High-Dimensional Statistics with a View Toward Applications in Biology

Figure 1: (a) A density function P. (b) Density clusters corresponding to Lt={x:p(x)>t}. (c) The density tree corresponding to P is shown under the density. The leaves of the tree correspond to modes....

Figure 2: (a,b) Density trees in panels a and b are homeomorphic; there exists a bicontinuous map from one tree to the other. (c) The third tree is not homeomorphic to the other two. Thus, the first t...

Figure 3: Example of density tree using the bootstrap from Chen et al. (2016), showing (a) the data and (b) the tree. The solid lines are the pruned trees; the dashed lines are leaves and branches tha...

Figure 4: (a) A density with four modes. (b) The partition (basins of attraction) of the space induced by the modes. These are the population clusters.

Figure 5: The mean shift algorithm. The data are represented by the black dots. The two modes of the density estimate are the two blue dots. The red curves show the mean shift paths; each data point m...

Figure 6: The Swiss-roll dataset. The ambient dimension is d=3, but the data are supported on a set S of dimension r=2.

Figure 7: These data are two dimensional, but there is a set S of dimension r=1 with a high concentration of data. The red lines show an estimate of S using the methods in Section 3.3.

Figure 8: After applying Isomap to the Swiss-roll dataset with ϵ=5, we recover the underlying two-dimensional structure.

Figure 9: This is a plot of a two-dimensional density function with a clearly defined one-dimensional ridge (blue circle). The density above the ridge is in red.

Figure 10: The data for this figure are generated as Yi=Xi+ϵi, where the Xi are sampled from a circle and ϵi are bivariate Gaussian. The ridge of the kernel density estimator is found using the subsp...

Figure 11: (a) The support S of the distribution. (b–d) Union of balls approximately 60 data points drawn from a uniform on S, with ϵ=0.03, 0.10, 0.30.

Figure 12: (a) The barcode plot corresponding to the data from Figure 11. The gray lines show the birth and death of each connected component as ϵ increases. The red line shows the birth and death of ...

Figure 13: (a) This set has one connected component and two holes, and hence β0=1 and β1=2. (b) This set has two connected components and one hole, and hence β0=2 and β1=1.

Figure 14: The bottleneck distance between two persistence diagrams is computed by finding the best matching between the two diagrams. This plot shows two diagrams that have been overlaid, with the po...

Figure 15: Data sampled from a circle, with Gaussian noise added. There are also 100 outliers sampled uniformly from the square.

Figure 16: (a) The kernel density estimator. (b) The persistence diagram corresponding to the upper level sets of the estimator. The points above the pink band are significant compared to the bootstra...

Figure 17: (a) The distance to a measure (DTM). (b) The persistence diagram corresponding to the lower level sets of the DTM. The points above the pink band are significant compared with the bootstrap...

Figure 18: (a) Starting at the top of the density and moving down, each mode has a birth time B and a death time d. (b) The persistence diagram plots the points (d1, b1), …, (d4, b4). Modes with a lon...

Figure 19: (a) A density P and a level set {p>t}. (b) The smoothed density Ph and the level set {ph>t}. The latter is biased but captures the shape of the true level set. The red line is t, which defi...

Figure 20: (a) A distribution with a continuous component and a point mass at 0. (b) The smoothed density Ph. The level set Lh is biased but is estimable, and it approximates the main features of L. T...

Figure 21: A filament map from Chen et al. (2015c). The data are galaxies from the Sloan Digital Sky Survey. The blue lines are detected filaments. The red dots are clusters. Right ascension and decli...

Figure 22: A three-dimensional image. Classifying such images requires features that are invariant to small deformations of the image. Topological data analysis can potentially provide such features.

Figure 23: An example of a histology image from http://medicalpicturesinfo.com/histology/.

Figure 24: A maltose binding protein. The image of the protein is from http://lilith.nec.aps.anl.gov/Structures/Publications.htm.

Figure 1: Spaghetti plot for sparsely recorded CD4 count data (the number of CD4 T lymphocytes in a sample of blood) for 25 subjects, each shown in a different color.

Figure 2: Pooled CD4 count data (the number of CD4 T lymphocytes in a sample of blood) and estimated mean function (black line) for 369 subjects.

Figure 3: Raw covariances (dots) and fitted smooth covariance surface, obtained by omitting the data on the diagonal, where the diagonal forms a ridge owing to the measurement errors in the data. The ...

Figure 4: First four eigenfunctions for the CD4 count data (the number of CD4 T lymphocytes in a sample of blood).

Figure 5: Observations superimposed on the estimated mean functions of daily vehicle speed recorded by a dual-loop vehicle detector station for (a) holidays and (b) nonholidays. The different colors i...

Figure 6: Histograms of cluster labels grouped by (a) day of the week and (b) holiday or nonholiday, respectively. Red shows nonholidays, and blue shows holidays. Of the 76 days, 20 belong to Cluster ...

Figure 7: (a) Median housing price index trajectories (red) for 20 US metropolitan regions, 2000–2009; values in the year 2000 are standardized at 100. The blue curve is the standard cross-sectional m...

Figure 1: Imago Mundi Babylonian map, which is the oldest known world map (sixth century BCE). The representation of the world is relatively abstract. The world is shown as a disc, with Babylon repres...

Figure 2: The Hertzsprung-Russell diagram (devised circa 1910) allowed astrophysicists to make the connection between a star's temperature, visual color, and size, enabling a better understanding of t...

Figure 3: Hurricane forecast map issued on October 9, 2018, shortly before Hurricane Michael made landfall in the Florida panhandle. The predicted hurricane path, wind field, size, and coastal weather...

Figure 4: A static reproduction of a birthday chart, which shows the average number of births on a yearly basis in the United States, by month and day of the month. The interactive version, available ...

Figure 5: From data to graphic: a schematic representation of the steps required to create a graphic from a data set, which can be used to specify the variable mapping, data transformations, coordinat...

Figure 6: Three different plots of the Swiss bank note data (Flury & Riedwyl 1988), created using the grammar of graphics as implemented in the ggplot2 code. The data consist of measurements of the di...

Figure 7: Callaghan (1984) showed that irrelevant variation increases response times, indicating that hue and brightness are integrated and preattentively perceived. The images shown here are examples...

Figure 8: Sample plots from Shah & Carpenter (1995), showing data where one independent variable (pupil size) has no effect on the y variable (depression) for one value of the independent variable (th...

Figure 9: A framed spine plot similar to those shown in the Statistical Atlas (Walker 1874), recreated using modern methods. The chart shows the proportion of church accommodations in Oregon, with the...

Figure 10: The original data (panel a) are embedded into a set of 19 generated under the null hypothesis (panel b). A visual hypothesis test is conducted by asking multiple users to select one (or mor...

Figure 11: Two different arrangements of the same data to illustrate how the proximity of elements makes a question easier to answer. Both plots show TB notification counts by demographics in Australi...

Figure 12: Four examples of pairs of time series displayed as line plots (top) and scatter plots (bottom). The gestalt principle of common fate may dominate the perception of correlation in the overla...

Figure 13: Examples of visual comparisons ranked by difficulty. (a) Two sets of points are plotted on a common scale; it is easy to compare the relative trajectories of the two lines. (b) This plot sh...

Figure 14: Examples of diverging color schemes that are perceptually suboptimal (no transition through a neutral color), good, and color-blind–accessible. The purple–orange gradient through white is d...

Figure 15: The CIE 1931 color space chromaticity diagram. The CIE 1931 color spaces defined the color space based on physiologically perceived colors in human color vision. The outer boundary of the c...

Figure 1: February 2013 Bank of England forecast of inflation in the United Kingdom as a percentage increase in the consumer price index (Bank of England 2013, with permission). The shaded bands in th...

Figure 2: Two-hour-ahead regime-switching space-time forecasts of hourly average wind speed at the Stateline wind energy center for the 7-day period beginning July 5, 2003. The predictive mean is show...

Figure 3: (a) Probability integral transform (PIT) histogram, (b) sample autocorrelation function (ACF) for the PIT values, and (c) calibration curve for exceedance of 10 m/s with bootstrap confidence...

Figure 4: Twenty-four-hour-ahead temperature forecasts valid April 4, 2011, at 2:00 am local time over Germany for four randomly selected members of the 50-member European Center for Medium-Range Weat...

Figure 5: Twenty-four-hour-ahead nonhomogeneous regression (NR) postprocessed predictive distributions for temperature valid (a) April 4, 2011 and (b) April 1–14, 2011 at 2:00 am local time in Frankfu...

Figure 6: Twenty-four-hour-ahead ensemble forecasts valid April 4, 2011, at 2:00 am local time in Frankfurt, Germany, for temperature, pressure, and six-hour accumulated precipitation. (a,b) Subpanels...