Numerosity Reduction in Data Mining
The data reduction process reduces data size and makes it suitable and feasible for analysis. In the reduction process, the integrity of the data must be preserved, and data volume is reduced. Many techniques can be used for data reduction, and two primary methods of Data Reduction are Dimensionality Reduction and Numerosity Reduction.
What is Numerosity Reduction?
In the numerosity reduction, the data volume is decreased by selecting an alternative, smaller form of data representation. These techniques can be parametric or nonparametric. For parametric methods, a model can estimate the data so that only the data parameters need to be saved, instead of the actual data, for example, Loglinear models. Nonparametric methods are used to store a reduced representation of the data, including histograms, clustering, and sampling.
Types of Numerosity Reduction
This method uses an alternate, small forms of data representation, thus reducing data volume. There are two types of Numerosity reduction, such as:
1. Parametric
This method assumes a model into which the data fits. Data model parameters are estimated, and only those parameters are stored, and the rest of the data is discarded. Regression and LogLinear methods are used for creating such models. For example, a regression model can be used to achieve parametric reduction if the data fits the Linear Regression model.
 Regression: Regression can be a simple linear regression or multiple linear regression. When there is only a single independent attribute, such a regression model is called simple linear regression. If there are multiple independent attributes, then such regression models are called multiple linear regression.
Linear Regression models a linear relationship between two attributes of the data set. We need to fit a linear regression model between two attributes, x, and y, where y is the dependent attribute, and x is the independent attribute or predictor attribute. The model can be represented by the equation y=wx+b, where w and b are regression coefficients. A multiple linear regression model lets us express the attribute y in terms of multiple predictor attributes.
 LogLinear Model: Loglinear model can be used to estimate the probability of each data point in a multidimensional space for a set of discretized attributes based on a smaller subset of dimensional combinations. This allows a higherdimensional data space to be constructed from lowerdimensional attributes. The LogLinear model discovers the relationship between two or more discrete attributes. Assume we have a set of tuples in ndimensional space; the loglinear model helps derive each tuple's probability in this ndimensional space.
NOTE: Regression and loglinear models can both be used on sparse data, although their application may be limited.
2. NonParametric
A nonparametric numerosity reduction technique does not assume any model. The nonParametric technique results in a more uniform reduction, irrespective of data size, but it may not achieve a high volume of data reduction like the Parametric one. There are at least four types of NonParametric data reduction techniques, Histogram, Clustering, Sampling, Data Cube Aggregation, Data Compression.
 Histograms: A histogram is the data representation in terms of frequency. It uses binning to approximate data distribution and is a popular form of data reduction. Suppose a histogram for an attribute A and divisions the data distribution of A into disjoint subsets or buckets. If each bucket defines only an individual attributevalue or frequency pair, the buckets are known as singleton buckets.
 Clustering: Clustering techniques consider data tuples as objects. They partition the objects into groups or clusters so that objects within a cluster are "similar" to one another and "dissimilar" to objects in other clusters. It is commonly defined in terms of how "close" the objects are in space, based on a distance function.
The quality of a cluster can be defined by its diameter, the maximum distance between any two objects in the cluster. Centroid distance is an alternative measure of cluster quality. It is represented as the average distance of each cluster object from the cluster centroid denoting the "average object," or average point in the area for the cluster.
 Sampling: Sampling can be used as a data reduction approach because it enables a huge data set to be defined by a much smaller random sample or a subset of the information. Sampling can reduce large data sets into smaller sample data sets to represent the original data set. There are four types of sampling data reduction methods.
 Simple Random Sample Without Replacement of sizes
 Simple Random Sample with Replacement of sizes
 Cluster Sample
 Stratified Sample
 Data Cube Aggregation: Data cube aggregation involves moving the data from a detailed level to fewer dimensions. The resulting data set is smaller in volume, without loss of information necessary for the analysis task.
Data Cube Aggregation is a multidimensional aggregation that uses aggregation at various levels of a data cube to represent the original data set, thus achieving data reduction. Data Cube Aggregation, where the data cube is a much more efficient way of storing data, thus achieving data reduction, besides faster aggregation operations.
 Data Compression: It employs modification, encoding, or converting the structure of data in a way that consumes less space. Data compression involves building a compact representation of information by removing redundancy and representing data in binary form. Data that can be restored successfully from its compressed form is called Lossless compression. In contrast, the opposite, where it is not possible to restore the original form from the compressed form, is Lossy compression.
Difference between Numerosity Reduction and Dimensionality Reduction
There are two primary methods of Data Reduction, Dimensionality Reduction, and Numerosity Reduction.
Data encoding or transformations are used to access a reduced or "compressed" depiction of the original data in dimensionality reduction. If the original data can be regenerated from the compressed data without any loss of data, the data reduction is known as lossless. If data reconstructed is only approximated by the original data, the data reduction is called lossy.
Let's see the comparison between Dimensionality Reduction and Numerosity Reduction, such as:
Numerosity Reduction 
Dimensionality Reduction 
In numerosity reduction, data volume is reduced by choosing alternating, smaller forms of data representation. 
In dimensionality reduction, data encoding or transformation are applied to obtain a reduced or compressed representation of original data. 
In numerosity reduction, regression and loglinear models can be used to approximate the given data. In linear regression, the data are modeled to fit a straight line.
For example, a random variable, y (known as the response variable), can be modeled as a linear function of another random variable, x (known as a predictor variable), with the equation y = wx+b, where the variance of y is assumed to be constant. 
In dimensionality reduction, the discrete wavelet transform (DWT) is a linear signal processing technique that changes it to a numerically different vector, X', of wavelet coefficients when used to a data vector X.
The two vectors are of the same length. When applying this technique to data reduction, it can consider each tuple as an ndimensional data vector, that is, X=(x_{1},x_{2},…x_{n})depicting n measurements made on the tuple from n database attributes. 
It is merely a representation technique of original data to a smaller form. 
It can be used for removing irrelevant and redundant attributes. 
There is no loss of data in this method, but the whole data is represented in a smaller form. 
In this technique, some data can be lost, which is inappropriate. 
