# Advancing Real-Estate Forecasting: A Novel Approach Using Kolmogorov-Arnold Networks
## 3 Materials and Methods  
### Problem Formulation and Proposed Model Description  
The primary goal of this research is to predict housing prices based on a set of input features derived from real-estate datasets. Formally, we define the problem as follows:  
Let \(X=\{x_{1},x_{2},\ldots,x_{n}\}\) represent the input features, where \(x_{i}\) corresponds to specific characteristics of a property (e.g., size, location, and number of rooms). The target variable \(y\) represents the predicted price of the property. The objective is to learn a mapping function \(f:X\to y\), such that the prediction error \(L(y,\hat{y})\), where \(\hat{y}=f(x)\), is minimized. Here, \(L\) is the loss function used to quantify the prediction error. To ensure the reliability and generalizability of the proposed KAN model, we explicitly state the following assumptions:  
1. Independence: The input features are assumed to be conditionally independent given the target variable. 2. Sufficient data: The dataset is assumed to be sufficiently large and diverse to capture the underlying patterns in housing prices. 3. Feature engineering: The input features are assumed to be preprocessed (e.g., normalized) to ensure compatibility with the neural network. These assumptions are tested and validated during the experimental phases. KAN is a neural network architecture inspired by the Kolmogorov-Arnold representation theorem. This property allows KAN to decompose complex multivariate functions into simpler univariate components, making it uniquely suited for capturing nonlinear interactions in high-dimensional data. The Kolmogorov-Arnold representation theorem provides the theoretical foundation for KAN. For a continuous function \(f:\mathbb{R}^{n}\rightarrow\mathbb{R}^{f}\), the theorem states the following:  
\[f(x_{1},x_{2},\ldots,x_{n})=\sum_{i=1}^{2n+1}g_{i}\left(\sum_{j=1}^{n}h_{ij}( x_{j})\right) \tag{1}\]  
where  
\(g_{i}\) and \(h_{ij}\) are continuous univariate functions. \(x_{1},x_{2},\ldots,x_{n}\) are the input variables. In the KAN model, this decomposition is approximated using neural network layers. The input features are transformed into intermediate representations:  
\[z_{i}=\sum_{j=1}^{n}h_{ij}(x_{j}) \tag{2}\]  
where \(h_{ij}\) are univariate transformations modeled by spline functions. The intermediate features \(z_{i}\) are passed through another set of univariate functions \(\mathrm{g}_{i}\) to compute the final output:  
\[\hat{y}=\sum_{i=1}^{2n+1}g_{i}(z_{i}) \tag{3}\]  
The model minimizes a loss function \(L\), such as the Mean Squared Error (MSE):  
\[L(y,\hat{y})=\frac{1}{m}\sum_{i=1}^{m}(y_{i}-\hat{y_{i}})^{2} \tag{4}\]  
where \(m\) is the number of training samples, \(y_{i}\) is the true value, and \(\hat{y_{i}}\) is the predicted value. To prevent overfitting, regularization terms are added:  
\[L_{\text{total}}=L(y,\hat{y})+\lambda_{1}\Big{|}\theta|_{1}+\lambda_{2}\Big{|} \theta|_{2}^{2} \tag{5}\]  
Below, a detailed architectural description is provided:  
**Input Layer**: The input layer accepts a vector \(\mathsf{X}\) of size \(\mathsf{n}\), where \(\mathsf{n}\) corresponds to the number of features. Each feature is standard-scaled to ensure consistent scaling across all inputs. **Intermediate Layers:**  
* Three KAN layers are sequentially stacked, each performing a decomposition of the input into univariate components using splines. Each KAN layer introduces nonlinear transformations, enabling the model to capture intricate feature interactions. * The splines are defined by a set number of knots and a specific order, which control the granularity and smoothness of the approximations. The number of knots determines how many polynomials are strung together. The grid range and epsilon define the bounds and resolution of the decomposition. * Fully connected layers refine the outputs of the KAN layers. * Activation functions introduce nonlinearity and enhance the model's expressiveness.