MRC {stepR}

Compute Multiresolution Criterion

Description

Computes multiresolution coefficients, the corresponding criterion, simulates these for Gaussian white or coloured noise, based on which p-values and quantiles are obtained.

Deprecation warning: The function MRC.simul is deprecated, but still working, however, may be defunct in a future version. Please use instead the function monteCarloSimulation. An example how to reproduce results is given below. Some other functions are help function and might be removed, too.

Usage

MRC(x, lengths = 2^(floor(log2(length(x))):0), norm = sqrt(lengths),
  penalty = c("none", "log", "sqrt"))
MRCoeff(x, lengths = 2^(floor(log2(length(x))):0), norm = sqrt(lengths), signed = FALSE)
MRC.simul(n, r, lengths = 2^(floor(log2(n)):0), penalty = c("none", "log", "sqrt"))
MRC.pvalue(q, n, r, lengths = 2^(floor(log2(n)):0), penalty = c("none", "log", "sqrt"),
  name = ".MRC.table", pos = .MCstepR, inherits = TRUE)
MRC.FFT(epsFFT, testFFT, K = matrix(TRUE, nrow(testFFT), ncol(testFFT)), lengths,
  penalty = c("none", "log", "sqrt"))
MRC.quant(p, n, r, lengths = 2^(floor(log2(n)):0), penalty = c("none", "log", "sqrt"),
  name = ".MRC.table", pos = .MCstepR, inherits = TRUE, ...)
kMRC.simul(n, r, kern, lengths = 2^(floor(log2(n)):ceiling(log2(length(kern)))))
kMRC.pvalue(q, n, r, kern, lengths = 2^(floor(log2(n)):ceiling(log2(length(kern)))),
  name = ".MRC.ktable", pos = .MCstepR, inherits = TRUE)
kMRC.quant(p, n, r, kern, lengths = 2^(floor(log2(n)):ceiling(log2(length(kern)))),
  name = ".MRC.ktable", pos = .MCstepR, inherits = TRUE, ...)

Arguments

x

a vector of numerical observations

lengths

vector of interval lengths to use, dyadic intervals by default

signed

whether signed coefficients should be returned

q

quantile

n

length of data set

r

number of simulations to use

name, pos, inherits

under which name and where precomputed results are stored, or retrieved, see assign

K

a logical matrix indicating the set of valid intervals

epsFFT

a vector containg the FFT of the data set

testFFT

a matrix containing the FFTs of the intervals

kern

a filter kernel

penalty

penalty term in the multiresolution statistic: "none" for no penalty, "log" for penalizing the log-length of an interval, and "sqrt" for penalizing the square root of the MRC; or a function taking two arguments, the first being the multiresolution coefficients, the second the interval lenghts

norm

how the partial sums should be normalised, by default sqrt(lengths), so they are normalised to equal variance across all interval lengths

p

p-value

...

further arguments passed to function quantile

Value

MRC

a vector giving the maximum as well as the indices of the corresponding interval's start and length

MRCoeff

a matrix giving the multiresolution coefficients for all test intervals

MRC.pvalue, MRC.quant, MRC.simul

the corresponding p-value / quantile / vector of simulated values under the assumption of standard Gaussian white noise

kMRC.pvalue, kMRC.simul, kMRC.simul

the corresponding p-value / quantile / vector of simulated values under the assumption of filtered Gaussian white noise

References

Davies, P. L., Kovac, A. (2001) Local extremes, runs, strings and multiresolution. The Annals of Statistics 29, 1–65.

Dümbgen, L., Spokoiny, V. (2001) Multiscale testing of qualitative hypotheses. The Annals of Statistics 29, 124–152.

Siegmund, D. O., Venkatraman, E. S. (1995) Using the generalized likelihood ratio statistic for sequential detection of a change-point. The Annals of Statistics 23, 255–271.

Siegmund, D. O., Yakir, B. (2000) Tail probabilities for the null distribution of scanning statistics. Bernoulli 6, 191–213.

See Also

monteCarloSimulation, smuceR, jsmurf, stepbound, stepsel, quantile

Examples

set.seed(100)
all.equal(MRC.simul(100, r = 100),
          sort(monteCarloSimulation(n = 100, r = 100, output = "maximum",
                                    penalty = "none", intervalSystem = "dyaLen")),
          check.attributes = FALSE)

# simulate signal of 100 data points
set.seed(100)
f <- rep(c(0, 2, 0), c(60, 10, 30))
# add gaussian noise
x <- f + rnorm(100)
# compute multiresolution criterion
m <- MRC(x)
# compute Monte-Carlo p-value based on 100 simulations
MRC.pvalue(m["max"], length(x), 100)
# compute multiresolution coefficients
M <- MRCoeff(x)

# plot multiresolution coefficients, colours show p-values below 5% in 1% steps
op <- par(mar = c(5, 4, 2, 4) + 0.1)
image(1:length(x), seq(min(x), max(x), length = ncol(M)), apply(M[,ncol(M):1], 1:2,
  MRC.pvalue, n = length(x), r = 100), breaks = (0:5) / 100,
  col = rgb(1, seq(0, 1, length = 5), 0, 0.75),
  xlab = "location / left end of interval", ylab ="measurement",
  main = "Multiresolution Coefficients",
  sub = paste("MRC p-value =", signif(MRC.pvalue(m["max"], length(x), 100), 3)))
axis(4, min(x) + diff(range(x)) * ( pretty(1:ncol(M) - 1) ) / dim(M)[2],
  2^pretty(1:ncol(M) - 1))
mtext("interval lengths", 4, 3)
# plot signal and its mean
points(x)
lines(f, lty = 2)
abline(h = mean(x))
par(op)