Framework Overview on The Example of Prague

Using an example of the capital city Prague, the percentage of variability cap-tured by the first eight columns of Umatrix is present in the figure 6.2. It is evidently distinguishable that the first eight columns ofUmatrix are explaining a reasonable amount of variability within the GWR’s coefficients.

0.0%

Percentage of variability explained by each Principle Component

Figure 6.2: Fraction of Variability Captured by the PCs (Case of Prague)

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To provide a reasonable form of interpretability of the results of PCA, the first four loading vectors (columns of theV matrix) are inspected. It should also be noted that prior to the PCA the GWR’s coefficient matrix was standardized such that each column would have mean equal to zero and standard deviation one (z-standardization) (James et al.2013).

In the figure 6.3, we can see that the PC1 places the biggest weight to the interaction termBrick estate × New Estateand to theBrick Estateterm, with much smaller weights placed on the structural characteristics such as e.g. Garage, Kitchenette, Room. Therefore, the first Principal Component corresponds ap-proximately to the Building Type. The second vector tends to put most of the weight on the structural characteristics of an estate e.g. theRoom,Kitchenette, Floor,Meters. Note that the effect ofPrivateownership has a relatively strong negative effect on the PC1 space. Then, the effect on other PCs seems to be rather positive and slowly getting more bigger weight.

3 4 After ReconstructionBrick X New EstateBalcony/TerraceFloor TopVery goodPrivateFloor Concrete X New EstateKitcheneteFloor ZeroRoom

Balcony/TerraceMeters Concrete X New EstateAfter ReconstructionBrick X New EstateBrick EstateKitcheneteFloor ZeroFloor TopVery goodPrivateGarageRoomFloor Brick EstateFloor ZeroPrivateFloor

Concrete X New EstateAfter ReconstructionBrick X New EstateBalcony/TerraceKitcheneteVery goodFloor TopGarageMetersRoom

Floor TopGarage KitcheneteRoom Balcony/TerraceBrick EstateFloor ZeroMetersFloor Brick X New EstateVery goodPrivate After Reconstruction Concrete X New Estate

value

element

Capital City Prague

First four principal components of flats estates

Figure 6.3: The First Four Loading Vectors of V Matrix (Case of Prague)

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The Housing Submarkets6.1 Framework Overview on The Example of Prague

When the low-dimensional representation of the GWR’s coefficients is con-structed, the k-means clustering of the coefficients is performed. In the case of Prague, k was determined to be equal to ten. This leaves us with discretely estimated and distributed housing submarkets. Then, similarly as in the case of ”grandiose clusters”, the spatial kriging interpolation to obtain continuous spatial distribution is performed using all similar steps described in the sections 4.7 and 5.3.

In the case of the housing submarkets, unlike in the case of "grandiose clus-ters" where the continuous variable (percentage differences of the residuals) is interpolated, the interpolated values are the categorical variables. And thus the kriging interpolation, which is based on the regression model and not the classification model, can be, to a certain extend limiting. This is fairly similar to the Linear probability model (LPM), which frequently serves as a baseline model in the classification problems. Nonetheless, this approach can still be utilized and assumed to provide us with a reasonably acceptable identification of the housing submarkets. However, the kriging interpolation techniques in the classification framework have also been developed.

In figure 6.4, we can inspect the spatial distribution of the housing submarkets.

It is expected that these submarkets, by the definition of both GWR and PCA, dispose of highly heterogenic characteristics between each other. In the figure 6.5, the distribution of effects on the price of selected flat characteristics, which are considered as important price determinants, can be compared within each housing submarket separately.

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Submarket 1 2 3 4 5 6 7 8 9 10 11

Capital city Prague

Housing Submarkets: Spatial Distribution

Figure 6.4: Housing Submarkets: Capital City Prague

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The Housing Submarkets6.1 Framework Overview on The Example of Prague

percentage effect of Concrete building

Submarket

Housing Submarkets: Effect of Concrete building

-10%

percentage effect of Garage

Submarket

Housing Submarkets: Effect of Garage

0.50%

1.00%

8 3 2 6 7 4 10 5 1 9

Submarket

percentage effect of Meters

Submarket

Housing Submarkets: Effect of Meters

-10%

percentage effect of After reconstruction

Submarket

Housing Submarkets: Effect of After reconstruction

Figure 6.5: Housing Submarkets: Capital City Prague

The main advantage of described methodology is that it is easily expandable to the time dimension. If real estate data are collected over time and housing submarkets are constructed for each time period of interest (say, one year), the growing and decreasing tendencies of each particular estate characteristic can be directly compared between each of the time periods.

From the figure above, a few interesting observations can be concluded. First and foremost this provides huge support for our hypothesis that the effects of key price determinants are not fixed in space but rather very quite signifi-cantly.

For example, the effect of Concrete building type has quite a large effect on the Price. In submarket 6, we see the effect to be in the magnitudes of almost one-third of the entire real estate price (Given the relative price level in that submarket area). A very interesting effect seems to be given by the presence of the Garage. In the submarket 1, which according to the figure 6.4 is mostly

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associated with the very historical part of Prague, the Garage is perceived as an extensively luxury feature, clearly visible in the Garage effect distribution in the submarket 1.

Note, that distribution of the effect (via boxplot) can be constructed for every single regressor used in the model specification form 3.1. Here, we, however, only plot the most distinctive ones in order to provide an overview of the housing submarkets.