| Title: | Graph-Based Change-Point Detection (g-Segmentation) |
|---|---|
| Description: | Using an approach based on similarity graph to estimate change-point(s) and the corresponding p-values. Can be applied to any type of data (high-dimensional, non-Euclidean, etc.) as long as a reasonable similarity measure is available. |
| Authors: | Hao Chen, Nancy R. Zhang, Lynna Chu, and Hoseung Song |
| Maintainer: | Hao Chen <[email protected]> |
| License: | GPL (>= 2) |
| Version: | 1.0 |
| Built: | 2024-12-03 06:43:33 UTC |
| Source: | CRAN |
This is the variable name for an edge matrix in the "Example" data. It is constructed from a sequence of n=200 observations with a change in mean at t = 100. E1 is a matrix with the number of rows the number of edges in the similarity graph and 2 columns. Each row contains the node indices of an edge.
This is the variable name for an edge matrix in the "Example" data. It is constructed from a sequence of n=200 observations with a change in mean starting at t=45. E2 is a matrix with the number of rows the number of edges in the similarity graph and 2 columns. Each row contains the node indices of an edge.
This is the variable name for an edge matrix in the "Example" data. It is constructed from a sequence of n=200 observations with a change in mean and variance starting at t = 145. E3 is a matrix with the number of rows the number of edges in the similarity graph and 2 columns. Each row contains the node indices of an edge.
This is the variable name for an edge matrix in the "Example" data. It is constructed from a sequence of n=200 observations with a change in mean and variance starting at t=50. E4 is a matrix with the number of rows the number of edges in the similarity graph and 2 columns. Each row contains the node indices of an edge.
This is the variable name for an edge matrix in the "Example" data. It is constructed from a sequence of n=200 observations with a change in mean on the interval t= 155 to t=185. E5 is a matrix with the number of rows the number of edges in the similarity graph and 2 columns. Each row contains the node indices of an edge.
This package can be used to estimate change-points in a sequence of observations, where the observation can be a vector or a data object, e.g., a network. A similarity graph is required. It can be a minimum spanning tree, a minimum distance pairing, a nearest neighbor graph, or a graph based on domain knowledge.
For sequence with no repeated observations, if you believe the sequence has at most one change point, the function gseg1 should be used; if you believe an interval of the sequence has a changed distribution, the function gseg2 should be used. If you feel the sequence has multiple change-points, you can use gseg1 and gseg2 multiple times. See gseg1 and gseg2 for the details of these two function.
If you believe the sequence has repeated observations, the function gseg1_discrete should be used for single change-point. For a changed interval of the sequence, the function gseg2_discrete should be used. The function nnl can be used to construct the nearest neighbor link.
Hao Chen, Nancy R. Zhang, Lynna Chu, and Hoseung Song
Maintainer: Hao Chen ([email protected])
Chen, Hao, and Nancy Zhang. (2015). Graph-based change-point detection. The Annals of Statistics, 43(1), 139-176.
Chu, Lynna, and Hao Chen. (2019). Asymptotic distribution-free change-point detection for modern data. The Annals of Statistics, 47(1), 382-414.
Song, Hoseung, and Hao Chen (2020). Asymptotic distribution-free change-point detection for data with repeated observations. arXiv:2006.10305
gseg1, gseg2, gseg1_discrete, gseg2_discrete, nnl
data(Example) # Five examples, each example is a n-length sequence. # Ei (i=1,...,5): an edge matrix representing a similarity graph constructed on the # observations in the ith sequence. # The following code shows how the Ei's were constructed. require(ade4) # For illustration, we use 'mstree' in this package to construct the similarity graph. # You can use other ways to construct the graph. ## Sequence 1: change in mean in the middle of the sequence. d = 50 mu = 2 tau = 100 n = 200 set.seed(500) y = rbind(matrix(rnorm(d*tau),tau), matrix(rnorm(d*(n-tau),mu/sqrt(d)), n-tau)) y.dist = dist(y) E1 = mstree(y.dist) # For illustration, we constructed the minimum spanning tree. # You can use other ways to construct the graph. r1 = gseg1(n,E1, statistics="all") # output results based on all four statistics # the scan statistics can be found in r1$scanZ r1_a = gseg1(n,E1, statistics="w") # output results based on the weighted edge-count statistic r1_b = gseg1(n,E1, statistics=c("w","g")) # output results based on the weighted edge-count statistic # and generalized edge-count statistic # image(as.matrix(y.dist)) # run this if you would like to have some idea on the pairwise distance ## Sequence 2: change in mean away from the middle of the sequence. d = 50 mu = 2 tau = 45 n = 200 set.seed(500) y = rbind(matrix(rnorm(d*tau),tau), matrix(rnorm(d*(n-tau),mu/sqrt(d)), n-tau)) y.dist = dist(y) E2 = mstree(y.dist) r2 = gseg1(n,E2,statistic="all") # image(as.matrix(y.dist)) ## Sequence 3: change in both mean and variance away from the middle of the sequence. d = 50 mu = 2 sigma=0.7 tau = 145 n = 200 set.seed(500) y = rbind(matrix(rnorm(d*tau),tau), matrix(rnorm(d*(n-tau),mu/sqrt(d),sigma), n-tau)) y.dist = dist(y) E3 = mstree(y.dist) r3=gseg1(n,E3,statistic="all") # image(as.matrix(y.dist)) ## Sequence 4: change in both mean and variance away from the middle of the sequence. d = 50 mu = 2 sigma=1.2 tau = 50 n = 200 set.seed(500) y = rbind(matrix(rnorm(d*tau),tau), matrix(rnorm(d*(n-tau),mu/sqrt(d),sigma), n-tau)) y.dist = dist(y) E4 = mstree(y.dist) r4=gseg1(n,E4,statistic="all") # image(as.matrix(y.dist)) ## Sequence 5: change in both mean and variance happens on an interval. d = 50 mu = 2 sigma=0.5 tau1 = 155 tau2 = 185 n = 200 set.seed(500) y1 = matrix(rnorm(d*tau1),tau1) y2 = matrix(rnorm(d*(tau2-tau1),mu/sqrt(d),sigma), tau2-tau1) y3 = matrix(rnorm(d*(n-tau2)), n-tau2) y = rbind(y1, y2, y3) y.dist = dist(y) E5 = mstree(y.dist) r5=gseg2(n,E5,statistics="all") # image(as.matrix(y.dist)) ## Sequence 6: change in mean away from the middle of the sequence ## when data has repeated observations. d = 50 mu = 2 tau = 100 n = 200 set.seed(500) y1_temp = matrix(rnorm(d*tau),tau) sam1 = sample(1:tau, replace = TRUE) y1 = y1_temp[sam1,] y2_temp = matrix(rnorm(d*(n-tau),mu/sqrt(d)), n-tau) sam2 = sample(1:tau, replace = TRUE) y2 = y2_temp[sam2,] y = rbind(y1, y2) # Data y has repeated observations y_uni = unique(y) E6 = nnl(dist(y_uni), 1) cha = do.call(paste, as.data.frame(y)) id = match(cha, unique(cha)) r6 = gseg1_discrete(n, E6, id, statistics="all") # image(as.matrix(y.dist))data(Example) # Five examples, each example is a n-length sequence. # Ei (i=1,...,5): an edge matrix representing a similarity graph constructed on the # observations in the ith sequence. # The following code shows how the Ei's were constructed. require(ade4) # For illustration, we use 'mstree' in this package to construct the similarity graph. # You can use other ways to construct the graph. ## Sequence 1: change in mean in the middle of the sequence. d = 50 mu = 2 tau = 100 n = 200 set.seed(500) y = rbind(matrix(rnorm(d*tau),tau), matrix(rnorm(d*(n-tau),mu/sqrt(d)), n-tau)) y.dist = dist(y) E1 = mstree(y.dist) # For illustration, we constructed the minimum spanning tree. # You can use other ways to construct the graph. r1 = gseg1(n,E1, statistics="all") # output results based on all four statistics # the scan statistics can be found in r1$scanZ r1_a = gseg1(n,E1, statistics="w") # output results based on the weighted edge-count statistic r1_b = gseg1(n,E1, statistics=c("w","g")) # output results based on the weighted edge-count statistic # and generalized edge-count statistic # image(as.matrix(y.dist)) # run this if you would like to have some idea on the pairwise distance ## Sequence 2: change in mean away from the middle of the sequence. d = 50 mu = 2 tau = 45 n = 200 set.seed(500) y = rbind(matrix(rnorm(d*tau),tau), matrix(rnorm(d*(n-tau),mu/sqrt(d)), n-tau)) y.dist = dist(y) E2 = mstree(y.dist) r2 = gseg1(n,E2,statistic="all") # image(as.matrix(y.dist)) ## Sequence 3: change in both mean and variance away from the middle of the sequence. d = 50 mu = 2 sigma=0.7 tau = 145 n = 200 set.seed(500) y = rbind(matrix(rnorm(d*tau),tau), matrix(rnorm(d*(n-tau),mu/sqrt(d),sigma), n-tau)) y.dist = dist(y) E3 = mstree(y.dist) r3=gseg1(n,E3,statistic="all") # image(as.matrix(y.dist)) ## Sequence 4: change in both mean and variance away from the middle of the sequence. d = 50 mu = 2 sigma=1.2 tau = 50 n = 200 set.seed(500) y = rbind(matrix(rnorm(d*tau),tau), matrix(rnorm(d*(n-tau),mu/sqrt(d),sigma), n-tau)) y.dist = dist(y) E4 = mstree(y.dist) r4=gseg1(n,E4,statistic="all") # image(as.matrix(y.dist)) ## Sequence 5: change in both mean and variance happens on an interval. d = 50 mu = 2 sigma=0.5 tau1 = 155 tau2 = 185 n = 200 set.seed(500) y1 = matrix(rnorm(d*tau1),tau1) y2 = matrix(rnorm(d*(tau2-tau1),mu/sqrt(d),sigma), tau2-tau1) y3 = matrix(rnorm(d*(n-tau2)), n-tau2) y = rbind(y1, y2, y3) y.dist = dist(y) E5 = mstree(y.dist) r5=gseg2(n,E5,statistics="all") # image(as.matrix(y.dist)) ## Sequence 6: change in mean away from the middle of the sequence ## when data has repeated observations. d = 50 mu = 2 tau = 100 n = 200 set.seed(500) y1_temp = matrix(rnorm(d*tau),tau) sam1 = sample(1:tau, replace = TRUE) y1 = y1_temp[sam1,] y2_temp = matrix(rnorm(d*(n-tau),mu/sqrt(d)), n-tau) sam2 = sample(1:tau, replace = TRUE) y2 = y2_temp[sam2,] y = rbind(y1, y2) # Data y has repeated observations y_uni = unique(y) E6 = nnl(dist(y_uni), 1) cha = do.call(paste, as.data.frame(y)) id = match(cha, unique(cha)) r6 = gseg1_discrete(n, E6, id, statistics="all") # image(as.matrix(y.dist))
This function finds a break point in the sequence where the underlying distribution changes. It provides four graph-based test statistics.
gseg1(n, E, statistics=c("all","o","w","g","m"), n0=0.05*n, n1=0.95*n, pval.appr=TRUE, skew.corr=TRUE, pval.perm=FALSE, B=100)gseg1(n, E, statistics=c("all","o","w","g","m"), n0=0.05*n, n1=0.95*n, pval.appr=TRUE, skew.corr=TRUE, pval.perm=FALSE, B=100)
n |
The number of observations in the sequence. |
E |
The edge matrix (a "number of edges" by 2 matrix) for the similarity graph. Each row contains the node indices of an edge. |
statistics |
The scan statistic to be computed. A character indicating the type of of scan statistic desired. The default is
|
n0 |
The starting index to be considered as a candidate for the change-point. |
n1 |
The ending index to be considered as a candidate for the change-point. |
pval.appr |
If it is TRUE, the function outputs p-value approximation based on asymptotic properties. |
skew.corr |
This argument is useful only when pval.appr=TRUE. If skew.corr is TRUE, the p-value approximation would incorporate skewness correction. |
pval.perm |
If it is TRUE, the function outputs p-value from doing B permutations, where B is another argument that you can specify. Doing permutation could be time consuming, so use this argument with caution as it may take a long time to finish the permutation. |
B |
This argument is useful only when pval.perm=TRUE. The default value for B is 100. |
Returns a list scanZ with tauhat, Zmax, and a vector of the scan statistics for each type of scan statistic specified. See below for more details.
tauhat |
An estimate of the location of the change-point. |
Zmax |
The test statistic (maximum of the scan statistics). |
Z |
A vector of the original scan statistics (standardized counts) if statistic specified is "all" or "o". |
Zw |
A vector of the weighted scan statistics (standardized counts) if statistic specified is "all" or "w". |
S |
A vector of the generalized scan statistics (standardized counts) if statistic specified is "all" or "g". |
M |
A vector of the max-type scan statistics (standardized counts) if statistic specified is "all" or "m". |
R |
A vector of raw counts of the original scan statistic. This output only exists if the statistic specified is "all" or "o". |
Rw |
A vector of raw counts of the weighted scan statistic. This output only exists if statistic specified is "all" or "w". |
pval.appr |
The approximated p-value based on asymptotic theory for each type of statistic specified. |
pval.perm |
This output exists only when the argument pval.perm is TRUE . It is the permutation p-value from B permutations and appears for each type of statistic specified (same for perm.curve, perm.maxZs, and perm.Z). |
perm.curve |
A B by 2 matrix with the first column being critical values corresponding to the p-values in the second column. |
perm.maxZs |
A sorted vector recording the test statistics in the B permutaitons. |
perm.Z |
A B by n matrix with each row being the scan statistics from each permutaiton run. |
data(Example) # Five examples, each example is a n-length sequence. # Ei (i=1,...,5): an edge matrix representing a similarity graph constructed on the # observations in the ith sequence. # Check '?gSeg' to see how the Ei's were constructed. ## E1 is an edge matrix representing a similarity graph. # It is constructed on a sequence of length n=200 with a change in mean # in the middle of the sequence (tau = 100). r1 = gseg1(n,E1, statistics="all") # output results based on all four statistics # the scan statistics can be found in r1$scanZ r1_a = gseg1(n,E1, statistics="w") # output results based on the weighted edge-count statistic r1_b = gseg1(n,E1, statistics=c("w","g")) # output results based on the weighted edge-count statistic # and generalized edge-count statistic ## E2 is an edge matrix representing a similarity graph. # It is constructed on a sequence of length n=200 with a change in mean # away from the middle of the sequence (tau=45). r2 = gseg1(n,E2,statistic="all") ## E3 is an edge matrix representing a similarity graph. # It is constructed on a sequence of length n=200 with a change in both mean # and variance away from the middle of the sequence (tau = 145). r3=gseg1(n,E3,statistic="all") ## E4 is an edge matrix representing a similarity graph. # It is constructed on a sequence of length n=200 with a change in both mean # and variance away from the middle of the sequence (tau = 50). r4=gseg1(n,E4,statistic="all")data(Example) # Five examples, each example is a n-length sequence. # Ei (i=1,...,5): an edge matrix representing a similarity graph constructed on the # observations in the ith sequence. # Check '?gSeg' to see how the Ei's were constructed. ## E1 is an edge matrix representing a similarity graph. # It is constructed on a sequence of length n=200 with a change in mean # in the middle of the sequence (tau = 100). r1 = gseg1(n,E1, statistics="all") # output results based on all four statistics # the scan statistics can be found in r1$scanZ r1_a = gseg1(n,E1, statistics="w") # output results based on the weighted edge-count statistic r1_b = gseg1(n,E1, statistics=c("w","g")) # output results based on the weighted edge-count statistic # and generalized edge-count statistic ## E2 is an edge matrix representing a similarity graph. # It is constructed on a sequence of length n=200 with a change in mean # away from the middle of the sequence (tau=45). r2 = gseg1(n,E2,statistic="all") ## E3 is an edge matrix representing a similarity graph. # It is constructed on a sequence of length n=200 with a change in both mean # and variance away from the middle of the sequence (tau = 145). r3=gseg1(n,E3,statistic="all") ## E4 is an edge matrix representing a similarity graph. # It is constructed on a sequence of length n=200 with a change in both mean # and variance away from the middle of the sequence (tau = 50). r4=gseg1(n,E4,statistic="all")
This function finds a break point in the sequence where the underlying distribution changes when data has repeated observations. It provides four graph-based test statistics.
gseg1_discrete(n, E, id, statistics=c("all","o","w","g","m"), n0=0.05*n, n1=0.95*n, pval.appr=TRUE, skew.corr=TRUE, pval.perm=FALSE, B=100)gseg1_discrete(n, E, id, statistics=c("all","o","w","g","m"), n0=0.05*n, n1=0.95*n, pval.appr=TRUE, skew.corr=TRUE, pval.perm=FALSE, B=100)
n |
The number of observations in the sequence. |
E |
The edge matrix (a "number of edges" by 2 matrix) for the similarity graph. Each row contains the node indices of an edge. |
id |
The index of observations (order of observations). |
statistics |
The scan statistic to be computed. A character indicating the type of of scan statistic desired. The default is
|
n0 |
The starting index to be considered as a candidate for the change-point. |
n1 |
The ending index to be considered as a candidate for the change-point. |
pval.appr |
If it is TRUE, the function outputs p-value approximation based on asymptotic properties. |
skew.corr |
This argument is useful only when pval.appr=TRUE. If skew.corr is TRUE, the p-value approximation would incorporate skewness correction. |
pval.perm |
If it is TRUE, the function outputs p-value from doing B permutations, where B is another argument that you can specify. Doing permutation could be time consuming, so use this argument with caution as it may take a long time to finish the permutation. |
B |
This argument is useful only when pval.perm=TRUE. The default value for B is 100. |
Returns a list scanZ with tauhat, Zmax, and a vector of the scan statistics for each type of scan statistic specified. See below for more details.
tauhat_a |
An estimate of the location of the change-point for averaging approach. |
tauhat_u |
An estimate of the location of the change-point for union approach. |
Z_a_max |
The test statistic (maximum of the scan statistics) for averaging approach. |
Z_u_max |
The test statistic (maximum of the scan statistics) for union approach. |
Zo_a |
A vector of the original scan statistics (standardized counts) for averaging approach if statistic specified is "all" or "o". |
Zo_u |
A vector of the original scan statistics (standardized counts) for union approach if statistic specified is "all" or "o". |
Zw_a |
A vector of the weighted scan statistics (standardized counts) for averaging approach if statistic specified is "all" or "w". |
Zw_u |
A vector of the weighted scan statistics (standardized counts) for union approach if statistic specified is "all" or "w". |
S_a |
A vector of the generalized scan statistics (standardized counts) for averaging approach if statistic specified is "all" or "g". |
S_u |
A vector of the generalized scan statistics (standardized counts) for union approach if statistic specified is "all" or "g". |
M_a |
A vector of the max-type scan statistics (standardized counts) for averaging appraoch if statistic specified is "all" or "m". |
M_u |
A vector of the max-type scan statistics (standardized counts) for union appraoch if statistic specified is "all" or "m". |
Ro_a |
A vector of raw counts of the original scan statistic for averaging approach. This output only exists if the statistic specified is "all" or "o". |
Ro_u |
A vector of raw counts of the original scan statistic for union approach. This output only exists if the statistic specified is "all" or "o". |
Rw_a |
A vector of raw counts of the weighted scan statistic for averaging appraoch. This output only exists if statistic specified is "all" or "w". |
Rw_u |
A vector of raw counts of the weighted scan statistic for union appraoch. This output only exists if statistic specified is "all" or "w". |
pval.appr |
The approximated p-value based on asymptotic theory for each type of statistic specified. |
pval.perm |
This output exists only when the argument pval.perm is TRUE . It is the permutation p-value from B permutations and appears for each type of statistic specified (same for perm.curve, perm.maxZs, and perm.Z). |
perm.curve |
A B by 2 matrix with the first column being critical values corresponding to the p-values in the second column. |
perm.maxZs |
A sorted vector recording the test statistics in the B permutaitons. |
perm.Z |
A B by n matrix with each row being the scan statistics from each permutaiton run. |
gSeg, gseg1, nnl, gseg2_discrete
d = 50 mu = 2 tau = 100 n = 200 set.seed(500) y1_temp = matrix(rnorm(d*tau),tau) sam1 = sample(1:tau, replace = TRUE) y1 = y1_temp[sam1,] y2_temp = matrix(rnorm(d*(n-tau),mu/sqrt(d)), n-tau) sam2 = sample(1:tau, replace = TRUE) y2 = y2_temp[sam2,] y = rbind(y1, y2) # This data y has repeated observations y_uni = unique(y) E = nnl(dist(y_uni), 1) cha = do.call(paste, as.data.frame(y)) id = match(cha, unique(cha)) r1 = gseg1_discrete(n, E, id, statistics="all") # output results based on all four statistics # the scan statistics can be found in r1$scanZ r1_a = gseg1_discrete(n, E, id, statistics="w") # output results based on the weighted edge-count statistic r1_b = gseg1_discrete(n, E, id, statistics=c("w","g")) # output results based on the weighted edge-count statistic # and generalized edge-count statisticd = 50 mu = 2 tau = 100 n = 200 set.seed(500) y1_temp = matrix(rnorm(d*tau),tau) sam1 = sample(1:tau, replace = TRUE) y1 = y1_temp[sam1,] y2_temp = matrix(rnorm(d*(n-tau),mu/sqrt(d)), n-tau) sam2 = sample(1:tau, replace = TRUE) y2 = y2_temp[sam2,] y = rbind(y1, y2) # This data y has repeated observations y_uni = unique(y) E = nnl(dist(y_uni), 1) cha = do.call(paste, as.data.frame(y)) id = match(cha, unique(cha)) r1 = gseg1_discrete(n, E, id, statistics="all") # output results based on all four statistics # the scan statistics can be found in r1$scanZ r1_a = gseg1_discrete(n, E, id, statistics="w") # output results based on the weighted edge-count statistic r1_b = gseg1_discrete(n, E, id, statistics=c("w","g")) # output results based on the weighted edge-count statistic # and generalized edge-count statistic
This function finds an interval in the sequence where their underlying distribution differs from the rest of the sequence. It provides four graph-based test statistics.
gseg2(n, E, statistics=c("all","o","w","g","m"), l0=0.05*n, l1=0.95*n, pval.appr=TRUE, skew.corr=TRUE, pval.perm=FALSE, B=100)gseg2(n, E, statistics=c("all","o","w","g","m"), l0=0.05*n, l1=0.95*n, pval.appr=TRUE, skew.corr=TRUE, pval.perm=FALSE, B=100)
n |
The number of observations in the sequence. |
E |
The edge matrix (a "number of edges" by 2 matrix) for the similarity graph. Each row contains the node indices of an edge. |
statistics |
The scan statistic to be computed. A character indicating the type of of scan statistic desired. The default is
|
l0 |
The minimum length of the interval to be considered as a changed interval. |
l1 |
The maximum length of the interval to be considered as a changed interval. |
pval.appr |
If it is TRUE, the function outputs p-value approximation based on asymptotic properties. |
skew.corr |
This argument is useful only when pval.appr=TRUE. If skew.corr is TRUE, the p-value approximation would incorporate skewness correction. |
pval.perm |
If it is TRUE, the function outputs p-value from doing B permutations, where B is another argument that you can specify. Doing permutation could be time consuming, so use this argument with caution as it may take a long time to finish the permutation. |
B |
This argument is useful only when pval.perm=TRUE. The default value for B is 100. |
Returns a list scanZ with tauhat, Zmax, and a matrix of the scan statistics for each type of scan statistic specified. See below for more details.
tauhat |
An estimate of the two ends of the changed interval. |
Zmax |
The test statistic (maximum of the scan statistics). |
Z |
A matrix of the original scan statistics (standardized counts) if statistic specified is "all" or "o". |
Zw |
A matrix of the weighted scan statistics (standardized counts) if statistic specified is "all" or "w". |
S |
A matrix of the generalized scan statistics (standardized counts) if statistic specified is "all" or "g". |
M |
A matrix of the max-type scan statistics (standardized counts) if statistic specified is "all" or "m". |
R |
A matrix of raw counts of the original scan statistic. This output only exists if the statistic specified is "all" or "o". |
Rw |
A matrix of raw counts of the weighted scan statistic. This output only exists if statistic specified is "all" or "w". |
pval.appr |
The approximated p-value based on asymptotic theory for each type of statistic specified. |
pval.perm |
This output exists only when the argument pval.perm is TRUE . It is the permutation p-value from B permutations and appears for each type of statistic specified (same for perm.curve, perm.maxZs, and perm.Z). |
perm.curve |
A B by 2 matrix with the first column being critical values corresponding to the p-values in the second column. |
perm.maxZs |
A sorted vector recording the test statistics in the B permutaitons. |
perm.Z |
A B by n-squared matrix with each row being the vectorized scan statistics from each permutaiton run. |
data(Example) # Five examples, each example is a n-length sequence. # Ei (i=1,...,5): an edge matrix representing a similarity graph constructed on the # observations in the ith sequence. # Check '?gSeg' to see how the Ei's were constructed. ## E5 is an edge matrix representing a similarity graph. # It is constructed on a sequence of length n=200 with a change in both mean # and variance on an interval (tau1 = 155, tau2 = 185). r5=gseg2(n,E5,statistics="all")data(Example) # Five examples, each example is a n-length sequence. # Ei (i=1,...,5): an edge matrix representing a similarity graph constructed on the # observations in the ith sequence. # Check '?gSeg' to see how the Ei's were constructed. ## E5 is an edge matrix representing a similarity graph. # It is constructed on a sequence of length n=200 with a change in both mean # and variance on an interval (tau1 = 155, tau2 = 185). r5=gseg2(n,E5,statistics="all")
This function finds an interval in the sequence where their underlying distribution differs from the rest of the sequence when data has repeated observations. It provides four graph-based test statistics.
gseg2_discrete(n, E, id, statistics=c("all","o","w","g","m"), l0=0.05*n, l1=0.95*n, pval.appr=TRUE, skew.corr=TRUE, pval.perm=FALSE, B=100)gseg2_discrete(n, E, id, statistics=c("all","o","w","g","m"), l0=0.05*n, l1=0.95*n, pval.appr=TRUE, skew.corr=TRUE, pval.perm=FALSE, B=100)
n |
The number of observations in the sequence. |
E |
The edge matrix (a "number of edges" by 2 matrix) for the similarity graph. Each row contains the node indices of an edge. |
id |
The index of observations (order of observations). |
statistics |
The scan statistic to be computed. A character indicating the type of of scan statistic desired. The default is
|
l0 |
The minimum length of the interval to be considered as a changed interval. |
l1 |
The maximum length of the interval to be considered as a changed interval. |
pval.appr |
If it is TRUE, the function outputs p-value approximation based on asymptotic properties. |
skew.corr |
This argument is useful only when pval.appr=TRUE. If skew.corr is TRUE, the p-value approximation would incorporate skewness correction. |
pval.perm |
If it is TRUE, the function outputs p-value from doing B permutations, where B is another argument that you can specify. Doing permutation could be time consuming, so use this argument with caution as it may take a long time to finish the permutation. |
B |
This argument is useful only when pval.perm=TRUE. The default value for B is 100. |
Returns a list scanZ with tauhat, Zmax, and a vector of the scan statistics for each type of scan statistic specified. See below for more details.
tauhat_a |
An estimate of the two ends of the changed interval for averaging approach. |
tauhat_u |
An estimate of the two ends of the changed interval for union approach. |
Z_a_max |
The test statistic (maximum of the scan statistics) for averaging approach. |
Z_u_max |
The test statistic (maximum of the scan statistics) for union approach. |
Zo_a |
A matrix of the original scan statistics (standardized counts) for averaging approach if statistic specified is "all" or "o". |
Zo_u |
A matrix of the original scan statistics (standardized counts) for union approach if statistic specified is "all" or "o". |
Zw_a |
A matrix of the weighted scan statistics (standardized counts) for averaging approach if statistic specified is "all" or "w". |
Zw_u |
A matrix of the weighted scan statistics (standardized counts) for union approach if statistic specified is "all" or "w". |
S_a |
A matrix of the generalized scan statistics (standardized counts) for averaging approach if statistic specified is "all" or "g". |
S_u |
A matrix of the generalized scan statistics (standardized counts) for union approach if statistic specified is "all" or "g". |
M_a |
A matrix of the max-type scan statistics (standardized counts) for averaging approach if statistic specified is "all" or "m". |
M_u |
A matrix of the max-type scan statistics (standardized counts) for union approach if statistic specified is "all" or "m". |
Ro_a |
A matrix of raw counts of the original scan statistic for averaging approach. This output only exists if the statistic specified is "all" or "o". |
Ro_u |
A matrix of raw counts of the original scan statistic for union approach. This output only exists if the statistic specified is "all" or "o". |
Rw_a |
A matrix of raw counts of the weighted scan statistic for averaging approach. This output only exists if statistic specified is "all" or "w". |
Rw_a |
A matrix of raw counts of the weighted scan statistic for union approach. This output only exists if statistic specified is "all" or "w". |
pval.appr |
The approximated p-value based on asymptotic theory for each type of statistic specified. |
pval.perm |
This output exists only when the argument pval.perm is TRUE . It is the permutation p-value from B permutations and appears for each type of statistic specified (same for perm.curve, perm.maxZs, and perm.Z). |
perm.curve |
A B by 2 matrix with the first column being critical values corresponding to the p-values in the second column. |
perm.maxZs |
A sorted vector recording the test statistics in the B permutaitons. |
perm.Z |
A B by n-squared matrix with each row being the vectorized scan statistics from each permutaiton run. |
gSeg, gseg2, nnl, gseg1_discrete
d = 50 mu = 2 tau = 100 n = 200 set.seed(500) y1_temp = matrix(rnorm(d*tau),tau) sam1 = sample(1:tau, replace = TRUE) y1 = y1_temp[sam1,] y2_temp = matrix(rnorm(d*(n-tau),mu/sqrt(d)), n-tau) sam2 = sample(1:tau, replace = TRUE) y2 = y2_temp[sam2,] y = rbind(y1, y2) # This data y has repeated observations y_uni = unique(y) E = nnl(dist(y_uni), 1) cha = do.call(paste, as.data.frame(y)) id = match(cha, unique(cha)) r1 = gseg2_discrete(n, E, id, statistics="all")d = 50 mu = 2 tau = 100 n = 200 set.seed(500) y1_temp = matrix(rnorm(d*tau),tau) sam1 = sample(1:tau, replace = TRUE) y1 = y1_temp[sam1,] y2_temp = matrix(rnorm(d*(n-tau),mu/sqrt(d)), n-tau) sam2 = sample(1:tau, replace = TRUE) y2 = y2_temp[sam2,] y = rbind(y1, y2) # This data y has repeated observations y_uni = unique(y) E = nnl(dist(y_uni), 1) cha = do.call(paste, as.data.frame(y)) id = match(cha, unique(cha)) r1 = gseg2_discrete(n, E, id, statistics="all")
This is the variable name for the number of observations in the sequences in the "Example" data.
This function provides a method to construct the NNL.
nnl(distance, K)nnl(distance, K)
distance |
The distance matrix on the distinct values (a "number of unique observations" by "number of unique observations" matrix). |
K |
The value of k in "k-MST" or "k-NNL" to construct the similarity graph. |
E |
The edge matrix representing the similarity graph on the distinct values with the number of edges in the similarity graph being the number of rows and 2 columns. Each row records the subject indices of the two ends of an edge in the similarity graph. |
gSeg, gseg1_discrete, gseg2_discrete
n = 50 d = 10 dat = matrix(rnorm(d*n),n) sam = sample(1:n, replace = TRUE) dat = dat[sam,] # This data has repeated observations dat_uni = unique(dat) E = nnl(dist(dat_uni), 1)n = 50 d = 10 dat = matrix(rnorm(d*n),n) sam = sample(1:n, replace = TRUE) dat = dat[sam,] # This data has repeated observations dat_uni = unique(dat) E = nnl(dist(dat_uni), 1)