# Analyzing Data Reduction Techniques: An Experimental Perspective
## 7 Data Reduction Techniques
### Lossy Data Reduction Algorithms  
We evaluated two types of lossy data reduction algorithms: lossy data compression techniques (Random Projection and Quantization) and numerosity data reduction algorithms. Numerosity data reduction algorithms are divided into parametric (Linear Regression and Principal Component Analysis) and non-parametric (K-Means and Simple Random Sampling). **The Random Projection** algorithm is a dimensionality reduction technique that uses random matrices to transform high-dimensional data into a lower-dimensional space [34; 35]. Unlike deterministic methods, Random Projection selects matrix elements from probability distributions, often Gaussian or Rademacher distributions. Despite its seemingly arbitrary nature, Random Projections can, under certain conditions, preserve pairwise distances between data points with high probability. This method is particularly advantageous for efficiently handling large datasets. By sacrificing the need for exact distance preservation, Random Projection provides a computationally advantageous approach to reducing dimensionality while preserving relative distances, making it applicable in various fields, including machine learning and signal processing. **Quantization** is another studied lossy data compression algorithm, used in signal processing and data compression to reduce the precision of numerical data representation [36]. In signal processing, the algorithm involves mapping continuous values to discrete levels, typically by dividing a range into intervals and assigning specific values to each interval. This results in a representation with a slightly lower depth. In data compression, Quantization reduces the number of bits needed to represent numerical values, thereby reducing storage or transmission requirements. However, this reduction introduces quantization error as an approximation to the original values. Achieving a balance between efficient data representation and minimizing quantization error is critical in several applications, including image and audio compression, where optimization is essential to maintain an acceptable level of quality. **Linear Regression** is a statistical technique that models the relationship between a dependent variable and one or more independent variables using a linear equation [37]. The goal is to find coefficients that minimize the difference between predicted and actual values. This model identifies significant variables and their impact on the dependent variable. In data reduction, the technique helps simplify and represent the underlying patterns in the data by highlighting the most influential variables. Setting the coefficients allows researchers to determine whether independent variables contribute significantly to the variability of the dependent variable, and if not, dimensionality can be reduced by removing or merging these less influential variables. **Principal Component Analysis** (PCA) is a data reduction technique that is used to simplify high-dimensional datasets while preserving the most significant information. It achieves this by identifying the principal components, which are the eigenvectors of the covariance matrix sorted by their corresponding eigenvalues. These principal components represent the directions of maximum variance in the original dataset. By selecting a subset of these components, it is possible to project the data into a lower-dimensional subspace, effectively reducing the dimensionality of the dataset [38]. This process preserves as much of the original variance as possible, allowing for a more concise representation of the data. PCA is valuable for several tasks, such as feature extraction, noise reduction, and visualization of complex datasets, allowing for more efficient analysis and modeling. **K-Means** is a clustering algorithm that focuses on grouping similar observations or data points into distinct clusters, thereby simplifying the complexity of the dataset [39]. This technique organizes data based on inherent patterns or similarities to reveal underlying structures.