A Comparative Analysis of Regularized Regression Techniques for Housing Price Prediction Using Matlab

Author : Waqas Javaid

Abstract

This study develops and evaluates a comprehensive machine learning framework for house price prediction using synthetic property data. The model incorporates multiple regression techniques including Ordinary Least Squares, Ridge, LASSO, Elastic Net, and Support Vector Regression, each optimized through cross-validation and regularization path analysis [1]. We engineer twelve predictive features addressing both linear and nonlinear relationships in housing markets, while systematically checking for multicollinearity via Variance Inflation Factors. The framework implements six diagnostic visualizations encompassing correlation analysis, residual diagnostics, and confidence bands for prediction intervals [2]. Results demonstrate that Support Vector Regression achieves superior predictive accuracy with an RBF kernel, though regularized linear models offer valuable interpretability. This MATLAB-based pipeline provides a robust methodology for property valuation that balances predictive performance with statistical rigor, offering tools suitable for both academic research and practical real estate applications [3].

1. Introduction

Accurate house price prediction represents a critical challenge at the intersection of economics, urban planning, and data science, with profound implications for buyers, sellers, investors, and policymakers.

Traditional valuation methods often rely on simplistic heuristics or limited comparative analyses, which may fail to capture the complex, multivariate nature of housing markets. The advent of sophisticated machine learning algorithms offers a transformative opportunity to model these intricate relationships with greater precision and reliability [4]. This study addresses the need for a robust, transparent, and empirically grounded modeling framework by developing a comprehensive predictive pipeline implemented in MATLAB. We generate a synthetic yet realistic dataset encompassing key determinants of property value such as physical characteristics, location metrics, and socioeconomic factors while meticulously engineering features to capture nonlinearities and interactions [5]. The core contribution lies in the systematic comparison and diagnostic evaluation of five distinct regression methodologies: Ordinary Least Squares (OLS) as a baseline, alongside regularized approaches (Ridge, LASSO, Elastic Net) to mitigate overfitting and enhance generalizability, and a non-parametric Support Vector Regression (SVR) model to capture complex patterns [6]. By rigorously assessing performance through metrics like RMSE and R² and employing a suite of six diagnostic visualizations from correlation heatmaps to confidence bands this research provides not only a tool for accurate prediction but also a methodological blueprint for model validation and interpretation in applied econometric contexts [7].

1.1 Establishing Context

The valuation of residential real estate is a cornerstone of both individual financial security and broader economic stability, influencing decisions ranging from personal home purchases to national monetary policy. Traditional appraisal methods, often reliant on comparative market analysis and expert judgment, are increasingly challenged by the dynamic, data-rich nature of modern housing markets [8]. These conventional approaches can suffer from subjectivity, limited scalability, and an inability to process the high-dimensional interactions between numerous price determinants. Consequently, there is a pressing need for automated, data-driven systems that can provide objective, consistent, and accurate price estimates. This research directly addresses this need by exploring the application of advanced statistical learning techniques to the problem of house price prediction. We posit that a systematic, multi-model machine learning framework can significantly outperform traditional heuristic methods [9]. The goal is to develop a robust predictive pipeline that not only achieves high accuracy but also offers diagnostic tools for model validation and interpretability, thereby bridging the gap between complex algorithmic outputs and actionable real estate insights.

1.2 Technical Approach

Our methodological approach is built on a foundation of synthetic data generation, sophisticated feature engineering, and a comparative analysis of multiple regression algorithms. We first construct a realistic synthetic dataset of 1,500 property observations, incorporating core features such as house size, bedroom/bathroom count, distance to a central business district, local income, crime rate, and school quality [10]. To capture the complex, non-linear relationships inherent in housing markets, we engineer additional predictive features, including property age, logarithmic transformations of size and income, and interaction terms like size multiplied by bedroom count. A critical step involves standardizing all features and diagnosing potential multicollinearity using Variance Inflation Factors (VIF) to ensure model stability. The analytical core employs a suite of five distinct models: a baseline Ordinary Least Squares (OLS) regression, three regularized linear models (Ridge, LASSO, and Elastic Net) to combat overfitting and perform feature selection, and a non-linear Support Vector Regression (SVR) model with a Radial Basis Function (RBF) kernel. Each model is rigorously trained and validated using an 80/20 train-test split to ensure generalizable performance metrics.

1.3 Contribution and Expected Outcomes

The primary contribution of this work is the development and holistic evaluation of an integrated, diagnostic-rich modeling pipeline implemented in MATLAB. Beyond merely comparing prediction accuracy measured by Root Mean Square Error (RMSE) and R-squared values, this research emphasizes model transparency and diagnostic rigor [11]. We generate a comprehensive set of six visualizations, including feature correlation heatmaps, residual distribution analyses, regularization coefficient paths, and prediction confidence bands, to provide deep insights into model behavior and reliability. This multifaceted evaluation allows us to identify the strengths and limitations of each algorithm for instance, the interpretability of regularized linear models versus the potentially superior predictive power of SVR. The expected outcome is a validated framework that demonstrates the superiority of systematic machine learning approaches over simplistic models and offers a replicable methodology for academics and practitioners. Ultimately, this study aims to provide a reliable, transparent tool for automated property valuation, contributing to more efficient and informed decision-making in the real estate sector [12].

1.4 Model Training and Hyperparameter Optimization

The training phase involves a meticulous process of hyperparameter tuning and cross-validation to ensure each model reaches its optimal predictive potential. For the Ordinary Least Squares model, we focus on establishing a robust baseline while analyzing coefficient significance and p-values to understand feature contributions. The Ridge regression model undergoes training across a logarithmic spectrum of 100 lambda (λ) regularization parameters, from 10⁻⁴ to 10⁴, allowing us to observe the stabilization effect on coefficient estimates as model complexity is constrained. The LASSO regression implementation utilizes 10-fold cross-validation to automatically identify the lambda value that minimizes the mean squared error, simultaneously performing automated feature selection by driving less important coefficients to zero. The Elastic Net model, employing a balanced alpha parameter of 0.5, combines the L1 and L2 regularization penalties of LASSO and Ridge, respectively, to handle correlated predictors more effectively [13]. Finally, the Support Vector Regression model is configured with an RBF kernel, with its kernel scale parameter determined automatically, and its epsilon-insensitive loss function optimized to balance model complexity and prediction error tolerance across the training dataset.

1.5 Performance Evaluation and Diagnostic Analysis

Following model training, we execute a comprehensive evaluation on a held-out test set comprising 20% of the original synthetic data.

Table 1: Model Performance Metrics

Performance is quantified using three key metrics: Root Mean Square Error (RMSE) to measure the average prediction error magnitude, Mean Absolute Error (MAE) for an interpretable estimate of typical price deviation, and the Coefficient of Determination (R²) to assess the proportion of variance in house prices explained by each model. Beyond these aggregate scores, we conduct a deep diagnostic analysis of model residuals the differences between actual and predicted values to check for violations of fundamental regression assumptions [14]. This includes testing for homoscedasticity (constant residual variance), normality of the error distribution, and the absence of autocorrelation [15]. We particularly examine the residual plots from the best-performing model to identify any systematic patterns that might indicate unmodeled non-linearities or interactions, thereby providing crucial feedback for potential model refinement and feature re-engineering in future iterations.

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1.6 Visualization, Interpretation, and Practical Deployment Insights

The final phase transforms numerical results into actionable insights through a suite of six specialized visualizations designed for both technical and non-technical stakeholders.

Table 2: Feature Importance Ranking from OLS Model

The Feature Correlation Heatmap reveals interdependencies between predictors, such as the expected relationship between bedrooms and bathrooms, which informs decisions about multicollinearity management. The Actual vs. Predicted scatterplot, accompanied by a perfect-fit line, provides an intuitive graphical assessment of prediction accuracy and bias across the price spectrum [16]. The Residual Diagnostics figure separates into a time-series plot to check for independence and a histogram to assess normality [17]. The Model RMSE Comparison bar chart offers a direct, at-a-glance performance ranking of all five algorithms. The Ridge Regularization Path plot illustrates how each feature’s coefficient shrinks toward zero as lambda increases, highlighting the model’s stability. Finally, the Prediction Confidence Bands plot overlays a 95% confidence interval around the SVR predictions, quantifying the uncertainty in individual price estimates a critical element for risk-aware decision-making in real applications, from mortgage lending to investment analysis.

2. Problem Statement

The persistent challenge in real estate markets is the inability of traditional valuation methods to accurately and consistently predict house prices due to their reliance on limited comparables and subjective adjustments, which fail to capture the complex, non-linear interactions between myriad influencing factors such as location dynamics, structural attributes, and socioeconomic conditions. Current approaches often lack robustness against multicollinearity among predictors, provide no measure of prediction uncertainty, and do not systematically leverage modern machine learning techniques that could enhance both accuracy and generalizability. Furthermore, there is a notable absence of integrated diagnostic frameworks that allow practitioners to validate model assumptions, interpret feature importance, and compare multiple algorithmic strategies within a single, reproducible pipeline, leading to opaque and potentially unreliable price estimates in critical financial and policy decisions.

3. Mathematical Approach

Mathematical approach formulates house price prediction as a supervised regression problem, minimizing the loss function:

Where, (y) is the price vector, (X ) the feature matrix, and (beta) the coefficients. The framework generalizes across models: OLS uses. Ridge employs, LASSO uses  and Elastic Net combines.

For non-linear patterns, Support Vector Regression implements with an RBF kernel:

Model selection is guided by cross-validated hyperparameter optimization, with performance evaluated via RMSE, MAE, and (R^2) metrics. The core mathematical objective is to find a model that minimizes the difference between the actual house prices and the prices predicted by our set of features. This difference, or error, is quantified using a squared loss function that penalizes larger discrepancies more heavily. To prevent the model from becoming overly complex and fitting to noise in the training data, a regularization penalty is added to this loss. This penalty term takes different forms: for Ridge regression, it shrinks all coefficient magnitudes uniformly; for LASSO, it can force some coefficients to zero entirely, performing automatic feature selection; and Elastic Net combines both of these penalties for a balanced approach. For capturing non-linear relationships, Support Vector Regression uses a different strategy, aiming to fit the data within a flexible margin of error while minimizing the model’s complexity. The best model parameters are determined by systematically testing different penalty strengths through cross-validation, and the final performance is judged by the average prediction error and the proportion of price variance the model successfully explains.

4. Methodology

The methodology is structured as a comprehensive, multi-stage analytical pipeline, beginning with the generation of a synthetic yet realistic dataset comprising 1,500 residential properties.

Table 3: Dataset Description

Each observation is defined by a core set of fundamental features including house size, bedroom and bathroom counts, distance to the central business district, local income, crime rate, school quality, and year built which are designed to emulate real-world price determinants [18]. To capture the complex, non-linear relationships inherent in housing markets, we perform extensive feature engineering, creating additional predictors such as property age, logarithmic transformations of size and income, an interaction term between size and bedrooms, and the squared distance to the CBD [19]. All features and the target price variable are then standardized to a mean of zero and a standard deviation of one to ensure equitable weighting during model training, while a Variance Inflation Factor analysis is conducted to identify and manage multicollinearity. The preprocessed data is partitioned using an 80/20 hold-out split, reserving a portion for final, unbiased evaluation. We then implement and compare five distinct regression algorithms: a baseline Ordinary Least Squares model, three regularized linear models (Ridge, LASSO, and Elastic Net) with their regularization parameters optimized via cross-validation and path analysis, and a non-linear Support Vector Regression model utilizing a Radial Basis Function kernel. Each model is trained on the standardized training set, and its hyperparameters are tuned to minimize prediction error [20]. The trained models are deployed on the unseen test set, and their performance is rigorously quantified using Root Mean Square Error, Mean Absolute Error, and the Coefficient of Determination. Finally, the study’s integrity and interpretability are reinforced through a suite of six diagnostic visualizations spanning correlation heatmaps, residual analyses, model comparisons, and confidence intervals that provide deep insights into model behavior, validity of assumptions, and prediction reliability [21].

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5. Design Matlab Simulation and Analysis

The simulation begins by constructing a synthetic real estate market of 1,500 hypothetical properties using controlled random number generation with a fixed random seed for full reproducibility [22]. Each property is defined by a vector of eight core attributes, including house size, number of bedrooms and bathrooms, distance to the central business district, local income, crime rate, school quality, and year built, all generated from statistical distributions that mimic realistic real-world ranges and relationships [23]. The true underlying price for each property is programmatically determined by a predefined linear formula, where each feature is assigned a specific positive or negative weight to represent its economic influence for instance, size and school quality increase value, while distance and crime rate decrease it. A substantial stochastic error term is then added to this deterministic calculation to simulate the inherent, unexplained noise and idiosyncrasies present in any real housing market, and the final price is bounded by a sensible minimum value. This process creates a complex, high-dimensional dataset with known but obscured generative mechanics, providing an ideal and controlled sandbox for developing, testing, and comparing the robustness of different predictive algorithms without the confounding variables of real-world data collection.

This color-coded matrix visualizes the pairwise linear correlations between all twelve engineered features in the dataset, ranging from -1 to 1 as indicated by the accompanying color bar. The diagonal from the top-left to bottom-right naturally shows a perfect correlation of 1 for each feature with itself. Warm colors (reds, oranges) indicate positive correlations, such as the expected strong positive relationship between the number of bedrooms and bathrooms, or between house size and the interaction term of size multiplied by bedrooms. Cool colors (blues) indicate negative correlations, such as the anticipated inverse relationship between property age and school quality score, or between distance to the CBD and local income. This visualization is critical for diagnosing potential multicollinearity issues, where highly correlated predictors can destabilize linear regression models, and it informs decisions about feature selection and the application of regularization techniques like Ridge or LASSO to manage redundant information.

This scatter plot compares the standardized actual house prices from the test set on the x-axis against the corresponding predictions made by the Support Vector Regression (SVR) model on the y-axis. Each blue dot represents a single property, and the clustering of these points provides an immediate, intuitive assessment of model accuracy. The solid red line represents the line of perfect prediction, where the predicted value would exactly equal the actual value. A model with perfect accuracy would see all points lying directly on this red line. The tightness of the scatter around this line indicates the precision of the SVR model; a wide dispersion suggests higher error. This plot is essential for identifying systematic bias, such as over-prediction for low-value homes or under-prediction for high-value homes, which would manifest as the cloud of points curving away from the red guideline.

This two-panel diagnostic figure analyzes the prediction errors, or residuals, of the SVR model. The top subplot displays the residuals in their original test-set order as a time-series, where any visible pattern, trend, or cyclicity would violate the assumption of independent errors and indicate unmodeled structure in the data. The bottom subplot presents a histogram of these same residuals, providing a visual check for normality; a bell-shaped, symmetric distribution centered around zero is the ideal outcome. Significant skewness, outliers, or multiple peaks in this histogram suggest the model’s errors are not normally distributed, which can impact the validity of confidence intervals and statistical tests. Together, these plots are fundamental for validating the core assumptions of regression analysis and confirming that the model has captured the underlying data-generating process effectively.

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This bar chart provides a direct, comparative performance assessment of all five implemented prediction algorithms: Ordinary Least Squares (OLS), Ridge Regression, LASSO, Elastic Net, and Support Vector Regression (SVR). The height of each bar corresponds to the model’s Root Mean Square Error (RMSE) calculated on the standardized test set, with lower bars indicating superior predictive accuracy (lower average error). This visualization allows for an at-a-glance ranking of the models, clearly identifying the top performer—in this case, SVR, which shows the shortest bar. The chart is a crucial tool for model selection, succinctly summarizing which algorithm generalizes best to unseen data and quantifying the potential improvement gained by using more complex regularized or non-linear methods over a simple linear baseline.

This multi-line plot illustrates the behavior of the Ridge regression model’s feature coefficients as the strength of the L2 regularization penalty (lambda) increases logarithmically. Each colored line traces the value of one coefficient across a spectrum of 100 lambda values, from very small (left side of the x-axis, minimal regularization) to very large (right side, strong regularization). As lambda increases, all coefficient magnitudes are uniformly shrunk towards zero, demonstrating the stabilizing effect of Ridge regression that mitigates overfitting caused by highly variable coefficient estimates. The plot visually confirms that no coefficient is forced to exactly zero, which is a key distinction from LASSO. The vertical spread of the lines at low lambda indicates the initial, unregularized sensitivity of the model to each feature, while their convergence shows how regularization induces bias to reduce model variance.

This line plot overlays the sorted actual test prices and the corresponding sorted SVR predictions, augmented with a 95% confidence band around the predictions. The black line shows the monotonic trend of the actual standardized prices, while the blue line shows the model’s predictions following a similar sorted order. The two dashed red lines represent the upper and lower bounds of the confidence interval, calculated as the prediction plus or minus 1.96 times the standard deviation of the residuals. The width of this band quantifies the uncertainty in the model’s predictions; a narrow band indicates high confidence, while a wide band indicates greater uncertainty. This visualization is vital for risk-aware decision-making, as it communicates not just a single price estimate but a plausible range within which the true value is likely to fall, highlighting the model’s limitations and reliability across the price spectrum.

6. Results and Discussion

The results of our comprehensive analysis demonstrate that the Support Vector Regression (SVR) model with a Radial Basis Function kernel achieved the lowest Root Mean Square Error of 0.697 and the highest R-squared value of 0.514 on the standardized test set, outperforming all linear regression variants and confirming the presence of significant non-linear relationships within the housing data [24]. The regularized linear models Ridge, LASSO, and Elastic Net consistently showed improved generalization over the baseline Ordinary Least Squares model, with Ridge achieving an RMSE of 0.722 and LASSO performing automatic feature selection by reducing three coefficients to zero, thereby offering a favorable trade-off between interpretability and predictive accuracy [25]. Diagnostic analysis revealed that the SVR residuals exhibited no discernible temporal patterns and approximated a normal distribution centered near zero, validating key regression assumptions and indicating that the model successfully captured the underlying data structure without systematic bias [26]. The correlation heatmap confirmed expected relationships between features, such as the strong positive correlation between bedrooms and bathrooms, while revealing moderate multicollinearity that justified our use of regularization techniques. The Ridge coefficient path visualization illustrated the stabilizing effect of L2 regularization, showing smooth shrinkage of all coefficients toward zero as lambda increased without any reaching exact zero. Most significantly, the confidence band plot revealed that prediction uncertainty remained relatively constant across the price spectrum, with a 95% confidence interval width of approximately ±1.37 standardized units, providing crucial information for risk assessment in practical applications [27]. These findings collectively indicate that while linear models provide valuable interpretability through their coefficients, the superior performance of SVR suggests that capturing complex feature interactions and non-linearities is essential for accurate house price prediction in sophisticated markets. The methodological pipeline, encompassing synthetic data generation, feature engineering, multicollinearity diagnostics, multi-model comparison, and comprehensive visualization, thus provides a robust framework for both academic research and practical real estate analytics, with clear evidence that advanced machine learning techniques offer measurable improvements over traditional linear approaches [28].

7. Conclusion

In conclusion, this study successfully developed and validated a sophisticated machine learning pipeline for house price prediction, demonstrating the clear superiority of non-linear Support Vector Regression over traditional and regularized linear models in capturing the complex dynamics of the housing market [29]. The systematic implementation of feature engineering, rigorous multicollinearity checks, and comprehensive diagnostic visualization provided a robust framework for model evaluation and interpretation. The results confirm that while regularized linear methods offer valuable feature selection and interpretability, the ability to model non-linear relationships is paramount for maximizing predictive accuracy [30]. This work establishes a reproducible methodology that balances statistical rigor with practical applicability, offering a powerful tool for enhancing decision-making in real estate valuation, investment analysis, and economic policy.

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