## 8.1 Assignment: CRQA and Diagonal Profile

- Create two variables for CRQA analysis, or use the \(x\) and \(y\) coordinates we roecorded during the lecture:

```
y1 <- sin(1:900*2*pi/67)
y2 <- sin(.01*(1:900*2*pi/67)^2)
# Here are the circle trace data
xy <- import("https://raw.githubusercontent.com/FredHasselman/DCS/master/assignmentData/RQA_circletrace/mouse_circle_xy.csv")
y1 <- xy$x
y2 <- xy$y
```

You have just created two sine(-like) waves. We’ll examine if and how they are coupled in a shared phase space. As a first step plot them.

Find an embedding delay (using mutual information) and an embedding dimension (if you calculate an embedding dimension using package

`fractal`

for each signal seperately, as a rule of thumb use the highest embedding dimension you find in further analyses).

```
# General settings for `crqa()`
par0 <- list(rescale = 0,
normalize = 0,
mindiagline = 2,
minvertline = 2,
tw = 0,
whiteline = FALSE,
recpt = FALSE,
side = "both",
checkl = list(do = FALSE, thrshd = 3, datatype = "categorical",pad = TRUE)
)
# Settings for `optimizeParam()`
par <- list(lgM = 20, steps = seq(1, 6, 1),
radiusspan = 100, radiussample = 40,
normalize = par0$normalize,
rescale = par0$rescale,
mindiagline = par0$mindiagline, minvertline = par0$minvertline,
tw = par0$tw,
whiteline = par0$whiteline,
recpt = par0$recpt,
fnnpercent = 10, typeami = "mindip")
```

- We can now create a cross recurrence matrix. Fill in the values you decided on. You can choose a radius automatically, look in the
`crqa`

manual.

- Get the optimal parameters using a radius which will give us 2%-5% recurrent points.

Note: There is no rescaling of data, the sines were created in the same range. You can plot a matrix using

`image()`

. You could also check package`nonlinearTseries`

. If you sourced the`nlRtsa`

library, use`plotRP.crqa()`

`(ans <- optimizeParam(ts1 = y1, ts2 = y2, par = par, min.rec = 2, max.rec = 5))`

Run the CRQA and produce a plot of the recurrence matrix.

- Can you understand what is going on?

- For the simulated data: Explain the the lack of recurrent points at the beginning of the time series.
- For the circle trace: How could one see these are not determisnistic sine waves?

- Examine the synchronisation under the diagonal LOS. Look in the manual of
`crqa`

or Coco & Dale (2014).

- To get the diagonal profile from the recurrence plot, use
`spdiags()`

. - Make a plot of the diagonal profile. How far is the peak in RR removed from 0 (Line of Synchronisation)?

- Perform the same steps with a shuffled version (or use surrogate analysis!) of the data of timeseries \(y1\). You can use the embedding parameters you found earlier.

NOTE: If you generate surrogate timeseries, make sure the RR is the same for all surrogates. Try to keep the RR in the same range by using the

`min.rec`

and`max.rec`

settings of`optimizeParam`

for each surrogate.