The adept
package implements ADaptive Empirical Pattern Transformation (ADEPT) method1 for pattern segmentation from a time-series. ADEPT is optimized to perform fast, accurate walking strides segmentation from high-density data collected with a wearable accelerometer during walking. The method was validated using data collected with sensors worn at left wrist, left hip and both ankles.
This vignette introduces ADEPT algorithm and demonstrates the usage of segmentPattern
function which implements ADEPT approach. Here, we focus on examples with simulated data; see the Walking strides segmentation with adept2 for the example of walking stride segmentation in real-life data.
ADEPT identifies patterns in a time-series x
via maximization of chosen similarity statistic (correlation, covariance, etc.) between a time-series x
and a pattern template(s). It accounts for variability in both (1) pattern duration and (2) pattern shape.
We define a pattern template as a 1-dimensional numeric vector which values represent the pattern of interest (e.g. accelerometry data of a human stride).
In this vignette, a pattern template is a simulated data vector.
adept
packageInstall adept
package from GitHub.
Load adept
and other packages.
adept
packageThe examples below are organized into suites. Examples within one suite share data simulation settings, for example: Examples 1: signal with no noise, same-length pattern.
We simulate a time-series x
. We assume that
x
is collected at a frequency of 100 Hz,x
,true.pattern <- cos(seq(0, 2 * pi, length.out = 100))
x <- c(true.pattern[1], replicate(10, true.pattern[-1]))
data.frame(x = seq(0, 1, length.out = 100), y = true.pattern) %>%
ggplot() + geom_line(aes(x = x, y = y), color = "red") +
theme_bw(base_size = 9) + labs(x = "Phase", y = "Value", title = "Pattern")
We use segmentPattern
to segment pattern from a time-series x
. We assume that a perfect template is available. We use a grid of potential pattern durations of {0.9, 0.95, 1.03, 1.1} seconds; the grid is imperfect in a sense it does not contain the duration of the true pattern used in x
simulation.
out <- segmentPattern(
x = x,
x.fs = 100,
template = true.pattern,
pattern.dur.seq = c(0.9, 0.95, 1.03, 1.1),
similarity.measure = "cor",
compute.template.idx = TRUE)
out
#> tau_i T_i sim_i template_i
#> 1 4 95 0.9987941 1
#> 2 98 103 0.9992482 1
#> 3 202 95 0.9987941 1
#> 4 296 103 0.9992482 1
#> 5 400 95 0.9987941 1
#> 6 494 103 0.9992482 1
#> 7 598 95 0.9987941 1
#> 8 692 103 0.9992482 1
#> 9 796 95 0.9987941 1
#> 10 895 95 0.9987941 1
segmentPattern
outputThe segmentation result is a data frame, where each row describes one identified pattern occurrence:
tau_i
- index of x
where pattern starts,T_i
- pattern duration, expressed in x
vector length,sim_i
- similarity between a template and x
,template_i
- index of a template best matched to a time-series x
(here: one template was used, hence all template_i
’s equal 1).See ?segmentPattern
for details.
pattern.dur.seq
to modify a grid of pattern durationWe next generate a dense grid of potential pattern durations, including value 1.0
seconds used in the x
simulation. A perfect match (sim_i = 1
) between a time-series x
and a template is obtained.
out <- segmentPattern(
x = x,
x.fs = 100,
template = true.pattern,
pattern.dur.seq = c(0.9, 0.95, 1, 1.03, 1.1),
similarity.measure = "cor",
compute.template.idx = TRUE)
out
#> tau_i T_i sim_i template_i
#> 1 1 100 1 1
#> 2 100 100 1 1
#> 3 199 100 1 1
#> 4 298 100 1 1
#> 5 397 100 1 1
#> 6 496 100 1 1
#> 7 595 100 1 1
#> 8 694 100 1 1
#> 9 793 100 1 1
#> 10 892 100 1 1
x.fs
to modify x
time-series frequencyWe use x.fs
to modify x
time-series frequency, expressed in a number of observations in seconds, and we adjust pattern.dur.seq
accordingly. We observe that results are the same as in Example 1(b).
out <- segmentPattern(
x = x,
x.fs = 10, ## Assumed data frequency of 10 observations per second
template = true.pattern,
pattern.dur.seq = c(0.9, 0.95, 1, 1.03, 1.1) * 10, ## Adjusted accordingly
similarity.measure = "cor",
compute.template.idx = TRUE)
out
#> tau_i T_i sim_i template_i
#> 1 1 100 1 1
#> 2 100 100 1 1
#> 3 199 100 1 1
#> 4 298 100 1 1
#> 5 397 100 1 1
#> 6 496 100 1 1
#> 7 595 100 1 1
#> 8 694 100 1 1
#> 9 793 100 1 1
#> 10 892 100 1 1
We simulate a time-series x
. We assume that
x
is collected at a frequency of 100 Hz,x
,set.seed(1)
true.pattern <- cos(seq(0, 2 * pi, length.out = 200))
x <- numeric()
for (vl in seq(70, 130, by = 10)){
true.pattern.s <- approx(
seq(0, 1, length.out = 200),
true.pattern, xout = seq(0, 1, length.out = vl))$y
x <- c(x, true.pattern.s[-1])
if (vl == 70) x <- c(true.pattern.s[1], x)
}
data.frame(x = seq(0, by = 0.01, length.out = length(x)), y = x) %>%
ggplot() + geom_line(aes(x = x, y = y)) + theme_bw(base_size = 9) +
labs(x = "Time [s]", y = "Value", title = "Time-series x")
## Function to plot segmentation results with ggplot2
library(ggplot2)
out.plot1 <- function(val, out, fs = 100){
yrange <- c(-1, 1) * max(abs(val))
y.h <- 0
plt <- ggplot()
for (i in 1:nrow(out)){
tau1_i <- out[i, "tau_i"]
tau2_i <- tau1_i + out[i, "T_i"] - 1
tau1_i <- tau1_i/fs
tau2_i <- tau2_i/fs
plt <-
plt +
geom_vline(xintercept = tau1_i, color = "red") +
geom_vline(xintercept = tau2_i, color = "red") +
annotate(
"rect",
fill = "pink",
alpha = 0.3,
xmin = tau1_i,
xmax = tau2_i,
ymin = yrange[1],
ymax = yrange[2]
)
}
geom_line.df <- data.frame(x = seq(0, by = 1/fs, length.out = length(val)), y = val)
plt <-
plt +
geom_line(data = geom_line.df,
aes(x = x, y = y),
color = "black",
size = 0.3) +
theme_bw(base_size = 9) +
labs(x = "Time [s]", y = "Black line: x",
title = "Black line: signal x\nRed vertical lines: start and end points of identified pattern occurrence\nRed shaded area: area corresponding to identified pattern occurrence")
plot(plt)
}
We use a dense grid of potential pattern duration, including all values used in the x
simulation to again obtain the perfect match (sim_i = 1
). In this example, the start and the end points of identified patterns are connected (see figure below).
out <- segmentPattern(
x = x,
x.fs = 100,
template = true.pattern,
pattern.dur.seq = 60:130 * 0.01,
similarity.measure = "cor",
compute.template.idx = TRUE)
out
#> tau_i T_i sim_i template_i
#> 1 1 70 1 1
#> 2 70 80 1 1
#> 3 149 90 1 1
#> 4 238 100 1 1
#> 5 337 110 1 1
#> 6 446 120 1 1
#> 7 565 130 1 1
out.plot1(x, out)
pattern.dur.seq
to modify a grid of pattern durationNext, we use a less dense grid of potential pattern duration. We observe that perfect match (sim_i = 1
) between a template and time-series x
is no longer obtained. Note:
pattern.dur.seq
grid yields a shorter time of method execution.out <- segmentPattern(
x = x,
x.fs = 100,
template = true.pattern,
pattern.dur.seq = c(0.6, 0.9, 1.2),
similarity.measure = "cor",
compute.template.idx = TRUE)
out
#> tau_i T_i sim_i template_i
#> 1 6 60 0.9954038 1
#> 2 80 60 0.9887682 1
#> 3 149 90 1.0000000 1
#> 4 243 90 0.9976255 1
#> 5 347 90 0.9934059 1
#> 6 446 120 1.0000000 1
#> 7 570 120 0.9985549 1
out.plot1(x, out)
similarity.measure
to modify similarity statisticWe use similarity.measure
to modify the similarity statistic. We observe that sim_i
values in the result data frame change and the segmentation results change slightly too. The explanation is that a change of similarity statistic takes an effect on ADEPT iterative maximization procedure.
out <- segmentPattern(
x = x,
x.fs = 100,
template = true.pattern,
pattern.dur.seq = c(0.6, 0.9, 1.2),
similarity.measure = "cov", ## Use covariance as a similarity statistic
compute.template.idx = TRUE)
out
#> tau_i T_i sim_i template_i
#> 1 6 60 0.6440590 1
#> 2 65 90 0.7338618 1
#> 3 154 90 0.6711920 1
#> 4 243 90 0.6675955 1
#> 5 332 120 0.7320470 1
#> 6 451 120 0.6883499 1
#> 7 570 120 0.6786581 1
out.plot1(x, out)
We simulate a time-series x
. We assume that
x
is collected at a frequency of 100 Hz,x
,true.pattern.1 <- cos(seq(0, 2 * pi, length.out = 200))
true.pattern.2 <- true.pattern.1
true.pattern.2[70:130] <- 2 * true.pattern.2[min(70:130)] + abs(true.pattern.2[70:130])
x <- numeric()
for (vl in seq(70, 130, by = 10)){
true.pattern.1.s <- approx(
seq(0, 1, length.out = 200),
true.pattern.1, xout = seq(0, 1, length.out = vl))$y
true.pattern.2.s <- approx(
seq(0, 1, length.out = 200),
true.pattern.2, xout = seq(0, 1, length.out = vl))$y
x <- c(x, true.pattern.1.s[-1], true.pattern.2.s[-1])
if (vl == 70) x <- c(true.pattern.1.s[1], x)
}
data.frame(x = seq(0, by = 0.01, length.out = length(x)), y = x) %>%
ggplot() + geom_line(aes(x = x, y = y)) + theme_bw(base_size = 9) +
labs(x = "Time [s]", y = "Value", title = "Time-series x")
plt1 <-
data.frame(x = seq(0, 1, length.out = length(true.pattern.1)), y = true.pattern.1) %>%
ggplot() + geom_line(aes(x = x, y = y), color = "red") +
theme_bw(base_size = 9) + labs(x = "Phase", y = "Value", title = "Pattern 1") +
scale_y_continuous(limits = c(-1,1))
plt2 <-
data.frame(x = seq(0, 1, length.out = length(true.pattern.2)), y = true.pattern.2) %>%
ggplot() + geom_line(aes(x = x, y = y), color = "red") +
theme_bw(base_size = 9) + labs(x = "Phase", y = "Value", title = "Pattern 2") +
scale_y_continuous(limits = c(-1,1))
plt1;plt2
To segment pattern from x
, we use a dense grid of potential pattern duration. We use a template
object consisting of one of the two true patterns used in x
simulation.
out <- segmentPattern(
x = x,
x.fs = 100,
template = true.pattern.1, ## Template consisting of one out of two true patterns
pattern.dur.seq = 60:130 * 0.01,
similarity.measure = "cor",
compute.template.idx = TRUE)
out
#> tau_i T_i sim_i template_i
#> 1 1 70 1.0000000 1
#> 2 70 70 0.9290889 1
#> 3 139 80 1.0000000 1
#> 4 218 80 0.9291321 1
#> 5 297 90 1.0000000 1
#> 6 386 90 0.9287727 1
#> 7 475 100 1.0000000 1
#> 8 574 100 0.9287316 1
#> 9 673 110 1.0000000 1
#> 10 782 110 0.9285349 1
#> 11 891 120 1.0000000 1
#> 12 1010 120 0.9284384 1
#> 13 1129 130 1.0000000 1
#> 14 1258 130 0.9283518 1
out.plot1(x, out)
similarity.measure.thresh
to modify the threshold of minimal similarityWe use a similarity threshold to segment only those patterns for which similarity (here: correlation) is greater than 0.95. Note that using the threshold may substantially speed up method execution when working with a large data set.
out <- segmentPattern(
x = x,
x.fs = 100,
template = true.pattern.1,
pattern.dur.seq = 60:130 * 0.01,
similarity.measure = "cor",
similarity.measure.thresh = 0.95,
compute.template.idx = TRUE)
out
#> tau_i T_i sim_i template_i
#> 1 1 70 1 1
#> 2 139 80 1 1
#> 3 297 90 1 1
#> 4 475 100 1 1
#> 5 673 110 1 1
#> 6 891 120 1 1
#> 7 1129 130 1 1
out.plot1(x, out)
We next use a template
object consisting of both true patterns used in x
simulation. We observe that the index of a pattern template best matched to a pattern in the time-series x
is 1
and 2
interchangeably.
out <- segmentPattern(
x = x,
x.fs = 100,
template = list(true.pattern.1, true.pattern.2),
pattern.dur.seq = 60:130 * 0.01,
similarity.measure = "cor",
compute.template.idx = TRUE)
out
#> tau_i T_i sim_i template_i
#> 1 1 70 1 1
#> 2 70 70 1 2
#> 3 139 80 1 1
#> 4 218 80 1 2
#> 5 297 90 1 1
#> 6 386 90 1 2
#> 7 475 100 1 1
#> 8 574 100 1 2
#> 9 673 110 1 1
#> 10 782 110 1 2
#> 11 891 120 1 1
#> 12 1010 120 1 2
#> 13 1129 130 1 1
#> 14 1258 130 1 2
out.plot1(x, out)
We simulate a time-series x
. We assume that
x
is collected at a frequency of 100 Hz,x
,## Generate `x` as a noisy version of a time-series generated in *Examples 3*.
set.seed(1)
x <- x + rnorm(length(x), sd = 0.5)
data.frame(x = seq(0, by = 0.01, length.out = length(x)), y = x) %>%
ggplot() + geom_line(aes(x = x, y = y), size = 0.3) + theme_bw(base_size = 9) +
labs(x = "Time [s]", y = "Value", title = "Time-series x")
We use two templates simultaneously in segmentation.
out <- segmentPattern(
x = x,
x.fs = 100,
template = list(true.pattern.1, true.pattern.2),
pattern.dur.seq = 60:130 * 0.01,
similarity.measure = "cor",
compute.template.idx = TRUE)
out
#> tau_i T_i sim_i template_i
#> 1 4 63 0.8615665 1
#> 2 76 62 0.7351057 2
#> 3 137 84 0.7708233 1
#> 4 220 76 0.6768561 2
#> 5 295 92 0.8507212 1
#> 6 391 83 0.7404364 2
#> 7 473 106 0.8300020 1
#> 8 582 87 0.6515298 2
#> 9 668 118 0.8047385 1
#> 10 785 106 0.6958617 2
#> 11 890 123 0.7927484 1
#> 12 1015 113 0.6789908 1
#> 13 1129 130 0.7938183 1
#> 14 1265 123 0.7686607 2
out.plot1(x, out)
x.adept.ma.W
to smooth x
for similarity matrix computationWe use x.adept.ma.W
to define a length of a smoothing window to smooth x
for similarity matrix computation; x.adept.ma.W
is expressed in seconds and the default is NULL
(no smoothing applied).
Smoothing of a time-series x
Function windowSmooth
allows observing the effect of smoothing for different values of smoothing window length W
. W
is expressed in seconds. Here, W = 0.1
seconds seems to be a plausible choice.
x.smoothed <- windowSmooth(x = x, x.fs = 100, W = 0.1)
data.frame(x = seq(0, by = 0.01, length.out = length(x.smoothed)), y = x.smoothed) %>%
ggplot() + geom_line(aes(x = x, y = y)) + theme_bw(base_size = 9) +
labs(x = "Time [s]", y = "Value", title = "Time-series x smoothed")
#> Warning: Removed 8 row(s) containing missing values (geom_path).
Use x.adept.ma.W = 0.1
and compare with results from Example 4(a). Observe that using a smoothed version of x
in similarity matrix computation is pronounced in sim_i
values in the output data frame, as well as in a slight change in tau_i
and T_i
values.
out <- segmentPattern(
x = x,
x.fs = 100,
template = list(true.pattern.1, true.pattern.2),
pattern.dur.seq = 60:130 * 0.01,
similarity.measure = "cor",
x.adept.ma.W = 0.1,
compute.template.idx = TRUE)
out
#> tau_i T_i sim_i template_i
#> 1 4 63 0.9931174 1
#> 2 75 63 0.9683646 2
#> 3 139 79 0.9683054 1
#> 4 217 80 0.9748040 2
#> 5 296 94 0.9802473 1
#> 6 391 82 0.9462213 2
#> 7 472 106 0.9855837 1
#> 8 578 93 0.9608881 2
#> 9 670 115 0.9887225 1
#> 10 784 107 0.9562694 2
#> 11 896 113 0.9734575 1
#> 12 1008 127 0.9703118 1
#> 13 1134 116 0.9606235 1
#> 14 1266 122 0.9593345 2
out.plot1(x, out)
We employ a fine-tuning procedure for stride identification.
Fine-tune procedure "maxima"
Fine-tune procedure "maxima"
tunes preliminarily identified start and end of a pattern occurrence so as they correspond to local maxima of x
found within neighborhoods of the preliminary locations.
finetune.maxima.nbh.W
, expressed in seconds, defines a length of the two neighborhoods within which we search for local maxima.x
may be used for local maxima search (as presented later).out <- segmentPattern(
x = x,
x.fs = 100,
template = list(true.pattern.1, true.pattern.2),
pattern.dur.seq = 60:130 * 0.01,
x.adept.ma.W = 0.1,
finetune = "maxima",
finetune.maxima.nbh.W = 0.3,
compute.template.idx = TRUE)
out
#> tau_i T_i sim_i template_i
#> 1 4 67 0.7183698 1
#> 2 70 78 0.4250660 2
#> 3 147 60 0.6161509 1
#> 4 206 98 0.5407526 2
#> 5 303 78 0.7570218 1
#> 6 380 83 0.3459633 2
#> 7 462 117 0.7566122 1
#> 8 578 93 0.5045821 2
#> 9 670 105 0.7153597 1
#> 10 774 113 0.5760463 2
#> 11 893 128 0.6794591 1
#> 12 1020 112 0.4026054 1
#> 13 1131 123 0.6736391 1
#> 14 1253 125 0.5570096 2
out.plot1(x, out)
We observe that almost all identified pattern occurrence start/end points are hitting the time point which our eyes identify as local x
maxima.
We smooth x
for both similarity matrix computation (set x.adept.ma.W = 0.1
) and for fine-tune peak detection procedure (set finetune.maxima.nbh.W = 0.3
).
W = 0.5
seconds seems to be a plausible choice for fine-tune peak detection procedure as it removes (“smooth together”) multiple local maxima of x
, leaving out a single one.x.smoothed.2 <- windowSmooth(x = x, x.fs = 100, W = 0.5)
data.frame(x = seq(0, by = 0.01, length.out = length(x.smoothed.2)), y = x.smoothed.2) %>%
ggplot() + geom_line(aes(x = x, y = y)) + theme_bw(base_size = 9) +
labs(x = "Time [s]", y = "Value", title = "Time-series x smoothed aggresively")
#> Warning: Removed 48 row(s) containing missing values (geom_path).