Time series analysis - CNR...• Linear / nonlinear time series analysis • Uni- / Bivariate...

• Linear / nonlinear time series analysis

• Uni- / Bivariate (Synchronization)

• Continuous / discrete time series

• Exemplary application to medical data

- EEG and neuronal recordings

- Epilepsy (“window to the brain”)

6 lectures of 2 hours: Thu, May 12 – Tue, May 31, 2016

Thomas Kreuz (ISC-CNR)(thomas.kreuz@cnr.it; http://www.fi.isc.cnr.it/users/thomas.kreuz/)

Time series analysis

• Lecture 1: Example (Epilepsy & spike train synchrony),

Data acquisition, Dynamical systems

• Lecture 2: Linear measures, Introduction to non-linear

dynamics, Non-linear measures I

• Lecture 3: Non-linear measures II

• Lecture 4: Measures of continuous synchronization

• Lecture 5: Measures of discrete synchronization

(spike trains)

• Lecture 6: Measure comparison & Application to epileptic

seizure prediction

Schedule

(spike trains)

• Lecture 6: Measure comparison & Application to epileptic seizure prediction

Schedule

• General Introduction

• Example: Epileptic seizure prediction

• Data acquisition

• Introduction to dynamical systems

First lecture: Introduction

Non-linear model systems

Linear measures

Introduction to non-linear dynamics

Non-linear measures

- Introduction to phase space reconstruction

- Lyapunov exponent

Second lecture

Non-linear measures

- Dimension

[ Excursion: Fractals ]

- Entropies

- Relationships among non-linear measures

Third lecture

Motivation

Measures of synchronization for continuous data

• Linear measures: Cross correlation, coherence

• Mutual information

• Phase synchronization (Hilbert transform)

• Non-linear interdependences

Measure comparison on model systems

Measures of directionality

• Granger causality

• Transfer entropy

Fourth lecture

Motivation and examples

Measures of synchronization for discrete data (here: spike trains, but in principle can be any other kind of discrete data)

• Victor-Purpura distance

• Van Rossum distance

• ISI-distance

• SPIKE-distance (& Applications)

Fifth lecture

Spikes / Spike trains

Spike: Action potential (event in which the membrane potential of a neuron rapidly rises and falls.)

Spike train: Temporal sequence of spikes.

Basic assumptions:

All-or-non law: “There is no such thing as half a spike.”Either full response or no response at all(depending on whether firing threshold is crossed or not)

Spikes are stereotypical. Shape does not carry information.

Background activity carries minimal information. Only spike times matter.

Motivation: Spike train (dis)similarity

Three different scenarios:

1. Simultaneous recording of population

Neuronal correlations, pathology (e.g. epilepsy)

2. Repeated presentation of just one stimulus

Reliability

3. Repeated presentation of different stimuli

Stimulus discrimination, neural coding

• Monkey retina (functioning in vitro for ~ 15h)

• Multi-Electrode Array (MEA) recordings (512 electrodes)

• Complete populations of retinal ganglion cells (~ 100 RGCs)

1. Simultaneous recording: Example

Time [s]

One neuron, 60 repetitions: High reliability

2. Repeated stimulus presentation: Example

3. Different stimuli: Neural coding

Neural coding:Relationship between the stimulus and the individual or ensemble neuronal responses

Neural encoding: Map from stimulus to responseAim: Response prediction

Neural decoding: Map from response to stimulusAim: Stimulus reconstruction

Encoding Decoding

Stimulus

Response

Neural coding schemes I

Labelled line coding: Individual neurons code on their own.Identity of neuron that fires a spike matters.

Population coding: Joint activities of a number of neurons.Identity of the neuron is irrelevant. All that is important is that the spike is fired as part of the population response, not which neuron fired it.Advantages: Individual neurons are noisy, summed population is robust. Multi-coding possible. More spikes, thus faster.

See also: Sparseness vs. distributed representation in memory and recognition

Extreme sparseness: Grandmother cellJennifer Aniston neuron (concept cell)

Jennifer Aniston neuron

[Quian Quiroga et al. Nature (2005)]

Sensory-motor system: Cortical homunculus

[Wilder Penfield: Epilepsy and the Functional Anatomy of the Human Brain. 1954]

Primary somatosensory cortex Primary motor cortex

Neural coding schemes II

Rate coding: Most (if not all) information about the stimulusis contained in the firing rate of the neuron

Edgar Adrian 1929 (NP 1932): Firing rate of stretch receptor neurons in the muscles is related to the force applied to the muscle.

Temporal coding: Precise spike timing carries information

Many studies: Temporal resolution on millisecond time scale

No absolute time reference in the nervous systemRelative timing to stimulus onset / other spikes, but also with

respect to ongoing brain oscillation

(Special cases: Latency code, Pattern code, Coincidence code)

Measures of spike train (dis)similarity

- Victor-Purpura distance (Victor & Purpura, 1996)

- van Rossum distance (van Rossum, 2001)

- Event synchronization (Quian Quiroga et al., 2002)

- Schreiber correlation measure (Schreiber et al., 2003)

- Hunter-Milton similarity (Hunter & Milton, 2003)

- ISI-distance (ISI = Inter-spike interval) (Kreuz et al., 2007)

- SPIKE-distance (Kreuz et al., 2013)

Overview and comparison: Kreuz et al. JNeurosci Methods, 2007; JNeurophysiol (2013)

Victor-Pupura: Sequence of elementary steps

0 1 2 3 4 5 6 7 8 9

Output

Time [s]

ISI-distance: DI=0.06

0 100 200 300 400 500 600 700 800

Time [ms]

trains

Motivation: SPIKE-distance

Distance

SPIKE-

Distance

0 1 2 3 4 5 6 7 8 9 10 11

trains

Time [arbitrary unit]

P (t) t

ISI (t)

P (t) x

(1) (t)

(2) (t) t

(2) (t)

SPIKE-distance

Visualization: Dissimilarity profile

0 200 400 600 800 1000 1200

1Spike

trains

Time [ms]

0 500 1000 1500 2000 2500 3000 3500 4000

trains

Time [arbitrary units]

Causal (real-time) SPIKE-distance

Population averages

0 500 1000 1500 2000 2500 3000 3500 4000

Time [ms]

10 20 30 40

Spike trains

10 20 30 40

Spike trains

G1 G2 G3 G4

Spike trains

< S >G

2 3 1 4

Spike train groups

G1 G2 G3 G4

Spike trains

< Sr >

2 3 1 4

Spike train groups

Advantages

• Perfect time resolution, no binning, no parameter

• Not invariant to shuffling of spikes among spike trains(in contrast to peri-stimulus time histogram, PSTH)

• Time-scale independence

• Computational efficiency

• Online monitoring (Real-time SPIKE-distance)

Applications: - Epilepsy

- Brain-machine interfacing

• Application to continuous data (e.g. EEG)

• Papers and Matlab source codes:

http://www.fi.isc.cnr.it/users/thomas.kreuz/sourcecode.html

Leaders and followers

• Quantifying consistency in spatio-temporal propagation patterns

Comparison of continuous measure of synchronization

• Application to epileptic seizure prediction

• Predictive performance

• Statistical validation (Measure profile surrogates)

Today’s lecture

Examples of propagation patterns

• Avalanches• Tsunamis• Chemical waves and diffusion

processes• Epileptic seizures• Epidemic spread of diseases

Setup: Neuronal recordings, set of spike trains

Task: Sort spike trains from

Leader

Follower

(in terms of temporal sequence, not causality)

Finding propagation patterns in spike trains

Context

matters!

Coincidence detection

• Counts number of coincidences

• Maximum time lag adapted to local spike rate (scale-free):

},,,{min

tttttttt No parameter!

Quian Quiroga, Kreuz, Grassberger, PRE 2002

x = min{ti+1

x - ti

x - ti-1

Coincidence detection

x = min{ti+1

x - ti

x - ti-1

y = min{t j+1

y - t j

y, t j

y - t j-1

2t ij = min{t i

},,,{min

tttttttt

SPIKE-synchronization

0 100 200 300 400 500 600 700 800 900 1000

Spike trainsa

0 100 200 300 400 500 600 700 800 900 1000

C=0.769

Profileb

(1,2) =1 if min j (| ti

(1) - t j

(2) |) < t ij

0 else

Mean value: Fraction

of coincident spikes

Kreuz et al. 2015; Mulansky et al. 2015

+1+1 0

0 200 400 600 800 1000

Reliable

trains

0 200 400 600 800 1000

Bursts

trains

0 200 400 600 800 1000

Random

trains

0.3289

Symmetric measure of coincidence: invariant to order of spikes and spike trains

0 100 200 300 400 500 600 700 800 900 1000

Spike trainsa

0 100 200 300 400 500 600 700 800 900 1000

C=0.769

Profileb

SPIKE-order

• Asymmetric order indicator:

SPIKE-order : Is this spike leading or following?

(n,m) = Ci

(n,m) ×sign(t j

(m) - ti

(m,n) = Cj

(m,n) ×sign(ti

(n) - t j

(m)) = -Di

)( ktD

SPIKE-order profile D

by construction D = 𝐷(𝑡) = 0

Very nice for color-coding of spikes, but not meaningful as a profile.

SPIKE-order and spike train order

(n,m) = Ci

(n,m) ×sign(t j

(m) - ti

(m,n) = Cj

(m,n) ×sign(ti

(n) - t j

(m)) = -Di

)( ktD

SPIKE-order and spike train order

• Symmetric order indicator:

Spike train order : Are the two spike trains in the correct order?

(n,m) = Ci

(n,m) ×sign(t j

(m) - ti

(m,n) = Cj

(m,n) ×sign(ti

(n) - t j

(m)) = -Di

(n,m) = Ci

(n,m) ×sign(t j

(m) - ti

(m,n) = Cj

(m,n) ×sign(t j

(m) - ti

(n)) = Ei

)( ktD

)( ktE

𝑛 < 𝑚

Spike train order profile E

0+1 -1 -1+1

Multivariate profiles

SPIKE-order

Spike train order-profile

SPIKE-order profile

Synfire pattern

Synfire indicator F

𝑭 =𝑫 𝒏<𝒎

ቀ𝑵 − 𝟏)σ𝒏𝑴𝒏

𝑫 𝒏,𝒎 =

𝑫𝒊𝒏,𝒎

𝑫 𝒏<𝒎 =

𝒏<𝒎

𝑫 𝒏,𝒎

Finding the best spike train order: Maximal F

Simulated annealing

• Heuristic optimization algorithm

• Useful when the search space it too large for brute force (spike train permutations)

• Only tries small portion of possibilities: uses cost function and random variable to determine where to head next

• Search space for order of spike trains grows very fast when more spike trains are added

Number of spike trains N

Number of permutations N!

8 40320

10 3628800

𝑭 =𝑫 𝒏<𝒎

ቀ𝑵 − 𝟏)σ𝒏𝑴𝒏

Optimal order and significance

Poisson test sets

Neuronal data

• Giant depolarizing potentials

• Calcium imaging

• Rat hippocampus

Global event detection

Neuronal data

Climate science data: El Nino

• Sea Surface Temperatures (SST)• Deviations from long term mean in degrees• Setting threshold to gaussian filtered data gives spikes

Results of the method

• Synfire indicator Fu for the unsorted spike trains

• Synfire indicator Fs for the sorted spike trains

• Significance of Fs for the sorted spike trains

• Order of the sorted spike trains (leaders – followers)

Conclusions

• Leader to follower dynamics and the significance of the result

• Parameter free and time scale adaptive

• Conseptually intuitive interpretation

• Low computational cost

• Can be applied to all kinds of discrete events

Outreach

• Available online: • ISI-distance• SPIKE-distance• SPIKE-synchronization

SPIKY (Matlab GUI) and

PySpike (Python) packages

cSPIKE (Matlab cmd, MEX) coming soon

Download-page • Papers and Matlab source codes:

http://www.fi.isc.cnr.it/users/thomas.kreuz/sourcecode.html• Python source codes:

http://mariomulansky.github.io/PySpike/

Py Spike

Introduction and motivation

Comparitive investigation:

Predictive performance of measures of synchronization

Statistical validation of seizure predictions:

The method of measure profile surrogates

Summary and outlook

Predictability of epileptic seizures

- Content -

~ 1 % of world population suffers from epilepsy

~ 22 % cannot be treated sufficiently

~ 70 % can be treated with antiepileptic drugs

~ 8 % might profit from epilepsy surgery

Exact localization of seizure generating area

Delineation from functionally relevant areas

Aim: Tailored resection of epileptic focus

- Introduction: Epilepsy -

Intracranially implanted electrodes

TBARTBPR TBAL

FLRFPR

FPLFLL

EEG containing onset of a seizure (preictal and ictal)

EEG in the seizure-free period (interictal)

- Motivation I -

Open questions:

Does a preictal state exist?

Do characterizing measures allow a reliable detection of this

state?

Goals / Perspectives:

Increasing the patient‘s quality of life

Therapy on demand (Medication, Prevention)

Understanding seizure generating processes

- Motivation II -

State of the art:

Reports on the existence of a preictal state, mainly based on

univariate measures

Gradual shift towards the application of bivariate measures

Little experience with continuous multi-day recordings

No comparison of different characterizing measures

Mostly no statistical validation of results

- Motivation III -

Why bivariate measures?

Synchronization phenomena key feature for establishing the

communication between different regions of the brain

Epileptic seizure: Abnormal synchronization of neuronal ensembles

First promising results on short datasets:

“Drop of synchronization” before epileptic seizures *

* Mormann, Kreuz, Andrzejak et al., Epilepsy Research, 2003; Mormann, Andrzejak, Kreuz et al., Phys. Rev. E, 2003

I. Continuous EEG – multichannel recordings

II. Calculation of a characterizing measure

III. Investigation of suitability for prediction by means of a

seizure prediction statistics

- Sensitivity

Performance

- Specificity

IV. Estimation of statistical significance

- Procedure -

1 2 3 4 5 6 7 80

20 40 60 80 100 120 140 160

- Moving window analysis -

Window

Chan. 1

Chan. 2

1 2 3 4 5 6 7 80

20 40 60 80 100 120 140 160

Window

Chan. 1

Chan. 2

1 2 3 4 5 6 7 80

20 40 60 80 100 120 140 160

Window

Chan. 1

Chan. 2

1 2 3 4 5 6 7 80

20 40 60 80 100 120 140 160

Window

Chan. 1

Chan. 2

1 2 3 4 5

Zeit [Tage]

sensitivenot

specific specific

For this channel combination:

Reliable seperation preictal interictal impossible !

- Example: Drop of synchronization as a predictor -

Time [Days]

1 2 3 4 5

Zeit [Tage]

- Example: Drop of synchronization as a predictor -

Selection of best channel combination :

Clearly improved seperation preictal interictal

Significant ? Seizure times surrogates

Time [Days]

Summary and outlook

- Content -

- Sensitivity

Performance

- Specificity

- Procedure -

Anfälle

Zeit [Std.]

30 60 90 120 150 180

I. Database

Seizures

Time [h]

- Sensitivity

Performance

- Specificity

- Procedure -

• Cross Correlation Cmax

• Mutual Information I

• Indices of phase synchronization

based on

and using

• Nonlinear interdependencies Ss and Hs

• Event synchronization Q

Synchronization Directionality

• Nonlinear interdependencies Sa and Ha

• Delay asymmetry q

- Shannon entropy (se)

- Conditional probabilty (cp)

- Circular variance (cv)

- Hilbert phase (H)

- Wavelet phase (W)

se , , , , ,

II. Bivariate measures

- Overview -

- Cross correlation and mutual information -

Cmax I

* *1.0

Cmax I

-1 -0.5 0 0.5 1-1

- Phase synchronization -

- Nonlinear interdependencies -

No coupling:

- Nonlinear interdependencies -

Strong coupling:

Kanal 2

Kanal 1

Zeit [s]

- Event synchronization and Delay asymmetry I -

Time [s]

Chan. 1

Chan. 2

- Sensitivity

Performance

- Specificity

- Procedure -

III. Seizure prediction statistics

- Steps of analysis -

Measure profiles of all neighboring channel combinations

Statistical approach:

Comparison of preictal and interictal

amplitude distributions

Measure of discrimination: Area below the

Receiver-Operating-Characteristics (ROC) - Curve

Mormann, Kreuz, Rieke et al., Clin Neurophysiol 2005

-4 -2 0 2 40

0 0.5 10

Sensitiv

1-Spezifizität

III. Seizure prediction statistics: ROC

1 - Specificity

-4 -2 0 2 40

0 0.5 10

nsitiv

1-Spezifizität

1 - Specificity

-4 -2 0 2 40

0 0.5 10

nsitiv

1-Spezifizität

1 - Specificity

-4 -2 0 2 40

0 0.5 10

nsitiv

1-Spezifizität

1 - Specificity

-4 -2 0 2 40

0 0.5 10

nsitiv

1-Spezifizität

1 - Specificity

-4 -2 0 2 40

0 0.5 10

Sensitiv

1-Spezifizität

1 - Specificity

-4 -2 0 2 40

0 0.5 10

nsitiv

1-Spezifizität

1 - Specificity

-4 -2 0 2 40

0 0.5 10

Sensitiv

1-Spezifizität

1 - Specificity

-4 -2 0 2 40

0 0.5 10

nsitiv

1-Spezifizität

1 - Specificity

-4 -2 0 2 40

0 0.5 10

ROC-Fläche: 0.86

Sensitiv

1-Spezifizität

1 - Specificity

ROC-Area

ROC-Fläche: 0Se

nsitiv

ROC-Fläche: 1Se

nsitiv

ROC-Fläche: -1Se

nsitiv

0 0.5 10

ROC-Fläche: -0.003Se

nsitiv

t1-Spezifizität

ROC-Area

1 - SpecificityS

1 2 3 4 5

Zeit [Tage]

Time Profile : BON , TR08-TR09

0.5 0.6 0.7 0.8 0.90

InterPrä

0 0.2 0.4 0.6 0.8 10

nsitiv

1 - Spezifizität

ROC-Fläche: 0.67

III. Seizure prediction statistics: Example

1 - Specificity

ROC-Area

Time [days]

For each channel combination 2 * 4 * 2 = 16 combinations

III. Seizure prediction statistics

- Parameter of analysis -

• Smoothing of measure profiles (s = 0; 5 min)

• Length of the preictal interval (d = 5; 30; 120; 240 min)

• ROC hypothesis H

- Preictal drop (ROC-Area > 0, )

- Preictal peak (ROC-Area < 0, )

Optimization criterion for each measure: Best mean over patients

- Sensitivity

Performance

- Specificity

- Procedure -

IV. Statistical Validation

- Problem: Over-optimization -

Given performance: Significant or statistical fluctuation?

Good measure: „Correspondence“ seizure times - measure profile

To test against null hypothesis:

Correspondence has to be destroyed

I. Seizure times surrogates II. Measure profile surrogates

Randomization

of measure profiles

Randomization

of seizure times

- Seizure times surrogates -

Random permutation of the time intervals between actual

seizures: Seizure times surrogates

Calculation of the seizure prediction statistics for the original as well

as for 19 surrogate seizure times ( p=0.05)

Andrzejak, Mormann, Kreuz et al., Phys Rev E, 2003

1 2 3 4 5

TL01-TL02

TL02-TL03

TL03-TL04

TL04-TL05

TL05-TL06

TL06-TL07

TL07-TL08

TL08-TL09

TL09-TL10

TR01-TR02

TR02-TR03

TR03-TR04

TR04-TR05

TR05-TR06

TR06-TR07

TR07-TR08

TR08-TR09

TR09-TR10

Zeit [Tage]

- Results: Measure profiles of phase synchronization -

Time [days]

Discrimination of amplitude distributions Interictal Preictal

1. Global effect:

All Interictal All Preictal (1)

2. Local effect:

Interictal per channel comb Preictcal per channel comb (#comb)

Results

- Evaluation schemes -

1 2 3 4 5

TL01-TL02

TL02-TL03

TL03-TL04

TL04-TL05

TL05-TL06

TL06-TL07

TL07-TL08

TL08-TL09

TL09-TL10

TR01-TR02

TR02-TR03

TR03-TR04

TR04-TR05

TR05-TR06

TR06-TR07

TR07-TR08

TR08-TR09

TR09-TR10

Zeit [Tage]

- First evaluation scheme -

Time [days]

n.s.p = n.s. n.s. n.s. n.s. n.s. n.s. n.s. n.s. n.s. n.s. n.s. n.s. n.s.

Results: First evaluation scheme

Measures

Discrimination of amplitude distributions Interictal Preictal

1. Global effect:

All Interictal All Preictal (1)

2. Local effect:

Interictal per channel comb Preictcal per channel comb (#comb)

Results

- Evaluation schemes -

1 2 3 4 5

TL01-TL02

TL02-TL03

TL03-TL04

TL04-TL05

TL05-TL06

TL06-TL07

TL07-TL08

TL08-TL09

TL09-TL10

TR01-TR02

TR02-TR03

TR03-TR04

TR04-TR05

TR05-TR06

TR06-TR07

TR07-TR08

TR08-TR09

TR09-TR10

Zeit [Tage]

- Second evaluation scheme -

Time [days]

1 2 3 4 5

TL01-TL02

TL02-TL03

TL03-TL04

TL04-TL05

TL05-TL06

TL06-TL07

TL07-TL08

TL08-TL09

TL09-TL10

TR01-TR02

TR02-TR03

TR03-TR04

TR04-TR05

TR05-TR06

TR06-TR07

TR07-TR08

TR08-TR09

TR09-TR10

Zeit [Tage]

Time [days]

1 2 3 4 5

TL01-TL02

TL02-TL03

TL03-TL04

TL04-TL05

TL05-TL06

TL06-TL07

TL07-TL08

TL08-TL09

TL09-TL10

TR01-TR02

TR02-TR03

TR03-TR04

TR04-TR05

TR05-TR06

TR06-TR07

TR07-TR08

TR08-TR09

TR09-TR10

Zeit [Tage]

Time [days]

0.03 TL01-TL02

TL02-TL03

0.03 TL03-TL04

0.04 TL04-TL05

0.06 TL05-TL06

TL06-TL07

0.08 TL07-TL08

TL08-TL09

0.06 TL09-TL10

0.03 TR01-TR02

TR02-TR03

TR03-TR04

TR04-TR05

TR05-TR06

0.08 TR06-TR07

TR07-TR08

TR08-TR09

TR09-TR10

1 1.5 2-1

Results: Preictal and interictal distributions

.05p = .05 .05 .05 .05 .05 .05 .05 n.s. n.s. n.s. .05 n.s. .05

Results: Second evaluation scheme

Measures

- Summary I: Comparison of measures -

General tendency regarding predictive performance:

- Phase synchronization based on Hilbert Transform

- Mutual Information, cross correlation

- Nonlinear interdependencies

Measures of directionality among measures of synchronization

No global effect, but significant local effects

Summary and outlook

- Content -

* Kreuz, Andrzejak, Mormann et al., Phys. Rev. E (2004)

Mostly not sufficient data for „Out of sample“ – study (Separation in

training- and test sample)

„In sample“ – Optimization (Selection)

(Best parameter, best measure, best channel, best patient, …)

Statistical fluctuations difficult to estimate

Seizure prediction

- Problem : Statistical validation -

I. Continuous EEG multi channel recordings

II. Calculation of characterizing measures

- Procedure -

- Patient A (18 channel combinations)

- Phase synchronization and event synchronization Q

- ROC, same optimization, for every channel combination

- Method of measure profile surrogates

- Problem: Over-optimization -

Given performance: Significant or statistical fluctuation?

Good measure: „Correspondence“ seizure times - measure profile

To test against null hypothesis:

Correspondence has to be destroyed

I. Seizure times surrogates II. Measure profile surrogates

Randomization

of measure profiles

Randomization

of seizure times

1 2 3 4 50

Zeit [Tage]

Measure profile surrogates

Zeit [Tage]

Time [days]

• Formulation of constraints in cost function E

• Minimization among all permutations of the original measure profile

• Iterative scheme: Exchange of randomly chosen pairs

- Simulated Annealing I -

Schreiber, Phys. Rev. Lett., 1998

• Cooling scheme (Temp. T→0), abort at desired precision

Probability of acceptance:

Iterationsschritte

- Simulated Annealing II -

18 channel combinations

(Phase synchronization)

Iteration steps

- Simulated Annealing III -

Properties to maintain:

Recording gaps are not permuted

Ictal and postictal intervals are not permuted

Amplitude distribution Permutation

Autocorrelation Cost function

nn xxN

)]()([

OriSurr CCE

1 2 3 4-1

Zeit [Tage]

- Original autocorrelation functions (Phase sync.) -

Time [days]

1 2 3 4-1

Zeit [Tage]

- Original autocorrelation functions (Phase sync.) -

Time [days]

1 2 3 4 50

Zeit [Tage]

Time [days]

1 2 3 4 50

Zeit [Tage]

Time [days]

- Two evaluation schemes -

• Each channel combination separately

• Selection of best channel combination

1 TL01-TL02

1 TL02-TL03

1 TL03-TL04

1 TL04-TL05

1 TL05-TL06

1 TL06-TL07

1 TL07-TL08

1 TL08-TL09

1 TL09-TL10

1 TR01-TR02

1 TR02-TR03

1 TR03-TR04

1 TR04-TR05

1 TR05-TR06

1 TR06-TR07

1 TR07-TR08

1 TR08-TR09

1 TR09-TR10

Results: Phase synchronization

1 TL01-TL02

1 TL02-TL03

1 TL03-TL04

1 TL04-TL05

1 TL05-TL06

1 TL06-TL07

1 TL07-TL08

1 TL08-TL09

1 TL09-TL10

1 TR01-TR02

1 TR02-TR03

1 TR03-TR04

1 TR04-TR05

1 TR05-TR06

1 TR06-TR07

1 TR07-TR08

1 TR08-TR09

1 TR09-TR10

Results: Event synchronization

1 TL01-TL02

1 TL02-TL03

1 TL03-TL04

1 TL04-TL05

1 TL05-TL06

1 TL06-TL07

1 TL07-TL08

1 TL08-TL09

1 TL09-TL10

1 TR01-TR02

1 TR02-TR03

1 TR03-TR04

1 TR04-TR05

1 TR05-TR06

1 TR06-TR07

1 TR07-TR08

1 TR08-TR09

1 TR09-TR10

Results: Phase synchronization

1 TL01-TL02

1 TL02-TL03

1 TL03-TL04

1 TL04-TL05

1 TL05-TL06

1 TL06-TL07

1 TL07-TL08

1 TL08-TL09

1 TL09-TL10

1 TR01-TR02

1 TR02-TR03

1 TR03-TR04

1 TR04-TR05

1 TR05-TR06

1 TR06-TR07

1 TR07-TR08

1 TR08-TR09

1 TR09-TR10

Results: Event synchronization

Results

- Each channel combination separately -

Phase synchronization:

Event synchronization:

Nominal size: p = 0.05 (One-sided test with 19 surrogates)

Independent tests: q = 18 (18 channel combinations)

At least r rejections:

Significant,

Null hypothesis rejected !

0000011.0)8( rP

0015.0)5( rP

1 Phasensynchronisationa)

1 Event Synchronisationb)

Results

- ES II: Selection of best channel combination -

Event synchronization

Phase synchronization

- Two Evaluation schemes -

Null hypothesis H0 I :

Measure not suitable to find significant number of local effects

predictive of epileptic seizures.

Null hypothesis H0 II :

Measure not suitable to find maximum local effects

- Two Evaluation schemes -

Null hypothesis H0 I :

Measure not suitable to find significant number of local effects

Null hypothesis H0 II :

Measure not suitable to find maximum local effects

1 Phasensynchronisationa)

1 Event Synchronisationb)

Results

- ES II: Selection of best channel combination -

Phase synchronization

1 PhasensynchronisationR

1 Event Synchronisation

Results

- Selection of best channel combination -

Significant!

Null hypothesis H0 II rejected

Not significant!

Null hypothesis H0 II accepted

Phase synchronization| R

- Summary II: Measure profiles surrogates -

Method for statistical validation of seizure predictions

Test against null hypothesis Level of significance

Estimating the effect of „In sample“ – optimization

Phase synchronization more significant than event synchronization.

Given example:

Discrimination of pre- and interictal intervals:

Summary and outlook

- Content -

- Summary and outlook -

Retrospective investigation:

Evidence of significant changes before seizures

Measures good enough for prospective application ???

(spike trains)

• Lecture 6: Measure comparison & Application to epileptic

seizure prediction

Schedule

Time series analysis - CNR...• Linear / nonlinear time series analysis • Uni- / Bivariate...

Documents

Transcript of Time series analysis - CNR...• Linear / nonlinear time series analysis • Uni- / Bivariate...

Exploratory Causal Analysis in Bivariate Time Series Databobweigel.net/projects/images/JMMcCracken_thesis_submitteddraft.… · Exploratory Causal Analysis in Bivariate Time Series

Bivariate Regression

Bayesian VARs · The Bivariate VAR Model Suppose we have observed two time series, Y1,t and Y2,t, over time. A priori, we wish to allow for the two time series to co-move. That is,

Bivariate data.

Decision maps for bivariate time series with potential ...

Cristina Masoller - fisica.edu.uycris/teaching/mvd_masoller_bi.pdf · Nonlinear time series analysis Bivariate analysis Cristina Masoller Universitat Politecnica de Catalunya, Terrassa,

NONPARAMETRIC ESTIMATION OF BIVARIATE FAILURE TIME … · 2017. 2. 12. · 1 NONPARAMETRIC ESTIMATION OF BIVARIATE FAILURE TIME ASSOCIATIONS IN THE PRESENCE OF A COMPETING RISK B

Bivariate Data Analysis Bivariate Data analysis 4.

Scatter Diagram of Bivariate Measurement Data. Bivariate Measurement Data Example of Bivariate Measurement:

Bivariate Populations

An Analysis of a Bivariate Time Series in Which the ...brill/Papers/eeden19.pdfAn Analysis of a Bivariate Time Series in Which the Components Are Sampled at Di ... The genesis of our

Spectral Analysis of Univariate and Bivariate Time Series · Spectral Analysis of Univariate and Bivariate Time Series DonaldB.Percival SeniorMathematician ... due both to the wide

Bivariate analysis

Bivariate Data

BIVARIATE DISTRIBUTION

Poisson bivariate

Chapter 3 Bivariate Time Series Analysis with Rlegacy.fordham.edu/economics/vinod/chap3BivTimeSer.pdf · Chapter 3 Bivariate Time Series Analysis with R 3.1 Autoregressive Distributed

Kocherlakota-Bivariate Discrete Distributions (Statistics_ a Series of Textbooks and Monographs) (1992).pdf

Bivariate time-series analysis of the relationship between

A Gini Autocovariance Function for Time Series Modelingserfling/papers/Gini_autocov_fcn_final.pdf2.2 The Gini covariance For a bivariate random vector (X,Y) with joint distribution