Deciding truncation threshold

Qu Tang

Jul 04, 2018

Source: vignettes/truncation_threshold.Rmd

Unlike Actigraph count algorithm, MIMS unit algorithm truncates non-zero values caused by floating number precision limit and the imperfect filtering transition band during MIMS unit computation. This article describes the process of deciding the truncation threshold.

MIMS unit value of constant signal

The constant signal has zero variance.

st = Sys.time()
ts = seq(st, st + 300, length = 300 * 80)
x = rep(1, length(ts))
y = rep(0, length(ts))
z = rep(0, length(ts))
df = data.frame(HEADER_TIME_STAMP = ts, X=x, Y=y, Z=z)

Input data

Plot input accelerometer’s raw signal

MIMSunit::generate_interactive_plot(df, "Acceleration (g)", c(2,3,4))

Compute MIMS-unit values

mims = MIMSunit::custom_mims_unit(df, epoch = '1 sec', dynamic_range = c(-6, 6), allow_truncation = FALSE, output_mims_per_axis = TRUE)

Output: MIMS unit values

Plot MIMS-unit values

MIMSunit::generate_interactive_plot(mims, "MIMS-unit values", c(2,3,4,5))

The output MIMS unit values have high shoot up at the beginning due to the nonlinear phase response of butterworth filter. It takes about 1 min for the output signal to stabilize. Although the signal is constant, the MIMS unit values are not zeros, which is caused by floating number precision limit during MIMS unit computation (filtering).

The median value after stabilization is

## [1] 3.188147e-10

And the standard deviation is

## [1] 7.586365e-14

MIMS unit value of accelerometer signal when a device is resting on a table

The original signal has variance, first import data

df = MIMSunit::rest_on_table
## [1] "X axis variance: 1.997448e-03"
## [1] "Y axis variance: 2.012883e-03"
## [1] "Z axis variance: 1.732148e-03"

Input data

Plot signal of input data

MIMSunit::generate_interactive_plot(df, "Acceleration (g)", c(2,3,4))

Compute MIMS-unit values

mims = MIMSunit::custom_mims_unit(df, epoch = '1 sec', dynamic_range = c(-6, 6), allow_truncation=FALSE, output_mims_per_axis = TRUE)

Output: MIMS unit values

Plot MIMS-unit values

MIMSunit::generate_interactive_plot(mims, "MIMS-unit values", c(2,3,4,5))

The output MIMS unit values have high shoot up at the beginning due to the nonlinear phase response of butterworth filter. It takes about 1 min for the output signal to stabilize. The MIMS unit algorithm outputs non-zero values, which is caused by fluctuation of electronic current of hardware.

The median value after stabilization is

## [1] "1.791693e-03"

And the standard deviation is

## [1] "2.653232e-04"

Conclusion

According to the analysis, a proper truncation threshold for MIMS unit value (1 second bout) can be one numerical magnitude above the larger value of the two cases above, which may be set at 1e-4.

Therefore, the truncation threshold for a given MIMS unit at t epoch length will be,

\[0.0001 \times t\]