Day 3- RQA, CRQA and Entropy

Question 1

Recurrence Rate

Answer 1

% of the plot occupied by points

Question 2

% Determinism

Answer 2

% of points that fall on a diagonal line structure

Question 3

Average line length

Answer 3

Average length of time the system repeats

Question 4

Max line length

Answer 4

Longest diagonal line; strength of attractor

Question 5

Entropy (Shannon)

Answer 5

The probability that the length of a line will be repeated

Question 6

To create plot in RQA (3 steps)

Answer 6

1. Embedding dimension (FNN)
2. Time delay (AMI)
3. Radius (define a space to determine what points are recurrent

Question 7

Unweighted RQA plot

Answer 7

Binary matrix that indicates that two points are recurrent, once below a threshold

Question 8

Weighted RQA plot

Answer 8

Based on the distances between the points in phase space

Question 9

Cross RQA (CRQA)

Answer 9

Coupling of two different TS in one time scale

Question 10

Neighbors of that region

Answer 10

When some occupied region of phase space is revisited within the same “radius” of that space

Question 11

Rescaling (What do you rescale to? 2 Answers)

Answer 11

Rescaling both TS to mean or max distance separating points in the reconstructed phase space

Question 12

Radius

Answer 12

Define a space to determine what points are recurrent

Question 13

% Recurrence

Select radius between what criteria?

Answer 13

How often do two systems occupy the same space? Select radius between ~2-5%

Question 14

CRQA Plot characteristics

Answer 14

1. Rescaling
2. Radius
3. % recurrence
4. % determinism
5. Max line length

Question 15

Entropy (2 answers)

Answer 15

1. Loss of information in a TS or signal

2. Amount of uncertainty

Question 16

Put in order from most ordered to least ordered (low to high entropy):

1. Gas
2. Solid
3. Liquid

Answer 16

1. Solid (least entropy)
2. Liquid (middle entropy)
3. Gas (most entropy)

Question 17

Low probability = how much information?

Answer 17

Low probability -> Surprise -> More information

Question 18

High probability = how much information?

Answer 18

High probability -> No Surprise -> No new information

Question 19

The amount of info =

Answer 19

Some function of probability

Question 20

Shannon’s entropy equation

Answer 20

H = - ( n SUM i = 1 ((Pi)log2(Pi)))

Question 21

Max H =

Answer 21

log P

Question 22

0 < H < log2P

How does H affect outcome? More or less certain?

Answer 22

Outcome is more uncertain if H is larger

Question 23

What are the single scale input parameters (3)?

Answer 23

N = data length
m = pattern length (length of vector)
r = tolerance (criteria of similarity, filter)

Question 24

General procedures of single scale entropy (7 Steps, 1-3 repeat to 4-6; DCC).

Answer 24

1. Divide data up into vectors length m
2. Count like matches
3. Calculate conditional probability
4. Divide data up into vectors length m+1
5. Count your like matches
6. Calculate conditional probability
7. Entropy is a ratio of conditional probabilities

Question 25

Parameters: r

Answer 25

Allowable difference when determining similarity between elements in vector (tolerance)

Question 26

Too small r

Answer 26

Similar data points are not seen as similar

Question 27

Too large r

Answer 27

Details of the system dynamics are lost

Question 28

Selection of r: Experimental error is known

Answer 28

Set r 3 times larger than noise amplitude

Question 29

Selection of r: Experimental error is unknown

Answer 29

0.1~0.25 times the SD (common in literature at 0.20 SD)

Question 30

Definition of complexity (DIS)

Answer 30

Very many definitions: Probability, predictability, regularity?

1. Originates from deterministic origin ( 1 to 1)
2. Infinitely entangled
3. Structural richness

Question 31

Permutation entropy

Answer 31

Temporal order of data within vectors.

Question 32

What is the only parameter needed for permutation entropy?

Answer 32

m = pattern length

Question 33

Steps to permutation entropy

Answer 33

1. Data
2. Divide into length m vectors
3. Convert vectors to ordinal data
4. Count like patterns

Question 34

Steps to symbolic entropy

Answer 34

1. Convert time series to binary symbol on threshold value
2. Form words from symbols (typically L = 3)
3. Convert binary to decimal
4. Calculate Shannon entropy from word symbol series
5. Normalize result by dividing max entropy for the possible number of words

Question 35

How to select threshold?

Answer 35

Mean: Half of symbols zeroes and half ones.