Package 'TSclust' reference manual

Title:	Time Series Clustering Utilities
Description:	A set of measures of dissimilarity between time series to perform time series clustering. Metrics based on raw data, on generating models and on the forecast behavior are implemented. Some additional utilities related to time series clustering are also provided, such as clustering algorithms and cluster evaluation metrics.
Authors:	Pablo Montero Manso [cre], Jose Vilar Fernandez [aut]
Maintainer:	Pablo Montero Manso <[email protected]>
License:	GPL-2
Version:	1.3.1
Built:	2024-10-27 04:04:35 UTC
Source:	https://github.com/pmontman/tsclust

Clustering Evaluation Index Based on Known Ground Truth

Description

Computes the similarity between the true cluster solution and the one obtained with a method under evaluation.

Usage

cluster.evaluation(G, S)
cluster.evaluation(G, S)

Arguments

`G`	Integer vector with the labels of the true cluster solution. Each element of the vector specifies the cluster 'id' that the element belongs to.
`S`	Integer vector with the labels of the cluster solution to be evaluated. Each element of the vector specifies the cluster 'id' that the element belongs to.

Details

The measure of clustering evaluation is defined as

$Sim(G,C) = 1/k \sum_{i=1}^k \max_{1\leq j\leq k} Sim(G_i,C_j),$

where

$Sim(G_i, C_j) = \frac{ 2 | G_i \cap C_j|}{ |G_i| + |C_j|}$

with |.| denoting the cardinality of the elements in the set. This measure has been used for comparing different clusterings, e.g. in Kalpakis et al. (2001) and Pértega and Vilar (2010).

Value

The computed index.

Note

This index is not simmetric.

Author(s)

Pablo Montero Manso, José Antonio Vilar.

References

Larsen, B. and Aone, C. (1999) Fast and effective text mining using linear-time document clustering. Proc. KDD' 99.16–22.

Kalpakis, K., Gada D. and Puttagunta, V. (2001) Distance measures for effective clustering of arima time-series. Proceedings 2001 IEEE International Conference on Data Mining, 273–280.

Pértega S. and Vilar, J.A (2010) Comparing several parametric and nonparametric approaches to time series clustering: A simulation study. J. Classification, 27(3), 333-362.

Montero, P and Vilar, J.A. (2014) TSclust: An R Package for Time Series Clustering. Journal of Statistical Software, 62(1), 1-43. http://www.jstatsoft.org/v62/i01/.

Examples


 #create a true cluster 
 #(first 4 elements belong to cluster '1', next 4 to cluster '2' and the last 4 to cluster '3'.
 true_cluster <- c(1, 1, 1, 1, 2, 2, 2, 2, 3, 3, 3, 3)
 #the cluster to be tested
 new_cluster <- c( 2, 1, 2, 3, 3, 2, 2, 1, 3, 3, 3, 3)
 
 #get the index
 cluster.evaluation(true_cluster, new_cluster)
 
 #it can be seen that the index is not simmetric
 cluster.evaluation(new_cluster, true_cluster)
#create a true cluster 
 #(first 4 elements belong to cluster '1', next 4 to cluster '2' and the last 4 to cluster '3'.
 true_cluster <- c(1, 1, 1, 1, 2, 2, 2, 2, 3, 3, 3, 3)
 #the cluster to be tested
 new_cluster <- c( 2, 1, 2, 3, 3, 2, 2, 1, 3, 3, 3, 3)
 
 #get the index
 cluster.evaluation(true_cluster, new_cluster)
 
 #it can be seen that the index is not simmetric
 cluster.evaluation(new_cluster, true_cluster)

TSclust Dissimilarity Computation

Description

Computes the dissimilarity matrix of the given numeric matrix, list, data.frame or mts object using the selected TSclust dissimilarity method.

Usage

diss(SERIES, METHOD, ...)
diss(SERIES, METHOD, ...)

Arguments

`SERIES`	Numeric matrix, `list`, `data.frame` or `mts` object. Numeric matrices are interpreted row-wise (one series per row) meanwhile `data.frame` and `mts` objects are interpredted column-wise.
`METHOD`	the dissimilarity measure to be used. This must be one of "ACF", "AR.LPC.CEPS", "AR.MAH", "AR.PIC", "CDM", "CID", "COR", "CORT", "DTWARP", "DWT", "EUCL", "FRECHET", INT.PER", "NCD", "PACF", "PDC", PER", "PRED", "MINDIST.SAX", "SPEC.LLR", "SPEC.GLK" or "SPEC.ISD". Any unambiguous substring can be given. See details for individual usage.
`...`	Additional arguments for the selected method.

Details

SERIES argument can be a numeric matrix, with one row per series, a list object with one numeric vector per element, a data.frame or a mts object. Some methods can have additional arguments. See the individual help page for each dissimilarity method, detailed below. Methods that have arguments that require one value per time series in series must provide so using a vector, a matrix (in the case of a multivalued argument) or a list when appropiate. In the case of a matrix, the values are conveyed row-wise. See the AR.LPC.CEPS example below.

"ACF" Autocorrelation-based method. See diss.ACF.
"AR.LPC.CEPS" Linear Predictive Coding ARIMA method. This method has two value-per-series arguments, the ARIMA order, and the seasonality.See diss.AR.LPC.CEPS.
"AR.MAH" Model-based ARMA method. See diss.AR.MAH.
"AR.PIC" Model-based ARMA method. This method has a value-per-series argument, the ARIMA order. See diss.AR.PIC.
"CDM" Compression-based dissimilarity method. See diss.CDM.
"CID" Complexity-Invariant distance. See diss.CID.
"COR" Correlation-based method. See diss.COR.
"CORT" Temporal Correlation and Raw values method. See diss.CORT.
"DTWARP" Dynamic Time Warping method. See diss.DTWARP.
"DWT" Discrete wavelet transform method. See diss.DWT.
"EUCL" Euclidean distance. See diss.EUCL. For many more convetional distances, see link[stats]{dist}, though you may need to transpose the dataset.
"FRECHET" Frechet distance. See diss.FRECHET.
"INT.PER" Integrate Periodogram-based method. See diss.INT.PER.
"NCD" Normalized Compression Distance. See diss.NCD.
"PACF" Partial Autocorrelation-based method. See diss.PACF.
"PDC" Permutation distribution divergence. Uses the pdc package. See pdcDist for additional arguments and details. Note that series given by numeric matrices are interpreted row-wise and not column-wise, opposite as in pdcDist.
"PER" Periodogram-based method. See diss.PER.
"PRED" Prediction Density-based method. This method has two value-per-series agument, the logarithm and difference transform. See diss.PRED.
"MINDIST.SAX" Distance that lower bounds the Euclidean, based on the Symbolic Aggregate approXimation measure. See diss.MINDIST.SAX.
"SPEC.LLR" Spectral Density by Local-Linear Estimation method. See diss.SPEC.LLR.
"SPEC.GLK" Log-Spectra Generalized Likelihood Ratio test method. See diss.SPEC.GLK.
"SPEC.ISD" Intregated Squared Differences between Log-Spectras method. See diss.SPEC.ISD.

Value

dist

A dist object with the pairwise dissimilarities between series.

Some methods produce additional output, see their respective documentation pages for more information.

Author(s)

Pablo Montero Manso, José Antonio Vilar.

References

Montero, P and Vilar, J.A. (2014) TSclust: An R Package for Time Series Clustering. Journal of Statistical Software, 62(1), 1-43. http://www.jstatsoft.org/v62/i01/.

Examples

data(electricity)
diss(electricity, METHOD="INT.PER", normalize=FALSE)

## Example of multivalued, one per series argument
## The AR.LPC.CEPS dissimilarity allows the specification of the ARIMA model for each series
## Create three sample time series and a mts object
x <- arima.sim(model=list(ar=c(0.4,-0.1)), n =100, n.start=100)
y <- arima.sim(model=list(ar=c(0.9)), n =100, n.start=100)
z <- arima.sim(model=list(ar=c(0.5, 0.2)), n =100, n.start=100)
seriests <- rbind(x,y,z)

## If we want to provide the ARIMA order for each series
## and use it with AR.LPC.CEPS, we create a matrix with the row-wise orders
orderx <- c(2,0,0) 
ordery <- c(1,0,0)
orderz <- c(2,0,0)
orders = rbind(orderx, ordery, orderz)

diss( seriests, METHOD="AR.LPC.CEPS", k=30, order= orders )

##other examples
diss( seriests, METHOD="MINDIST.SAX", w=10, alpha=4 )
diss( seriests, METHOD="PDC" )
data(electricity)
diss(electricity, METHOD="INT.PER", normalize=FALSE)

## Example of multivalued, one per series argument
## The AR.LPC.CEPS dissimilarity allows the specification of the ARIMA model for each series
## Create three sample time series and a mts object
x <- arima.sim(model=list(ar=c(0.4,-0.1)), n =100, n.start=100)
y <- arima.sim(model=list(ar=c(0.9)), n =100, n.start=100)
z <- arima.sim(model=list(ar=c(0.5, 0.2)), n =100, n.start=100)
seriests <- rbind(x,y,z)

## If we want to provide the ARIMA order for each series
## and use it with AR.LPC.CEPS, we create a matrix with the row-wise orders
orderx <- c(2,0,0) 
ordery <- c(1,0,0)
orderz <- c(2,0,0)
orders = rbind(orderx, ordery, orderz)

diss( seriests, METHOD="AR.LPC.CEPS", k=30, order= orders )

##other examples
diss( seriests, METHOD="MINDIST.SAX", w=10, alpha=4 )
diss( seriests, METHOD="PDC" )

Autocorrelation-based Dissimilarity

Description

Computes the dissimilarity between two time series as the distance between their estimated simple (ACF) or partial (PACF) autocorrelation coefficients.

Usage

diss.ACF(x, y, p = NULL, omega=NULL, lag.max=50)
diss.PACF(x, y, p = NULL, omega=NULL, lag.max=50)
diss.ACF(x, y, p = NULL, omega=NULL, lag.max=50)
diss.PACF(x, y, p = NULL, omega=NULL, lag.max=50)

Arguments

`x`	Numeric vector containing the first of the two time series.
`y`	Numeric vector containing the second of the two time series.
`p`	If not NULL, sets the weight for the geometric decaying of the autocorrelation coefficients. Ranging from `0` to `1`.
`lag.max`	Maximum number of simple or partial autocorrelation coefficients to be considered.
`omega`	If not NULL, completely specifies the weighting matrix for the autocorrelation coefficients. `p` is ignored if `omega` is used.

Details

Performs the weighted Euclidean distance between the simple autocorrelation ( dist.ACF) or partial autocorrelation ( dist.PACF ) coefficients. If neither p nor omega are specified, uniform weighting is used. If p is specified, geometric wights decaying with the lag in the form $p(1-p)^i$ are applied. If omega ( $\Omega$ ) is specified,

$d(x,y) = {\{ ( \hat{\rho}_{x} - \hat{\rho}_{y} )^t \bm{\Omega} (\hat{\rho}_{x} - \hat{\rho}_{y} ) \}}^\frac{1}{2}$

with $\hat{\rho}_{x}$ and $\hat{\rho}_{y}$ the respective (partial) autocorrelation coefficient vectors.

Value

The computed distance.

Author(s)

Pablo Montero Manso, José Antonio Vilar.

References

Galeano, P. and Peña, D. (2000). Multivariate analysis in vector time series. Resenhas, 4 (4), 383–403.

Montero, P and Vilar, J.A. (2014) TSclust: An R Package for Time Series Clustering. Journal of Statistical Software, 62(1), 1-43. http://www.jstatsoft.org/v62/i01/.

Examples

## Create three sample time series
x <- cumsum(rnorm(100))
y <- cumsum(rnorm(100))
z <- sin(seq(0, pi, length.out=100))
## Compute the distance and check for coherent results
diss.PACF(x, y)
diss.ACF(x, z)
diss.PACF(y, z)
#create a dist object for its use with clustering functions like pam or hclust
diss( rbind(x,y,z), "ACF", p=0.05)

## Create three sample time series
x <- cumsum(rnorm(100))
y <- cumsum(rnorm(100))
z <- sin(seq(0, pi, length.out=100))
## Compute the distance and check for coherent results
diss.PACF(x, y)
diss.ACF(x, z)
diss.PACF(y, z)
#create a dist object for its use with clustering functions like pam or hclust
diss( rbind(x,y,z), "ACF", p=0.05)

Dissimilarity Based on LPC Cepstral Coefficients

Description

Computes the dissimilarity between two time series in terms of their Linear Predicitive Coding (LPC) ARIMA processes.

Usage

diss.AR.LPC.CEPS(x, y, k = 50, order.x=NULL, order.y=NULL,
 seasonal.x=list(order=c(0, 0, 0), period=NA),
 seasonal.y=list(order=c(0, 0, 0), period=NA),
 permissive=TRUE)

diss.AR.LPC.CEPS(x, y, k = 50, order.x=NULL, order.y=NULL,
 seasonal.x=list(order=c(0, 0, 0), period=NA),
 seasonal.y=list(order=c(0, 0, 0), period=NA),
 permissive=TRUE)

Arguments

`x`	Numeric vector containing the first of the two time series.
`y`	Numeric vector containing the second of the two time series.
`k`	Number of cepstral coefficients to be considered.
`order.x`	Numeric matrix. Specifies the ARIMA models to be fitted for the series x. When using `diss` wrapper, use `order` argument instead. See details.
`order.y`	Numeric matrix. Specifies the ARIMA ARIMA models to be fitted for the series y. When using `diss` wrapper, use `order` argument instead. See details.
`seasonal.x`	A list of `arima` seasonal elements for series x. When using `diss` wrapper, use `seasonal` argument instead. See details.
`seasonal.y`	A list of `arima` seasonal elements for series x. When using `diss` wrapper, use `seasonal` argument instead. See details.
`permissive`	Specifies whether to force an AR order of 1 if no order is found. Ignored if neither order.x or order.y are NULL

Details

If order.x or order.y are NULL, their respective series will be fitted automatically using a AR model. order.x and order.y contain the three components of the ARIMA model: the AR order, the degree of differencing and the MA order, specified as in the function arima.

seasonal.x and seasonal.y are lists with two components: 'order' and 'period'. See seasonal parameter of arima, except that specification using a numeric vector of length 3 is not allowed.

If using diss function with "AR.LPC.CEPS" method, the argument order must be used instead of order.x and order.y. order is a matrix with one row per series, specified as in arima. If order is NULL, automatic fitting imposing a AR model is performed. The argument seasonal is used instead of seasonal.x and seasonal.y. seasonal is a list of elements, one per series in the same order that the series are input. Each element of seasonal must have the same format as the one in arima.

Value

The computed distance.

Author(s)

Pablo Montero Manso, José Antonio Vilar.

References

Kalpakis, K., Gada D. and Puttagunta, V. (2001) Distance measures for effective clustering of arima time-series. Proceedings 2001 IEEE International Conference on Data Mining, 273–280.

Montero, P and Vilar, J.A. (2014) TSclust: An R Package for Time Series Clustering. Journal of Statistical Software, 62(1), 1-43. http://www.jstatsoft.org/v62/i01/.

Examples

## Create three sample time series
x <- arima.sim(model=list(ar=c(0.4,-0.1)), n =100, n.start=100)
y <- arima.sim(model=list(ar=c(0.9)), n =100, n.start=100)
z <- arima.sim(model=list(ar=c(0.5, 0.2)), n =100, n.start=100)
## Compute the distance and check for coherent results
diss.AR.LPC.CEPS(x, y, 25) #impose an AR automatically selected for both series
#impose an ARIMA(2,0,0) for series x and an AR automatically selected for z
diss.AR.LPC.CEPS(x, z, 25, order.x = c(2,0,0), order.y = NULL ) 
diss.AR.LPC.CEPS(y, z, 25)
#create a dist object for its use with clustering functions like pam or hclust

diss( rbind(x,y,z), METHOD="AR.LPC.CEPS", k=20, order=rbind(c(2,0,0), c(1,0,0), c(2,0,0)),
 seasonal=list( list(order=c(1,0,0), period=1), list(order=c(2,0,0), period=3),
  list(order=c(1,0,0), period=1)) )

## Create three sample time series
x <- arima.sim(model=list(ar=c(0.4,-0.1)), n =100, n.start=100)
y <- arima.sim(model=list(ar=c(0.9)), n =100, n.start=100)
z <- arima.sim(model=list(ar=c(0.5, 0.2)), n =100, n.start=100)
## Compute the distance and check for coherent results
diss.AR.LPC.CEPS(x, y, 25) #impose an AR automatically selected for both series
#impose an ARIMA(2,0,0) for series x and an AR automatically selected for z
diss.AR.LPC.CEPS(x, z, 25, order.x = c(2,0,0), order.y = NULL ) 
diss.AR.LPC.CEPS(y, z, 25)
#create a dist object for its use with clustering functions like pam or hclust

diss( rbind(x,y,z), METHOD="AR.LPC.CEPS", k=20, order=rbind(c(2,0,0), c(1,0,0), c(2,0,0)),
 seasonal=list( list(order=c(1,0,0), period=1), list(order=c(2,0,0), period=3),
  list(order=c(1,0,0), period=1)) )

Model-based Dissimilarity Proposed by Maharaj (1996, 2000)

Description

Computes the dissimilarity between two time series by testing whether both series are or not generated by the same ARMA model.

Usage

diss.AR.MAH(x, y, dependence=FALSE, permissive=TRUE)
diss.AR.MAH(x, y, dependence=FALSE, permissive=TRUE)

Arguments

`x`	Numeric vector containing the first of the two time series.
`y`	Numeric vector containing the second of the two time series.
`dependence`	Boolean for considering dependence between observations of the series at the same point in time.
`permissive`	Boolean for continuing with the method even if no valid order is selected by AIC.

Details

Assuming that the time series x and y belong to the class of invertible and stationary ARMA processes, this dissimilarity measure is based on checking the equality of their underlying ARMA models by following the testing procedures proposed by Maharaj (1996,2000). The ARMA structures are approximated by truncated AR( $\infty$ ) models with a common order $k = \max{(k_x, k_y)}$ , where $k_x$ and $k_y$ are determined by the AIC criterion. The AR coefficients are automatically fitted. The dissimilarity can be evaluated by using the value of the test statistic or alternatively the associated p-value. If dependence is FALSE, the dissimilarity measure is constructed by following the procedure introduced by Maharaj (1996), which is designed to compare independent time series. Otherwise, a more general testing procedure is used (Maharaj, 2000), which assumes that both models are correlated at the same time points but uncorrelated across observations (Maharaj, 2000). When permissive argument is TRUE, if the automatic fitting of the AR order fails, the method shows a warning and then forces an AR of order 1. If permissive is FALSE the method produces an error if no AR order is found by AIC.

Value

`statistic`	The statistic of the homogeneity test.
`p_value`	The p-value of the homogeneity test.

Author(s)

Pablo Montero Manso, José Antonio Vilar.

References

Maharaj, E.A. (1996) A significance test for classifying ARMA models. J. Statist. Comput. Simulation, 54(4), 305–331.

Maharaj E.A. (2000) Clusters of time series. J. Classification, 17(2), 297–314.

Montero, P and Vilar, J.A. (2014) TSclust: An R Package for Time Series Clustering. Journal of Statistical Software, 62(1), 1-43. http://www.jstatsoft.org/v62/i01/.

Examples

## Create three sample time series
x <- arima.sim(model=list(ar=c(0.4,-0.1)), n =100, n.start=100)
y <- arima.sim(model=list(ar=c(0.9)), n =100, n.start=100)
z <- arima.sim(model=list(ar=c(0.5, 0.2)), n =100, n.start=100)
## Compute the distance and check for coherent results
diss.AR.MAH(x, y)
diss.AR.MAH(x, z)
diss.AR.MAH(y, z)

#create a dist object for its use with clustering functions like pam or hclust
diss( rbind(x,y,z), "AR.MAH")$statistic

## Create three sample time series
x <- arima.sim(model=list(ar=c(0.4,-0.1)), n =100, n.start=100)
y <- arima.sim(model=list(ar=c(0.9)), n =100, n.start=100)
z <- arima.sim(model=list(ar=c(0.5, 0.2)), n =100, n.start=100)
## Compute the distance and check for coherent results
diss.AR.MAH(x, y)
diss.AR.MAH(x, z)
diss.AR.MAH(y, z)

#create a dist object for its use with clustering functions like pam or hclust
diss( rbind(x,y,z), "AR.MAH")$statistic

Model-based Dissimilarity Measure Proposed by Piccolo (1990)

Description

Computes the distance between two time series as the Euclidean distance between the truncated AR operators approximating their ARMA structures.

Usage

diss.AR.PIC(x, y, order.x=NULL, order.y=NULL, permissive=TRUE)

diss.AR.PIC(x, y, order.x=NULL, order.y=NULL, permissive=TRUE)

Arguments

`x`	Numeric vector containing the first of the two time series.
`y`	Numeric vector containing the second of the two time series.
`order.x`	Specifies the ARIMA model to be fitted for the series x. When using `diss` wrapper, use `order` argument instead. See details.
`order.y`	Specifies the ARIMA model to be fitted for the series y. When using `diss` wrapper, use `order` argument instead. See details.
`permissive`	Specifies whether to force an AR order of 1 if no order is found. Ignored if neither order.x or order.y are NULL

Details

If order.x or order.y are NULL, their respective series will be fitted automatically using a AR model. If permissive is TRUE and no AR order is found automatically, an AR order of 1 will be imposed, if this case fails, then no order can be found and the function produces an error. order.x and order.y contain the three components of the ARIMA model: the AR order, the degree of differencing and the MA order, specified as in the function arima.

If using diss function with "AR.PIC" method, the argument order must be used instead of order.x and order.y. orders is a matrix with one row per ARIMA, specified as in arima. If order is NULL, automatic fitting imposing a AR model is performed.

Value

The computed distance.

Author(s)

Pablo Montero Manso, José Antonio Vilar.

References

Piccolo, D. (1990) A distance measure for classifying arima models. J. Time Series Anal., 11(2), 153–164.

Montero, P and Vilar, J.A. (2014) TSclust: An R Package for Time Series Clustering. Journal of Statistical Software, 62(1), 1-43. http://www.jstatsoft.org/v62/i01/.

Examples

## Create three sample time series
x <- arima.sim(model=list(ar=c(0.4,-0.1)), n =100, n.start=100)
y <- arima.sim(model=list(ar=c(0.9)), n =100, n.start=100)
z <- arima.sim(model=list(ar=c(0.5, 0.2)), n =100, n.start=100)
## Compute the distance and check for coherent results
#ARIMA(2,0,0) for x and ARIMA(1,0,0) for y
diss.AR.PIC( x, y, order.x = c(2,0,0), order.y = c(1,0,0) )
diss.AR.PIC( x, z, order.x = c(2,0,0), order.y = c(2,0,0) )
# AR for y (automatically selected) and ARIMA(2,0,0) for z
diss.AR.PIC( y, z, order.x=NULL, order.y=c(2,0,0) ) 
#create a dist object for its use with clustering functions like pam or hclust
diss( rbind(x,y,z), METHOD="AR.PIC", order=rbind(c(2,0,0), c(1,0,0), c(2,0,0)) )

## Create three sample time series
x <- arima.sim(model=list(ar=c(0.4,-0.1)), n =100, n.start=100)
y <- arima.sim(model=list(ar=c(0.9)), n =100, n.start=100)
z <- arima.sim(model=list(ar=c(0.5, 0.2)), n =100, n.start=100)
## Compute the distance and check for coherent results
#ARIMA(2,0,0) for x and ARIMA(1,0,0) for y
diss.AR.PIC( x, y, order.x = c(2,0,0), order.y = c(1,0,0) )
diss.AR.PIC( x, z, order.x = c(2,0,0), order.y = c(2,0,0) )
# AR for y (automatically selected) and ARIMA(2,0,0) for z
diss.AR.PIC( y, z, order.x=NULL, order.y=c(2,0,0) ) 
#create a dist object for its use with clustering functions like pam or hclust
diss( rbind(x,y,z), METHOD="AR.PIC", order=rbind(c(2,0,0), c(1,0,0), c(2,0,0)) )

Compression-based Dissimilarity measure

Description

Computes the dissimilarity based on the sizes of the compressed time series.

Usage

diss.CDM(x, y, type = "min")

diss.CDM(x, y, type = "min")

Arguments

`x`	Numeric vector containing the first of the two time series.
`y`	Numeric vector containing the second of the two time series.
`type`	Character string, the type of compression. May be abbreviated to a single letter, defaults to the first of the alternatives.

Details

The compression based dissimilarity is calculated:

$d(x,y) = C(xy) / ( C(x) + C(y) )$

where $C(x)$ , $C(y)$ are the sizes in bytes of the compressed series $x$ and $y$ . $C(xy)$ is the size in bytes of the series $x$ and $y$ concatenated. The algorithm used for compressing the series is chosen with type. type can be "gzip", "bzip2" or "xz", see memCompress. "min" selects the best separately for x, y and the concatenation. Since the compression methods are character-based, a symbolic representation can be used, see details for an example using SAX as the symbolic representation. The series are transformed to a text representation prior to compression using as.character, so small numeric differences may produce significantly different text representations. While this dissimilarity is asymptotically symmetric, for short series the differences between diss.CDM(x,y) and diss.CDM(y,x) may be noticeable.

Value

The computed dissimilarity.

Author(s)

Pablo Montero Manso, José Antonio Vilar.

References

Keogh, E., Lonardi, S., & Ratanamahatana, C. A. (2004). Towards parameter-free data mining. Proceedings of the tenth ACM SIGKDD international conference on Knowledge discovery and data mining (pp. 206-215).

Montero, P and Vilar, J.A. (2014) TSclust: An R Package for Time Series Clustering. Journal of Statistical Software, 62(1), 1-43. http://www.jstatsoft.org/v62/i01/.

Examples

n = 50
x <- rnorm(n)  #generate sample series, white noise and a wiener process
y <- cumsum(rnorm(n))

diss.CDM(x, y)

z <- rnorm(n)
w <- cumsum(rnorm(n))
series = rbind(x, y, z, w)
diss(series, "CDM", type="bzip2")

################################################################
#####symbolic representation prior to compression, using SAX####
####simpler symbolization, such as round() could also be used###
################################################################
#normalization function, required for SAX
z.normalize = function(x) {
    (x - mean(x)) / sd(x)
}

sx <- convert.to.SAX.symbol( z.normalize(x), alpha=4 )
sy <- convert.to.SAX.symbol( z.normalize(y), alpha=4 )
sz <- convert.to.SAX.symbol( z.normalize(z), alpha=4 )
sw <- convert.to.SAX.symbol( z.normalize(w), alpha=4 )

diss(rbind(sx, sy, sz, sw), "CDM", type="bzip2")
n = 50
x <- rnorm(n)  #generate sample series, white noise and a wiener process
y <- cumsum(rnorm(n))

diss.CDM(x, y)

z <- rnorm(n)
w <- cumsum(rnorm(n))
series = rbind(x, y, z, w)
diss(series, "CDM", type="bzip2")

################################################################
#####symbolic representation prior to compression, using SAX####
####simpler symbolization, such as round() could also be used###
################################################################
#normalization function, required for SAX
z.normalize = function(x) {
    (x - mean(x)) / sd(x)
}

sx <- convert.to.SAX.symbol( z.normalize(x), alpha=4 )
sy <- convert.to.SAX.symbol( z.normalize(y), alpha=4 )
sz <- convert.to.SAX.symbol( z.normalize(z), alpha=4 )
sw <- convert.to.SAX.symbol( z.normalize(w), alpha=4 )

diss(rbind(sx, sy, sz, sw), "CDM", type="bzip2")

Complexity-Invariant Distance Measure For Time Series

Description

Computes the distance based on the Euclidean distance corrected by the complexity estimation of the series.

Usage

diss.CID(x, y)
diss.CID(x, y)

Arguments

`x`	Numeric vector containing the first of the two time series.
`y`	Numeric vector containing the second of the two time series.

Details

This distance is defined

$CID(x,y) = ED(x,y) \times CF(x,y)$

where $CF(x,y)$ is a complexity correction factor defined as:

$max(CE(x), CE(y)) / min(CE(x), CE(y))$

and $CE(x)$ is a compexity estimate of a time series $x$ . diss.CID therefore increases the distance between series with different complexities. If the series have the same complexity estimate, the distance defenerates Euclidean distance. The complexity is defined in diss.CID as:

$CE(x) = \sqrt{ \sum_{t=1} (x_{t+1} - x_t)^2 }$

Value

The computed dissimilarity.

Author(s)

Pablo Montero Manso, José Antonio Vilar.

References

Batista, G. E., Wang, X., & Keogh, E. J. (2011). A Complexity-Invariant Distance Measure for Time Series. In SDM (Vol. 31, p. 32).

Montero, P and Vilar, J.A. (2014) TSclust: An R Package for Time Series Clustering. Journal of Statistical Software, 62(1), 1-43. http://www.jstatsoft.org/v62/i01/.

Examples

n = 100
x <- rnorm(n)  #generate sample series, white noise and a wiener process
y <- cumsum(rnorm(n))

diss.CID(x, y)

z <- rnorm(n)
w <- cumsum(rnorm(n))
series = rbind(x, y, z, w)
diss(series, "CID")


n = 100
x <- rnorm(n)  #generate sample series, white noise and a wiener process
y <- cumsum(rnorm(n))

diss.CID(x, y)

z <- rnorm(n)
w <- cumsum(rnorm(n))
series = rbind(x, y, z, w)
diss(series, "CID")

Correlation-based Dissimilarity

Description

Computes dissimilarities based on the estimated Pearson's correlation of two given time series.

Usage

diss.COR(x, y, beta = NULL)
diss.COR(x, y, beta = NULL)

Arguments

`x`	Numeric vector containing the first of the two time series.
`y`	Numeric vector containing the second of the two time series.
`beta`	If not NULL, specifies the regulation of the convergence in the second method.

Details

Two different measures of dissimilarity between two time series based on the estimated Pearson's correlation can be computed. If beta is not specified, the value $d_1 = \sqrt{ 2 ( 1 - \rho) }$ is computed, where $(\rho)$ denotes the Pearson's correlation between series x and y. If beta is specified, the function $d_2 = \sqrt{ (\frac{ 1 - \rho}{ 1 + \rho})^\beta }$ is used, where $\beta$ is beta .

Value

The computed distance.

Author(s)

Pablo Montero Manso, José Antonio Vilar.

References

Golay, X., Kollias, S., Stoll, G., Meier, D., Valavanis, A., and Boesiger, P. (2005) A new correlation-based fuzzy logic clustering algorithm for FMRI. Magnetic Resonance in Medicine, 40.2, 249–260.

Montero, P and Vilar, J.A. (2014) TSclust: An R Package for Time Series Clustering. Journal of Statistical Software, 62(1), 1-43. http://www.jstatsoft.org/v62/i01/.

Examples

## Create three sample time series
x <- cumsum(rnorm(100))
y <- cumsum(rnorm(100))
z <- sin(seq(0, pi, length.out=100))
## Compute the distance and check for coherent results
diss.COR(x, y)
diss.COR(x, z)
#create a dist object for its use with clustering functions like pam or hclust

diss( rbind(x,y,z), "COR")

## Create three sample time series
x <- cumsum(rnorm(100))
y <- cumsum(rnorm(100))
z <- sin(seq(0, pi, length.out=100))
## Compute the distance and check for coherent results
diss.COR(x, y)
diss.COR(x, z)
#create a dist object for its use with clustering functions like pam or hclust

diss( rbind(x,y,z), "COR")

Dissimilarity Index Combining Temporal Correlation and Raw Values Behaviors

Description

Computes an adaptive dissimilarity index between two time series that covers both dissimilarity on raw values and dissimilarity on temporal correlation behaviors.

Usage

diss.CORT(x, y, k = 2, deltamethod="Euclid")
diss.CORT(x, y, k = 2, deltamethod="Euclid")

Arguments

`x`	Numeric vector containing the first of the two time series.
`y`	Numeric vector containing the second of the two time series.
`k`	Parameter controlling the weight of the dissimilarity between dynamic behaviors (See Details).
`deltamethod`	Defines the method for the raw data discrepancy. Either `"Euclid"`, `"Frechet"` or `"DTW"`.

Details

The dissimilarity between time series x and y is given by:

$d(x,y) = \Phi[CORT(x,y)] \delta(x,y)$

where:

CORT(x,y) measures the proximity between the dynamic behaviors of x and y by means of the first order temporal correlation coefficient defined by:

$CORT(x,y) = \frac{ \sum_{t=1} (x_{t+1} - x_t) ( y_{t+1} - y_t) }{ \sqrt{ \sum_{t=1} (x_{t+1} - x_t)^2} \sqrt{ \sum_{t=1} (y_{t+1} - y_t)^2 } }$

$\Phi[u]$ is an adaptive tuning function taking the form:

$\frac{2}{1+e^{ku}}$

with $k \geq 0$ so that both $\Phi$ and k modulate the weight that CORT(x,y) has on d(x,y).

$\delta(x,y)$ denotes a dissimilarity measure between the raw values of series x and y, such as the Euclidean distance, the Frechet distance or the Dynamic Time Warping distance. Note that $d(x,y) = \delta(x,y)$ if k=0.

More details of the procedure can be seen in Chouakria-Douzal and Nagabhushan (2007).

deltamethod ( $\delta$ ) can be either Euclidean (deltamethod = "Euclid"), Frechet ( deltamethod = "Frechet") or Dynamic Time Warping (deltamethod ="DTW") distances. When calling from dis.CORT, DTW uses Manhattan as local distance.

Value

The computed distance.

Author(s)

Pablo Montero Manso, José Antonio Vilar.

References

Chouakria-Douzal, A. and Nagabhushan P. N. (2007) Adaptive dissimilarity index for measuring time series proximity. Adv. Data Anal. Classif., 1(1), 5–21.

Montero, P and Vilar, J.A. (2014) TSclust: An R Package for Time Series Clustering. Journal of Statistical Software, 62(1), 1-43. http://www.jstatsoft.org/v62/i01/.

Examples

## Create three sample time series
x <- cumsum(rnorm(100))
y <- cumsum(rnorm(100))
z <- sin(seq(0, pi, length.out=100))
## Compute the distance and check for coherent results
diss.CORT(x, y, 2)
diss.CORT(x, z, 2)
diss.CORT(y, z, 2)
#create a dist object for its use with clustering functions like pam or hclust

diss( rbind(x,y,z), "CORT", k=3, deltamethod="DTW")

## Create three sample time series
x <- cumsum(rnorm(100))
y <- cumsum(rnorm(100))
z <- sin(seq(0, pi, length.out=100))
## Compute the distance and check for coherent results
diss.CORT(x, y, 2)
diss.CORT(x, z, 2)
diss.CORT(y, z, 2)
#create a dist object for its use with clustering functions like pam or hclust

diss( rbind(x,y,z), "CORT", k=3, deltamethod="DTW")

Dynamic Time Warping Distance

Description

Computes Dynamic Time Warping distance between time series.

Usage

diss.DTWARP(x, y, ...)
diss.DTWARP(x, y, ...)

Arguments

`x`	Numeric vector containing the first of the two time series.
`y`	Numeric vector containing the second of the two time series.
`...`	Additional parameters for the function. See `link[dtw]{dtw}`.

Details

This is basically a wrapper for dtw of the pdc package, intended for an easier discovery of the functionalities used in TSclust.

Value

The computed distance.

Examples

## Create three sample time series
x <- cumsum(rnorm(100))
y <- cumsum(rnorm(100))
diss.DTWARP(x, y)

## Create three sample time series
x <- cumsum(rnorm(100))
y <- cumsum(rnorm(100))
diss.DTWARP(x, y)

Dissimilarity for Time Series Based on Wavelet Feature Extraction

Description

Performs an unsupervised feature extration using orthogonal wavelets on the series and returns the Euclidean distance between the wavelet approximations in an appropriate scale.

Usage

diss.DWT(series)
diss.DWT(series)

Arguments

series

Numeric matrix with row order time series

Details

This method differs from other dissimilarities in that pairwise dissimilaries depend on the whole dataset that is given to diss.DWT, hence, there is no pairwise version of the function defined, only accepts whole datasets. The set of original series is replaced by their wavelet approximation coefficients in an appropriate scale, and the dissimilarity between two series is computed as the Euclidean distance between these coefficients. The appropriate scale is automatically determined by using an algorithm addressed to obtain an efficient reduction of the dimensionality but preserving as much information from the original data as possible. The algorithm is introduced by Zhang, Ho, Zhang, and Lin (2006).

Value

Returns an object of type dist with the pairwise distances.

Author(s)

Pablo Montero Manso, José Antonio Vilar.

References

Zhang, H., Ho, T. B., Zhang, Y., and Lin, M. (2006) Unsupervised feature extraction for time series clustering using orthogonal wavelet transform. INFORMATICA-LJUBLJANA-, 30(3), 305.

Montero, P and Vilar, J.A. (2014) TSclust: An R Package for Time Series Clustering. Journal of Statistical Software, 62(1), 1-43. http://www.jstatsoft.org/v62/i01/.

Examples

## Create three sample time series
x <- cumsum(rnorm(100))
y <- cumsum(rnorm(100))
z <- sin(seq(0, pi, length.out=100))

#compute the distance
diss.DWT(rbind(x, y, z))
## Create three sample time series
x <- cumsum(rnorm(100))
y <- cumsum(rnorm(100))
z <- sin(seq(0, pi, length.out=100))

#compute the distance
diss.DWT(rbind(x, y, z))

Euclidean Distance

Description

Computes Euclidean distance between time series.

Usage

diss.EUCL(x, y)
diss.EUCL(x, y)

Arguments

`x`	Numeric vector containing the first of the two time series.
`y`	Numeric vector containing the second of the two time series.

Value

The computed distance.

Examples

## Create three sample time series
x <- cumsum(rnorm(100))
y <- cumsum(rnorm(100))
diss.EUCL(x, y)
## Create three sample time series
x <- cumsum(rnorm(100))
y <- cumsum(rnorm(100))
diss.EUCL(x, y)

Frechet Distance

Description

Computes the Frechet distance between time series.

Usage

diss.FRECHET(x, y, ...)
diss.FRECHET(x, y, ...)

Arguments

`x`	Numeric vector containing the first of the two time series.
`y`	Numeric vector containing the second of the two time series.
`...`	Additional parameters for the function. See `link[longitudinalData]{distFrechet}`.

Details

This is basically a wrapper for distFrechet of the longitudinalData package, intended for an easier discovery of the functionalities used in TSclust.

Value

The computed distance.

Examples

## Create three sample time series
x <- cumsum(rnorm(100))
y <- cumsum(rnorm(100))
diss.FRECHET(x, y)

## Create three sample time series
x <- cumsum(rnorm(100))
y <- cumsum(rnorm(100))
diss.FRECHET(x, y)

Integrated Periodogram Based Dissimilarity

Description

Computes the dissimilarity between two time series in terms of the distance between their integrated periodograms.

Usage

diss.INT.PER(x, y, normalize=TRUE)
diss.INT.PER(x, y, normalize=TRUE)

Arguments

`x`	Numeric vector containing the first of the two time series.
`y`	Numeric vector containing the second of the two time series.
`normalize`	If `TRUE` the normalized version is computed.

Details

The distance is computed as:

$d(x,y) = \int_{-\pi}^{\pi} | F_x(\lambda) - F_y(\lambda) | \, d\lambda,$

where $F_x(\lambda_j) = C_x^{-1} \sum_{i=1}^{j} I_x(\lambda_i)$ and $F_y(\lambda_j) = C_y^{-1} \sum_{i=1}^{j} I_y(\lambda_i)$ , with $C_x = \sum_i I_x(\lambda_i)$ and $C_y = \sum_i I_y(\lambda_i)$ in the normalized version. $C_x = 1$ and $C_y = 1$ in the non-normalized version. $I_x(\lambda_k)$ and $I_y(\lambda_k)$ denote the periodograms of x and y, respectively.

Value

The computed distance.

Author(s)

Pablo Montero Manso, José Antonio Vilar.

References

Casado de Lucas, D. (2010) Classification techniques for time series and functional data.

Montero, P and Vilar, J.A. (2014) TSclust: An R Package for Time Series Clustering. Journal of Statistical Software, 62(1), 1-43. http://www.jstatsoft.org/v62/i01/.

Examples

## Create three sample time series
x <- cumsum(rnorm(100))
y <- cumsum(rnorm(100))
z <- sin(seq(0, pi, length.out=100))
## Compute the distance and check for coherent results
diss.INT.PER(x, y, normalize=TRUE)
diss.INT.PER(x, y, normalize=TRUE)
diss.INT.PER(x, y, normalize=TRUE)

diss( rbind(x,y,z), "INT.PER", normalize=FALSE )

## Create three sample time series
x <- cumsum(rnorm(100))
y <- cumsum(rnorm(100))
z <- sin(seq(0, pi, length.out=100))
## Compute the distance and check for coherent results
diss.INT.PER(x, y, normalize=TRUE)
diss.INT.PER(x, y, normalize=TRUE)
diss.INT.PER(x, y, normalize=TRUE)

diss( rbind(x,y,z), "INT.PER", normalize=FALSE )

Symbolic Aggregate Aproximation related functions

Description

diss.MINDIST.SAX computes a dissimilarity that lower bounds the Euclidean on the discretized, dimensionality reduced series. Function PAA produces the dimension reduction. Function convert.to.SAX.symbol produces the discretization.

Usage

diss.MINDIST.SAX(x, y, w, alpha=4, plot=FALSE)
PAA(x, w)
convert.to.SAX.symbol(x, alpha)
MINDIST.SAX(x, y, alpha, n)
SAX.plot(series, w, alpha, col.ser=rainbow(ncol(as.matrix(series))))
diss.MINDIST.SAX(x, y, w, alpha=4, plot=FALSE)
PAA(x, w)
convert.to.SAX.symbol(x, alpha)
MINDIST.SAX(x, y, alpha, n)
SAX.plot(series, w, alpha, col.ser=rainbow(ncol(as.matrix(series))))

Arguments

`x`	Numeric vector containing the first of the two time series.
`y`	Numeric vector containing the second of the two time series.
`w`	The amount of equal sized frames that the series will be reduced to.
`alpha`	The size of the alphabet, the amount of symbols used to represents the values of the series.
`plot`	If `TRUE`, plot a graphic of the reduced series, with their corresponding symbols.
`n`	The original size of the series.
`series`	A `ts` or `mts` object with the series to plot.
`col.ser`	Colors for the series. One per series.

Details

SAX is a symbolic representation of continuous time series.

w must be an integer but it does not need to divide the length of the series. If w divides the length of the series, the diss.MINDIST.SAX plot uses this to show the size of the frames.

PAA performs the Piecewise Aggregate Approximation of the series, reducing it to w elements, called frames. Each frame is composed by $n/w$ observations of the original series, averaged. Observations are weighted when w does not divide n.

convert.to.SAX.symbol performs SAX discretization: Discretizes the series x to an alphabet of size alpha, x should be z-normalized in this case. The $N(0,1)$ distribution is divided in alpha equal probability parts, if an observation falls into the $i$ th part (starting from minus infinity), it is assigned the $i$ symbol.

MINDIST.SAX calculates the MINDIST dissimilarity between symbolic representations.

diss.MINDIST.SAX combines the previous procedures to compute a dissimilarity between series. The series are z-normalized at first. Then the dimensionality is reduced uusin PAA to produce series of length w. The series are discretized to an alphabet of size alpha using convert.to.SAX.symbol. Finally the dissimilarity value is produced using MINDIST.SAX.

SAX.plot produces a plot of the SAX representation of the given series.

Value

The computed dissimilarity.

Author(s)

Pablo Montero Manso, José Antonio Vilar.

References

Lin, J., Keogh, E., Lonardi, S. & Chiu, B. (2003) A Symbolic Representation of Time Series, with Implications for Streaming Algorithms. In Proceedings of the 8th ACM SIGMOD Workshop on Research Issues in Data Mining and Knowledge Discovery.

Keogh, E., Chakrabarti, K., Pazzani, M., & Mehrotra, S. (2001). Dimensionality reduction for fast similarity search in large time series databases. Knowledge and information Systems, 3(3), 263-286.

Montero, P and Vilar, J.A. (2014) TSclust: An R Package for Time Series Clustering. Journal of Statistical Software, 62(1), 1-43. http://www.jstatsoft.org/v62/i01/.

Examples

set.seed(12349)
n = 100
x <- rnorm(n)  #generate sample series, white noise and a wiener process
y <- cumsum(rnorm(n))
w <- 20 #amount of equal-sized frames to divide the series, parameters for PAA
alpha <- 4 #size of the alphabet, parameter for SAX

#normalize
x <- (x - mean(x)) /sd(x)
y <- (y - mean(y)) /sd(y)

paax <- PAA(x, w) #generate PAA reductions
paay <- PAA(y, w)

plot(x, type="l", main="PAA reduction of series x") #plot an example of PAA reduction
p <- rep(paax,each=length(x)/length(paax)) #just for plotting the PAA
lines(p, col="red")

#repeat the example with y
plot(y, type="l", main="PAA reduction of series y") 
py <- rep(paay,each=length(y)/length(paay))
lines(py, col="blue")

#convert to SAX representation
SAXx <- convert.to.SAX.symbol( paax, alpha)
SAXy <- convert.to.SAX.symbol( paay, alpha)

#CALC THE SAX DISTANCE
MINDIST.SAX(SAXx, SAXy, alpha, n)

#this whole process can be computed using diss.MINDIST.SAX
diss.MINDIST.SAX(x, y, w, alpha, plot=TRUE)

z <- rnorm(n)^2

diss(rbind(x,y,z), "MINDIST.SAX", w, alpha)

SAX.plot( as.ts(cbind(x,y,z)), w=w, alpha=alpha)

set.seed(12349)
n = 100
x <- rnorm(n)  #generate sample series, white noise and a wiener process
y <- cumsum(rnorm(n))
w <- 20 #amount of equal-sized frames to divide the series, parameters for PAA
alpha <- 4 #size of the alphabet, parameter for SAX

#normalize
x <- (x - mean(x)) /sd(x)
y <- (y - mean(y)) /sd(y)

paax <- PAA(x, w) #generate PAA reductions
paay <- PAA(y, w)

plot(x, type="l", main="PAA reduction of series x") #plot an example of PAA reduction
p <- rep(paax,each=length(x)/length(paax)) #just for plotting the PAA
lines(p, col="red")

#repeat the example with y
plot(y, type="l", main="PAA reduction of series y") 
py <- rep(paay,each=length(y)/length(paay))
lines(py, col="blue")

#convert to SAX representation
SAXx <- convert.to.SAX.symbol( paax, alpha)
SAXy <- convert.to.SAX.symbol( paay, alpha)

#CALC THE SAX DISTANCE
MINDIST.SAX(SAXx, SAXy, alpha, n)

#this whole process can be computed using diss.MINDIST.SAX
diss.MINDIST.SAX(x, y, w, alpha, plot=TRUE)

z <- rnorm(n)^2

diss(rbind(x,y,z), "MINDIST.SAX", w, alpha)

SAX.plot( as.ts(cbind(x,y,z)), w=w, alpha=alpha)

Normalized Compression Distance

Description

Computes the distance based on the sizes of the compressed time series.

Usage

diss.NCD(x, y, type = "min")

diss.NCD(x, y, type = "min")

Arguments

`x`	Numeric vector containing the first of the two time series.
`y`	Numeric vector containing the second of the two time series.
`type`	Character string, the type of compression. May be abbreviated to a single letter, defaults to the first of the alternatives.

Details

The compression based dissimilarity is calculated:

$d(x,y) = C(xy) - max(C(x),C(y))/ min(C(x),C(y))$

Value

The computed distance.

Author(s)

Pablo Montero Manso, José Antonio Vilar.

References

Cilibrasi, R., & Vitányi, P. M. (2005). Clustering by compression. Information Theory, IEEE Transactions on, 51(4), 1523-1545.

Montero, P and Vilar, J.A. (2014) TSclust: An R Package for Time Series Clustering. Journal of Statistical Software, 62(1), 1-43. http://www.jstatsoft.org/v62/i01/.

Examples

n = 50
x <- rnorm(n)  #generate sample series, white noise and a wiener process
y <- cumsum(rnorm(n))

diss.NCD(x, y)

z <- rnorm(n)
w <- cumsum(rnorm(n))
series = rbind(x, y, z, w)
diss(series, "NCD", type="bzip2")

################################################################
#####symbolic representation prior to compression, using SAX####
####simpler symbolization, such as round() could also be used###
################################################################
#normalization function, required for SAX
z.normalize = function(x) {
    (x - mean(x)) / sd(x)
}

sx <- convert.to.SAX.symbol( z.normalize(x), alpha=4 )
sy <- convert.to.SAX.symbol( z.normalize(y), alpha=4 )
sz <- convert.to.SAX.symbol( z.normalize(z), alpha=4 )
sw <- convert.to.SAX.symbol( z.normalize(w), alpha=4 )

diss(rbind(sx, sy, sz, sw), "NCD", type="bzip2")

n = 50
x <- rnorm(n)  #generate sample series, white noise and a wiener process
y <- cumsum(rnorm(n))

diss.NCD(x, y)

z <- rnorm(n)
w <- cumsum(rnorm(n))
series = rbind(x, y, z, w)
diss(series, "NCD", type="bzip2")

################################################################
#####symbolic representation prior to compression, using SAX####
####simpler symbolization, such as round() could also be used###
################################################################
#normalization function, required for SAX
z.normalize = function(x) {
    (x - mean(x)) / sd(x)
}

sx <- convert.to.SAX.symbol( z.normalize(x), alpha=4 )
sy <- convert.to.SAX.symbol( z.normalize(y), alpha=4 )
sz <- convert.to.SAX.symbol( z.normalize(z), alpha=4 )
sw <- convert.to.SAX.symbol( z.normalize(w), alpha=4 )

diss(rbind(sx, sy, sz, sw), "NCD", type="bzip2")

Permutation Distribution Distance

Description

Computes the Permutation Distribution distance between time series.

Usage

diss.PDC(x, y, ...)
diss.PDC(x, y, ...)

Arguments

`x`	Numeric vector containing the first of the two time series.
`y`	Numeric vector containing the second of the two time series.
`...`	Additional parameters for the function. See `link[pdc]{pdcDist}`.

Details

This is basically a wrapper for pdcDist of the pdc package, intended for an easier discovery of the functionalities used in TSclust.

Value

The computed distance.

Examples

## Create three sample time series
x <- cumsum(rnorm(100))
y <- cumsum(rnorm(100))
diss.PDC(x, y)

## Create three sample time series
x <- cumsum(rnorm(100))
y <- cumsum(rnorm(100))
diss.PDC(x, y)

Periodogram Based Dissimilarity

Description

Computes the distance between two time series based on their periodograms.

Usage

diss.PER(x, y, logarithm=FALSE, normalize=FALSE)
diss.PER(x, y, logarithm=FALSE, normalize=FALSE)

Arguments

`x`	Numeric vector containing the first of the two time series.
`y`	Numeric vector containing the second of the two time series.
`logarithm`	Boolean. If `TRUE` logarithm of the periodogram coefficients will be taken.
`normalize`	Boolean. If `TRUE`, the periodograms will be normalized by the variance of their respective series.

Details

Computes the Euclidean distance between the periodogram coefficients of the series x and y. Additional transformations can be performed on the coefficients depending on the values of logarithm and normalize.

Value

The computed distance.

Author(s)

Pablo Montero Manso, José Antonio Vilar.

References

Caiado, J., Crato, N. and Peña, D. (2006) A periodogram-based metric for time series classification. Comput. Statist. Data Anal., 50(10), 2668–2684.

Montero, P and Vilar, J.A. (2014) TSclust: An R Package for Time Series Clustering. Journal of Statistical Software, 62(1), 1-43. http://www.jstatsoft.org/v62/i01/.

Examples

## Create three sample time series
x <- cumsum(rnorm(100))
y <- cumsum(rnorm(100))
z <- sin(seq(0, pi, length.out=100))
## Compute the distance and check for coherent results
diss.PER(x, y)
diss.PER(x, z)
diss.PER(y, z)
diss.PER(x, y, TRUE, TRUE)
diss.PER(x, z, TRUE, TRUE)
diss.PER(y, z, TRUE, TRUE)
#create a dist object for its use with clustering functions like pam or hclust
diss( rbind(x,y,z), "PER", logarithm=TRUE, normalize=TRUE)
## Create three sample time series
x <- cumsum(rnorm(100))
y <- cumsum(rnorm(100))
z <- sin(seq(0, pi, length.out=100))
## Compute the distance and check for coherent results
diss.PER(x, y)
diss.PER(x, z)
diss.PER(y, z)
diss.PER(x, y, TRUE, TRUE)
diss.PER(x, z, TRUE, TRUE)
diss.PER(y, z, TRUE, TRUE)
#create a dist object for its use with clustering functions like pam or hclust
diss( rbind(x,y,z), "PER", logarithm=TRUE, normalize=TRUE)

Dissimilarity Measure Based on Nonparametric Forecast

Description

Computes the dissimilarity between two time series as the L1 distance between the kernel estimators of their forecast densities at a pre-specified horizon.

Usage

diss.PRED(x, y, h, B=500, logarithm.x=FALSE, logarithm.y=FALSE,
differences.x=0, differences.y=0, plot=FALSE, models = NULL)
diss.PRED(x, y, h, B=500, logarithm.x=FALSE, logarithm.y=FALSE,
differences.x=0, differences.y=0, plot=FALSE, models = NULL)

Arguments

`x`	Numeric vector containing the first of the two time series.
`y`	Numeric vector containing the second of the two time series.
`h`	The horizon of interest, i.e the number of steps-ahead where the prediction is evaluated.
`B`	The amount of bootstrap resamples.
`logarithm.x`	Boolean. Specifies whether to transform series x by taking logarithms or not. When using `diss` wrapper, use `logarithms` argument instead. See details.
`logarithm.y`	Boolean. Specifies whether to transform series y by taking logarithms or not. When using `diss` wrapper, use `logarithms` argument instead. See details.
`differences.x`	Specifies the amount of differences to apply to series x. When using `diss` wrapper, use `differences` argument instead. See details.
`differences.y`	Specifies the amount of differences to apply to series y. When using `diss` wrapper, use `differences` argument instead. See details.
`plot`	If `TRUE`, plot the resulting forecast densities.
`models`	A list containing either `"ets"`, `"arima"` or a fitted model object from the `forecast` package. The list must have one element per series. In the case of the `x` and `y` version, a list with two elements. If models is not null `logarithm` and `differences` parameters are ignored.

Details

The dissimilarity between the time series x and y is given by

$d(x,y) = \int{ | f_{x,h}(u) - f_{y,h}(u) | du}$

where $f_{x,h}$ and $f_{y,h}$ are kernel density estimators of the forecast densities h-steps ahead of x and y, respectively. The horizon of interest h is pre-specified by the user. If models is specified, the given model for each series is used for obtaining the forecast densities. Currently, each element of the models list can be the string "ets", which will fit a ets model using the function ets in the forecast package. If the element of models is the string "arima", an ARIMA model using auto.arima from the forecast package will be used. Finally, the elements of models can be a fitted model on the series using a method from the forecast package which can be simulated, see link[forecast]{simulate.ets}. The kernel density estimators are based on B bootstrap replicates obtained by using a resampling procedure that mimics the generating processes, which are assumed to follow an arbitrary autoregressive structure (parametric or non-parametric). The procedure is completely detailed in Vilar et al. (2010). This function has high computational cost due to the bootstrapping procedure.

The procedure uses a bootstrap method that requires stationary time series. In order to support a wider range of time series, the method allows some transformations on the series before proceeding with the bootstrap resampling. This transformations are inverted before calculating the densities. The transformations allowed are logarithm and differenciation. The parameters logarithm.x, logarithm.y, differences.x, differences.y can be specified with this purpose.

If using diss function with "PRED" method, the argument logarithms must be used instead of logarithm.x and logarithm.y. logarithms is a boolean vector specifying if the logarithm transform should be taken for each one of the series. The argument differences, a numeric vector specifying the amount of differences to apply the series, is used instead of differences.x and differences.y. The plot is also different, showing all the densities in the same plot.

Value

diss.PRED returns a list with the following components.

`L1dist`	The computed distance.
`dens.x`	A 2-column matrix with the density of predicion of series `x`. First column is the base (x) and the second column is the value (y) of the density.
`dens.y`	A 2-column matrix with the density of predicion of series `y`. First column is the base (x) and the second column is the value (y) of the density.

When used from the diss wrapper function, it returns a list with the following components.

`dist`	A `dist` object with the pairwise L1 distances between series.
`densities`	A list of 2-column matrices containing the densities of each series, in the same format as 'dens.x' or 'dens.y' of `diss.PRED`.

Author(s)

José Antonio Vilar, Pablo Montero Manso.

References

Alonso, A.M., Berrendero, J.R., Hernandez, A. and Justel, A. (2006) Time series clustering based on forecast densities. Comput. Statist. Data Anal., 51,762–776.

Vilar, J.A., Alonso, A. M. and Vilar, J.M. (2010) Non-linear time series clustering based on non-parametric forecast densities. Comput. Statist. Data Anal., 54 (11), 2850–2865.

Montero, P and Vilar, J.A. (2014) TSclust: An R Package for Time Series Clustering. Journal of Statistical Software, 62(1), 1-43. http://www.jstatsoft.org/v62/i01/.

Examples

x <- (rnorm(100))
x <- x + abs(min(x)) + 1 #shift to produce values greater than 0, for a correct logarithm transform
y <- (rnorm(100))
z <- sin(seq(0, pi, length.out=100))
## Compute the distance and check for coherent results
diss.PRED(x, y, h=6, logarithm.x=FALSE, logarithm.y=FALSE, differences.x=1, differences.y=0)
#create a dist object for its use with clustering functions like pam or hclust
diss( rbind(x,y,z), METHOD="PRED", h=3, B=200,
 logarithms=c(TRUE,FALSE, FALSE), differences=c(1,1,2) )
#test the forecast package predictions
diss.PRED(x,y, h=5, models = list("ets", "arima"))

x <- (rnorm(100))
x <- x + abs(min(x)) + 1 #shift to produce values greater than 0, for a correct logarithm transform
y <- (rnorm(100))
z <- sin(seq(0, pi, length.out=100))
## Compute the distance and check for coherent results
diss.PRED(x, y, h=6, logarithm.x=FALSE, logarithm.y=FALSE, differences.x=1, differences.y=0)
#create a dist object for its use with clustering functions like pam or hclust
diss( rbind(x,y,z), METHOD="PRED", h=3, B=200,
 logarithms=c(TRUE,FALSE, FALSE), differences=c(1,1,2) )
#test the forecast package predictions
diss.PRED(x,y, h=5, models = list("ets", "arima"))

Dissimilarity based on the Generalized Likelihood Ratio Test

Description

The dissimilarity between two time series is computed by using an adaptation of the generalized likelihood ratio test to check the equality of two log-spectra.

Usage

diss.SPEC.GLK(x, y, plot=FALSE)
diss.SPEC.GLK(x, y, plot=FALSE)

Arguments

`x`	Numeric vector containing the first of the two time series.
`y`	Numeric vector containing the second of the two time series.
`plot`	If `TRUE`, the smoothed spectral densities of the two series are plotted.

Details

The dissimilarity between two series x and y is measured in terms of the vaue of a test statistic to check the equality of their log-spectra, $m_X(\lambda)$ and $m_Y(\lambda)$ respectivelty. The test statistic is constructed by using the generalized likelihood ratio test criterion (Fan and Zhang, 2004). Specifically, the test statistic takes the form:

$d(x,y) = \sum_{k=1}^T [ Z_k - \hat{\mu}( \lambda_k) - 2 \log( 1 + e^{ \{ Z_k - \hat{\mu}(\lambda_k)\}})] - \sum_{k=1}^T[Z_k - 2 \log(1 + e^{Z_k})],$

where $I_x(\lambda_k)$ and $I_y(\lambda_k)$ are the periodograms of x and y, $Z_k = \log(I_x(\lambda_k)) - \log( I_y(\lambda_k))$ , and $\hat{\mu}(\lambda_k)$ is the local maximum log-likelihood estimator of $\mu(\lambda_k)= m_x(\lambda_k) - m_y(\lambda_k)$ computed by local linear fitting.

Value

The computed distance.

Author(s)

Pablo Montero Manso, José Antonio Vilar.

References

Fan, J. and Zhang, W. (2004) Generalised likelihood ratio tests for spectral density. Biometrika, 195–209.

Pértega, S. and Vilar, J.A. (2010) Comparing several parametric and nonparametric approaches to time series clustering: A simulation study. J. Classification, 27(3), 333–362.

Montero, P and Vilar, J.A. (2014) TSclust: An R Package for Time Series Clustering. Journal of Statistical Software, 62(1), 1-43. http://www.jstatsoft.org/v62/i01/.

Examples

## Create two sample time series
x <- cumsum(rnorm(50))
y <- cumsum(rnorm(50))
z <- sin(seq(0, pi, length.out=50))
## Compute the distance and check for coherent results
diss.SPEC.GLK(x, y, plot=TRUE)
#create a dist object for its use with clustering functions like pam or hclust

diss( rbind(x,y,z), "SPEC.GLK" )

## Create two sample time series
x <- cumsum(rnorm(50))
y <- cumsum(rnorm(50))
z <- sin(seq(0, pi, length.out=50))
## Compute the distance and check for coherent results
diss.SPEC.GLK(x, y, plot=TRUE)
#create a dist object for its use with clustering functions like pam or hclust

diss( rbind(x,y,z), "SPEC.GLK" )

Dissimilarity Based on the Integrated Squared Difference between the Log-Spectra

Description

Computes the dissimilarity between two time series in terms of the integrated squared difference between non-parametric estimators of their log-spectra.

Usage

diss.SPEC.ISD(x, y, plot=FALSE, n=length(x))
diss.SPEC.ISD(x, y, plot=FALSE, n=length(x))

Arguments

`x`	Numeric vector containing the first of the two time series.
`y`	Numeric vector containing the second of the two time series.
`plot`	If `TRUE`, plot the smoothed spectral densities of the two series.
`n`	The number of points to use for the linear interpolation. A value of n=0 uses numerical integration instead of linear interpolation. See details.

Details

$d(x,y) = \int ( \hat{m}_x(\lambda) - \hat{m}_y(\lambda))^2 \, d\lambda,$

where $\hat{m}_x(\lambda)$ and $\hat{m}_y(\lambda)$ are local linear smoothers of the log-periodograms, obtained using the maximum local likelihood criterion.

Value

The computed distance.

Author(s)

Pablo Montero Manso, José Antonio Vilar.

References

Pértega, S. and Vilar, J.A. (2010) Comparing several parametric and nonparametric approaches to time series clustering: A simulation study. J. Classification, 27(3), 333–362.

Montero, P and Vilar, J.A. (2014) TSclust: An R Package for Time Series Clustering. Journal of Statistical Software, 62(1), 1-43. http://www.jstatsoft.org/v62/i01/.

Examples

## Create two sample time series
x <- cumsum(rnorm(50))
y <- cumsum(rnorm(50))
z <- sin(seq(0, pi, length.out=50))
## Compute the distance and check for coherent results
diss.SPEC.ISD(x, y, plot=TRUE) 
#create a dist object for its use with clustering functions like pam or hclust

diss.SPEC.ISD(x, y, plot=TRUE, n=0)#try integrate instead of interpolation
diss( rbind(x,y,z), "SPEC.ISD" )


## Create two sample time series
x <- cumsum(rnorm(50))
y <- cumsum(rnorm(50))
z <- sin(seq(0, pi, length.out=50))
## Compute the distance and check for coherent results
diss.SPEC.ISD(x, y, plot=TRUE) 
#create a dist object for its use with clustering functions like pam or hclust

diss.SPEC.ISD(x, y, plot=TRUE, n=0)#try integrate instead of interpolation
diss( rbind(x,y,z), "SPEC.ISD" )

General Spectral Dissimilarity Measure Using Local-Linear Estimation of the Log-Spectra

Description

Computes a general dissimilarity measure based on the ratio of local linear spectral estimators.

Usage

diss.SPEC.LLR(x, y, alpha=0.5, method="DLS", plot=FALSE, n=length(x))
diss.SPEC.LLR(x, y, alpha=0.5, method="DLS", plot=FALSE, n=length(x))

Arguments

`x`	Numeric vector containing the first of the two time series.
`y`	Numeric vector containing the second of the two time series.
`alpha`	Power for the ratio of densities in the Chernoff information measure. Between 0 and 1.
`method`	`"DLS"` for least squares estimation of the spectral density and `"LK"` for maximum likelihood estimation.
`plot`	if `TRUE`, plot the smoothed spectral densities of the two series.
`n`	The number of points to use for the linear interpolation. A value of n=0 uses numerical integration instead of linear interpolation. See details.

Details

$d_W = \int_{-\pi}^{\pi} W^{\prime}\Bigg( \frac{f_x(\lambda) } { f_y(\lambda) } \Bigg) d\lambda$

where:

$f_x$ and $f_y$ are nonparametric approximations of spectral densities of x and y respectively.
$W^{\prime}(x) = W(x) + W(1/x)$ with $W(x) = \log( \alpha x + (1- \alpha) x) - \alpha \log(x)$ , so that $W(.)$ is a divergence function depending on $\alpha$ .

This dissimilarity measure corresponds to the limiting spectral approximation of the Chernoff information measure in the time domain (see Kakizawa et al., 1998). The spectral densities are approximated by using local linear fitting by generalized least squared if method=”DLS” or by maximum likelihood if method=”LK” (in this case, higher computational cost is required).

By default, for performance reasons, the spectral densities are estimated using linear interpolation using n points. If n is 0, no linear interpolation is performed, and integrate is used to calculate the integral, using as many points as integrate sees fit. If the dissimilarity will be calculated for more than two series, calling SPEC.LLR from the diss wrapper function is preferred, since it saves some computations.

Value

The computed distance.

Author(s)

Pablo Montero Manso, José Antonio Vilar.

References

Vilar, J.A. and Pértega, S. (2004) Discriminant and cluster analysis for gaussian stationary processes: local linear fitting approach. J. Nonparametr. Stat., 16(3-4) 443–462.

Kakizawa, Y., Shumway, R. H. and Taniguchi M. (1998) Discrimination and clustering for multivariate time series. J. Amer. Statist. Assoc., 93(441), 328– 340.

Montero, P and Vilar, J.A. (2014) TSclust: An R Package for Time Series Clustering. Journal of Statistical Software, 62(1), 1-43. http://www.jstatsoft.org/v62/i01/.

Examples

## Create three sample time series
x <- cumsum(rnorm(50))
y <- cumsum(rnorm(50))
z <- sin(seq(0, pi, length.out=50))
## Compute the distance and check for coherent results
diss.SPEC.LLR(x, y, plot=TRUE)
diss.SPEC.LLR(x, z, n=0) #try integrate instead of interpolation
diss.SPEC.LLR(y, z, method="LK", n=0) #maximum likelihood with integration
#create a dist object for its use with clustering functions like pam or hclust
diss(rbind(x,y,z), METHOD="SPEC.LLR", method="DLS", alpha=0.5, n=50)

## Create three sample time series
x <- cumsum(rnorm(50))
y <- cumsum(rnorm(50))
z <- sin(seq(0, pi, length.out=50))
## Compute the distance and check for coherent results
diss.SPEC.LLR(x, y, plot=TRUE)
diss.SPEC.LLR(x, z, n=0) #try integrate instead of interpolation
diss.SPEC.LLR(y, z, method="LK", n=0) #maximum likelihood with integration
#create a dist object for its use with clustering functions like pam or hclust
diss(rbind(x,y,z), METHOD="SPEC.LLR", method="DLS", alpha=0.5, n=50)

Hourly Electricity Prices in the Spanish Market

Description

Partial realizations of time series of hourly electricity prices (Cent/kWh) in the Spanish market.

Usage

data(electricity)data(electricity)

Format

A matrix with 365 observations of the hourly electricity cost readings.

Details

The dataset consists of hourly electricity prices in the Spanish market during weekdays of the period December 31, 2007 - May 25, 2009.

References

Montero, P and Vilar, J.A. (2014) TSclust: An R Package for Time Series Clustering. Journal of Statistical Software, 62(1), 1-43. http://www.jstatsoft.org/v62/i01/.

Long-Term Interest Rates from 1995 to 2012

Description

Partial realizations of time series of long-term interest rates (10-year bonds) for several countries.

Usage

data(interest.rates)data(interest.rates)

Format

A ts object with 215 observations of the monthly long-term interest rates (10-year bonds) from January 1995 to November 2012 of several countries.

References

Montero, P and Vilar, J.A. (2014) TSclust: An R Package for Time Series Clustering. Journal of Statistical Software, 62(1), 1-43. http://www.jstatsoft.org/v62/i01/.

Clustering Evaluation Index Based on Leave-one-out One-nearest-neighbor Evaluation

Description

Computes the leave-one-out one-nearest-neighbor cross-validation of an arbitrary distance matrix.

Usage

loo1nn.cv(d, G)
loo1nn.cv(d, G)

Arguments

`d`	A `dist` object.
`G`	Integer vector with the labels of the true cluster solution. Each element of the vector specifies the cluster 'id' that the element belongs to.

Details

Computes the proportion of succesful clusters that the given distance matrix produces using leave-one-out one-nearest-neighbor cross-validation. Distance ties are solved by majority vote. A tie while voting produces a warning and is solved by selecting a candidate cluster at random.

Value

The computed proportion.

Author(s)

Pablo Montero Manso, José Antonio Vilar.

Examples

 data(synthetic.tseries)
 
 #create the ground thruth cluster 
 G <- rep(1:6, each = 3)
 
 #obtain candidate distance matrix (dist object)
 dACF <- diss(synthetic.tseries, "ACF")
 
 #calculate the cross-validation
 loo1nn.cv(dACF, G)
 
data(synthetic.tseries)
 
 #create the ground thruth cluster 
 G <- rep(1:6, each = 3)
 
 #obtain candidate distance matrix (dist object)
 dACF <- diss(synthetic.tseries, "ACF")
 
 #calculate the cross-validation
 loo1nn.cv(dACF, G)

Pairs of Time Series from Different Domains

Description

Dataset formed by pairs of time series from different domains. Series were selected from the UCR Time Series Archive.

Usage

data(paired.tseries)data(paired.tseries)

Format

A mts object with 36 series of length 1000.

Details

Each pair of series in the dataset (Series 1 and 2, Series 3 and 4, etc.) comes from the same domain, so this pairing could constitute a possible ground truth solution.

Note

abbreviate can be used on the colnames.

Source

http://www.cs.ucr.edu/~eamonn/SIGKDD2004/All_datasets/

References

Montero, P and Vilar, J.A. (2014) TSclust: An R Package for Time Series Clustering. Journal of Statistical Software, 62(1), 1-43. http://www.jstatsoft.org/v62/i01/.

Examples

data(paired.tseries)
#Create the true solution, the pairs
true_cluster <- rep(1:18, each=2)
#test a dissimilarity metric and a cluster algorithm
intperdist <- diss( paired.tseries, "INT.PER") #create the distance matrix
#use hierarchical clustering and divide the tree in 18 clusters
intperclust <- cutree( hclust(intperdist), k=18 )
#use a cluster simmilarity index to rate the solution
cluster.evaluation( true_cluster, intperclust)

#### other evaluation criterion used in this dataset  consist in counting the correct pairs
#### formed during agglomerative hierarchical cluster (see references)
true_pairs = (-matrix(1:36, ncol=2, byrow=TRUE))
hcintper <- hclust(intperdist, "complete")
#count within the hierarchical cluster the pairs
sum( match(data.frame(t(true_pairs)), data.frame(t(hcintper$merge)), nomatch=0) > 0 ) / 18

data(paired.tseries)
#Create the true solution, the pairs
true_cluster <- rep(1:18, each=2)
#test a dissimilarity metric and a cluster algorithm
intperdist <- diss( paired.tseries, "INT.PER") #create the distance matrix
#use hierarchical clustering and divide the tree in 18 clusters
intperclust <- cutree( hclust(intperdist), k=18 )
#use a cluster simmilarity index to rate the solution
cluster.evaluation( true_cluster, intperclust)

#### other evaluation criterion used in this dataset  consist in counting the correct pairs
#### formed during agglomerative hierarchical cluster (see references)
true_pairs = (-matrix(1:36, ncol=2, byrow=TRUE))
hcintper <- hclust(intperdist, "complete")
#count within the hierarchical cluster the pairs
sum( match(data.frame(t(true_pairs)), data.frame(t(hcintper$merge)), nomatch=0) > 0 ) / 18

Clustering Algorithm Based on p-values.

Description

Clustering algorithm based on p-values. Each group in the cluster solution is formed by series with associated p-values greater than a pre-specified level of significance.

Usage

pvalues.clust(pvalues, significance)
pvalues.clust(pvalues, significance)

Arguments

`pvalues`	A `dist` object containing the p-values from testing the equality of each pair of time series under study.
`significance`	The significance level.

Details

Each element (i,j) in pvalues corresponds to the p-value obtained from checking whether or not the $i$ -th and $j$ -th series come from the same generating model. The clustering algorithm will only group together those series whose associated p-values are greater than the pre-specified significance level. The algorithm was originally developed for its use with the p-values obtained with in diss.AR.MAH (see Maharaj, 2000), but it can be applied to any similar test.

Value

An integer vector of length n, the number of observations, giving for each observation the number (id) of the cluster to which it belongs.

Author(s)

Pablo Montero Manso, José Antonio Vilar.

References

Maharaj E.A. (2000) Clusters of time series. J. Classification, 17(2), 297–314.

Montero, P and Vilar, J.A. (2014) TSclust: An R Package for Time Series Clustering. Journal of Statistical Software, 62(1), 1-43. http://www.jstatsoft.org/v62/i01/.

Examples

## Create three sample time series
x <- cumsum(rnorm(100))
y <- cumsum(rnorm(100))
z <- sin(seq(0, pi, length.out=100))
##

## Compute the distance and check for coherent results
dd <- diss( rbind(x,y,z), "AR.MAH")
pvalues.clust( dd$p_value, 0.05 )

## Create three sample time series
x <- cumsum(rnorm(100))
y <- cumsum(rnorm(100))
z <- sin(seq(0, pi, length.out=100))
##

## Compute the distance and check for coherent results
dd <- diss( rbind(x,y,z), "AR.MAH")
pvalues.clust( dd$p_value, 0.05 )

Synthetic Time Series for Clustering Performace Comparisons.

Description

This dataset features three repetitions of several models of time series.

Usage

data(synthetic.tseries)data(synthetic.tseries)

Details

The dataset is a mts object, formed by several repetitions of each of the following models.

M1	AR	$X_t = 0.6 X_{t-1} + \varepsilon_{t}$
M2	Bilinear	$X_t = \left( 0.3 -0.2 \varepsilon_{t-1} \right) X_{t-1} + 1.0 +\varepsilon_{t}$
M3	EXPAR	$X_t =\left( 0.9 \exp \left( - X_{t-1}^2 \right) -0.6 \right) X_{t-1} + 1.0 + \varepsilon_{t}$
M4	SETAR	$X_t =\left( 0.3 X_{t-1} +1.0 \right) I \left( X_{t-1} \geq 0.2 \right) -$
		$\left( 0.3 X_{t-1} -1.0 \right) I \left( X_{t-1} < 0.2 \right) + \varepsilon_{t}$
M5	NLAR	$X_t = 0.7 \left\| X_{t-1} \right\| \left( 2 + \left\| X_{t-1} \right\| \right)^{-1} + \varepsilon_{t}$
M6	STAR	$X_t = 0.8 X_{t-1} -0.8 X_{t-1} \left( 1 + \exp \left( -10 X_{t-1} \right) \right)^{-1} + \varepsilon_{t}$

Three simulations of each model are included. This dataset can be used for comparing the performance of different dissimilarity measures between time series or clustering algorithms.

References

Montero, P and Vilar, J.A. (2014) TSclust: An R Package for Time Series Clustering. Journal of Statistical Software, 62(1), 1-43. http://www.jstatsoft.org/v62/i01/.

Examples

data(synthetic.tseries)
#Create the true solution, for this dataset, there are three series of each model
true_cluster <- c(1, 1, 1, 2, 2, 2, 3, 3, 3, 4, 4, 4, 5, 5, 5, 6, 6, 6)
#test a dissimilarity metric and a cluster algorithm
intperdist <- diss( synthetic.tseries, "INT.PER") #create the distance matrix
#use hierarchical clustering and divide the tree in 6 clusters
intperclust <- cutree( hclust(intperdist), 6 ) 
#use a cluster simmilarity index to rate the solution
cluster.evaluation( true_cluster, intperclust)

#test another dissimilarity metric and a cluster algorithm
acfdist <- diss( synthetic.tseries, "ACF", p=0.05) 
acfcluster <- pam( acfdist, 6 )$clustering #use pam clustering to form 6 clusters
cluster.evaluation( true_cluster, acfcluster)

#test another dissimilarity metric and a cluster algorithm
chernoffdist <- diss( synthetic.tseries, "SPEC.LLR")
chernoffclust <- pam( chernoffdist, 6 )$clustering 
cluster.evaluation( true_cluster, chernoffclust)


data(synthetic.tseries)
#Create the true solution, for this dataset, there are three series of each model
true_cluster <- c(1, 1, 1, 2, 2, 2, 3, 3, 3, 4, 4, 4, 5, 5, 5, 6, 6, 6)
#test a dissimilarity metric and a cluster algorithm
intperdist <- diss( synthetic.tseries, "INT.PER") #create the distance matrix
#use hierarchical clustering and divide the tree in 6 clusters
intperclust <- cutree( hclust(intperdist), 6 ) 
#use a cluster simmilarity index to rate the solution
cluster.evaluation( true_cluster, intperclust)

#test another dissimilarity metric and a cluster algorithm
acfdist <- diss( synthetic.tseries, "ACF", p=0.05) 
acfcluster <- pam( acfdist, 6 )$clustering #use pam clustering to form 6 clusters
cluster.evaluation( true_cluster, acfcluster)

#test another dissimilarity metric and a cluster algorithm
chernoffdist <- diss( synthetic.tseries, "SPEC.LLR")
chernoffclust <- pam( chernoffdist, 6 )$clustering 
cluster.evaluation( true_cluster, chernoffclust)

Package for Time Series Clustering.

Description

This package contains several measures of dissimilarity between time series, some examples of time series datasets, specific clustering algorithms, and dimension reduction algorithms. dissimilarities begin with diss.*, and a wrapper function diss is available. Cluster evaluation methods include cluster.evaluation and loo1nn.cv. A clustering algorithm based on pairwise p-values is implemented in pvalues.clust. The package should be used along with other existing clustering packages and function such as hclust, packages cluster, ...

Author(s)

Pablo Montero Manso, José Antonio Vilar.

References

Montero, P and Vilar, J.A. (2014) TSclust: An R Package for Time Series Clustering. Journal of Statistical Software, 62(1), 1-43. http://www.jstatsoft.org/v62/i01/.

Examples

#the available dissimilarities can be found in the diss help, page (?diss)
#and their individual pages from there.

### The most common use case begins with a set of time series we want to cluster.
### This package includes several example datasets.
### 
data(interest.rates)
###transformation of the interest rates
trans.inter.rates <- log(interest.rates[2:215,]) - log(interest.rates[1:214,])

##use the dist function of the proxy package to easily create the dist object
#applying ACF with geometric decaying to each pair of time series
tsdist <- diss( t(trans.inter.rates) , "ACF", p=0.05)

names(tsdist) <- colnames(interest.rates)

#perform hierachical clustering to the dist object
hc <- hclust(tsdist)

#show the results
plot(hc)

mahdist <- diss( t(trans.inter.rates) , "AR.MAH", p=0.05)$p_value

pvalues.clust(mahdist, 0.05)




#the available dissimilarities can be found in the diss help, page (?diss)
#and their individual pages from there.

### The most common use case begins with a set of time series we want to cluster.
### This package includes several example datasets.
### 
data(interest.rates)
###transformation of the interest rates
trans.inter.rates <- log(interest.rates[2:215,]) - log(interest.rates[1:214,])

##use the dist function of the proxy package to easily create the dist object
#applying ACF with geometric decaying to each pair of time series
tsdist <- diss( t(trans.inter.rates) , "ACF", p=0.05)

names(tsdist) <- colnames(interest.rates)

#perform hierachical clustering to the dist object
hc <- hclust(tsdist)

#show the results
plot(hc)

mahdist <- diss( t(trans.inter.rates) , "AR.MAH", p=0.05)$p_value

pvalues.clust(mahdist, 0.05)

Package 'TSclust'

Help Index

Clustering Evaluation Index Based on Known Ground Truth

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TSclust Dissimilarity Computation

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Autocorrelation-based Dissimilarity

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Dissimilarity Based on LPC Cepstral Coefficients

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Model-based Dissimilarity Proposed by Maharaj (1996, 2000)

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Model-based Dissimilarity Measure Proposed by Piccolo (1990)

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Compression-based Dissimilarity measure

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Complexity-Invariant Distance Measure For Time Series

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