. The study area except the eastern districts show negative relationship between employment and PIR. The increment of employment indicates a more mixed land use pattern and such economic development might mitigate housing affordability by providing more housing units and increasing household income.

This study discusses beyond categorization of urban compactness but comparing urban compactness across different time periods. The results show that high compact cities show an improvement on public infrastructure and become more livable. Medium and low compact city stay similar urban features such as manufacturing and residential. In addition, the results of GWR show various relationships between urban compactness and PIR. Among them, population density, building density, residential density have positive effect indicating the more people aggregate might increase the housing pressure. On the contrary, mixed land use and employment have negative effect indicating a more mixed-use environment might attract diverse industries to increase household income.