Visualization of uncertainty_without_a_mean
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The document discusses methods for visualizing uncertainty in categorical data, emphasizing that traditional mean and standard deviation measures are not applicable. It introduces entropy as a measure of uncertainty and explains its limitations, as it only captures randomness without indicating actual values. The proposed visualization technique colors voxels based on their maximum probability and entropy values, blending color with white to represent varying degrees of uncertainty.
![Entropy as a Measure of Uncertainty
• Let 𝑋 be a random variable with probability density 𝑝 on discrete sample
space 𝑝: {x1, …, xn} R+
• Mean: 𝐸 𝑋 = 𝑥𝑖 𝑝(𝑥𝑖)𝑛
𝑖=1
• Variance: Var 𝑋 = 𝐸 𝑋 − 𝑥𝑖
2𝑛
𝑖=1
• Entropy: 𝐻 𝑋 = − 𝑝 𝑥𝑖 log(𝑝 𝑥𝑖 )𝑛
𝑖=1
• Discrete Distributions’ Entropy Range : [0, - log(1/n)]
44/16/2014](https://image.slidesharecdn.com/visualizationofuncertaintywithoutamean-140418210926-phpapp02/85/Visualization-of-uncertainty_without_a_mean-4-320.jpg)
![Information Theory Perspective
• Entropy can be thought of as the minimal expected number of binary
questions (question with yes/no answer) we can ask to determine the value of
an observation of X. #Q
• H[X] <= E[#Q] < H[X] + 1 (in log2)
• p1 = {1, 0, 0, 0} H[X] = 0.00
p2 = {0.85, 0.15, 0, 0} H[X] = 0.61
p3 = {0.85, 0.05, 0.05, 0.05} H[X] = 0.85
p4 = {0.25, 0.25, 0.25, 0.25} H[X] = 2.00
74/16/2014](https://image.slidesharecdn.com/visualizationofuncertaintywithoutamean-140418210926-phpapp02/85/Visualization-of-uncertainty_without_a_mean-7-320.jpg)
![Visualization
• Find the maximal probability tissue type and take its color tag.
– arg maxx = X(i,j)p(i,j)(x)
• Blend this color through white on the basis of that locations entropy value.
– H[X(i,j)]
• Maximal entropy voxel will appear white and minimal entropy voxel will
have the characteristic color tag.
84/16/2014](https://image.slidesharecdn.com/visualizationofuncertaintywithoutamean-140418210926-phpapp02/85/Visualization-of-uncertainty_without_a_mean-8-320.jpg)
Visualization of uncertainty_without_a_mean
- 1. Visualization of Uncertainty without a Mean Authors: Kristin Potter Samuel Gerber Erik W. Anderson Presented by: Subhashis Hazarika, Ohio State University
- 2. Motivation • Mean and Standard Deviation are the most common ways of quantifying and visualizing uncertainty when we have a probability distribution function for the dataset. • For Categorical data which is inherently discrete, we cannot define mean at a voxel by using data-points from the different categories. 24/16/2014
- 4. Entropy as a Measure of Uncertainty • Let 𝑋 be a random variable with probability density 𝑝 on discrete sample space 𝑝: {x1, …, xn} R+ • Mean: 𝐸 𝑋 = 𝑥𝑖 𝑝(𝑥𝑖)𝑛 𝑖=1 • Variance: Var 𝑋 = 𝐸 𝑋 − 𝑥𝑖 2𝑛 𝑖=1 • Entropy: 𝐻 𝑋 = − 𝑝 𝑥𝑖 log(𝑝 𝑥𝑖 )𝑛 𝑖=1 • Discrete Distributions’ Entropy Range : [0, - log(1/n)] 44/16/2014
- 6. Limitations • Information about the value of the random variable is lost. • Provides only a measure of randomness. • Whereas mean and variance indicates a range of most likely values. • Only the probability density influences the entropy not the actual value of the random variables 64/16/2014
- 7. Information Theory Perspective • Entropy can be thought of as the minimal expected number of binary questions (question with yes/no answer) we can ask to determine the value of an observation of X. #Q • H[X] <= E[#Q] < H[X] + 1 (in log2) • p1 = {1, 0, 0, 0} H[X] = 0.00 p2 = {0.85, 0.15, 0, 0} H[X] = 0.61 p3 = {0.85, 0.05, 0.05, 0.05} H[X] = 0.85 p4 = {0.25, 0.25, 0.25, 0.25} H[X] = 2.00 74/16/2014
- 8. Visualization • Find the maximal probability tissue type and take its color tag. – arg maxx = X(i,j)p(i,j)(x) • Blend this color through white on the basis of that locations entropy value. – H[X(i,j)] • Maximal entropy voxel will appear white and minimal entropy voxel will have the characteristic color tag. 84/16/2014
- 10. In 3D Spatial Domain 104/16/2014