# pdfCluster: Clustering via nonparametric density estimation In pdfCluster: Cluster Analysis via Nonparametric Density Estimation

 pdfCluster R Documentation

## Clustering via nonparametric density estimation

### Description

Cluster analysis is performed by the density-based procedures described in Azzalini and Torelli (2007) and Menardi and Azzalini (2014), and summarized in Azzalini and Menardi (2014).

### Usage

```## S4 method for signature 'numeric'
pdfCluster(x, graphtype, hmult, Q = "QJ", lambda = 0.1,
grid.pairs = 10, n.grid = min(round((5 + sqrt(NROW(x))) * 4), NROW(x)), ...)

## S4 method for signature 'matrix'
pdfCluster(x, graphtype, hmult, Q = "QJ", lambda = 0.1,
grid.pairs = 10, n.grid = min(round((5 + sqrt(NROW(x))) * 4), NROW(x)), ...)

## S4 method for signature 'data.frame'
pdfCluster(x, graphtype, hmult, Q = "QJ", lambda = 0.1,
grid.pairs = 10, n.grid = min(round((5 + sqrt(NROW(x))) * 4), NROW(x)), ...)

## S4 method for signature 'pdfCluster'
pdfCluster(x, graphtype, hmult, Q, lambda = 0.1,
grid.pairs, n.grid = min(round((5 + sqrt(NROW(x@x))) * 4), NROW(x@x)), ...)
```

### Arguments

 `x` A vector, a matrix or a data frame of numeric data to be partitioned. Since density-based clustering is designed for continuous data only, if discrete data are provided, a warning message is displayed. Alternatively, `x` can be an object of `pdfCluster-class` itself, obtained when `graphtype` is set to "pairs". See Section Details below. `graphtype` Either "unidimensional", "delaunay" or "pairs", it defines the procedure used to build the graph associated with the data. If missing, a "delaunay" graph is built for data having dimension less than 7, otherwise a "pairs" graph is built. See details below. This argument has not to be set when `x` is of `pdfCluster-class`. `hmult` A shrink factor to be multiplied by the smoothing parameter `h` of function `kepdf`. If missing, it is taken to be 1 when data have dimension greater than 6, 0.75 otherwise. `Q` Optional arguments to be given when `graphtype = "delaunay"`. See `delaunayn` in package `geometry` for further details. This argument has not to be set when `graphtype = "pairs"`, when `graphtype = "unidimensional"` or when `x` is of `pdfCluster-class`. `lambda` Tolerance threshold to be used when `graphtype = "pairs"`. An edge is set between two observations if the density function, evaluated along the segment linking them, does not exhibit any valley having a measure exceeding `lambda`. Its range is [0,1) but a value larger than 0.3 is not recommended; default value is set to 0.10. This argument has not to be set when `graphtype = "delaunay"` or `graphtype = "unidimensional"`. `grid.pairs` When `graphtype = "pairs"`, this arguments defines the length of the grid of points along the segment linking each pair of observations, on which the density is evaluated. Default is 10. This argument has not to be set when `graphtype = "delaunay"`, when `graphtype = "unidimensional"` or when `x` is of `pdfCluster-class`. `n.grid` Defines the length of the grid on which evaluating the connected components of the density level sets. The default value is set to the minimum between the number of data rows n and \lfloor{(5 + √(n))4 + 0.5}\rfloor, an empirical rule of thumb which indicates that the length of the grid grows with the square root of the number of rows data.
 `...` Further arguments to be passed to `kepdf` or to `pdfClassification`.

### Details

Clusters are associated to the connected components of the level sets of the density underlying the data. Density estimation is performed by kernel methods and the connected regions are approximated by the connected components of a graph built on data. Three alternative procedures to build the graph are adopted:

Unidimensional procedure

When data are univariate an edge is set between two observations when all the data points included in the segment between the two candidate observations belong to the same level set.

Delaunay triangulation

An edge is set between two observations when they are contiguous in the Voronoi diagram; see Azzalini and Torelli (2007).

Pairs evaluation

An edge is set between two observations when the density function, evaluated along the segment joining them, does not exhibit any valley having a relative amplitude greater than a tolerance threshold 0 ≤ λ < 1. Being a tolerance threshold, sensible values of λ are, in practice, included in [0, 0.3]; see Menardi and Azzalini (2013).

As the level set varies, the number of detected components gives rise to the tree of clusters, where each leave corresponds to a mode of the density function. Observations around these modes form a number of cluster cores, while the lower density observations are allocated according to a classification procedure; see also `pdfClassification`.

### Value

An S4 object of `pdfCluster-class` with slots:

 `call ` The matched call. `x ` The matrix of data input. If a vector of data is provided as input, a one-column matrix is returned as output. `pdf ` An object of class `list` providing information about the density estimate. It includes: `kernel` character vector defining the kernel function used to estimate the density; `bwtype` character vector defining if a fixed or an adaptive kernel estimator has been used; `par` list of components defining the parameters used in density estimation; `estimate` vector of density estimates evaluated at the data points. See `kepdf` for further details. `nc ` An object of class `list` defining details about the identification of the connected regions. It includes: `nc` number of connected sets for each density level set. `p` vector of level sets, giving the proportions of data with estimated density below a threshold. `id` group label of each point at different sections of the density estimate. Negative values of `id` mean that the estimated density is below the considered threshold. `pq` for each `p` gives the corresponding quantile `q` of the values of the density. `graph` An object of class `list` defining details about the graph built to find the connected sets of high density regions. Its length depends on the value of its first element: `type` either "unidimensional", "delaunay" or "pairs", defines the procedure used to set edges among the observations. In the last case only, the list includes also the following elements: `comp.area` a list containing the vector `area` and the matrix `pairs.ord`. The element i of vector `area` is the measure of the maximum valley in the density function linking the observations having row position as given in column i of `pairs.ord`. `lambda` tolerance threshold. `cluster.cores` A vector with the same length as `NROW(x)`, defining the cluster cores membership. `NA` values correspond to low density, unlabeled data, to be classified in the second phase of the procedure by the intarnal call of `pdfClassification`. `tree ` Cluster tree with leaves corresponding to the connected components associated to different sections of the density estimate. The object is of class `dendrogram`. `noc ` Number of clusters. `stages ` List with elements corresponding to the data allocation to groups at the different stages of the classification procedure. `NA` values correspond to unlabeled data. `clusters ` Set to `NULL` if `n.stages` = 0, that is, if data belonging to the cluster cores only have been allocated. Otherwise it reports the final label groups. This component is obsolete. Use function `groups`, instead.

### Methods

`signature(x="data.frame")`

This method applies the `pdfCluster` procedure to objects of class `data.frame`.

`signature(x="matrix")`

This method applies the `pdfCluster` procedure to objects of class `matrix`.

`signature(x="numeric")`

This method applies the `pdfCluster` procedure to objects of class `numeric`.

`signature(x="pdfCluster")`

This method applies to objects of `pdfCluster-class` when the graph has been built according to the "pairs" procedure. It allows to save time and computations if the user wants to compare results of cluster analysis for different values of the `lambda` parameter. See examples below.

### Warning

It may happen that the variability of the estimated density is so high that not all jumps in the mode function can be detected by the selected grid scanning the density function. In that case, no output is produced and a message is displayed. As this may be associated to the occurrence of some spurious connected components, which appear and disappear within the range between two subsequent values of the grid, a natural solution is to increase the value of `n.grid`. Alternatively either `lambda` or `hmult` may be increased to alleviate the chance of detecting spurious connected components.

Using `graphtype= 'delaunay'` when the dimensionality d of data is greater than 6 is highly time-consuming unless the number of rows n is fairly small, since the number of operations to run the Delaunay triangulation grows exponentially with d. Use `graphtype= "pairs"`, instead, whose computational complexity grows quadratically with the number of observations.

### References

Azzalini, A., Menardi, G. (2014). Clustering via nonparametric density estimation: the R package pdfCluster. Journal of Statistical Software, 57(11), 1-26, URL http://www.jstatsoft.org/v57/i11/.

Azzalini A., Torelli N. (2007). Clustering via nonparametric density estimation. Statistics and Computing. 17, 71-80.

Menardi, G., Azzalini, A. (2014). An advancement in clustering via nonparametric density estimation. Statistics and Computing. DOI: 10.1007/s11222-013-9400-x.

`kepdf`, `pdfCluster-class`, `pdfClassification`.

### Examples

```##########
#example 1
###########
# not run here for time reasons
data(oliveoil)

#preparing data
olive1 <- 1 + oliveoil[, 3:10]
margin <- apply(data.matrix(olive1),1,sum)
olive1 <- olive1/margin
alr <- (-log( olive1[, -4]/olive1[, 4]))
#select the first 5 principal components
x <- princomp(alr, cor=TRUE)\$scores[, 1:5]

#clustering
# not run here for time reasons
#cl <- pdfCluster(x, h = h.norm(x), hmult=0.75)
#summary(cl)
#plot(cl)

#comparing groups with original macro-area membership
#groups <- groups(cl)
#table(oliveoil\$macro.area, groups)

#cluster cores
#table(groups(cl, stage = 0))

##########
#example 2
###########
# not run here for time reasons
#data(wine)
#x <-wine[ ,-1]
#gr <- wine[ ,1]

# when data are high-dimensional, an adaptive kernel estimator is preferable
# building the Delaunay graph entails a too high computational effort
# use option "pairs" to build the graph
# it is the default option for dimension >6

# cl <- pdfCluster(x, graphtype="pairs", bwtype="adaptive")
# summary(cl)
# plot(cl)

#comparison with original groups
#table(groups(cl),gr)

# a better classification is obtained with larger value of lambda
# not necessary to run the whole procedure again
# a pdfCluster method applies on pdfCluster-class objects!

#cl1 <- pdfCluster(cl, lambda=0.25)
#table(gr, groups(cl1))
```

pdfCluster documentation built on Dec. 2, 2022, 5:14 p.m.