Topology and fmri Data

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Pearl Allison
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1 Topology and fmri Data Adam Jaeger Statistical and Applied Mathematical Sciences Institute & Duke University May 5, 2016

2 fmri and Classical Methodology Most statistical analyses examine data with a variable by variable approach. Assume each variable describes an exact phenomenon which can be assessed by itself. Problematic with fmri; processingiuj can inadvertently create false structure. Brain structures are described by multiple voxels, but each voxel is not really a variable unto itself.

3 Advantages of Topology Not limited to standard metrics. Robust feature preservation. Dimensional reduction of data. Voxel alignment not critical.

4 Data Homology How do we determine the homology using data?

5 Data Homology How do we determine the homology using data? Use the filtered space X 1 X 2... X m X which we then use to generate the persistent homology H i X 1 H i X 2... H i X m. We will examine time series and spatial fields; build homologies based on proximity of voxels. Will only examine 0 dimensional homology (connected components).

6 Morse Filtration Capture trends in functions. Can filter space bottom-up or top-down. The summary (birth death plot, bar plot, persistence landscape, persistent intensity etc.) describes the length (time) a feature exists. More precisely describe number of local maximum and minimum along with persistence of a feature.

7 Example Plot on right tracks when a feature (local maximum) is born and how long feature persists.

8 Morse Filtration Algorithm 1. Sort data largest to smallest by response 2. Set number of features f = 0 3. For i = 1,..., n: Check if x i is neighbor of x 1,..., x i 1 If false then x i denotes start of feature f + 1 starting at x i Else If x i is neighbor of single feature f then x i absorbed into feature f Else x i is neighbor of multiple features then all features are combined. Oldest feature remains while all other features end at max value in feature set

9 Data are generated using the sin curve as the mean. Will fill space along y axis from top to bottom.

31 Capturing Features fmri data is not just preprocessed; sometimes it is post processed. The post processing is done to standardize the BOLD signal and force neighbor voxels to behave in a similar fashion. This imposes forced structure and we may not be aware of the post processing.

32 Time Series Since the intensity values of the bold signal have no real scale or meaning, values are usually standardized. How does standardization affect the topology?

33 Time Series; Standardize [0, 1]

34 Time Series; Standardize µ = 0 and σ = 1

35 Spatial Field Since we expect voxels next to each other to behave in a similar fashion signals are spatially smoothed. How does smoothing affect the topology?

36 Spatial Field; Smoothed using ±1

37 Spatial Field; Smoothed using ±2

38 Spatial Field; Smoothed using ±3

39 Additional Advantages of TDA Metrics more natural to the brain can be used. Voxels do not have to be structurally aligned.

40 Challenges Computationally intensive. Does not provide information on where features are. Determining probabilistic properties of topology for testing not developed.

41 Questions? Thank you.

Network Traffic Measurements and Analysis

Network Traffic Measurements and Analysis DEIB - Politecnico di Milano Fall, 2017 Introduction Often, we have only a set of features x = x 1, x 2,, x n, but no associated response y. Therefore we are not interested in prediction nor classification,