Big Data Analytics

Big Data Analytics: Views from Statistical ... 7
tome), complex graphical models, etc., new methods are being developed for sampling
large graphs to estimate a given network’s parameters, as well as the node-,
edge-, or subgraph-statistics of interest. For example, snowball sampling is a common
method that starts with an initial sample of seed nodes, and in each step i, it
includes in the sample all nodes that have edges with the nodes in the sample at step
i −1but were not yet present in the sample. Network sampling also includes degreebased
or PageRank-based methods and different types of random walks. Finally, the
statistics are aggregated from the sampled subnetworks. For dynamic data streams,
the task of statistical summarization is even more challenging as the learnt models
need to be continuously updated. One approach is to use a “sketch”, which is
not a sample of the data but rather its synopsis captured in a space-efficient representation,
obtained usually via hashing, to allow rapid computation of statistics
therein such that a probability bound may ensure that a high error of approximation
is unlikely. For instance, a sublinear count-min sketch could be used to determine
the most frequent items in a data stream [9]. Histograms and wavelets are also used
for statistically summarizing data in motion [8].
The most popular approach for summarizing large datasets is, of course, clustering.
Clustering is the general process of grouping similar data points in an unsupervised
manner such that an overall aggregate of distances between pairs of points
within each cluster is minimized while that across different clusters is maximized.
Thus, the cluster-representatives, (say, the k means from the classical k-means clustering
solution), along with other cluster statistics, can offer a simpler and cleaner
view of the structure of the dataset containing a much larger number of points
(n ≫ k) and including noise. For big n, however, various strategies are being used
to improve upon the classical clustering approaches. Limitations, such as the need
for iterative computations involving a prohibitively large O(n 2 ) pairwise-distance
matrix, or indeed the need to have the full dataset available beforehand for conducting
such computations, are overcome by many of these strategies. For instance, a twostep
online–offline approach (cf. Aggarwal [10]) first lets an online step to rapidly
assign stream data points to the closest of the k ′ (≪ n) “microclusters.” Stored in
efficient data structures, the microcluster statistics can be updated in real time as
soon as the data points arrive, after which those points are not retained. In a slower
offline step that is conducted less frequently, the retained k ′ microclusters’ statistics
are then aggregated to yield the latest result on the k(< k ′ ) actual clusters in data.
Clustering algorithms can also use sampling (e.g., CURE [11]), parallel computing
(e.g., PKMeans [12] using MapReduce) and other strategies as required to handle
big data.
While subsampling and clustering are approaches to deal with the big n problem
of big data, dimensionality reduction techniques are used to mitigate the challenges
of big p. Dimensionality reduction is one of the classical concerns that can
be traced back to the work of Karl Pearson, who, in 1901, introduced principal component
analysis (PCA) that uses a small number of principal components to explain
much of the variation in a high-dimensional dataset. PCA, which is a lossy linear
model of dimensionality reduction, and other more involved projection models, typically
using matrix decomposition for feature selection, have long been the main-