IUP Seminar on
Nonlinear Statistics
Pernille Hansen
Medical Images
Alzheimer's disease (AD) Shrinkage and deformation of Corpus Callosum (CC)
Exercise: Discuss how the hypothesis can be tested from a data set of MRI's from healthy and non-healthy brains.
Hypothesis: (AD) can be detected from shape of (CC).
Mean value
Data set:
- Mean value:
\overline{x} = \frac{1}{N} (x_1+x_2+...+x_N) = \frac{1}{N} \sum_{i=1}^{N} x_i
x_1,x_2,....,x_N\in \mathbb{R}^n
\begin{pmatrix}2 \\ 2 \end{pmatrix}, \begin{pmatrix}2 \\ 3 \end{pmatrix}, \begin{pmatrix}-1 \\ 1 \end{pmatrix}
\overline{x} = \frac{1}{3} \begin{pmatrix} 2+2+(-1) \\ 2+3+1 \end{pmatrix} = \frac{1}{3} \begin{pmatrix} 3 \\ 6 \end{pmatrix} = \begin{pmatrix} 1 \\ 2 \end{pmatrix}
Example:
Choosing a location for a Tony's Pizza!
\\x_1 = \begin{pmatrix} 0.34 \\ 0.93 \end{pmatrix}\\ x_2 = \begin{pmatrix}-0.64 \\ 0.77 \end{pmatrix} \\ x_3 = \begin{pmatrix}-0.71 \\ -0.71 \end{pmatrix}
Data set:
x_1
x_2
x_3
\overline{x}= \begin{pmatrix} -0.33 \\ 0.33 \end{pmatrix}
Mean value:
Exercise: Cooperatively come up with two properties a mean value should have and based on these, suggest a new mean value for the data set.
Mean value from distance!
Idea: Mean value should:
- be within the space
- minimize distance to the data
x_1
x_2
x_3
Def: The mean value of the points is the point that minimizes
x_1,x_2,....,x_N\in X
\frac{1}{N} \sum_{i=1}^N dist(y,x_i)^2
Hypothesis: (AD) can be detected from shape of (CC).
Mean value for healthy and sick can be computed and compared!
Exercise: Come up with an example of data, that could be considered 'nonlinear'
Thank you for your attention!
ILO's
- Discuss how the Alzheimer hypothesis can be tested from a data set of MRI's from healthy and non-healthy brains.
- Cooperatively come up with two properties a mean value should have and based on these, suggest a new mean value for the data set.
- Be able to come up with an example of data, that could be considered 'nonlinear'
Mean value on circle?
Data set:
Mean value
\overline{x} = \frac{1}{N} (x_1+x_2+...+x_N) = \frac{1}{N} \sum_{i=1}^{N} x_i
x_1,x_2,....,x_N\in \quad \quad \enspace \subset \mathbb{R}^2
\begin{pmatrix} 0.34 \\ 0.93 \end{pmatrix}, \begin{pmatrix}-0.64 \\ 0.77 \end{pmatrix}, \begin{pmatrix}-0.71 \\ -0.71 \end{pmatrix}
Exercise
?
-
Compute mean value as usual and plot it in the coordinate system.
- Discuss whether the mean value is sensible as a mean value for data on the circle.
Data set:
Mean value on circle?
Limitations of
- Not an additive space
- Scaling is not allowed
Problems?
- Mean point is not located on the circle
Exercise continued :
3. Cooperatively come up with two properties a mean value should have.
4. Based on these, suggest a new point point for the data set.
(NOT A LINEAR SPACE)
Mean value from distance
Idea: Mean value should be the point that minimizes the distance to
x_1,x_2,....,x_N\in X
\overline{x} = \arg\min_{y\in X} \frac{1}{N} \sum_{i=1}^N dist(y,x_i)^2
Distance of two points :
x,y\in
dist(x,y) = \cos^{-1}(x\cdot y)
\overline{x} = \arg\min_{y\in \enspace} \frac{1}{N} \sum_{i=1}^N \cos^{-1}(y\cdot x_i)
Data set:
Mean value:
x_1,x_2,....,x_N\in \quad \quad \subset \mathbb{R}^2
Mean value on circle
Exercise revisited
\begin{pmatrix} 0.34 \\ 0.93 \end{pmatrix}, \begin{pmatrix}-0.64 \\ 0.77 \end{pmatrix}, \begin{pmatrix}-0.71 \\ -0.71 \end{pmatrix}
Data set:
Using only words, write down which steps are needed to find the distance mean value of a data set of points on a sphere.
Last exercise
IUP seminar talk
By pernilleehh
