In this study, both macro and micro-level approaches are used in order to examine the relationship between climateavariability and real estate prices in environmental and non-interior areas. At the macro level we use Wavelet coaches analyzes to examine the correlation between climateavariaability and average real estate prices in both regions. In order to alleviate the influence of outliers on the Wavelet analysis, this research uses the average unit price of all properties sold in the course of the analysis variables that reduce the effects of extreme price data. With this analytical method, you can comprehensively determine which real estate prices in the region are more sensitive to climateavarability.
At the micro level, this paper examines price pattern at real estate level in terms of climate variaability and compares the regions of the eco-community and not in the city center. We use the accumulated local effect method (ALE) based on Catboost technology. Taking into account the non -linears and complex associations between climate variaability and real estate dynamics, machine learning methods are particularly suitable for uncovering these patterns. Traditional approaches to machine learning are often faced with challenges with interpretability; However, the ALE method provides an intuitive visual representation of the mechanical learning process and its results, which is important for the analysis of the effects of climateavariability on the individual properties against different urban environmental backdrop.
Wavelet coherence analysis
The Wavelet analysis is an efficient and versatile mathematical method that is used in detail in the areas of economy and finance to analyze various data relationships (Magazzino and Golli 2024). It shows extraordinary skills in the analysis of non-stationary time series that process and interpret dynamic changes within data (Bilgili et al. 2024). By breaking down time data into different scales, the Wavelet analysis reveals structures and properties that are hidden in the data at different levels of detail, which means that multi-scale local features are presented in time and frequency dimensions.
The continuous Wavelet transformation is an exceptionally effective tool for recording and visualizing the development of non-in-stationary signals. The Wavelet coaches analysis, a form of application within the Wavelet analysis, is developed on the basis of the translation principles and scaling of Wavelet -based functions. When examining the relationship between climate varability and average real estate prices, we chose the Morlet -Wavelet. With its real and imaginary parts, the Morlet -Wavelet offers the ability to observe the phase and amplitude of signals and thus enable a more detailed analysis (Büssow 2007). For two time series X((T) And y((T) According to Torrence and Compo (1998) method, the Cross-Wavelet transformation can be used as:
$$ {w} _ {xy} (\ number, s) = {x Number} ^ {* x} (\ numbers, $$
(1)
Where \ ({W} _ {x} \ left (\ tau, s \ right) \) And \ ({W} _ {y} \ left (\ tau, s \ right) \) Represent the continuous Wavelet transformations of the two time series. T And S Design the translation parameter for the Wavelet position and the scale parameter, while * the complex conjugate means. Wavelet coherence can express various domains or intervals in which changes in the variance intensity between variables in the time series under consideration have a common movement. According to Torrence and Webster (1999), the adapted Wavelet -Quadrat -Coherence as:
$ {r} _ {xy}^} (\ tau, s) = \ frac {\ left \ vert s ({w} _ {xy} (\ tau, s) \ right \}^} \ right) \ cdot s \ left (| {w _ {y} (\ tau, s) {|}^{2} \ right)}, $$
(2)
Where S describes the smoothing operator over time and scale. \ ({R} _ {xy}^{2} \ left (\ tau, s \ right) \) can take values between 0 and 1 with greater benefits R2 Values that show a closer connection between climatic variables and average real estate prices. Regions affected by marginal effects are shown by the cone of influence (COI).
In order to interpret the lead or delay relationship between series, it introduces the Wavelet phase difference (Bloomfield et al. 2004), which can be defined:
Eight \ or [-\pi ,\pi ]$$
(3)
Where \({\I am}\) And \({\Re}\) Design the operators for the imaginary and real parts.
Accumulated local effects (ALE) method based on the Catboost -Algorithm
With socio -economic systems, climate variaability interacts in complex and non -linear ways with socio -economic systems, especially at an individual level, whereby linear modeling approaches can lead to considerable distortions. In such cases, model approaches for machine learning become more useful. Catboost represents a form of increasing implementation, its efficiency and performance distinguishes it from the large number of tree -based models (Prokhorenkova et al. 2018). This algorithm was first introduced by Yandex in 2017.
As an improved gradient -boosting decision -making tree (GBDT) -Algorithm, Catboost differs from conventional algorithms for the gradient model by using an orderly boosting to optimize the gradient estimation method for the gradient -boosting decision -making trees, which means that the associated with punctual gradient assessment Avoid over -adjustment questions. The advantages include fewer parameters, support for categorical variables and high accuracy.
In every iteration of traditional gradient boosting, the gradient of the current model is obtained based on the same data record, and the basic learners are trained on this gradient. This process can lead to a distortion in the event of a selective gradient estimate, which leads to an over -adaptation of the final ensemble model. Catboost optimizes the gradient assessment method for the gradient boosting decision -making trees by using the orderly increase. In this approach, models using various random permutations of the data are trained to ensure that information from the current tree is not used when calculating gradients, which contributes to reducing the distortion and reducing the risk of overhanging. The negative gradient of the loss function output by the previous decision tree serves as input for the MTH decision tree. The extended after the model forecast M Iterations are shown as a sum of the output values of several decision -making trees. New decision -making trees are continuously constructed to adapt the residues until the specified number of trees is reached or certain conditions are met. The characteristics of Catboost make it particularly effective when converting a variety of mechanical learning, such as classification and regression, especially if they are data records that have a large number of categorical features.
In this study we use RMSE (square error of the root middle value) as a loss function for the Catboost model. It can be expressed as:
$ \, {{{rmse}} \, = \ sqrt {\ frac {1} \ mathop {\ sum} \ limits_ {i = 1}^{n} {({y}}-{}} _ {i})}^{2}}, $$
(4)
Where yI Represents the ITH actual value and \ (\ Has {y} i \) Describes the ITH predicted value. In view of the fact that the relationship between climateavariability and the resilience of real estate prices in the eco-city is a regression problem, RMSE is suitable for predictive problems with continuous values, which is a common type of regression task. Therefore, the use of RMSE as a loss function is a very reasonable choice. Since RMSE is rooted in the terms of error, a greater punishment for major prediction errors is imposed, which supports the model in reducing the occurrence of errors. We divide the individual real estate prices into eco-city and non-hinting area into validation and training rates in a ratio of 20% to 80%. Then we create separate models for real estate prices for eco-city and real estate prices for non-income using MSE (medium square error) in order to evaluate the model output.
$ \, {{{mse}} \, = \ frac {1} \ mathop {\ sum} \ limits_ {i = 1}^{n} {({y}}-{{y}}}})
(5)
Although Catboost shows excellent performance in regression tasks, his predictive results suffer from poor interpretability. To address this problem, we present the ALE method (accumulated local effects) to tackle the type of “Black Box” models of machine learning. The core idea of ALE is to evaluate the average change in the model output if a function within its distribution varies slightly (Peely and ZHU 2020). In contrast to conventional PDP (partial dependency diagrams), which require strict independence assumptions between variables, the ALE method can provide precise interpretations that are not influenced by characteristic correlations, even if dependencies between characteristics are present.
The ALE calculation mainly includes the integration and accumulation of local effects for each data point in a certain characteristic. This process includes the calculation of the variations of the model predictions within each interval of the feature values and the accumulation of these differences in order to display the global effect of the characteristic (Okoli 2024). Especially for a specific function X and its corresponding model forecast function F((X), It is necessary to share X In several intervals K.
The local effects for each interval are calculated. For the KTh interall \(\left[{x}_{k},{x}_{k+1}\right]\)The average change in the predicted values for all data points within this interval is calculated. This process can be expressed as:
$ {\ text {ale} _ {k} = \ frac {1} {{n} _ _ \ mathop {\ sum} \ limits _ {i: {x} _ {i} \ in [{x}_{k},{x}_{k+1}]} \ left (f ({x} _ {ki}+\ delta {x} _ {k})-f ({x} _ {ki}) \ right) $$
(6)
Where NK is the number of data points that fall into the fall KTh interval and δXK Is the width of the KTh interall. Then all local effects are accumulated in order to obtain the overall value.
$$ \, {{ale}} (x) = \ mathop {\ sum} \ limits_ {k = 1}^{k (x)} {\ text {ale}} _ {k}, $$
(7)
Where K((X) is the number of intervals that are smaller than or the same X. After all, the ale values are so centered that the ale value at the average level of the function is zero.
$ \, {{{ale}} \, {(x)} _ {\ rm}}} = \, \ text {ale} (x)-\ frac {1} \ mathop {\ sum} \ limits_ {i = 1}^{n} {\ text {ale}} \, ({x} _ {i}), $$
(8)
Where N Is the total number of data points and XI Is the characteristic value of the ITh data point. By using the ALE method, we can clearly identify the specific relationships between each characteristic and the initial variables, which not only ensures the accuracy of the results of the machine learning model, but also significantly improves the interpretability of the model.