"To the untrained eye, randomness appears as regularity or tendency to cluster." (William Feller, "An Introduction to Probability Theory and its Applications", 1950)
"Sometimes clusters of variables tend to vary together in the normal course of events, thereby rendering it difficult to discover the magnitude of the independent effects of the different variables in the cluster. And yet it may be most desirable, from a practical as well as scientific point of view, to disentangle correlated describing variables in order to discover more effective policies to improve conditions. Many economic indicators tend to move together in response to underlying economic and political events."
"The logarithmic transformation serves several purposes: (1) The resulting regression coefficients sometimes have a more useful theoretical interpretation compared to a regression based on unlogged variables. (2) Badly skewed distributions - in which many of the observations are clustered together combined with a few outlying values on the scale of measurement - are transformed by taking the logarithm of the measurements so that the clustered values are spread out and the large values pulled in more toward the middle of the distribution. (3) Some of the assumptions underlying the regression model and the associated significance tests are better met when the logarithm of the measured variables is taken."
"The scatterplot is a useful exploratory method for providing a first look at bivariate data to see how they are distributed throughout the plane, for example, to see clusters of points, outliers, and so forth."
"The second type of uncertainty results from random causes that produce fluctuations in both sign and magnitude, the latter within well-defined limits, however. In the long run, the random error averages out to zero. The random error accounts for the variability of individual measurements and it will be shown that it can be statistically characterized by what is called a standard deviation. This term is thus a measure of the dispersion of the data around a mean or average value. When the value of the standard deviation is small, the data cluster closely around the mean; when it is large, the spread is greater." (Cheryl Cihon & John K Taylor, "Statistical Techniques for Data Analysis" 2nd. ed., 2005)
"Because the cluster solutions grow tree-like (starting with the branches and ending with the trunk) results are often displayed in a graphic called the dendrogram. Horizontal lines indicate linking of two samples or clusters, and thus the vertical axis presents the associated height or similarity as a measure of distance. The samples are arranged in such a way that the branches of the tree do not overlap. Linking of two groups at a large height indicates strong dissimilarity (and vice versa). Therefore, a clear cluster structure would be indicated if observations are linked at a very low height, and the distinct clusters are linked at a greater height (long branches of the tree). Cutting the dendrogram at such a greater height permits assigning the samples to the resulting distinct clusters. Visual inspection of a dendrogram is often helpful in obtaining an initial estimate of the number of clusters for partitioning methods." (Clemens Reimann et al, "Statistical Data Analysis Explained: Applied Environmental Statistics with R", 2008)
"Multivariate techniques often summarize or classify many variables to only a few groups or factors (e.g., cluster analysis or multi-dimensional scaling). Parallel coordinate plots can help to investigate the influence of a single variable or a group of variables on the result of a multivariate procedure. Plotting the input variables in a parallel coordinate plot and selecting the features of interest of the multivariate procedure will show the influence of different input variables." (Martin Theus & Simon Urbanek, "Interactive Graphics for Data Analysis: Principles and Examples", 2009)
"Parallel coordinate plots are often overrated concerning their ability to depict multivariate features. Scatterplots are clearly superior in investigating the relationship between two continuous variables and multivariate outliers do not necessarily stick out in a parallel coordinate plot. Nonetheless, parallel coordinate plots can help to find and understand features such as groups/clusters, outliers and multivariate structures in their multivariate context. The key feature is the ability to select and highlight individual cases or groups in the data, and compare them to other groups or the rest of the data." (Martin Theus & Simon Urbanek, "Interactive Graphics for Data Analysis: Principles and Examples", 2009)
"Be careful not to confuse clustering and stratification. Even though both of these sampling strategies involve dividing the population into subgroups, both the way in which the subgroups are sampled and the optimal strategy for creating the subgroups are different. In stratified sampling, we sample from every stratum, whereas in cluster sampling, we include only selected whole clusters in the sample. Because of this difference, to increase the chance of obtaining a sample that is representative of the population, we want to create homogeneous groups for strata and heterogeneous (reflecting the variability in the population) groups for clusters." (Roxy Peck et al, "Introduction to Statistics and Data Analysis" 4th Ed., 2012)
"Hierarchical clustering creates a hierarchy of clusters which can be represented in a treelike diagram, called a dendrogram. In the dendrogram, units in the same cluster are joined by a horizontal line, with the scale on the y-axis of the dendrogram reflecting a measure of the distances of the units within the cluster. The leaves at the bottom of the dendrogram represent the individual units; leaves are combined to form small branches, small branches are combined into larger branches, until one reaches the trunk or root of the tree that represents a single cluster containing all units. Dendrograms are quite useful as they give us a visual representation of the clusters." (Johannes Ledolter, "Data mining and business analytics with R", 2013)
"Different joining/linkage rules change how the final hierarchical clustering is presented. [...] Since the barrier for merging observations and clusters is lowest with the single linkage approach, the clustering dendrogram may contain chains of clusters as well as clusters that are spread out. The barrier to joining clusters is highest with complete linkage; however, it is possible that an observation is closer to observations in other clusters than the cluster to which it has been assigned. The average linkage approach moderates the tendencies of the single or complete linkage approaches." (Glenn J Myatt & Wayne P Johnson, "Making sense of data I: a practical guide to exploratory data analysis and data mining" 2nd. Ed., 2014)
"Determining the optimal number of clusters in a data set is a fundamental issue in partitioning clustering, such as k-means clustering, which requires the user to specify the number of clusters k to be generated. Unfortunately, there is no definitive answer to this question. The optimal number of clusters is somehow subjective and depends on the method used for measuring similarities and the parameters used for partitioning. A simple and popular solution consists of inspecting the dendrogram produced using hierarchical clustering to see if it suggests a particular number of clusters. Unfortunately, this approach is also subjective." (Alboukadel Kassambara, "Practical Guide To Cluster Analysis in R: Unsupervised Machine Learning", 2016)
"Linking is a powerful dynamic interactive graphics technique that can help us better understand high-dimensional data. This technique works in the following way: When several plots are linked, selecting an observation's point in a plot will do more than highlight the observation in the plot we are interacting with - it will also highlight points in other plots with which it is linked, giving us a more complete idea of its value across all the variables. Selecting is done interactively with a pointing device. The point selected, and corresponding points in the other linked plots, are highlighted simultaneously. Thus, we can select a cluster of points in one plot and see if it corresponds to a cluster in any other plot, enabling us to investigate the high-dimensional shape and density of the cluster of points, and permitting us to investigate the structure of the disease space."
"Hierarchical clustering is comprised of a general family of clustering algorithms that construct nested clusters by successive merging or splitting of data. The hierarchy of clusters is represented as a tree. The tree is usually called a dendrogram. The root of the dendrogram is the single cluster that contains all the samples; the leaves are the clusters containing only one sample each. This is a nice tool, since it can be straightforwardly interpreted: it 'explains' how clusters are formed and visualizes clusters at different scales. The tree that results from the technique shows the similarity between the samples. Partitioning is computed by selecting a cut on the tree at a certain level." (Laura Igual & Santi Seguí, "Introduction to Data Science: A Python Approach to Concepts, Techniques and Applications", 2017)
"Dimensionality reduction is a way of reducing a large number of different measures into a smaller set of metrics. The intent is that the reduced metrics are a simpler description of the complex space that retains most of the meaning. […] Clustering techniques are similarly useful for reducing a large number of items into a smaller set of groups. A clustering technique finds groups of items that are logically near each other and gathers them together." (Danyel Fisher & Miriah Meyer, "Making Data Visual", 2018)
"Unsupervised learning or clustering is a way of discovering hidden structures in unlabeled data. Clustering algorithms aim to discover latent patterns in unlabeled data using features to organize instances into meaningfully dissimilar groups." (Benjamin Bengfort et al, "Applied Text Analysis with Python: Enabling Language-Aware Data Products with Machine Learning", 2018)
"Important features to look for in a scatter plot are whether there is one cloud of dots or several clusters, whether there is an upward or downward slope to the cloud of dots, and whether there is any curvature to the slope." (Robert Grant, "Data Visualization: Charts, Maps and Interactive Graphics", 2019)
"[...] scatterplots had advantages over earlier graphic forms: the ability to see clusters, patterns, trends, and relations in a cloud of points. Perhaps most importantly, it allowed the addition of visual annotations (point symbols, lines, curves, enclosing contours, etc.) to make those relationships more coherent and tell more nuanced stories." (Michael Friendly & Howard Wainer, "A History of Data Visualization and Graphic Communication", 2021)
"Hierarchical clustering is summarised by a dendrogram, which sequentially shows points being joined to form a cluster, with the corresponding distances. Breaking the data into clusters is done by cutting the dendrogram at the long edges. [...] Plotting the dendrogram in the data space can help you understand how the hierarchical clustering has collected the points together into clusters. You can learn if the algorithm has been confused by nuisance patterns in the data, and how different choices of linkage method affect the result." (Dianne Cook & Ursula Laa, "Interactively Exploring High-Dimensional Data and Models in R", 2026)
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