## The Mode Tree: A Tool for Visualization of Nonparametric Density Features (1993)

Venue: | Journal of Computational and Graphical Statistics |

Citations: | 35 - 4 self |

### BibTeX

@ARTICLE{Minnotte93themode,

author = {Michael C. Minnotte and David W. Scott},

title = {The Mode Tree: A Tool for Visualization of Nonparametric Density Features},

journal = {Journal of Computational and Graphical Statistics},

year = {1993},

volume = {2},

pages = {51--68}

}

### Years of Citing Articles

### OpenURL

### Abstract

Recognition and extraction of features in a nonparametric density estimate is highly dependent on correct calibration. The data-driven choice of bandwidth h in kernel density estimation is a difficult one, compounded by the fact that the globally optimal h is not generally optimal for all values of x. In recognition of this fact, a new type of graphical tool, the mode tree, is proposed. The basic mode tree plot relates the locations of modes in density estimates with the bandwidths of those estimates. Additional information can be included on the plot indicating such factors as the size of modes, how modes split, and the locations of antimodes and bumps. The use of a mode tree in adaptive multimodality investigations is proposed, and an example is given to show the value in using a Normal kernel, as opposed to the biweight or other kernels, in such investigations. Examples of such investigations are provided for Ahrens' chondrite data and van Winkle's Hidalgo stamp data. Finally, the b...

### Citations

741 |
On estimation of a probability density function and mode
- Parzen
- 1962
(Show Context)
Citation Context ...e. Under such circumstances, a nonparametric density estimation technique can be highly valuable. Kernel density estimation is a popular example of nonparametric density estimation (Rosenblatt, 1956; =-=Parzen, 1962-=-). Given a sample fX 1 ; : : : ; X n g of size n, the kernel density estimate at x is computed as f h (x) = 1 nh n X i=1 K ` x \Gamma X i h ' = 1 n n X i=1 K h (x \Gamma X i ); (1) where K h (t) = K(t... |

573 |
Multivariate Density Estimation: Theory, Practice, and Visualization
- Scott
- 1992
(Show Context)
Citation Context ..., a criterion which is only loosely related to bumps and modes.] Even assuming the best global choice for h, the fact remains that no single value of h will perform well for all points x (Terrell and =-=Scott, 1992-=-), as we will demonstrate in Section 4. Jones (1990) and Terrell and Scott (1992) have investigated the theoretical and practical advantages of an adaptive kernel estimate introduced by Breiman, et al... |

290 |
Remarks on some nonparametric estimates of a density function
- Rosenblatt
- 1956
(Show Context)
Citation Context ... be too restrictive. Under such circumstances, a nonparametric density estimation technique can be highly valuable. Kernel density estimation is a popular example of nonparametric density estimation (=-=Rosenblatt, 1956-=-; Parzen, 1962). Given a sample fX 1 ; : : : ; X n g of size n, the kernel density estimate at x is computed as f h (x) = 1 nh n X i=1 K ` x \Gamma X i h ' = 1 n n X i=1 K h (x \Gamma X i ); (1) where... |

82 | LISP-STAT: An Object-Oriented Environment for Statistical Computing and Dynamic Graphics - Tierney - 1990 |

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10 |
A Test of Mode Existence with Applications to Multimodality
- Minnotte
- 1992
(Show Context)
Citation Context ...ed shaded areas. This value is representative of the "size" of the mode, and in fact can be used as a statistic to test the reality of the mode in question when the bandwidth is chosen appro=-=priately (Minnotte, 1992-=-). M j is the minimal L 1 distance from the density to the set of continuous functions without a local maximum between the observed antimodes in the density function. M j is also the single-mode equiv... |

8 | Density estimation and bump hunting by the penalized maximum likelihood method exempli¯ed by scattering and meteorite data (with discussion - Good, Gaskins - 1980 |

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3 |
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- Ahrens
- 1965
(Show Context)
Citation Context ...rties of the modes of f h (\Delta) have been studied by Parzen (1962) and Eddy (1980). In Figure 1, a kernel density estimate and its component kernels are displayed for the chondrite meteorite data (=-=Ahrens, 1965-=-). These data, the percentages of silica in 22 chondrite meteors, were first discussed in the bump-hunting context by Good and Gaskins (1980). The vertical lines below the x-axis represent the values ... |

2 | Plasma Lipids as Collateral Risk Factors - Scott, Gotto, et al. - 1978 |

1 | On Silverman's Test for the Number of Modes in a Univariate Density Function", unpublished B.A. honors thesis - Matthews - 1983 |

1 |
Variable Kernel Density Estimation," The Annals of Statistics
- Terrell, Scott
- 1992
(Show Context)
Citation Context ...ng L 2 error, a criterion which is only loosely related to bumps and modes.] Even assuming the best global choice for h, the fact remains that no single value of h will perform well for all points x (=-=Terrell and Scott, 1992-=-), as we will demonstrate in Section 4. Jones (1990) and Terrell and Scott (1992) have investigated the theoretical and practical advantages of an adaptive kernel estimate introduced by Breiman, et al... |

1 | Transformations in Density Estimation, " (with discussion - Wand, Marron, et al. - 1991 |