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

?

  1. Compute mean value as usual and plot it in the coordinate system.
     
  2. 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

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