stepR-package: Multiscale Change-Point Inference
In stepR: Multiscale Change-Point Inference
stepR-package R Documentation
Multiscale Change-Point Inference
Description
Allows fitting of step-functions to univariate serial data where neither the number of jumps nor their positions is known by implementing the multiscale regression estimators SMUCE (Frick et al., 2014) and HSMUCE (Pein et al., 2017). In addition, confidence intervals for the change-point locations and bands for the unknown signal can be obtained. This is implemented in the function stepFit. Moreover, technical quantities like the statistics itself, bounds or critical values can be computed by other functions of the package. More details can be found in the vignette.
Details
New in version 2.0-0:
stepFit Piecewise constant multiscale inference
critVal Critical values
computeBounds Computation of the bounds
computeStat Computation of the multiscale statistic
monteCarloSimulation Monte Carlo simulation
parametricFamily Parametric families
intervalSystem Interval systems
penalty Penalties
From version 1.0-0:
compareBlocks Compare fit blockwise with ground truth
neighbours Neighbouring integers
sdrobnorm Robust standard deviation estimate
stepblock Step function
stepcand Forward selection of candidate jumps
stepfit Fitted step function
steppath Solution path of step-functions
stepsel Automatic selection of number of jumps
Mainly used for patchclamp recordings and may be transferred to a specialised package:
BesselPolynomial Bessel Polynomials
contMC Continuous time Markov chain
dfilter Digital filters
jsmurf Reconstruct filtered piecewise constant functions with noise
transit TRANSIT algorithm for detecting jumps
Deprecated (please read the documentation of them theirself for more details):
MRC Compute Multiresolution Criterion
MRC.1000 Values of the MRC statistic for 1,000 observations (all intervals)
MRC.asymptotic "Asymptotic" values of the MRC statistic (all intervals)
MRC.asymptotic.dyadic "Asymptotic" values of the MRC statistic(dyadic intervals)
bounds Bounds based on MRC
family Family of distributions
smuceR Piecewise constant regression with SMUCE
Storing of Monte-Carlo simulations
If q == NULL in critVal, stepFit or computeBounds a Monte-Carlo simulation is required for computing critical values. Since a Monte-Carlo simulation lasts potentially much longer (up to several hours or days if the number of observations is in the millions) than the main calculations, this package offers multiple possibilities for saving and loading the simulations. Simulations can either be saved in the workspace in the variable critValStepRTab or persistently on the file system for which the package R.cache is used. Moreover, storing in and loading from variables and RDS files is supported. Finally, a pre-simulated collection of simulations can be accessed by installing the package stepRdata available from http://www.stochastik.math.uni-goettingen.de/stepRdata_1.0-0.tar.gz. By default simulations will be saved in the workspace and on the file system. For more details and for how simulation can be removed see Section Simulating, saving and loading of Monte-Carlo simulations in critVal.
References
Frick, K., Munk, A., Sieling, H. (2014) Multiscale change-point inference. With discussion and rejoinder by the authors. Journal of the Royal Statistical Society, Series B 76(3), 495–580.
Pein, F., Sieling, H., Munk, A. (2017) Heterogeneous change point inference. Journal of the Royal Statistical Society, Series B, 79(4), 1207–1227.
Pein, F., Tecuapetla-Gómez, I., Schütte, O., Steinem, C., Munk, A. (2017) Fully-automatic multiresolution idealization for filtered ion channel recordings: flickering event detection. arXiv:1706.03671.
Hotz, T., Schütte, O., Sieling, H., Polupanow, T., Diederichsen, U., Steinem, C., and Munk, A. (2013) Idealizing ion channel recordings by a jump segmentation multiresolution filter. IEEE Transactions on NanoBioscience 12(4), 376–386.
VanDongen, A. M. J. (1996) A new algorithm for idealizing single ion channel data containing multiple unknown conductance levels. Biophysical Journal 70(3), 1303–1315.
Futschik, A., Hotz, T., Munk, A., Sieling, H. (2014) Multiresolution DNA partitioning: statistical evidence for segments. Bioinformatics, 30(16), 2255–2262.
Boysen, L., Kempe, A., Liebscher, V., Munk, A., Wittich, O. (2009) Consistencies and rates of convergence of jump-penalized least squares estimators. The Annals of Statistics 37(1), 157–183.
Davies, P. L., Kovac, A. (2001) Local extremes, runs, strings and multiresolution. The Annals of Statistics 29, 1–65.
Friedrich, F., Kempe, A., Liebscher, V., Winkler, G. (2008) Complexity penalized M-estimation: fast computation. Journal of Computational and Graphical Statistics 17(1), 201–224.
See Also
stepFit, critVal, computeStat, computeBounds, jsmurf, transit, sdrobnorm, compareBlocks, parametricFamily, intervalSystem, penalty
Examples
# generate random observations
set.seed(1)
n <- 100L
x <- seq(1 / n, 1, 1 / n)
mu <- stepfit(cost = 0, family = "gauss", value = c(0, 3, 0, -2, 0), param = NULL,
leftEnd = x[c(1, 21, 26, 71, 81)],
rightEnd = x[c(20, 25, 70, 80, 100)], x0 = 0,
leftIndex = c(1, 21, 26, 71, 81),
rightIndex = c(20, 25, 70, 80, 100))
sigma0 <- 0.5
epsilon <- rnorm(n, 0, sigma0)
y <- fitted(mu) + epsilon
plot(x, y, pch = 16, col = "grey30", ylim = c(-3, 4))
lines(mu, lwd = 3)
# computation of SMUCE and its confidence statements
fit <- stepFit(y, x = x, alpha = 0.5, jumpint = TRUE, confband = TRUE)
lines(fit, lwd = 3, col = "red", lty = "22")
# confidence intervals for the change-point locations
points(jumpint(fit), col = "red", lwd = 3)
# confidence band
lines(confband(fit), lty = "22", col = "darkred", lwd = 2)
# higher significance level for larger detection power, but less confidence
# suggested for screening purposes
stepFit(y, x = x, alpha = 0.9, jumpint = TRUE, confband = TRUE)
# smaller significance level for the small risk that the number of
# change-points is overestimated with probability not more than 5%,
# but smaller detection power
stepFit(y, x = x, alpha = 0.05, jumpint = TRUE, confband = TRUE)
# different interval system, lengths, penalty and given parameter sd
stepFit(y, x = x, alpha = 0.5, intervalSystem = "dyaLen",
lengths = c(1L, 2L, 4L, 8L), penalty = "weights",
weights = c(0.4, 0.3, 0.2, 0.1), sd = sigma0,
jumpint = TRUE, confband = TRUE)
# the above calls saved and (attempted to) load Monte-Carlo simulations and
# simulated them for nq = 128 observations
# in the following call no saving, no loading and simulation for n = 100
# observations is required, progress of the simulation will be reported
stepFit(y, x = x, alpha = 0.5, jumpint = TRUE, confband = TRUE, messages = 1000L,
options = list(simulation = "vector", load = list(), save = list()))
# critVal was called in stepFit, can be called explicitly,
# for instance outside of a for loop to save computation time
qVector <- critVal(100L, alpha = 0.5)
identical(stepFit(y, x = x, q = qVector, jumpint = TRUE, confband = TRUE), fit)
qValue <- critVal(100L, alpha = 0.5, output = "value")
identical(stepFit(y, x = x, q = qValue, jumpint = TRUE, confband = TRUE), fit)
# computeBounds gives the multiscale contraint
computeBounds(y, alpha = 0.5)
# monteCarloSimulation will be called in critVal if required
# can be called explicitly
stat <- monteCarloSimulation(n = 100L)
identical(critVal(n = 100L, alpha = 0.5, stat = stat),
critVal(n = 100L, alpha = 0.5,
options = list(load = list(), simulation = "vector")))
identical(critVal(n = 100L, alpha = 0.5, stat = stat, output = "value"),
critVal(n = 100L, alpha = 0.5, output = "value",
options = list(load = list(), simulation = "vector")))
stat <- monteCarloSimulation(n = 100L, output = "maximum")
identical(critVal(n = 100L, alpha = 0.5, stat = stat),
critVal(n = 100L, alpha = 0.5,
options = list(load = list(), simulation = "vector")))
identical(critVal(n = 100L, alpha = 0.5, stat = stat, output = "value"),
critVal(n = 100L, alpha = 0.5, output = "value",
options = list(load = list(), simulation = "vector")))
# fit satisfies the multiscale contraint, i.e.
# the computed penalized multiscale statistic is not larger than the global quantile
computeStat(y, signal = fit, output = "maximum") <= qValue
# multiscale vector of statistics is componentwise not larger than
# the vector of critical values
all(computeStat(y, signal = fit, output = "vector") <= qVector)
# family "hsmuce"
set.seed(1)
y <- c(rnorm(50, 0, 1), rnorm(50, 1, 0.2))
plot(x, y, pch = 16, col = "grey30", ylim = c(-2.5, 2))
# computation of HSMUCE and its confidence statements
fit <- stepFit(y, x = x, alpha = 0.5, family = "hsmuce",
jumpint = TRUE, confband = TRUE)
lines(fit, lwd = 3, col = "red", lty = "22")
# confidence intervals for the change-point locations
points(jumpint(fit), col = "red", lwd = 3)
# confidence band
lines(confband(fit), lty = "22", col = "darkred", lwd = 2)
# for comparison SMUCE, not recommend to use here
lines(stepFit(y, x = x, alpha = 0.5,
jumpint = TRUE, confband = TRUE),
lwd = 3, col = "blue", lty = "22")
# family "mDependentPS"
# generate observations from a moving average process
set.seed(1)
y <- c(rep(0, 50), rep(2, 50)) +
as.numeric(arima.sim(n = 100, list(ar = c(), ma = c(0.8, 0.5, 0.3)), sd = sigma0))
correlations <- as.numeric(ARMAacf(ar = c(), ma = c(0.8, 0.5, 0.3), lag.max = 3))
covariances <- sigma0^2 * correlations
plot(x, y, pch = 16, col = "grey30", ylim = c(-2, 4))
# computation of SMUCE for dependent observations with given covariances
fit <- stepFit(y, x = x, alpha = 0.5, family = "mDependentPS",
covariances = covariances, jumpint = TRUE, confband = TRUE)
lines(fit, lwd = 3, col = "red", lty = "22")
# confidence intervals for the change-point locations
points(jumpint(fit), col = "red", lwd = 3)
# confidence band
lines(confband(fit), lty = "22", col = "darkred", lwd = 2)
# for comparison SMUCE for independent observations, not recommend to use here
lines(stepFit(y, x = x, alpha = 0.5,
jumpint = TRUE, confband = TRUE),
lwd = 3, col = "blue", lty = "22")
# with given correlations, standard deviation will be estimated by sdrobnorm
stepFit(y, x = x, alpha = 0.5, family = "mDependentPS",
correlations = correlations, jumpint = TRUE, confband = TRUE)
# examples from version 1.0-0
# estimating step-functions with Gaussian white noise added
# simulate a Gaussian hidden Markov model of length 1000 with 2 states
# with identical transition rates 0.01, and signal-to-noise ratio 2
sim <- contMC(1e3, 0:1, matrix(c(0, 0.01, 0.01, 0), 2), param=1/2)
plot(sim$data, cex = 0.1)
lines(sim$cont, col="red")
# maximum-likelihood estimation under multiresolution constraints
fit.MRC <- smuceR(sim$data$y, sim$data$x)
lines(fit.MRC, col="blue")
# choose number of jumps using BIC
path <- steppath(sim$data$y, sim$data$x, max.blocks=1e2)
fit.BIC <- path[[stepsel.BIC(path)]]
lines(fit.BIC, col="green3", lty = 2)
# estimate after filtering
# simulate filtered ion channel recording with two states
set.seed(9)
# sampling rate 10 kHz
sampling <- 1e4
# tenfold oversampling
over <- 10
# 1 kHz 4-pole Bessel-filter, adjusted for oversampling
cutoff <- 1e3
df.over <- dfilter("bessel", list(pole=4, cutoff=cutoff / sampling / over))
# two states, leaving state 1 at 10 Hz, state 2 at 20 Hz
rates <- rbind(c(0, 10), c(20, 0))
# simulate 0.5 s, level 0 corresponds to state 1, level 1 to state 2
# noise level is 0.3 after filtering
Sim <- contMC(0.5 * sampling, 0:1, rates, sampling=sampling, family="gaussKern",
param = list(df=df.over, over=over, sd=0.3))
plot(Sim$data, pch = ".")
lines(Sim$discr, col = "red")
# fit under multiresolution constraints using filter corresponding to sample rate
df <- dfilter("bessel", list(pole=4, cutoff=cutoff / sampling))
Fit.MRC <- jsmurf(Sim$data$y, Sim$data$x, param=df, r=1e2)
lines(Fit.MRC, col = "blue")
# fit using TRANSIT
Fit.trans <- transit(Sim$data$y, Sim$data$x)
lines(Fit.trans, col = "green3", lty=2)
stepR documentation built on Nov. 14, 2023, 1:09 a.m.
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| stepR-package | R Documentation |
Multiscale Change-Point Inference
Description
Allows fitting of step-functions to univariate serial data where neither the number of jumps nor their positions is known by implementing the multiscale regression estimators SMUCE (Frick et al., 2014) and HSMUCE (Pein et al., 2017). In addition, confidence intervals for the change-point locations and bands for the unknown signal can be obtained. This is implemented in the function stepFit. Moreover, technical quantities like the statistics itself, bounds or critical values can be computed by other functions of the package. More details can be found in the vignette.
Details
New in version 2.0-0:
stepFit | Piecewise constant multiscale inference |
critVal | Critical values |
computeBounds | Computation of the bounds |
computeStat | Computation of the multiscale statistic |
monteCarloSimulation | Monte Carlo simulation |
parametricFamily | Parametric families |
intervalSystem | Interval systems |
penalty | Penalties |
From version 1.0-0:
compareBlocks | Compare fit blockwise with ground truth |
neighbours | Neighbouring integers |
sdrobnorm | Robust standard deviation estimate |
stepblock | Step function |
stepcand | Forward selection of candidate jumps |
stepfit | Fitted step function |
steppath | Solution path of step-functions |
stepsel | Automatic selection of number of jumps |
Mainly used for patchclamp recordings and may be transferred to a specialised package:
BesselPolynomial | Bessel Polynomials |
contMC | Continuous time Markov chain |
dfilter | Digital filters |
jsmurf | Reconstruct filtered piecewise constant functions with noise |
transit | TRANSIT algorithm for detecting jumps |
Deprecated (please read the documentation of them theirself for more details):
MRC | Compute Multiresolution Criterion |
MRC.1000 | Values of the MRC statistic for 1,000 observations (all intervals) |
MRC.asymptotic | "Asymptotic" values of the MRC statistic (all intervals) |
MRC.asymptotic.dyadic | "Asymptotic" values of the MRC statistic(dyadic intervals) |
bounds | Bounds based on MRC |
family | Family of distributions |
smuceR | Piecewise constant regression with SMUCE |
Storing of Monte-Carlo simulations
If q == NULL in critVal, stepFit or computeBounds a Monte-Carlo simulation is required for computing critical values. Since a Monte-Carlo simulation lasts potentially much longer (up to several hours or days if the number of observations is in the millions) than the main calculations, this package offers multiple possibilities for saving and loading the simulations. Simulations can either be saved in the workspace in the variable critValStepRTab or persistently on the file system for which the package R.cache is used. Moreover, storing in and loading from variables and RDS files is supported. Finally, a pre-simulated collection of simulations can be accessed by installing the package stepRdata available from http://www.stochastik.math.uni-goettingen.de/stepRdata_1.0-0.tar.gz. By default simulations will be saved in the workspace and on the file system. For more details and for how simulation can be removed see Section Simulating, saving and loading of Monte-Carlo simulations in critVal.
References
Frick, K., Munk, A., Sieling, H. (2014) Multiscale change-point inference. With discussion and rejoinder by the authors. Journal of the Royal Statistical Society, Series B 76(3), 495–580.
Pein, F., Sieling, H., Munk, A. (2017) Heterogeneous change point inference. Journal of the Royal Statistical Society, Series B, 79(4), 1207–1227.
Pein, F., Tecuapetla-Gómez, I., Schütte, O., Steinem, C., Munk, A. (2017) Fully-automatic multiresolution idealization for filtered ion channel recordings: flickering event detection. arXiv:1706.03671.
Hotz, T., Schütte, O., Sieling, H., Polupanow, T., Diederichsen, U., Steinem, C., and Munk, A. (2013) Idealizing ion channel recordings by a jump segmentation multiresolution filter. IEEE Transactions on NanoBioscience 12(4), 376–386.
VanDongen, A. M. J. (1996) A new algorithm for idealizing single ion channel data containing multiple unknown conductance levels. Biophysical Journal 70(3), 1303–1315.
Futschik, A., Hotz, T., Munk, A., Sieling, H. (2014) Multiresolution DNA partitioning: statistical evidence for segments. Bioinformatics, 30(16), 2255–2262.
Boysen, L., Kempe, A., Liebscher, V., Munk, A., Wittich, O. (2009) Consistencies and rates of convergence of jump-penalized least squares estimators. The Annals of Statistics 37(1), 157–183.
Davies, P. L., Kovac, A. (2001) Local extremes, runs, strings and multiresolution. The Annals of Statistics 29, 1–65.
Friedrich, F., Kempe, A., Liebscher, V., Winkler, G. (2008) Complexity penalized M-estimation: fast computation. Journal of Computational and Graphical Statistics 17(1), 201–224.
See Also
stepFit, critVal, computeStat, computeBounds, jsmurf, transit, sdrobnorm, compareBlocks, parametricFamily, intervalSystem, penalty
Examples
# generate random observations
set.seed(1)
n <- 100L
x <- seq(1 / n, 1, 1 / n)
mu <- stepfit(cost = 0, family = "gauss", value = c(0, 3, 0, -2, 0), param = NULL,
leftEnd = x[c(1, 21, 26, 71, 81)],
rightEnd = x[c(20, 25, 70, 80, 100)], x0 = 0,
leftIndex = c(1, 21, 26, 71, 81),
rightIndex = c(20, 25, 70, 80, 100))
sigma0 <- 0.5
epsilon <- rnorm(n, 0, sigma0)
y <- fitted(mu) + epsilon
plot(x, y, pch = 16, col = "grey30", ylim = c(-3, 4))
lines(mu, lwd = 3)
# computation of SMUCE and its confidence statements
fit <- stepFit(y, x = x, alpha = 0.5, jumpint = TRUE, confband = TRUE)
lines(fit, lwd = 3, col = "red", lty = "22")
# confidence intervals for the change-point locations
points(jumpint(fit), col = "red", lwd = 3)
# confidence band
lines(confband(fit), lty = "22", col = "darkred", lwd = 2)
# higher significance level for larger detection power, but less confidence
# suggested for screening purposes
stepFit(y, x = x, alpha = 0.9, jumpint = TRUE, confband = TRUE)
# smaller significance level for the small risk that the number of
# change-points is overestimated with probability not more than 5%,
# but smaller detection power
stepFit(y, x = x, alpha = 0.05, jumpint = TRUE, confband = TRUE)
# different interval system, lengths, penalty and given parameter sd
stepFit(y, x = x, alpha = 0.5, intervalSystem = "dyaLen",
lengths = c(1L, 2L, 4L, 8L), penalty = "weights",
weights = c(0.4, 0.3, 0.2, 0.1), sd = sigma0,
jumpint = TRUE, confband = TRUE)
# the above calls saved and (attempted to) load Monte-Carlo simulations and
# simulated them for nq = 128 observations
# in the following call no saving, no loading and simulation for n = 100
# observations is required, progress of the simulation will be reported
stepFit(y, x = x, alpha = 0.5, jumpint = TRUE, confband = TRUE, messages = 1000L,
options = list(simulation = "vector", load = list(), save = list()))
# critVal was called in stepFit, can be called explicitly,
# for instance outside of a for loop to save computation time
qVector <- critVal(100L, alpha = 0.5)
identical(stepFit(y, x = x, q = qVector, jumpint = TRUE, confband = TRUE), fit)
qValue <- critVal(100L, alpha = 0.5, output = "value")
identical(stepFit(y, x = x, q = qValue, jumpint = TRUE, confband = TRUE), fit)
# computeBounds gives the multiscale contraint
computeBounds(y, alpha = 0.5)
# monteCarloSimulation will be called in critVal if required
# can be called explicitly
stat <- monteCarloSimulation(n = 100L)
identical(critVal(n = 100L, alpha = 0.5, stat = stat),
critVal(n = 100L, alpha = 0.5,
options = list(load = list(), simulation = "vector")))
identical(critVal(n = 100L, alpha = 0.5, stat = stat, output = "value"),
critVal(n = 100L, alpha = 0.5, output = "value",
options = list(load = list(), simulation = "vector")))
stat <- monteCarloSimulation(n = 100L, output = "maximum")
identical(critVal(n = 100L, alpha = 0.5, stat = stat),
critVal(n = 100L, alpha = 0.5,
options = list(load = list(), simulation = "vector")))
identical(critVal(n = 100L, alpha = 0.5, stat = stat, output = "value"),
critVal(n = 100L, alpha = 0.5, output = "value",
options = list(load = list(), simulation = "vector")))
# fit satisfies the multiscale contraint, i.e.
# the computed penalized multiscale statistic is not larger than the global quantile
computeStat(y, signal = fit, output = "maximum") <= qValue
# multiscale vector of statistics is componentwise not larger than
# the vector of critical values
all(computeStat(y, signal = fit, output = "vector") <= qVector)
# family "hsmuce"
set.seed(1)
y <- c(rnorm(50, 0, 1), rnorm(50, 1, 0.2))
plot(x, y, pch = 16, col = "grey30", ylim = c(-2.5, 2))
# computation of HSMUCE and its confidence statements
fit <- stepFit(y, x = x, alpha = 0.5, family = "hsmuce",
jumpint = TRUE, confband = TRUE)
lines(fit, lwd = 3, col = "red", lty = "22")
# confidence intervals for the change-point locations
points(jumpint(fit), col = "red", lwd = 3)
# confidence band
lines(confband(fit), lty = "22", col = "darkred", lwd = 2)
# for comparison SMUCE, not recommend to use here
lines(stepFit(y, x = x, alpha = 0.5,
jumpint = TRUE, confband = TRUE),
lwd = 3, col = "blue", lty = "22")
# family "mDependentPS"
# generate observations from a moving average process
set.seed(1)
y <- c(rep(0, 50), rep(2, 50)) +
as.numeric(arima.sim(n = 100, list(ar = c(), ma = c(0.8, 0.5, 0.3)), sd = sigma0))
correlations <- as.numeric(ARMAacf(ar = c(), ma = c(0.8, 0.5, 0.3), lag.max = 3))
covariances <- sigma0^2 * correlations
plot(x, y, pch = 16, col = "grey30", ylim = c(-2, 4))
# computation of SMUCE for dependent observations with given covariances
fit <- stepFit(y, x = x, alpha = 0.5, family = "mDependentPS",
covariances = covariances, jumpint = TRUE, confband = TRUE)
lines(fit, lwd = 3, col = "red", lty = "22")
# confidence intervals for the change-point locations
points(jumpint(fit), col = "red", lwd = 3)
# confidence band
lines(confband(fit), lty = "22", col = "darkred", lwd = 2)
# for comparison SMUCE for independent observations, not recommend to use here
lines(stepFit(y, x = x, alpha = 0.5,
jumpint = TRUE, confband = TRUE),
lwd = 3, col = "blue", lty = "22")
# with given correlations, standard deviation will be estimated by sdrobnorm
stepFit(y, x = x, alpha = 0.5, family = "mDependentPS",
correlations = correlations, jumpint = TRUE, confband = TRUE)
# examples from version 1.0-0
# estimating step-functions with Gaussian white noise added
# simulate a Gaussian hidden Markov model of length 1000 with 2 states
# with identical transition rates 0.01, and signal-to-noise ratio 2
sim <- contMC(1e3, 0:1, matrix(c(0, 0.01, 0.01, 0), 2), param=1/2)
plot(sim$data, cex = 0.1)
lines(sim$cont, col="red")
# maximum-likelihood estimation under multiresolution constraints
fit.MRC <- smuceR(sim$data$y, sim$data$x)
lines(fit.MRC, col="blue")
# choose number of jumps using BIC
path <- steppath(sim$data$y, sim$data$x, max.blocks=1e2)
fit.BIC <- path[[stepsel.BIC(path)]]
lines(fit.BIC, col="green3", lty = 2)
# estimate after filtering
# simulate filtered ion channel recording with two states
set.seed(9)
# sampling rate 10 kHz
sampling <- 1e4
# tenfold oversampling
over <- 10
# 1 kHz 4-pole Bessel-filter, adjusted for oversampling
cutoff <- 1e3
df.over <- dfilter("bessel", list(pole=4, cutoff=cutoff / sampling / over))
# two states, leaving state 1 at 10 Hz, state 2 at 20 Hz
rates <- rbind(c(0, 10), c(20, 0))
# simulate 0.5 s, level 0 corresponds to state 1, level 1 to state 2
# noise level is 0.3 after filtering
Sim <- contMC(0.5 * sampling, 0:1, rates, sampling=sampling, family="gaussKern",
param = list(df=df.over, over=over, sd=0.3))
plot(Sim$data, pch = ".")
lines(Sim$discr, col = "red")
# fit under multiresolution constraints using filter corresponding to sample rate
df <- dfilter("bessel", list(pole=4, cutoff=cutoff / sampling))
Fit.MRC <- jsmurf(Sim$data$y, Sim$data$x, param=df, r=1e2)
lines(Fit.MRC, col = "blue")
# fit using TRANSIT
Fit.trans <- transit(Sim$data$y, Sim$data$x)
lines(Fit.trans, col = "green3", lty=2)
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