Distance Measures for Time Series in R.

Description

A complete set of distance measures specifically designed to deal with time series.

Details

This package provides a comprehensive set of time series distance measures published in the literature and some additional functions which, although initially not designed for this purpose, can be used to measure the dissimilarity between time series. These measures can be used to perform clustering, classification or other data mining tasks which require the definition of a distance measure between time series. Some of the measures are specifically implemented for this package while other are originally hosted in other R packages. The measures included are:

• Lp distances LPDistance

• Distance based on the cross-correlation CCorDistance

• Short Time Series distance (STS) STSDistance

• Dynamic Time Warping (DTW) DTWDistance

• LB_Keogh lower bound for the Dynamic Time Warping distance LBKeoghDistance

• Edit Distance for Real Sequences (EDR) EDRDistance

• Longest Common Subsequence distance for real sequences(LCSS) LCSSDistance

• Edit Distance based on Real Penalty (ERP) ERPDistance

• Distance based on the Fourier Discrete Transform FourierDistance

• TQuest distance TquestDistance

• Dissim distance DissimDistance

• Autocorrelation-based dissimilarity ACFDistance.

• Partial autocorrelation-based dissimilarity PACFDistance.

• Dissimilarity based on LPC cepstral coefficients ARLPCCepsDistance.

• Model-based dissimilarity proposed by Maharaj (1996, 2000) ARMahDistance.

• Model-based dissimilarity proposed by Piccolo (1990) ARPicDistance.

• Compression-based dissimilarity measure CDMDistance.

• Complexity-invariant distance measure CIDDistance.

• Dissimilarities based on Pearson's correlation CorDistance.

• Dissimilarity index which combines temporal correlation and raw value behaviors CortDistance.

• Integrated periodogram based dissimilarity IntPerDistance.

• Periodogram based dissimilarity PerDistance.

• Symbolic Aggregate Aproximation based dissimilarity MindistSaxDistance.

• Normalized compression based distance NCDDistance.

• Dissimilarity measure cased on nonparametric forecasts PredDistance.

• Dissimilarity based on the integrated squared difference between the log-spectra SpecISDDistance.

• General spectral dissimilarity measure using local-linear estimation of the log-spectra SpecLLRDistance.

• Permutation Distribution Distance PDCDistance.

• Frechet distance FrechetDistance.

All the measures are implemented in separate functions but can also be invoked by means of the wrapper function TSDistances. Moreover, this distance enables the use of time series objects of type ts, zoo and xts.

As an additional functionality of the package, pairwise distances between all the time series in a database can be easily computed by using the dist function from the proxy package or the TSDatabaseDistances function included in the TSdist package.

Author(s)

Usue Mori, Alexander Mendiburu, Jose A. Lozano. Maintainer: <usue.mori@ehu.es>

References

Esling, P., & Agon, C. (2012). Time-series data mining. ACM Computing Surveys, 45(1), 1-34.

Liao, T. W. (2005). Clustering of time series data-a survey. Pattern Recognition, 38(11), 1857-1874.

Wang, X., Mueen, A., Ding, H., Trajcevski, G., Scheuermann, P., & Keogh, E. (2012). Experimental comparison of representation methods and distance measures for time series data. Data Mining and Knowledge Discovery, 26(2), 275-309.

David Meyer and Christian Buchta (2013). proxy: Distance and Similarity Measures. R package version 0.4-10. http://CRAN.R-project.org/package=proxy

Examples

 library(TSdist);

Autocorrelation-based Dissimilarity

Description

Computes the dissimilarity between a pair of numeric time series based on their estimated autocorrelation coefficients.

Usage

ACFDistance(x, y, ...)

Arguments

Details

This is simply a wrapper for the diss.ACF function of package TSclust. As such, all the functionalities of the diss.ACF function are also available when using this function.

Value

Author(s)

Usue Mori, Alexander Mendiburu, Jose A. Lozano.

References

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

Galeano, P., & Pella, D. (2000). Multivariate Analysis in Vector Time Series Pedro Galeano and Daniel Pella. Resenhas, the Journal of the Institute of Mathematics and Statistics of the University of Sao Paolo, 4, 383–403.

Lei, H., & Sun, B. (2007). A Study on the Dynamic Time Warping in Kernel Machines. In 2007 Third International IEEE Conference on Signal-Image Technologies and Internet-Based System (pp. 839–845).

See Also

To calculate this distance measure using ts, zoo or xts objects see TSDistances. To calculate distance matrices of time series databases using this measure see TSDatabaseDistances.

Examples

# The objects example.series3 and example.series4 are two 
# numeric series of length 100 and 120 contained in the 
# TSdist package. 

data(example.series3)
data(example.series4)

# For information on their generation and shape see 
# help page of example.series.

help(example.series)

# Calculate the autocorrelation based distance between the two series using
# the default parameters:

ACFDistance(example.series3, example.series4)

Dissimilarity Based on LPC Cepstral Coefficients

Description

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

Usage

ARLPCCepsDistance(x, y, ...)

Arguments

Details

This is simply a wrapper for the diss.AR.LPC.CEPS function of package TSclust. As such, all the functionalities of the diss.AR.LPC.CEPS function are also available when using this function.

Value

Author(s)

Usue Mori, Alexander Mendiburu, Jose A. Lozano.

References

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

See Also

To calculate this distance measure using ts, zoo or xts objects see TSDistances. To calculate distance matrices of time series databases using this measure see TSDatabaseDistances.

Examples

# The objects example.series3 and example.series4 are two 
# numeric series of length 100 and 120 contained in the 
# TSdist package obtained from an ARIMA(3,0,2) process. 

data(example.series3)
data(example.series4)

# For information on their generation and shape see 
# help page of example.series.

help(example.series)

# Calculate the ar.lpc.ceps distance between the two series using
# the default parameters. In this case an AR model is automatically 
# selected for each of the series:

ARLPCCepsDistance(example.series3, example.series4)

# Calculate the ar.lpc.ceps distance between the two series
# imposing the order of the ARIMA model of each series:

ARLPCCepsDistance(example.series3, example.series4, order.x=c(3,0,2), 
order.y=c(3,0,2))

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

Description

Computes the model based dissimilarity proposed by Maharaj.

Usage

ARMahDistance(x, y, ...)

Arguments

Details

This is simply a wrapper for the diss.AR.MAH function of package TSclust. As such, all the functionalities of the diss.AR.MAH function are also available when using this function.

Value

Author(s)

Usue Mori, Alexander Mendiburu, Jose A. Lozano.

References

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

See Also

To calculate this distance measure using ts, zoo or xts objects see TSDistances. To calculate distance matrices of time series databases using this measure see TSDatabaseDistances.

Examples

# The objects example.series3 and example.series4 are two 
# numeric series of length 100 and 120 contained in the 
# TSdist package obtained from an ARIMA(3,0,2) process. 

data(example.series3)
data(example.series4)

# For information on their generation and shape see 
# help page of example.series.

help(example.series)

# Calculate the ar.mah distance between the two series using
# the default parameters. 

ARMahDistance(example.series3, example.series4)

# The p-value is almost 1, which indicates that the two series come from the same 
# ARMA process. 

Model-based Dissimilarity Measure Proposed by Piccolo (1990)

Description

Computes the model based dissimilarity proposed by Piccolo.

Usage

ARPicDistance(x, y, ...)

Arguments

Details

This is simply a wrapper for the diss.AR.PIC function of package TSclust. As such, all the functionalities of the diss.AR.PIC function are also available when using this function.

Value

Author(s)

Usue Mori, Alexander Mendiburu, Jose A. Lozano.

References

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

See Also

To calculate this distance measure using ts, zoo or xts objects see TSDistances. To calculate distance matrices of time series databases using this measure see TSDatabaseDistances.

Examples

# The objects example.series3 and example.series4 are two 
# numeric series of length 100 and 120 contained in the 
# TSdist package obtained from an ARIMA(3,0,2) process. 

data(example.series3)
data(example.series4)

# For information on their generation and shape see 
# help page of example.series.

help(example.series)

# Calculate the Piccolo distance between the two series using
# the default parameters. In this case an AR model is automatically 
# selected for each of the series:

ARPicDistance(example.series3, example.series4)

# Calculate the Piccolo distance between the two series
# imposing the order of the ARMA model of each series:

ARPicDistance(example.series3, example.series4, order.x=c(3,0,2), 
order.y=c(3,0,2))

Cross-correlation based distance.

Description

Computes the distance measure based on the cross-correlation between a pair of numeric time series.

Usage

CCorDistance(x, y, lag.max=(min(length(x), length(y))-1))

Arguments

Details

The cross-correlation based distance between two numeric time series is calculated as follows:

D= \sqrt{ ((1 - CC(x, y, 0) ^ 2) / \sum (1 - CC(x, y, k) ^ 2)) }

where CC(x,y,k) is the cross-correlation between x and y at lag k.

The summatory in the denominator goes from 1 to lag.max. In view of this, the parameter must be a positive integer no larger than the length of the series.

Value

Author(s)

Usue Mori, Alexander Mendiburu, Jose A. Lozano.

References

Liao, T. W. (2005). Clustering of time series data-a survey. Pattern Recognition, 38(11), 1857-1874.

Pree, H., Herwig, B., Gruber, T., Sick, B., David, K., & Lukowicz, P. (2014). On general purpose time series similarity measures and their use as kernel functions in support vector machines. Information Sciences, 281, 478–495.

See Also

To calculate this distance measure using ts, zoo or xts objects see TSDistances. To calculate distance matrices of time series databases using this measure see TSDatabaseDistances.

Examples

# The objects example.series3 and example.series4 are two 
# numeric series of length 100 and 120 contained in the 
# TSdist package. 

data(example.series3)
data(example.series4)

# For information on their generation and shape see 
# help page of example.series.

help(example.series)

# Calculate the cross-correlation based distance 
# using the default lag.max.

CCorDistance(example.series3, example.series4)

# Calculate the cross-correlaion based distance 
# with lag.max=50.

CCorDistance(example.series3, example.series4, lag.max=50)

Compression-based Dissimilarity measure

Description

Computes the dissimilarity between two numeric series based on their size after compression.

Usage

CDMDistance(x, y, ...)

Arguments

Details

This is simply a wrapper for the diss.CDM function of package TSclust. As such, all the functionalities of the diss.CDM function are also available when using this function.

Value

Author(s)

Usue Mori, Alexander Mendiburu, Jose A. Lozano.

References

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

See Also

To calculate this distance measure using ts, zoo or xts objects see TSDistances. To calculate distance matrices of time series databases using this measure see TSDatabaseDistances.

Examples

# The objects example.series3 and example.series4 are two 
# numeric series of length 100 and 120.

data(example.series3)
data(example.series4)

# For information on their generation and shape see 
# help page of example.series.

help(example.series)

# Calculate the compression based distance between the two series using
# the default parameters. 

CDMDistance(example.series3, example.series4)

Complexity-Invariant Distance Measure For Time Series

Description

Computes the dissimilarity between two numeric series of the same length by calculating a correction of the Euclidean distance based on the complexity estimation of the series.

Usage

CIDDistance(x, y)

Arguments

Details

This is simply a wrapper for the diss.CID function of package TSclust. As such, all the functionalities of the diss.CID function are also available when using this function.

Value

Author(s)

Usue Mori, Alexander Mendiburu, Jose A. Lozano.

References

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

See Also

To calculate this distance measure using ts, zoo or xts objects see TSDistances. To calculate distance matrices of time series databases using this measure see TSDatabaseDistances.

Examples

# The objects example.series1 and example.series2 are two 
# numeric series of length 100.

data(example.series1)
data(example.series2)

# For information on their generation and shape see 
# help page of example.series.

help(example.series)

# Calculate the compression based distance between the two series using
# the default parameters. 

CIDDistance(example.series1, example.series2)

Dissimilarities based on Pearson's correlation

Description

Computes two different distance measure based on Pearson's correlation between a pair of numeric time series of the same length.

Usage

CorDistance(x, y, ...)

Arguments

Details

This is simply a wrapper for the diss.COR function of package TSclust. As such, all the functionalities of the diss.COR function are also available when using this function.

Value

Author(s)

Usue Mori, Alexander Mendiburu, Jose A. Lozano.

References

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

Golay, X., Kollias, S., Stoll, G., Meier, D., Valavanis, A., & Boesiger, P. (1998). A new correlation-based fuzzy logic clustering algorithm for FMRI. Magnetic Resonance in Medicine, 40(2), 249–260.

See Also

To calculate this distance measure using ts, zoo or xts objects see TSDistances. To calculate distance matrices of time series databases using this measure see TSDatabaseDistances.

Examples

# The objects example.series1 and example.series2 are two 
# numeric series of length 100.

data(example.series1)
data(example.series2)

# For information on their generation and shape see 
# help page of example.series.

help(example.series)

# Calculate the first correlation based distance between the series.

CorDistance(example.series1, example.series2)

# Calculate the second correlation based distance between the series
# by specifying \eqn{beta}.

CorDistance(example.series1, example.series2, beta=2)

Dissimilarity Index Combining Temporal Correlation and Raw Value Behaviors

Description

Computes the dissimilarity between two numeric series of the same length by combining the dissimilarity between the raw values and the dissimilarity between the temporal correlation behavior of the series.

Usage

CortDistance(x, y, ...)

Arguments

Details

This is simply a wrapper for the diss.CORT function of package TSclust. As such, all the functionalities of the diss.CORT function are also available when using this function.

Value

Author(s)

Usue Mori, Alexander Mendiburu, Jose A. Lozano.

References

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

Chouakria, A. D., & Nagabhushan, P. N. (2007). Adaptive dissimilarity index for measuring time series proximity. Advances in Data Analysis and Classification, 1(1), 5–21. http://doi.org/10.1007/s11634-006-0004-6

See Also

To calculate this distance measure using ts, zoo or xts objects see TSDistances. To calculate distance matrices of time series databases using this measure see TSDatabaseDistances.

Examples

# The objects example.series1 and example.series2 are two 
# numeric series of length 100.

data(example.series1)
data(example.series2)

# For information on their generation and shape see 
# help page of example.series.

help(example.series)

# Calculate the first correlation based distance between the series using the default 
# parameters.

CortDistance(example.series1, example.series2)

Dynamic Time Warping distance.

Description

Computes the Dynamic Time Warping distance between a pair of numeric time series.

Usage

DTWDistance(x, y, ...)

Arguments

Details

This is simply a wrapper for the dtw function of package dtw. As such, all the functionalities of the dtw function are also available when using this function.

Value

Author(s)

Usue Mori, Alexander Mendiburu, Jose A. Lozano.

References

Giorgino T (2009). Computing and Visualizing Dynamic Time Warping Alignments in R: The dtw Package. Journal of Statistical Software, 31(7), pp. 1-24. URL:http://www.jstatsoft.org/v31/i07/

Cuturi, M. (2011). Fast Global Alignment Kernels. In Proceedings of the 28th International Conference on Machine Learning (pp. 929–936).

Gaidon, A., Harchaoui, Z., & Schmid, C. (2011). A time series kernel for action recognition. In BMVC 2011 - British Machine Vision Conference (pp. 63.1–63.11).

Marteau, P.-F., & Gibet, S. (2014). On Recursive Edit Distance Kernels With Applications To Time Series Classification. IEEE Transactions on Neural Networks and Learning Systems, PP(6), 1–13.

Lei, H., & Sun, B. (2007). A Study on the Dynamic Time Warping in Kernel Machines. In 2007 Third International IEEE Conference on Signal-Image Technologies and Internet-Based System (pp. 839–845).

Pree, H., Herwig, B., Gruber, T., Sick, B., David, K., & Lukowicz, P. (2014). On general purpose time series similarity measures and their use as kernel functions in support vector machines. Information Sciences, 281, 478–495.

See Also

To calculate a lower bound of the DTW distance see LBKeoghDistance.

To calculate this distance measure using ts, zoo or xts objects see TSDistances. To calculate distance matrices of time series databases using this measure see TSDatabaseDistances.

Examples

# The objects example.series3 and example.series4 are two 
# numeric series of length 100 and 120 contained in the TSdist 
# package 


data(example.series3)
data(example.series4)

# For information on their generation and shape see 
# help page of example.series.

help(example.series)

# Calculate the basic DTW distance for two series of different length.

DTWDistance(example.series3, example.series4)

# Calculate the DTW distance for two series of different length
# with a sakoechiba window of size 30:

DTWDistance(example.series3, example.series4, window.type="sakoechiba", window.size=30)

# Calculate the DTW distance for two series of different length
# with an assymetric step pattern

DTWDistance(example.series3, example.series4, step.pattern=asymmetric)

The Dissim distance is calculated.

Description

Computes the Dissim distance between a pair of numeric series.

Usage

DissimDistance(x, y, tx, ty)

Arguments

Details

The Dissim distance is obtained by calculating the integral of the Euclidean distance between the two series. The series are assumed to be linear between sampling points.

The two series must start and end in the same interval but they may have different and non-constant sampling rates. These sampling rates must be positive and strictly increasing. For more information see reference below.

Value

Author(s)

Usue Mori, Alexander Mendiburu, Jose A. Lozano.

References

Frentzos, E., Gratsias, K., & Theodoridis, Y. (2007). Index-based Most Similar Trajectory Search. In Proceeding of the IEEE 23rd International Conference on Data Engineering (pp. 816-825).

Esling, P., & Agon, C. (2012). Time-series data mining. ACM Computing Surveys (CSUR), 45(1), 1–34.

See Also

To calculate this distance measure using ts, zoo or xts objects see TSDistances. To calculate distance matrices of time series databases using this measure see TSDatabaseDistances.

Examples

#The objects example.series1 and example.series2 are two 
#numeric series of length 100 contained in the TSdist package. 

data(example.series1)
data(example.series2)

#For information on their generation and shape see help 
#page of example.series.

help(example.series)

#Calculate the Dissim distance assuming even sampling:

DissimDistance(example.series1, example.series2)

#Calculate the Dissim distance assuming uneven sampling:

tx<-unique(c(seq(2, 175, 2), seq(7, 175, 7)))
tx <- tx[order(tx)]
ty <- tx
DissimDistance(example.series1, example.series2, tx, ty)

Edit Distance for Real Sequences (EDR).

Description

Computes the Edit Distance for Real Sequences between a pair of numeric time series.

Usage

EDRDistance(x, y, epsilon, sigma)

Arguments

Details

The basic Edit Distance for Real Sequences between two numeric series is calculated. The idea is to count the number of edit operations (insert, delete, replace) that are necessary to transform one series into the other.

For that, if the Euclidean distance between two points x_i and y_i is smaller that epsilon they will be considered equal (d=0) and if they are farther apart, they will be considered different (d=1). As a last detail, this distance permits gaps or sequences of points that are not matched with any other point.

The length of series x and y may be different. Furthermore, if desired, a temporal constraint may be added to the EDR distance. In this package, only the most basic windowing function, introduced by H.Sakoe and S.Chiba (1978), is implemented. This function sets a band around the main diagonal of the distance matrix and avoids the matching of the points that are farther in time than a specified \sigma.

The size of the window must be a positive integer value. Furthermore, the following condition must be fulfilled:

|length(x)-length(y)| < sigma

Value

Author(s)

Usue Mori, Alexander Mendiburu, Jose A. Lozano.

References

Chen, L., Ozsu, M. T., & Oria, V. (2005). Robust and Fast Similarity Search for Moving Object Trajectories. In Proceedings of the 2005 ACM SIGMOD International Conference on Management of Data (pp. 491-502).

Chen, L., & Ng, R. (2004). On The Marriage of Lp-norms and Edit Distance. In Proceedings of the Thirtieth International Conference on Very Large Data Bases (pp. 792–803).

Cuturi, M. (2011). Fast Global Alignment Kernels. In Proceedings of the 28th International Conference on Machine Learning (pp. 929–936).

Gaidon, A., Harchaoui, Z., & Schmid, C. (2011). A time series kernel for action recognition. In BMVC 2011 - British Machine Vision Conference (pp. 63.1–63.11).

Marteau, P.-F., & Gibet, S. (2014). On Recursive Edit Distance Kernels With Applications To Time Series Classification. IEEE Transactions on Neural Networks and Learning Systems, PP(6), 1–13.

Lei, H., & Sun, B. (2007). A Study on the Dynamic Time Warping in Kernel Machines. In 2007 Third International IEEE Conference on Signal-Image Technologies and Internet-Based System (pp. 839–845).

Pree, H., Herwig, B., Gruber, T., Sick, B., David, K., & Lukowicz, P. (2014). On general purpose time series similarity measures and their use as kernel functions in support vector machines. Information Sciences, 281, 478–495.

See Also

To calculate this distance measure using ts, zoo or xts objects see TSDistances. To calculate distance matrices of time series databases using this measure see TSDatabaseDistances.

Examples

# The objects example.series3 and example.series4 are two 
# numeric series of length 100 and 120 contained in the TSdist
# package. 

data(example.series3)
data(example.series4)

# For information on their generation and shape see 
# help page of example.series.

help(example.series)

# Calculate the EDR distance for two series of different length
# with no windowing constraint:

EDRDistance(example.series3, example.series4, epsilon=0.1)

# Calculate the EDR distance for two series of different length
# with a window of size 30:

EDRDistance(example.series3, example.series4, epsilon=0.1, sigma=30)

Edit Distance with Real Penalty (ERP).

Description

Computes the Edit Distance with Real Penalty between a pair of numeric time series.

Usage

ERPDistance(x, y, g, sigma)

Arguments

Details

The basic Edit Distance with Real Penalty between two numeric series is calculated. Unlike other edit based distances included in this package, this distance is a metric and fulfills the triangle inequality.

The idea is to search for the minimal path in a distance matrix that describes the mapping between the two series. This distance matrix is built by using the Euclidean distance. However, unlike DTW, this distance permits gaps or sequences of points that are not matched with any other point. These gaps will be penalized based on the distance of the unmatched points from a reference value g.

As with other edit based distances, the length of x and y may be different.

Furthermore, if desired, a temporal constraint may be added to the ERP distance. In this package, only the most basic windowing function, introduced by H.Sakoe and S.Chiba (1978), is implemented. This function sets a band around the main diagonal of the distance matrix and avoids the matching of the points that are farther in time than a specified \sigma.

The size of the window must be a positive integer value. Furthermore, the following condition must be fulfilled:

|length(x)-length(y)| < sigma

Value

Author(s)

Usue Mori, Alexander Mendiburu, Jose A. Lozano.

References

Chen, L., & Ng, R. (2004). On The Marriage of Lp-norms and Edit Distance. In Proceedings of the Thirtieth International Conference on Very Large Data Bases (pp. 792-803).

See Also

To calculate this distance measure using ts, zoo or xts objects see TSDistances. To calculate distance matrices of time series databases using this measure see TSDatabaseDistances.

Examples

#The objects example.series3 and example.series4 are two 
#numeric series of length 100 and 120 contained in the TSdist  
#package. 

data(example.series3)
data(example.series4)

#For information on their generation and shape see 
#help page of example.series.

help(example.series)

#Calculate the ERP distance for two series of different length
#with no windowing constraint:

ERPDistance(example.series3, example.series4, g=0)

#Calculate the ERP distance for two series of different length
#with a window of size 30:

ERPDistance(example.series3, example.series4, g=0, sigma=30)

Euclidean distance.

Description

Computes the Euclidean distance between a pair of numeric vectors.

Usage

EuclideanDistance(x, y)

Arguments

Details

The Euclidean distance is computed between the two numeric series using the following formula:

D=\sqrt{(x_i - y_i) ^ 2)}

The two series must have the same length. This distance is calculated with the help of the dist function of the proxy package.

Value

Author(s)

Usue Mori, Alexander Mendiburu, Jose A. Lozano.

References

David Meyer and Christian Buchta (2015). proxy: Distance and Similarity Measures. R package version 0.4-14. http://CRAN.R-project.org/package=proxy

See Also

This function can also be invoked by the wrapper function LPDistance.

Furthermore, to calculate this distance measure using ts, zoo or xts objects see TSDistances. To calculate distance matrices of time series databases using this measure see TSDatabaseDistances.

Examples

# The objects example.series1 and example.series2 are two 
# numeric series of length 100 contained in the TSdist package. 

data(example.series1)
data(example.series2)

# For information on their generation and shape see help 
# page of example.series.

help(example.series)

# Compute the Euclidean distance between them:

EuclideanDistance(example.series1, example.series2)

Fourier Coefficient based distance.

Description

Computes the distance between a pair of numerical series based on their Discrete Fourier Transforms.

Usage

FourierDistance(x, y, n = (floor(length(x) / 2) + 1))

Arguments

Details

The Euclidean distance between the first n Fourier coefficients of series x and y is computed. The series must have the same length. Furthermore, n should not be larger than the length of the series.

Value

Author(s)

Usue Mori, Alexander Mendiburu, Jose A. Lozano.

References

Agrawal, R., Faloutsos, C., & Swami, A. (1993). Efficient similarity search in sequence databases. In Proceedings of the 4th International Conference of Foundations of Data Organization and Algorithms (Vol. 5, pp. 69-84).

See Also

To calculate this distance measure using ts, zoo or xts objects see TSDistances. To calculate distance matrices of time series databases using this measure see TSDatabaseDistances.

Examples

# The objects example.series1 and example.series2 are two 
# numeric series of length 100 contained in the TSdist package. 

data(example.series1)
data(example.series2)

# For information on their generation and shape see help 
# page of example.series.

help(example.series)

# Calculate the Fourier coefficient based  distance using 
# the default number of coefficients:

FourierDistance(example.series1, example.series2)

# Calculate the Fourier coefficient based  distance using 
# only the first 20 Fourier coefficients:

FourierDistance(example.series1, example.series2, n=20)

Frechet distance

Description

Computes the Frechet distance between two numerical trajectories.

Usage

FrechetDistance(x, y, tx, ty, ...)

Arguments

Details

This is essentially a wrapper for the distFrechet function of package longitudinalData. As such, all the functionalities of the distFrechet function are also available when using this function.

Value

Author(s)

Usue Mori, Alexander Mendiburu, Jose A. Lozano.

References

Christophe Genolini (2014). longitudinalData: Longitudinal Data. R package version 2.2. http://CRAN.R-project.org/package=longitudinalData

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

Eiter, T., & Mannila, H. (1994). Computing Discrete Frechet Distance. Technical Report. Retrieved from http://www.kr.tuwien.ac.at/staff/eiter/et-archive/cdtr9464.pdf

See Also

To calculate this distance measure using ts, zoo or xts objects see TSDistances. To calculate distance matrices of time series databases using this measure see TSDatabaseDistances.

Examples

# The objects example.serie3 and example.series4 are two 
# numeric series of length 100 and 120, respectively. 

data(example.series3)
data(example.series4)

# For information on their generation and shape see 
# help page of example.series.

help(example.series)

# Calculate the distance based on wavelet feature extraction between the series.

## Not run: FrechetDistance(example.series3, example.series4)

The infinite norm distance.

Description

Computes the infinite norm distance between a pair of numeric vectors.

Usage

InfNormDistance(x, y)

Arguments

Details

The infinite norm distance is computed between the two numeric series using the following formula:

D=\max{|x_i-y_i|}

The two series must have the same length. This distance is calculated with the help of the dist function of the proxy package.

Value

Author(s)

Usue Mori, Alexander Mendiburu, Jose A. Lozano.

References

David Meyer and Christian Buchta (2015). proxy: Distance and Similarity Measures. R package version 0.4-14. http://CRAN.R-project.org/package=proxy

See Also

This function can also be invoked by the wrapper function LPDistance.

To calculate this distance measure using ts, zoo or xts objects see TSDistances. To calculate distance matrices of time series databases using this measure see TSDatabaseDistances.

Examples

# The objects example.series1 and example.series2 are two 
# numeric series of length 100 contained in the TSdist package. 

data(example.series1)
data(example.series2)

# For information on their generation and shape see help 
# page of example.series.

help(example.series)


# Compute the  infinite norm distance between them:

InfNormDistance(example.series1, example.series2)

Integrated Periodogram based dissimilarity

Description

Calculates the dissimilarity between two numerical series of the same length based on the distance between their integrated periodograms.

Usage

IntPerDistance(x, y, ...)

Arguments

Details

This is simply a wrapper for the diss.INT.PER function of package TSclust. As such, all the functionalities of the diss.INT.PER function are also available when using this function.

Value

Author(s)

Usue Mori, Alexander Mendiburu, Jose A. Lozano.

References

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

See Also

To calculate this distance measure using ts, zoo or xts objects see TSDistances. To calculate distance matrices of time series databases using this measure see TSDatabaseDistances.

Examples

# The objects example.series1 and example.series2 are two 
# numeric series of length 100. 

data(example.series1)
data(example.series2)

# For information on their generation and shape see 
# help page of example.series.

help(example.series)

# Calculate the ar.mah distance between the two series using
# the default parameters. 

IntPerDistance(example.series1, example.series2)

K medoids clustering for a time series database using the selected distance measure.

Description

Given a specific distance measure and a time series database, this function provides the K-medoids clustering result. Furthermore, if the ground truth clustering is provided, and the associated F-value is also provided.

Usage

KMedoids(data,  k, ground.truth, distance, ...)

Arguments

Details

This function is useful to evaluate the performance of different distance measures in the task of clustering time series.

Value

Author(s)

Usue Mori, Alexander Mendiburu, Jose A. Lozano.

See Also

To calculate the distance matrices of time series databases the TSDatabaseDistances is used.

Examples

# The example.database3 synthetic database is loaded
data(example.database3)
tsdata <- example.database3[[1]]
groundt <- example.database3[[2]]

# Apply K-medoids clusterning for different distance measures

KMedoids(data=tsdata, ground.truth=groundt, k=5, "euclidean")
KMedoids(data=tsdata, ground.truth=groundt, k=5, "cid")
KMedoids(data=tsdata, ground.truth=groundt, k=5, "pdc")

LB_Keogh for DTW.

Description

Computes the Keogh lower bound for the Dynamic Time Warping distance between a pair of numeric time series.

Usage

LBKeoghDistance(x, y, window.size)

Arguments

Details

The lower bound introduced by Keogh and Ratanamahatana (2005) is calculated for the Dynamic Time Warping distance. Given window.size, the width of a Sakoe-Chiba band, an upper and lower envelope of the query time series is calculated in the following manner:

U[i] = max(x[i - window.size], x[i + window.size])

L[i] = min(x[i - window.size], x[i + window.size])

Based on this, the Keogh_LB distance is calculated as the Euclidean distance between the points in the reference time series (y) that fall outside both the lower and upper envelopes, and their nearest point of the corresponding envelope.

The series must have the same length. Furthermore, the width of the window should be even in order to assure a symmetric band around the diagonal and should not exceed the length of the series.

Value

Author(s)

Usue Mori, Alexander Mendiburu, Jose A. Lozano.

References

Keogh, E., & Ratanamahatana, C. A. (2004). Exact indexing of dynamic time warping. Knowledge and Information Systems, 7(3), 358-386.

Sakoe, H., & Chiba, S. (1978). Dynamic programming algorithm optimization for spoken word recognition. IEEE Transactions on Acoustics, Speech, and Signal Processing, 26(1), 43-49.

Esling, P., & Agon, C. (2012). Time-series data mining. ACM Computing Surveys (CSUR), 45(1), 1–34.

See Also

To calculate the full DTW distance see DTWDistance.

To calculate this distance measure using ts, zoo or xts objects see TSDistances. To calculate distance matrices of time series databases using this measure see TSDatabaseDistances.

Examples

# The objects example.series1 and example.series2 are two 
# numeric series of length 100 contained in the TSdist package. 

data(example.series1)
data(example.series2)

# For information on their generation and shape see help 
# page of example.series.

help(example.series)

# Calculate the LB_Keogh distance measure for these two series
# with a window of band of width 11:

LBKeoghDistance(example.series1, example.series2, window.size=11)

Longest Common Subsequence distance for Real Sequences.

Description

Computes the Longest Common Subsequence distance between a pair of numeric time series.

Usage

LCSSDistance(x, y, epsilon, sigma)

Arguments

Details

The Longest Common Subsequence for two real sequences is computed.

For this purpose, the distances between the points of x and y are reduced to 0 or 1. If the Euclidean distance between two points x_i and y_j is smaller than epsilon they are considered equal and their distance is reduced to 0. In the opposite case, the distance between them is represented with a value of 1.

Once the distance matrix is defined in this manner, the maximum common subsequence is seeked. Of course, as in other Edit Based Distances, gaps or unmatched regions are permitted and they are penalized with a value proportional to their length.

Based on its definition, the length of series x and y may be different.

If desired, a temporal constraint may be added to the LCSS distance. In this package, only the most basic windowing function, introduced by H.Sakoe and S.Chiba (1978), is implemented. This function sets a band around the main diagonal of the distance matrix and avoids the matching of the points that are farther in time than a specified \sigma.

The size of the window must be a positive integer value. Furthermore, the following condition must be fulfilled:

|length(x)-length(y)| < sigma

Value

Author(s)

Usue Mori, Alexander Mendiburu, Jose A. Lozano.

References

Vlachos, M., Kollios, G., & Gunopulos, D. (2002). Discovering similar multidimensional trajectories. In Proceedings 18th International Conference on Data Engineering (pp. 673-684). IEEE Comput. Soc. doi:10.1109/ICDE.2002.994784

Chen, L., & Ng, R. (2004). On The Marriage of Lp-norms and Edit Distance. In Proceedings of the Thirtieth International Conference on Very Large Data Bases (pp. 792–803).

Cuturi, M. (2011). Fast Global Alignment Kernels. In Proceedings of the 28th International Conference on Machine Learning (pp. 929–936).

Gaidon, A., Harchaoui, Z., & Schmid, C. (2011). A time series kernel for action recognition. In BMVC 2011 - British Machine Vision Conference (pp. 63.1–63.11).

Marteau, P.-F., & Gibet, S. (2014). On Recursive Edit Distance Kernels With Applications To Time Series Classification. IEEE Transactions on Neural Networks and Learning Systems, PP(6), 1–13.

Lei, H., & Sun, B. (2007). A Study on the Dynamic Time Warping in Kernel Machines. In 2007 Third International IEEE Conference on Signal-Image Technologies and Internet-Based System (pp. 839–845).

Pree, H., Herwig, B., Gruber, T., Sick, B., David, K., & Lukowicz, P. (2014). On general purpose time series similarity measures and their use as kernel functions in support vector machines. Information Sciences, 281, 478–495.

See Also

To calculate this distance measure using ts, zoo or xts objects see TSDistances. To calculate distance matrices of time series databases using this measure see TSDatabaseDistances.

Examples

# The objects example.series3 and example.series4 are two 
# numeric series of length 100 and 120 contained in the TSdist 
# package. 


data(example.series3)
data(example.series4)

# For information on their generation and shape see 
# help page of example.series.

help(example.series)

# Calculate the LCSS distance for two series of different length
# with no windowing constraint:

LCSSDistance(example.series3, example.series4, epsilon=0.1)

# Calculate the LCSS distance for two series of different length
# with a window of size 30:

LCSSDistance(example.series3, example.series4, epsilon=0.1, sigma=30)

Lp distances.

Description

Computes the distance based on the chosen Lp norm between a pair of numeric vectors.

Usage

LPDistance(x, y, method="euclidean", ...)

Arguments

Details

The distances based on Lp norms are computed between two numeric vectors using the following formulas:

Euclidean distance: \sqrt{(x_i-y_i)^2)}

Manhattan distance: \sum{|x_i-y_i|}

Infinite norm distance: \max{|x_i-y_i|}

Minkowski distance: \sqrt[p]{(x_i-y_i)^p)}

The two series must have the same length. Furthermore, in the case of the Minkowski distance, p must be specified as a positive integer value.

Value

Author(s)

Usue Mori, Alexander Mendiburu, Jose A. Lozano.

See Also

These distances are also implemeted in separate functions. For more information see EuclideanDistance, ManhattanDistance, MinkowskiDistance and InfNormDistance

To calculate this distance measure using ts, zoo or xts objects see TSDistances. To calculate distance matrices of time series databases using this measure see TSDatabaseDistances.

Examples

# The objects example.series1 and example.series2 are two 
# numeric series of length 100 contained in the TSdist package. 

data(example.series1)
data(example.series2)

# For information on their generation and shape see help 
# page of example.series.

help(example.series)

# Compute the different Lp distances

# Euclidean distance

LPDistance(example.series1, example.series2, method="euclidean")

# Manhattan distance

LPDistance(example.series1, example.series2, method="manhattan")

# Infinite norm distance

LPDistance(example.series1, example.series2, method="infnorm")

# Minkowski distance with p=3.

LPDistance(example.series1, example.series2, method="minkowski", p=3)

Manhattan distance.

Description

Computes the Manhattan distance between a pair of numeric vectors.

Usage

ManhattanDistance(x, y)

Arguments

Details

The Manhattan distance is computed between the two numeric series using the following formula:

D=\sum{|x_i-y_i|}

The two series must have the same length. This distance is calculated with the help of the dist function of the proxy package.

Value

Author(s)

Usue Mori, Alexander Mendiburu, Jose A. Lozano.

References

David Meyer and Christian Buchta (2015). proxy: Distance and Similarity Measures. R package version 0.4-14. http://CRAN.R-project.org/package=proxy

See Also

This function can also be invoked by the wrapper function LPDistance.

Furthermore, to calculate this distance measure using ts, zoo or xts objects see TSDistances. To calculate distance matrices of time series databases using this measure see TSDatabaseDistances.

Examples

# The objects example.series1 and example.series2 are two 
# numeric series of length 100 contained in the TSdist package. 

data(example.series1)
data(example.series2)

# For information on their generation and shape see help 
# page of example.series.

help(example.series)

# Compute the Manhattan distance between them:

ManhattanDistance(example.series1, example.series2)

Symbolic Aggregate Aproximation based dissimilarity

Description

Calculates the dissimilarity between two numerical series based on the distance between their SAX representations.

Usage

MindistSaxDistance(x, y, w, ...)

Arguments

Details

This is simply a wrapper for the diss.MINDIST.SAX function of package TSclust. As such, all the functionalities of the diss.MINDIST.SAX function are also available when using this function.

Value

Author(s)

Usue Mori, Alexander Mendiburu, Jose A. Lozano.

References

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

See Also

To calculate this distance measure using ts, zoo or xts objects see TSDistances. To calculate distance matrices of time series databases using this measure see TSDatabaseDistances.

Examples

# The objects example.series3 and example.series4 are two 
# numeric series of length 100 and 120 respectively. 

data(example.series3)
data(example.series4)

# For information on their generation and shape see 
# help page of example.series.

help(example.series)

# Calculate the mindist.sax distance between the two series using
# 20 equal sized windows for each series. The rest of the parameters
# are left in their default mode.

MindistSaxDistance(example.series3, example.series4, w=20)

Minkowski distance.

Description

Computes the Minkowski distance between two numeric vectors for a given p.

Usage

MinkowskiDistance(x, y, p)

Arguments

Details

The Minkowski distance is computed between the two numeric series using the following formula:

D=\sqrt[p]{(x_i-y_i)^p)}

The two series must have the same length and p must be a positive integer value. This distance is calculated with the help of the dist function of the proxy package.

Value

Author(s)

Usue Mori, Alexander Mendiburu, Jose A. Lozano.

References

David Meyer and Christian Buchta (2015). proxy: Distance and Similarity Measures. R package version 0.4-14. http://CRAN.R-project.org/package=proxy

See Also

This function can also be invoked by the wrapper function LPDistance.

Furthermore, to calculate this distance measure using ts, zoo or xts objects see TSDistances. To calculate distance matrices of time series databases using this measure see TSDatabaseDistances.

Examples

# The objects example.series1 and example.series2 are two 
# numeric series of length 100 contained in the TSdist package. 

data(example.series1)
data(example.series2)

# For information on their generation and shape see help 
# page of example.series.

help(example.series)


# Compute the Minkowski distance between them:

MinkowskiDistance(example.series1, example.series2, p=3)

Normalized Compression based distance

Description

Calculates a normalized distance between two numerical series based on their compressed sizes.

Usage

NCDDistance(x, y, ...)

Arguments

Details

This is simply a wrapper for the diss.NCD function of package TSclust. As such, all the functionalities of the diss.NCD function are also available when using this function.

Value

Author(s)

Usue Mori, Alexander Mendiburu, Jose A. Lozano.

References

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

See Also

To calculate this distance measure using ts, zoo or xts objects see TSDistances. To calculate distance matrices of time series databases using this measure see TSDatabaseDistances.

Examples

# The objects example.series3 and example.series4 are two 
# numeric series of length 100 and 120 respectively. 

data(example.series3)
data(example.series4)

# For information on their generation and shape see 
# help page of example.series.

help(example.series)

# Calculate the normalized compression based distance between the two series 
# using default parameter.

NCDDistance(example.series3, example.series4)

1NN classification for a pair of train/test time series datasets.

Description

Given a specific distance measure, this function provides the 1NN classification values and the associated error for a specific train/test pair of time series databases.

Usage

OneNN(train, trainc, test, testc, distance, ...)

Arguments

Details

This function is useful to evaluate the performance of different distance measures in the task of classification of time series.

Value

Author(s)

Usue Mori, Alexander Mendiburu, Jose A. Lozano.

See Also

To calculate the distance matrices of time series databases the TSDatabaseDistances is used.

Examples

# The example.database2 synthetic database is loaded
data(example.database2)

# Create train/test by dividing the dataset 70%-30%
set.seed(100)
trainindex <- sample(1:100, 70, replace=FALSE)
train <- example.database2[[1]][trainindex, ]
test <- example.database2[[1]][-trainindex, ]
trainclass <- example.database2[[2]][trainindex]
testclass <- example.database2[[2]][-trainindex]

# Apply the 1NN classifier for different distance measures

OneNN(train, trainclass, test, testclass, "euclidean")
OneNN(train, trainclass, test, testclass, "pdc")

Partial Autocorrelation-based Dissimilarity

Description

Computes the dissimilarity between a pair of numeric time series based on their estimated partial autocorrelation coefficients.

Usage

PACFDistance(x, y, ...)

Arguments

Details

This is simply a wrapper for the diss.PACF function of package TSclust. As such, all the functionalities of the diss.PACF function are also available when using this function.

Value

Author(s)

Usue Mori, Alexander Mendiburu, Jose A. Lozano.

References

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

See Also

To calculate this distance measure using ts, zoo or xts objects see TSDistances. To calculate distance matrices of time series databases using this measure see TSDatabaseDistances.

Examples

# The objects example.series3 and example.series4 are two 
# numeric series of length 100 and 120 contained in the 
# TSdist package. 

data(example.series3)
data(example.series4)

# For information on their generation and shape see 
# help page of example.series.

help(example.series)

# Calculate the autocorrelation based distance between the two series using
# the default parameters:

PACFDistance(example.series3, example.series4)

Permutation Distribution Distance

Description

Calculates the permutation distribution distance between two numerical series of the same length.

Usage

PDCDistance(x, y, ...)

Arguments

Details

This is simply a wrapper for the pdcDist function of package pdc. As such, all the functionalities of the pdcDist function are also available when using this function.

Value

Author(s)

Usue Mori, Alexander Mendiburu, Jose A. Lozano.

References

Andreas M. Brandmaier (2015). pdc: An R package for Complexity-Based Clustering of Time Series. Journal of Statistical Software, Vol 67, Issue 5.

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

See Also

To calculate this distance measure using ts, zoo or xts objects see TSDistances. To calculate distance matrices of time series databases using this measure see TSDatabaseDistances.

Examples

# The objects example.series1 and example.series2 are two 
# numeric series of length 100. 

data(example.series1)
data(example.series2)

# For information on their generation and shape see 
# help page of example.series.

help(example.series)

# Calculate the normalized compression based distance between the two series 
# using the default parameters.

PDCDistance(example.series1, example.series2)

Periodogram based dissimilarity

Description

Calculates the dissimilarity between two numerical series of the same length based on the distance between their periodograms.

Usage

PerDistance(x, y, ...)

Arguments

Details

This is simply a wrapper for the diss.PER function of package TSclust. As such, all the functionalities of the diss.PER function are also available when using this function.

Value

Author(s)

Usue Mori, Alexander Mendiburu, Jose A. Lozano.

References

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

See Also

To calculate this distance measure using ts, zoo or xts objects see TSDistances. To calculate distance matrices of time series databases using this measure see TSDatabaseDistances.

Examples

# The objects example.series1 and example.series2 are two 
# numeric series of length 100. 

data(example.series1)
data(example.series2)

# For information on their generation and shape see 
# help page of example.series.

help(example.series)

# Calculate the ar.mah distance between the two series using
# the default parameters. 

PerDistance(example.series1, example.series2)

Dissimilarity Measure Based on Nonparametric Forecasts

Description

The dissimilarity of two numerical series of the same length is calculated based on the L1 distance between the kernel estimators of their forecast densities at a given time horizon.

Usage

PredDistance(x, y, h, ...)

Arguments

Details

This is simply a wrapper for the diss.PRED function of package TSclust. As such, all the functionalities of the diss.PRED function are also available when using this function.

Value

Author(s)

Usue Mori, Alexander Mendiburu, Jose A. Lozano.

References

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

See Also

To calculate this distance measure using ts, zoo or xts objects see TSDistances. To calculate distance matrices of time series databases using this measure see TSDatabaseDistances.

Examples

# The objects example.series1 and example.series2 are two 
# numeric series of length 100. 

data(example.series1)
data(example.series2)

# For information on their generation and shape see 
# help page of example.series.

help(example.series)

# Calculate the prediction based distance between the two series using
# the default parameters. 

PredDistance(example.series1, example.series2)

Short time series distance (STS).

Description

Computes the Short Time Series Distance between a pair of numeric time series.

Usage

STSDistance(x, y, tx, ty)

Arguments

Details

The short time series distance between two series is designed specially for series with an equal but uneven sampling rate. However, it can also be used for time series with a constant sampling rate. It is calculated as follows:

STS= \sqrt{\sum_{k=\{1,...,N-1\}} (((y_{k+1} - y_{k}) / (tx_{k+1} - tx_{k}) - (x_{k+1} - x_{k}) / (ty_{k+1} - ty_{k}))^2)}

where N is the length of series x and y.

tx and ty must be positive and strictly increasing. Furthermore, the sampling rate in both indexes must be equal:

tx[k+1]-tx[k]=ty[k+1]-ty[k], \, \, \, for \, \, \, k=0 ,..., N-1

Value

Author(s)

Usue Mori, Alexander Mendiburu, Jose A. Lozano.

References

Möller-Levet, C. S., Klawonn, F., Cho, K., & Wolkenhauer, O. (2003). Fuzzy Clustering of Short Time-Series and Unevenly Distributed Sampling Points. In Proceedings of the 5th International Symposium on Intelligent Data Analysis.

See Also

To calculate this distance measure using ts, zoo or xts objects see TSDistances. To calculate distance matrices of time series databases using this measure see TSDatabaseDistances.

Examples

# The objects example.series1 and example.series2 are two 
# numeric series of length 100 contained in the TSdist package. 

data(example.series1)
data(example.series2)

# For information on their generation and shape see help 
# page of example.series.

help(example.series)

# Calculate the STS distance assuming even sampling:

STSDistance(example.series1, example.series2)

# Calculate the STS distance providing an uneven sampling:

tx<-unique(c(seq(2, 175, 2), seq(7, 175, 7)))
tx <- tx[order(tx)]
ty <- tx
STSDistance(example.series1, example.series2, tx, ty)

Dissimilarity based on the Generalized Likelihood Ratio Test

Description

The dissimilarity of two numerical series of the same length is calculated based on an adaptation of the generalized likelihood ratio test.

Usage

SpecGLKDistance(x, y, ...)

Arguments

Details

This function simply intends to be a wrapper for the diss.SPEC.GLK function of package TSclust. However, in the 1.2.3 version of the TSclust package we have found an error in the call to this function. As such, in this version, the more general diss function, designed for distance matrix calculations of time series databases, is used to calculate the spec.glk distance between two series. Once this bug is fixed in the original package, we will update our call procedure.

Value

Author(s)

Usue Mori, Alexander Mendiburu, Jose A. Lozano.

References

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

See Also

To calculate this distance measure using ts, zoo or xts objects see TSDistances. To calculate distance matrices of time series databases using this measure see TSDatabaseDistances.

Examples

# The objects example.series1 and example.series2 are two 
# numeric series of length 100. 

data(example.series1)
data(example.series2)

# For information on their generation and shape see 
# help page of example.series.

help(example.series)

# Calculate the ar.mah distance between the two series using
# the default parameters. 

SpecGLKDistance(example.series1, example.series2)

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

Description

The dissimilarity of two numerical series of the same length is calculated based on the integrated squared difference between the non-parametric estimators of their log-spectra.

Usage

SpecISDDistance(x, y, ...)

Arguments

Details

This is simply a wrapper for the diss.SPEC.ISD function of package TSclust. As such, all the functionalities of the diss.SPEC.ISD function are also available when using this function.

Value

Author(s)

Usue Mori, Alexander Mendiburu, Jose A. Lozano.

References

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

See Also

To calculate this distance measure using ts, zoo or xts objects see TSDistances. To calculate distance matrices of time series databases using this measure see TSDatabaseDistances.

Examples

# The objects example.series1 and example.series2 are two 
# numeric series of length 100. 

data(example.series1)
data(example.series2)

# For information on their generation and shape see 
# help page of example.series.

help(example.series)

# Calculate the spec.isd distance between the two series using
# the default parameters. 

SpecISDDistance(example.series1, example.series2)

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

Description

The dissimilarity of two numerical series of the same length is calculated based on the ratio between local linear estimations of the log-spectras.

Usage

SpecLLRDistance(x, y, ...)

Arguments

Details

This is simply a wrapper for the diss.SPEC.LLR function of package TSclust. As such, all the functionalities of the diss.SPEC.LLR function are also available when using this function.

Value

Author(s)

Usue Mori, Alexander Mendiburu, Jose A. Lozano.

References

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

See Also

To calculate this distance measure using ts, zoo or xts objects see TSDistances. To calculate distance matrices of time series databases using this measure see TSDatabaseDistances.

Examples

# The objects example.series1 and example.series2 are two 
# numeric series of length 100. 

data(example.series1)
data(example.series2)

# For information on their generation and shape see 
# help page of example.series.

help(example.series)

# Calculate the spec.isd distance between the two series using
# the default parameters. 

SpecLLRDistance(example.series1, example.series2)

Time Alignment Measurement (TAM) distance.

Description

Computes the Time Alignment Measurement between a pair of numeric time series.

Usage

TAMDistance(x, y)

Arguments

Details

The Time Alignment Measurement (TAM) between two numeric series is calculated. Quantifies the degree of temporal distortion between two time series.

The main idea behind TAM is to measure the warping cost between a given time series and another. TAM is calculated from the optimal alignment warping path between two time series provided by dtw, which allows characterizing the intervals when the series are in phase, advance or delay. This distance penalizes signals where advance or delay is present and benefits series that are in phase with each other. As the distance increases, the dissimilarity between both signals also increases. The distance is bounded between 0 (both series are in phase) and 3 (both series are completely out-of-phase).

The length of series x and y may be different.

Value

Author(s)

Duarte Folgado

References

Duarte Folgado, Marília Barandas, Ricardo Matias, Rodrigo Martins, Miguel Carvalho, Hugo Gamboa (2018). Time Alignment Measurement for Time Series. Pattern Recognition 81, pp. 268-279.

See Also

To calculate this distance measure using ts, zoo or xts objects see TSDistances. To calculate distance matrices of time series databases using this measure see TSDatabaseDistances.

Examples

# The objects example.series1 and example.series2 are two 
# numeric series of length 100 contained in the TSdist package. 

data(example.series1)
data(example.series2)

# The objects example.series3 and example.series4 are two 
# numeric series of length 100 and 120 contained in the TSdist
# package. 

data(example.series3)
data(example.series4)

# For information on their generation and shape see 
# help page of example.series.

help(example.series)

# Calculate the TAM distance for two series of the same length:

TAMDistance(example.series1, example.series2)

# Calculate the TAM distance for two series of different length:

TAMDistance(example.series3, example.series4)

TSdist distance matrix computation.

Description

TSdist distance matrix computation for time series databases.

Usage

TSDatabaseDistances(X, Y=NULL, distance, ...)

Arguments

f

Details

The distance matrix of a time series database is calculated by providing the pair-wise distances between the series that conform it. x can be saved in a numeric matrix, a list or a mts, zoo or xts object. The following distance methods are supported:

• "euclidean": Euclidean distance. EuclideanDistance

• "manhattan": Manhattan distance. ManhattanDistance

• "minkowski": Minkowski distance. MinkowskiDistance

• "infnorm": Infinite norm distance. InfNormDistance

• "ccor": Distance based on the cross-correlation. CCorDistance

• "sts": Short time series distance. STSDistance

• "dtw": Dynamic Time Warping distance. DTWDistance. Uses the dtw package (see dtw).

• "lb.keogh": LB_Keogh lower bound for the Dynamic Time Warping distance. LBKeoghDistance

• "edr": Edit distance for real sequences. EDRDistance

• "erp": Edit distance with real penalty. ERPDistance

• "lcss": Longest Common Subsequence Matching. LCSSDistance

• "fourier": Distance based on the Fourier Discrete Transform. FourierDistance

• "tquest": TQuest distance. TquestDistance

• "dissim": Dissim distance. DissimDistance

• "acf": Autocorrelation-based dissimilarity ACFDistance. Uses the TSclust package (see diss.ACF).

• "pacf": Partial autocorrelation-based dissimilarity PACFDistance. Uses the TSclust package (see diss.PACF).

• "ar.lpc.ceps": Dissimilarity based on LPC cepstral coefficients ARLPCCepsDistance. Uses the TSclust package (see diss.AR.LPC.CEPS).

• "ar.mah": Model-based dissimilarity proposed by Maharaj (1996, 2000) ARMahDistance. Uses the TSclust package (see diss.AR.MAH).

• "ar.pic": Model-based dissimilarity measure proposed by Piccolo (1990) ARPicDistance. Uses the TSclust package (see diss.AR.PIC).

• "cdm": Compression-based dissimilarity measure CDMDistance. Uses the TSclust package (see diss.CDM).

• "cid": Complexity-invariant distance measure CIDDistance. Uses the TSclust package (see diss.CID).

• "cor": Dissimilarities based on Pearson's correlation CorDistance. Uses the TSclust package (see diss.COR).

• "cort": Dissimilarity index which combines temporal correlation and raw value behaviors CortDistance. Uses the TSclust package (see diss.CORT).

• "int.per": Integrated periodogram based dissimilarity IntPerDistance. Uses the TSclust package (see diss.INT.PER).

• "per": Periodogram based dissimilarity PerDistance. Uses the TSclust package (see diss.PER).

• "mindist.sax": Symbolic Aggregate Aproximation based dissimilarity MindistSaxDistance. Uses the TSclust package (see diss.MINDIST.SAX).

• "ncd": Normalized compression based distance NCDDistance. Uses the TSclust package (see diss.NCD).

• "pred": Dissimilarity measure cased on nonparametric forecasts PredDistance. Uses the TSclust package (see diss.PRED).

• "spec.glk": Dissimilarity based on the generalized likelihood ratio test SpecGLKDistance. Uses the TSclust package (see diss.SPEC.GLK).

• "spec.isd": Dissimilarity based on the integrated squared difference between the log-spectra SpecISDDistance. Uses the TSclust package (see diss.SPEC.ISD).

• "spec.llr": General spectral dissimilarity measure using local-linear estimation of the log-spectra SpecLLRDistance. Uses the TSclust package (see diss.SPEC.LLR).

• "pdc": Permutation Distribution Distance PDCDistance. Uses the pdc package (see pdcDist).

• "frechet": Frechet distance FrechetDistance. Uses the longitudinalData package (see distFrechet).

• "tam": Time Aligment Measurement TAMDistance.

Some distance measures may require additional arguments. See the individual help pages (detailed above) for more information about each method. These parameters should be named in order to avoid mismatches.

Finally, for options dissim, dissimapprox and sts, databases conformed of series with different sampling rates can be introduced as a list of zoo, xts or ts objects, where each element in the list is a time series with its own time index.

Value

Author(s)

Usue Mori, Alexander Mendiburu, Jose A. Lozano.

Examples

# The object example.database is a numeric matrix that saves
# 6 ARIMA time series in a row-wise format. For more information
# see help page of example.databases:

help(example.database)
data(example.database)

# To calculate the distance matrix of this database: 

TSDatabaseDistances(example.database, distance="manhattan")
TSDatabaseDistances(example.database, distance="edr", epsilon=0.2)
TSDatabaseDistances(example.database, distance="fourier", n=20)

# The object zoo.database is a zoo object that saves
# the same 6 ARIMA time series saved in example.database. 

data(zoo.database)

# To calculate the distance matrix of this database: 

TSDatabaseDistances(zoo.database, distance="manhattan")
TSDatabaseDistances(zoo.database, distance="edr", epsilon=0.2)
TSDatabaseDistances(zoo.database, distance="fourier", n=20)

TSdist distance computation.

Description

TSdist distance calculation between two time series.

Usage

TSDistances(x, y, tx, ty, distance, ...)

Arguments

Details

The distance between the two time series x and y is calculated. x and y can be saved in a numeric vector or a ts, zoo or xts object. The following distance methods are supported:

• "euclidean": Euclidean distance. EuclideanDistance

• "manhattan": Manhattan distance. ManhattanDistance

• "minkowski": Minkowski distance. MinkowskiDistance

• "infnorm": Infinite norm distance. InfNormDistance

• "ccor": Distance based on the cross-correlation. CCorDistance

• "sts": Short time series distance. STSDistance

• "dtw": Dynamic Time Warping distance. DTWDistance. Uses the dtw package (see dtw).

• "lb.keogh": LB_Keogh lower bound for the Dynamic Time Warping distance. LBKeoghDistance

• "edr": Edit distance for real sequences. EDRDistance

• "erp": Edit distance with real penalty. ERPDistance

• "lcss": Longest Common Subsequence Matching. LCSSDistance

• "fourier": Distance based on the Fourier Discrete Transform. FourierDistance

• "tquest": TQuest distance. TquestDistance

• "dissim": Dissim distance. DissimDistance

• "acf": Autocorrelation-based dissimilarity ACFDistance. Uses the TSclust package (see diss.ACF).

• "pacf": Partial autocorrelation-based dissimilarity PACFDistance. Uses the TSclust package (see diss.PACF).

• "ar.lpc.ceps": Dissimilarity based on LPC cepstral coefficients ARLPCCepsDistance. Uses the TSclust package (see diss.AR.LPC.CEPS).

• "ar.mah": Model-based dissimilarity proposed by Maharaj (1996, 2000) ARMahDistance. Uses the TSclust package (see diss.AR.MAH).

• "ar.pic": Model-based dissimilarity measure proposed by Piccolo (1990) ARPicDistance. Uses the TSclust package (see diss.AR.PIC).

• "cdm": Compression-based dissimilarity measure CDMDistance. Uses the TSclust package (see diss.CDM).

• "cid": Complexity-invariant distance measure CIDDistance. Uses the TSclust package (see diss.CID).

• "cor": Dissimilarities based on Pearson's correlation CorDistance. Uses the TSclust package (see diss.COR).

• "cort": Dissimilarity index which combines temporal correlation and raw value behaviors CortDistance. Uses the TSclust package (see diss.CORT).

• "int.per": Integrated periodogram based dissimilarity IntPerDistance. Uses the TSclust package (see diss.INT.PER).

• "per": Periodogram based dissimilarity PerDistance. Uses the TSclust package (see diss.PER).

• "mindist.sax": Symbolic Aggregate Aproximation based dissimilarity MindistSaxDistance. Uses the TSclust package (see diss.MINDIST.SAX).

• "ncd": Normalized compression based distance NCDDistance. Uses the TSclust package (see diss.NCD).

• "pred": Dissimilarity measure cased on nonparametric forecasts PredDistance. Uses the TSclust package (see diss.PRED).

• "spec.glk": Dissimilarity based on the generalized likelihood ratio test SpecGLKDistance. Uses the TSclust package (see diss.SPEC.GLK).

• "spec.isd": Dissimilarity based on the integrated squared difference between the log-spectra SpecISDDistance. Uses the TSclust package (see diss.SPEC.ISD).

• "spec.llr": General spectral dissimilarity measure using local-linear estimation of the log-spectra SpecLLRDistance. Uses the TSclust package (see diss.SPEC.LLR).

• "pdc": Permutation Distribution Distance PDCDistance. Uses the pdc package (see pdcDist).

• "frechet": Frechet distance FrechetDistance. Uses the longitudinalData package (see distFrechet).

• "tam": Time Aligment Measurement TAMDistance.

Some distance measures may require additional arguments. See the individual help pages (detailed above) for more information about each method.

Value

Author(s)

Usue Mori, Alexander Mendiburu, Jose A. Lozano.

Examples

# The objects zoo.series1 and zoo.series2 are two 
# zoo objects that save two series of length 100. 

data(zoo.series1)
data(zoo.series2)

# For information on their generation and shape see 
# help page of example.series.

help(example.series)

# The distance calculation for these two series is done
# as follows:

TSDistances(zoo.series1, zoo.series2, distance="infnorm")
TSDistances(zoo.series1, zoo.series2, distance="cor", beta=3)
TSDistances(zoo.series1, zoo.series2, distance="dtw", sigma=20)

Tquest distance.

Description

Computes the Tquest distance between a pair of numeric vectors.

Usage

TquestDistance(x, y, tx, ty, tau)

Arguments

Details

The TQuest distance represents the series based on a set of intervals that fulfill the following conditions:

1. All the values that the time series takes during these time intervals must be strictly above a user specified threshold tau.

2. They are the largest possible intervals that satisfy the previous condition.

The final distance between two series is defined in terms of the similarity between their threshold passing interval sets. For more information, see references.

Value

Author(s)

Usue Mori, Alexander Mendiburu, Jose A. Lozano.

References

Aßfalg, J., Kriegel, H., Kröger, P., Kunath, P., Pryakhin, A., & Renz, M. (2006). Similarity Search on Time Series based on Threshold Queries. In Proceedings of the 10th international conference on Advances in Database Technology (pp. 276-294).

Esling, P., & Agon, C. (2012). Time-series data mining. ACM Computing Surveys (CSUR), 45(1), 1–34.

See Also

To calculate this distance measure using ts, zoo or xts objects see TSDistances. To calculate distance matrices of time series databases using this measure see TSDatabaseDistances.

Examples

# The objects example.series1 and example.series2 are two 
# numeric series of length 100 contained in the TSdist package. 

data(example.series1)
data(example.series2)

# For information on their generation and shape see help 
# page of example.series.

help(example.series)


# Calculate the Tquest distance assuming even sampling:

TquestDistance(example.series1, example.series2, tau=2.5)

# The objects example.series3 and example.series4 are two 
# numeric series of length 100 and 120 contained in the TSdist 
# package.

data(example.series3)
data(example.series4)

# Calculate the Tquest distance for two series of different length:

TquestDistance(example.series3, example.series4, tau=2.5)

Example databases.

Description

Example database saved both as a numeric matrix and as a zoo object.

Usage

data(example.database);
data(zoo.database);

Format

example.database is saved in a numerical matrix.

zoo.database is saved as a zoo object with a given temporal index.

Details

example.database is a numerical matrix conformed by six ARMA(3,2) series of coefficients AR=(1, -0.24, 0.1) and MA=(1, 1.2) and length 100 that are situated in a row-wise format. They are generated from innovation vectors obtained randomly from a normal distribution of mean 0 and standard deviation 1, but by setting different random seeds.

zoo.database is a copy of example.database but saved in a zoo object with a specific time index. The series are set in a column-wise format.

Examples

data(example.database);
data(zoo.database);


## In example.database the series are set in a row-wise format.

plot(example.database)[1,]

## In zoo.database the series are set in a column-wise format.

plot(zoo.database)[,1]

Example synthetic database with series belonging to different classes.

Description

Example synthetic database with series belonging to 6 different classes.

Usage

data(example.database2);

Format

example.database2 a list conformed of the following two elements:

• data The 100 time series are stored in a numeric matrix, row-wise.

• classes A numerical vector of length 100 that takes values in {1,2,3,4,5,6}. Each element in the vector represents the class of one of the series.

Details

example.database2 is a database conformed of 100 series of length 100 obtained from 6 different classes. Each class is represented by the following function:

The class to which each series belongs is given in the classes vector.

• Class 1: random function

  f1(t)=80+r(t)+n(t)

• Class 2: periodic function

  f2(t)=80+15\sin(\frac{2\pi t + sh}{T})+n(t)

• Class 3: increasing linear trend

  f3(t)=f_3(t)=80+0.4t+n(t)+sh

• Class 4: decreasing linear trend

  f4(t)=80-0.4t+n(t)+sh

• Class 5: piecewise linear function which takes a value of 80+n(t) for the first L/2+sh of the series and a value of 90+n(t) for the rest of the points.

• Class 6: piecewise linear function which takes a value of 90+n(t) for the first L/2+sh of the series and a value of 80+n(t) for the rest of the points.

  r(t) is a random value issued from a N(0,3) distribution, L is the length of the series, 100 in this case, and T is the period and is defined as a third of the length of the series. n(t) is a random noise obtained from a N(0,2.8) distribution.. Finally, sh is an integer value that takes a random value between (-7,7) and shifts the series sh positions to the right or left, depending on the sign.

Examples

data(example.database2);


##  The "data" element of the list contains the time series, set in a row-wise format.

plot(example.database2$data)[1,]

##  The "classes" element in example.database2 contains the classes of the series: 

example.database2$classes

Example synthetic database with series belonging to different classes.

Description

Example synthetic database with ARMA series belonging to 5 different classes.

Usage

data(example.database3);

Format

example.database3 a list conformed of the following two elements:

• data The 50 time series are stored in a numeric matrix, row-wise.

• classes A numerical vector of length 50 that takes values in {1,2,3,4,5}. Each element in the vector represents the class of one of the series.

Details

example.database3 is a database conformed of 50 series of length 100 obtained from 5 different classes. Each class is obtained from a different initializations of an ARMA(3,2) process of coefficients AR=(1,-0.24,0.1) and MA=(1,1.2).

Random noise is added to all the series by sampling values from a N(0,1.7) distribution. R is obtained from the following formula:

Finally, all the series in the database are shifted sh positions to the right or left, sh being a random integer value extracted from {-15,...,15} in each case.

Examples

data(example.database3);


##  The "data" element of the list contains the time series, set in a row-wise format.

plot(example.database3$data)[1,]

##  The "classes" element in example.database3 contains the classes of the series: 

example.database3$classes

Example series.

Description

Example series saved as numeric vectors and as zoo objects.

Usage

data(example.series1);
data(example.series2);
data(example.series3);
data(example.series4);
data(zoo.series1);
data(zoo.series2);

Format

example.series1, example.series2, example.series3 and example.series4 are saved in numerical vectors.

zoo.series1 and zoo.series2 are saved as zoo objects with a given temporal index.

Details

example.series1 and example.series2 are generated based on the Two Patterns synthetic database introduced by Geurts (2002).

example.series3 and example.series4 are two ARMA(3,2) series of coefficients AR=(1, -0.24, 0.1) and MA=(1, 1.2) and length 100 and 120 respectively. They are generated from a pair of innovation vectors obtained randomly from a normal distribution of mean 0 and standard deviation 1, but by setting different random seeds.

zoo.series1 and zoo.series2 are copies of example.series1 and example.series2 but with a specific time index.

References

Geurts, P. (2002). Contributions to decision tree induction: bias/variance tradeoff and time series classification. University of Liege, Belgium.

Examples

data(example.series1);
data(example.series2);
data(example.series3);
data(example.series4);
data(zoo.series1);
data(zoo.series2);

## Plot series

plot(example.series1, type="l")
plot(example.series2, type="l")
plot(example.series3, type="l")
plot(example.series4, type="l")
plot(zoo.series1)
plot(zoo.series2)