gSeg (version 1.1)

gseg1_discrete: Graph-Based Change-Point Detection for Single Change-Point for Data with Repeated Observations

Description

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.

Usage

gseg1_discrete(n, E, id, statistics="m", n0=0.05*n, n1=0.95*n, 
   pval.appr=TRUE, skew.corr=TRUE, pval.perm=FALSE, B=100)

Value

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 approach if statistic specified is "all" or "m".

M_u

A vector of the max-type scan statistics (standardized counts) for union approach 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 approach. 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 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 matrix with each row being the scan statistics from each permutaiton run.

Arguments

n

The number of observations in the sequence; it must be at least 6.

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 statistics to be computed. The default is "m", which reports the max-type scan.

"all": specifies to compute all of the scan statistics: original, weighted, generalized, and max-type;

"o", "ori" or "original": specifies the original edge-count scan statistic;

"w" or "weighted": specifies the weighted edge-count scan statistic;

"g" or "generalized": specifies the generalized edge-count scan statistic; and

"m" or "max": specifies the max-type edge-count scan statistic.

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 a p-value approximation based on asymptotic properties. The analytic approximation requires n >= 9.

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.

See Also

gSeg, gseg1, nnl, gseg2_discrete

Examples

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 statistic