# Detrended Cross-Correlation Analysis

A module to perform **DCCA coefficients** analysis. The coefficient `rho`

describes the **correlation strength** between two **time-series** depending on **time scales**. It lies in [-1, 1], 1 being perfect correlations, and -1 perfect anticorrelations.

The package provides also functions returning a 95% confidence interval for the null-hypothesis (= "no-correlations").

Travis |
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The implementation is based on
Zebende G, Et al. *DCCA cross-correlation coefficient differentiation: Theoretical and practical approaches* (2013), and was tested by reproducing the results of *DCCA and DMCA correlations of cryptocurrency markets* (2020) from Paulo Ferreira, et al.

## Perform a DCCA coefficients computation:

To compute DCCA coefficients, call the `rhoDCCA`

function like: `pts, rho = rhoDCCA(timeSeries1, timeSeries2)`

. It has the following parameters:

`rhoDCCA(timeSeries1, timeSeries2; box_start = 3, box_stop = div(length(series1),10), nb_pts = 30, order = 1)`

**Input arguments**:

**timeSeries1, timeSeries2**(Array{Float64,1}): Time series to analyse, need to be of the**same length**.**box_start, box_stop**(Int): Start and end point of the analysis. defaults respectively to 3 (the minimal possible time-scale) and 1/10th of the data length (passed this size the variance gets large).**nb_pts**(Int): Number of points to carry the analysis onto. mostly relevant for plotting.**order**(Int): Order of the polynomial to use for detrending. If not given, defaults to 1 (linear detrending). If`order`

is too high, overfitting can happen, impacting the results.

**Returns**:

**pts**(Array{Int,1}): List of points (time-scales) where the analysis is carried out.**rho**(Array{Float64,1}): Value of the DCCA coefficient at each points in`pts`

.

## Get the 95% confidence interval

As a rule of thumb : values of `rho`

in [-0.1,0.1] usually aren't significant.

The confidence intervals provided by this package correspond to the **null-hypothesis** i.e **no correlations**. If `rho`

gets **outside** of this interval it can be considered **significant**.

To get a fast estimation of the confidence interval, call the `empirical_CI`

function like: `pts, ci = empirical_CI(dataLength)`

.

For a more accurate estimation, you can call `bootstrap_CI`

: `pts, ci = bootstrap_CI(timeSeries1, timeSeries2; iterations = 200)`

. This operation is much more demanding and can take up to several minutes. The `iterations`

argument controls the number of repetitions for the bootstrap procedure, the higher the value, the smoother and cleaner the estimation will be, but it will also take longer.

## Example of simple analysis:

calling the DCCA function with random white noise

```
julia> x1 = rand(2000); x2 = rand(2000)
x,y = rhoDCCA(x1,x2)
pts, ci = empirical_CI(length(x1))
```

Gave the following plot :

```
a = scatter(x,y, markersize = 7, xscale = :log, title = "Example of DCCA analysis : \n Correlations between two white noise time series", label = "rho coefficients", xlabel = "window sizes", ylabel = "Correlation strengh")
plot!(a,pts,ci, color = "red", linestyle = :dot, label = "limits of null-hypothesis")
plot!(a,pts,-ci, color = "red", linestyle = :dot, label = "")
display(a)
```

## Installation:

```
julia> Using Pkg
Pkg.add("DCCA")
```

## To-do:

- implement spline detrending?