COEFbothscale {costat} R Documentation

## Produces plots from output of findstysol that attempt to group different solutions.

### Description

Uses hierarchical clustering and multidimensional scaling to produce a plot of all the convergence stationary solutions. These plots are designed to aid the user in identifying ‘unique’ sets of stationary solutions.

### Usage

```COEFbothscale(l, plotclustonly = FALSE, StyPval=0.05, ...)
```

### Arguments

 `l` An object returned by `findstysols`, of class `csFSS`, which contains the results of an optimization to find solutions that correspond to stationary series which are the time-varying linear combination of two locally stationary time series. `plotclustonly` If `TRUE` then only produce the hierarchical clustering plot. `StyPval` The p-value by which solutions are deemed to be stationary or not for inclusion into plots. If the p-value for a particular solution is greater than `StyPval` then the solution is deemed stationary and included. `...` Additional arguments to the hierarchical clustering plot.

### Details

The function `findstysols` uses numerical optimization to try and discover time-varying linear combinations of two time series to find a combination which is stationary. Like many numerical optimizations the optimizer is supplied with starting coordinates and proceeds through an optimization routine to end coordinates which are located at the minimum (in this case). So, the user has a choice over where to start each optimization.

A priori there is no recipe for knowing where to start the optimizer, so such situations are usually handled by running the optimizer many time each time starting in a different position. The solution here is to start from a set of different randomly chosen starting points. After the optimizer is run from these different starting positions it ends up in the same number of potentially different ending positions.

However, some of the ending solutions might be identical, some might be very close, some might be reflections (e.g. the if the coefficients (a,b) result in a stationary solution then so does (-a, -b)). Morally, though, all of these cases would reference the same solution.

Hence, we require some method for identifying the set of unique solutions. We can be considerably aided in this task by multidimensional scaling (which uses inter-solution distances to produce a map of how close solution sets really are) or hierarchical clustering (which can produce a nice picture to indicate how the solutions might be related).

In other words, the solution vectors can be viewed as a multivariate data set where the cases correspond to the results of different optimization runs and the variables correspond to the coefficients of the time-varying linear combinations.

Both multidimensional scaling (`cmdscale`) and hiearchical clustering (`hclust`) are used to determine possible clusterings of solutions. Then, representative members from these clusters can be further investigated with a function such as `LCTSres`

### Value

An object of class `csFSSgr` is returned containing the following components: the results of the multidimensional scaling and hierarchical clustering are returned as list with two components `epscale` and `epclust` respectively, and the input `l` object is returned as component `x` and the `StyPval` object is returned as a component.

Guy Nason

### References

Cardinali, A. and Nason, Guy P. (2013) Costationarity of Locally Stationary Time Series Using costat. Journal of Statistical Software, 55, Issue 1.

Cardinali, A. and Nason, G.P. (2010) Costationarity of locally stationary time series. J. Time Series Econometrics, 2, Issue 2, Article 1.

`findstysols`, `LCTSres`
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