Chapter 6 GEOSTATISTICAL APPROACH
6.5 Predict House Price in 2012
Most of the interpolation methods provided by (ArcGIS) geostatistical analyst do not require the data to be normally distributed, although in this case the prediction map may not be optimal. That is, data transformations that change the shape (distribution) of the data are not required as part of the interpolation model7 (Mitchell, 2005).
6.5 Predict House Price in 2012
After getting the annual house prices designed by kriging, the house price in 2012 is predicted by using the cokriging technique mentioned above. The interpolation is performed by considering the house prices in 2011 and 2010. The equation for predicting is shown as equation (6.16).
2011
Cokriging is a multivariate spatial method to estimate spatial correlated variables.
This method allows spatial estimations to be made and interpolated maps of house price to be created.
6.6 Empirical Analysis
In order to check the anisotropy in the house price, the conventional approach is to compare semi-variograms in several directions (Chica-Olmo, 1995). Anisotropy isn’t usually a determinist process that can be described by a single mathematical formula.
It doesn’t have a single source or influence that is expected to affect all measured points. Anisotropy is a characteristic of a random process that shows higher autocorrelation in one direction than in another (Johnston et al., 2003). In this study,
7 In our study, we still attached LN_UP as data transformations for comparison. The results refer to Appendix 1.
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semi-variograms are calculated in different directions, starting at the angle of 0° with the intervals of 45° (clockwise from the north direction) to test whether the semi-variogram values would be different or not in the direction of 0°, 45°, 90° and 135°axes. For example, in 2011 (see as Fig.6.1), due to the anisotropy, the empirical semi-variogram for the measured points indicates that the spatial relationship is different among these four directions. In the 0° and 135°axes the shape of the semi-variogram curve increases more rapidly, while the other two perpendicular axes i.e. 45° and 90° have roughly similar semi-variogram. For anisotropy, the figure of the semi-variogram may be different with direction. If isotropy exists, the semi-variogram doesn’t vary with direction (Johnston et al., 2003).
Note: In the case, UP is used as house prices.
Figure 6. 1 Directional Semi-Variograms of House Prices in 2011
Hence, the anisotropy is exercised as the underlying semi-variogram in 2011 (see Fig.6.2). Spherical model is fitted with the weighted least squares error method, in which the weights are inversely proportional to the lag distance. As previously discussion, the semi-variogram describes the spatial autocorrelation of measured observation points. There are certain features regularly used to describe the models.
The distance where the model first flattens out is recognized as the range. Sample locations separated by distances closer than the range are spatially autocorrelated,
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whereas locations far than the range are not. The value that the semi-variogram model achieves at the range is called sill (the value on the y-axis). In theory, at zero separation distance, the semi-variogram value should be zero. However, at micro scales smaller than the sampling distances, the difference between measurements does not often lead to zero. This is called the nugget effect. The nugget effect can be attributed to measurement errors or spatial sources of variation at distances smaller than the sampling interval. Measurement error happens due to the intrinsic error in measuring devices. In general, the nugget-to-sill ratio can be used to classify the spatial dependence (Cambardella et al., 1994). A variable is considered to have strong spatial dependence if nugget-to-sill ratio is less than 0.25, and has a moderate spatial dependence if the ratio is between 0.25 and 0.75; otherwise the variable has a weak spatial dependence (Liu et al., 2006). Therefore, it is understood that house prices have moderate spatial dependence in 2011 (Table 6.1). Table 6.1 shows parameters of semi-variogram in our study area from 2002 to 2011. In 2005 and 2010, the sill variances are higher than the rest years, implying an increase in the variance of the house prices in the two years. According to the nugget-to-sill ratio, house prices have spatial dependence in our study period.
Note: In the case, UP is used as house prices.
Figure 6. 2 Experimental and Fitted Semi-Variograms for Anisotropy in 2011
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Table 6. 1 Changes in Parameters of Semi-Variogram in Our Study Area
Year nugget Sill Range angle nugget-to-sill ratio isotropy/anisotropy 2002 0.4642 0.7929 0.0338 85.4297 0.59 anisotropy 2003 0.6568 0.9824 0.0411 78.2227 0.67 anisotropy 2004 0.5240 0.9318 0.0312 72.9492 0.56 anisotropy 2005 0.9473 1.3926 0.0061 -- 0.68 isotropy 2006 0.4501 0.8883 0.0285 -- 0.51 isotropy 2007 0.6330 0.9448 0.0425 -- 0.67 isotropy 2008 0.6873 1.0296 0.0012 -- 0.67 isotropy 2009 0.7447 1.1291 0.0457 -- 0.66 isotropy 2010 0.6294 1.4035 0.0604 -- 0.45 isotropy 2011 0.5344 0.8153 0.0087 62.2266 0.66 anisotropy Note: In the case, UP is used as house prices. Units of nugget and sill are 10 thousand/NTD (per square meter). Unit of range is km.
On the basis of kriging approach, the study area is divided into a regular grid, the house prices and estimation variances are estimated at each of the grid nodes by using the finally selected spherical model. Taking into account the interpolated values only, the contour maps of house prices and estimation variances are drawn and shown in Fig. 6.3 and Fig. 6.4, respectively. Fig. 6.4 can be interpreted as the map of the reliability of the kriging house prices in Fig. 6.3. As shown in Fig. 6.4, the estimation variance is low in the middle of the study area, where most of the observation points are located, and increases rapidly towards the boundaries. This indicates that the estimated house prices are more reliable in the middle of the study area than the boundary; these are not reliable to the same extent. The highest estimation variances occur at the corners of the study area, where there are no observations.
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(a) 2002 (b) 2003
(c) 2004 (d) 2005
(e) 2006 (f) 2007
Figure 6. 3 Contour Maps Obtained through Kriging of House Prices
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(g) 2008 (h) 2009
(i) 2010 (j) 2011
Note: Landmark: THU means Tunghai university; FCU means Fengchia university; TBS1 means Taichung bus station 1 (Ubus); TBS2 means Taichung bus station 2 (Chaoma); TAIG means Taichung city government; CMM means CMP block museum of arts; TTS mans Taichung train station. In the case, UP is used as house prices. Units: 10 thousand/NTD
Figure 6. 3 Contour Maps Obtained through Kriging of House Prices (Continuous)
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(a) 2002 (b) 2003
(c) 2004 (d) 2005
(e) 2006 (f) 2007
Figure 6. 4 Contour Maps Obtained through Kriging of Estimation Variance
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(g) 2008 (h) 2009
(i) 2010 (j) 2011
Note: Landmark: THU means Tunghai university; FCU means Fengchia university; TBS1 means Taichung bus station 1 (Ubus); TBS2 means Taichung bus station 2 (Chaoma); TAIG means Taichung city government; CMM means CMP block museum of arts; TTS mans Taichung train station. In the case, UP is used as house prices. Units: 10 thousand/NTD
Figure 6.4 Contour Maps Obtained through Kriging of Estimation Variance (Continuous)
In order to check the accuracy of fitted semi-variogram, some steps to evaluate how well the model for the prediction values at unknown locations. Cross-validation can help us to make a well-versed decision of which model offers the best predictions.
The calculated statistics supply as diagnostics that specify whether the model or parameter values are reasonable or not. Cross-validation holds one sample and then makes a prediction to the same data location. In this way, the predicted value to the observed value is compared and helpful information of the kriging model is obtained.
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Fig. 6.5 shows a scatter plot of predicted versus measurement values (In case 2011).
It is expected that these should scatter around the line with the slope of one (dashed line). However, the slope is usually less than one. It is a characteristic of kriging that tends to under-predict large values and over-predict small values (Johnston et al., 2003). Fig. 6.6 shows the correlation between the measured values and the estimation error (the difference between measured and estimated values). It can be easily observed that values are distributed around a horizontal line ( y0 ), demonstrating that the mean estimation error is zero and satisfies the unbiased constraint of kriging (Goovaerts, 1997). The majority of values vary between ±2 (10 thousand/ NTD). Hence, on average, the deviation of kriging house price is only 2%
and reliable enough for the application of those regions without house price data. By using this technique, more information of house prices can be estimated in other parts of a region for monitoring and assessing the house price.
Figure 6. 5 Scatter plot of Predicted (kriged) Versus Measurement House Prices
y = 0.227x + 3.3464
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Figure 6. 6 Correlation between Measurement House Prices and Associated Estimation Error
According to Fig. 6.3, the house prices are higher in the vicinity of Phase VII, which includes new Taichung city government site. In order to further understanding the dynamical changes in house price, the interpolation is performed by considering the increased house price. However, a map based on the real scale appears to be more helpful for the immediate determination of the dynamic impact. Hence, the interpolated results are shown including Phase VII Area, which is considered to have experienced the most drastic changes in house price at the period of interest.
From fig. 6.7, the spatial distribution of the house price drastically changes from 2003 to 2004. The house price near Phase VII area, which contains south side of Xitun District and northeast side of Nantun District, begins to increase from 2005.
The increase in house price is considered to be the result of low interest rates and increased credit availability (Fig. 6.8). This results show that the impact is obvious around Phase VII area.
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(a) 2002 to 2003 (b) 2003 to 2004
(c) 2004 to 2005 (d) 2005 to 2006
(e) 2006 to 2007 (f) 2007 to 2008
(g) 2008 to 2009 (h) 2009 to 2010
(i) 2010 to 2011
Figure 6. 7 Yearly Increment Value of Interpolated House Price
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Figure 6. 8 Yearly Average Mortgage Rate
At the same time, the growth rate of the spatial distribution of the house prices moves toward west gradually. Fig. 6.9 shows the price falls of 2007; Taichung’s ever-volatile housing market experienced a quick recovery in 2008. After 2007, the price of Phase VII (high-quality) houses appreciated higher during the boom fell further during the depression and recovered more quickly than other area (low-quality). Smith and Tesarek (1994) report similar results for Houston. Since the introduction of a luxury tax8 and other anti-speculative measures in 2011 to cool the residential property market, house prices in Taichung city have revised downward slightly (except for Phase VII area). On a yearly average house price return rate, house price declined 6.8% in Taichung city; however, house price increased 12.5% in Phase VII area. The house prices of Phase VII area had become leading indicator in real estate market of Taichung city after 2007. Although house prices rise in Phase VII area, but the yearly average house price growth rate is slow down gradually in the whole Taichung city.
8 The luxury tax was implemented in June 2011, Taiwan. Property not used as a primary residence by the owner, and sold within two years will be levied a 10%-15% tax.
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2002 2003 2004 2005 2006 2007 2008 2009 2010 2011 Mortgage Rate
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Figure 6. 9 Changes in Yearly Average House Price Return Rate for Phase VII Area and Taichung City
Finally, Fig. 6.10 shows the prediction of house price map in 2012 by using cokriging with house prices in 2010 and 2011. From Fig. 6.10, the high price region coincides with the areas closest to the Phase VII area. In the district, bus Station (TBS2), Taichung city government (TCG) and Science Park (CTSP) predominate.
Beside, the spatial distribution of house price is localized and performs multi peaks and valleys in North District.
The cross-validation results are shown in Fig. 6.11 and Fig. 6.12. In this application, the cross-validation shows significant differences between kriging (Fig. 5 and Fig.6) and cokriging. The cokriging method has demonstrated a smaller error than the kriging method. Furthermore, the cokriging method also shows a better fit and, therefore, the predicted values are closer to the observed values (Table 6.2). If the errors are unbiased, Mean Error (ME) should be close to zero; and Root-Mean-Square Error (RMSE) is small, then predictions will be close to the observed values; if Mean Standardized Error (MStE) is minimal, the uncertainty of predictions will be small (Chica-Olmo, 2007).
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2003 2004 2005 2006 2007 2008 2009 2010 2011 Phase VII_% Taichung City_%
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Figure 6. 10 Map of Price Predicted by Cokriging
Figure 6. 11 Scatter plot of Predicted (Cokriged) Versus Measurement House Prices
Figure 6. 12 Correlation between Measurement House Prices and Associated Estimation Error
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Table 6. 2 Cross-Validation between Different Models
Kriging_2011 Kriging_2010 Cokriging_2011_2010
Mean Error 0.0119 -0.0029 0.0000
Root-Mean-Square Error 0.9079 0.9007 0.0806
Mean Standardized Error 0.0080 -0.0035 -0.0002
Note: Cross-validation allows selection between different models or methods. The three methods used for comparison are the summary statistics of predicted errors (measured-predicted): Mean Error (ME 0), Root-Mean-Square Error (RMSE min.) and Mean Standardized Error (MStE min.).
6.7 Summary
This chapter describes how ordinary kriging can be used to interpolate house prices in an area where measurements are made at random places. This technique is applied for the estimation of house prices in Taichung city, Taiwan from 2002 to 2011. The directional semi-variogram indicates no drift presence in house prices and adopts for ordinary kriging. To consider the semi-variogram with direction would be the underlying semi-variogram. In 2011, a directional semi-variogram analysis resulted in the identification of direction (62.2266°) along which the semi-variogram value is at a minimum. In this study area, the spherical model is found to be the best model for representing the spatial variability of house price data. Estimation errors from this analysis can provide guidance for the selection of new observation sites to reduce estimation error. Combining the kriging house prices with the ground elevation map will provide an estimation of the high price area and areas prone to rising. In this study area, the house prices are higher in the vicinity of HuiLai Urban Land Consolidation Area (Phase VII), which contains south side of Xitun District and northeast side of Nantun District, begins to increase from 2005. The spatial distribution of the house price drastically changes from 2003 to 2004. The increase of house price is considered to be the result of low interest rates and increased credit availability; the results show that the most influence is on Phase VII area. The kriging only deviates the house prices by 2% in spatial analysis, and it is acceptable for
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house price estimation. Finally, by using cokriging to predict house price in 2012, the high price region coincides with the areas closest to the Phase VII area. In the district, bus Station, Taichung city government and Science Park predominate. The house prices are higher in the vicinity of the TBS2, TCG and CTSP, which indicate that the facilities raise neighboring housing prices. The house price is distributed with spatial location and the prediction of house price is also spatial dependent. Sometime the average house price cannot stand for the real social development or local area distribution of house price. The traffic, department store or other attractive cause may induce population migration and contribute to the distribution of house price. This behavior will induce opposite effect in localized house price and disturber the prediction of average house price by traditional methods. The key purpose of this study is to deliver a more precise prediction of house price with spatial information.
The prediction of house price in both time domain and spatial domain can apply the reasonable information for buyer, investigator, and government. The over estimation or housing-market hype of house price can be avoided.
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