motus.org

knitr::opts_chunk$set(comment="")
options(warn=0)
library(dplyr)
library(lattice)

We need some reasonable upper bound for the lifetimes of tags, by battery and BI.

Lotek provides a short table here:

http://www.lotek.com/bird+bat-nano.pdf (As of 1 April, 2016)

and there are additional models listed in "ANTC Spec Sheet.pdf" (no source), including the only numbers for battery type "3-1". Since lifetime is listed as depending only on the battery (numeric portion of model string), we model on that.

lt = as.tbl(read.csv("./lotekTagLifespanByBatteryAndBI.csv"))

Add inverses of columns bi and dutyCycle, for modelling

ls = lt %>% mutate(biInv = 1.0 / bi, dcInv = 1.0 / dutyCycle)

print(ls)

A simple model assumes that the battery capacity, $K$, depends on the battery model, and that tag power consumption $r$ is a sum of a baseline rate $r_0$ plus a pulse-dependent rate $r_1$. All tags transmit 4 pulses per burst, so the pulse-dependent rate $r_1$ depends only on duty cycle and burst interval like so: $r_1 = r_p * \frac{dutyCyle}{BI}$

The full non-linear model is :

[ lifespan = \frac{K}{r_0 + \frac{r_p * dutyCycle}{BI}} ]

This is over-parameterized (we can divide top and bottom by $r_0$ to get a model with two parameters), so we simplify by rewriting $K/r_0$ as $D$, the number of days of battery life at the baseline rate, and $r_p / r_0$ be $r_t$, the relative rate of power consumption during transmission, versus baseline. The new model is:

[ lifespan = \frac{D}{1 + \frac{r_t * dutyCycle}{BI}} ]

We fit the model to each type of battery:

par = res = pred = NULL
bnames = unique(ls$battery)
for (m in bnames) {
    res[[m]] = nls(lifespan~D / (1 + rt * dutyCycle / bi),
                   subset(ls, battery==m), list(D = 500, rt = 5))
    par = rbind(par, c(coefficients(res[[m]]), max(residuals(res[[m]]))))
    pred = rbind(pred,
                 data.frame(
                     battery=m,
                     bi=1:40,
                     lifespan=predict(res[[m]], list(bi=1:40, dutyCycle=1))
                 ))
}
rownames(par) = bnames
colnames(par) = c("D", "rt", "Max Residual (days)")

Stu Mackenzie pointed out there is a light-weight version of the NTQB-1 which he's dubbed NTQB-1-LW, with "~2/3 the lifetime of the NTQB-1". For now we'll assume that means the $D$ parameter for that model is 2/3 that for the NTQB-1

par = rbind(c(par[["1","D"]] * 2 / 3, par[["1", "rt"]], NA), par)
rownames(par)[1] = "1-LW"
pred = rbind(data.frame(
    battery="1-LW",
    bi=1:40,
    lifespan=par[["1-LW", "D"]] / (1 + par[["1-LW", "rt"]] / (1:40)))
  , pred)

## mark which rows are measurements
pred = cbind(pred, isMeasured = paste(pred$battery, pred$bi) %in% paste(lt$battery, lt$bi))

Now map model names to batteries. We do this as a simple table because the naming scheme isn't consistent enough to bother doing it programmatically.

## Models and the batteries they correspond to.  Note that
## we include a model called 'unknown' which corresponds to
## a battery called 'unknown' that is assigned a life equal
## to the average of those for NTQB-2, NTQB-3-2.
## This forces a warning to be emailed to the motus admin whenever
## the metadata cache is updated, because an unmodelled tag
## is a serious metadata issue.

modelBattery = list(
"NTQB-1-LW"  = "1-LW",
"NTQB-1"     = "1",
"NTQB-2"     = "2",
"NTQB-3-2"   = "3-2",
"NTQB-4-2"   = "4-2",
"NTQB-6-1"   = "6-1",
"NTQB-6-2"   = "6-2",
"NTQBW-3-2"  = "3-2",
"NTQBW-2"    = "2",
"NTQBW-4-2"  = "4-2",
"NTQBW-6-2"  = "6-2",
"ANTC-M1-1"  = "1",
"ANTC-M2-1"  = "2",
"ANTC-M3-1"  = "3-1",
"ANTC-M3-2"  = "3-2",
"ANTC-M4-2"  = "4-2",
"ANTC-M4-2S" = "4-2",
"ANTC-M4-2L" = "4-2",
"ANTC-M6-1"  = "6-1",
"ANTC-M6-2"  = "6-2",
"ANTCW-M1-1"  = "1",
"ANTCW-M2-1"  = "2",
"ANTCW-M3-1"  = "3-1",
"ANTCW-M3-2"  = "3-2",
"ANTCW-M4-2"  = "4-2",
"ANTCW-M4-2S" = "4-2",
"ANTCW-M4-2L" = "4-2",
"ANTCW-M6-1"  = "6-1",
"ANTCW-M6-2"  = "6-2",
"ACT-521" = "521",
"ACT-626" = "626",
"ACT-393" = "393",
"NTQB2-1" = "2-1",
"NTQB2-2" = "2-2",
"NTQB2-3-2" = "2-3-2",
"NTQB2-4-2" = "2-4-2",
"NTQB2-4-2S" = "2-4-2",
"NTQB2-5-1" = "2-5-1",
"NTQB2-6-1" = "2-6-1",
"NTQB2-6-2" = "2-6-2"
)

modPar = NULL
for (b in names(modelBattery))
    modPar = rbind(modPar, par[modelBattery[[b]],])
rownames(modPar) = names(modelBattery)

The results show good agreement with the data table from Lotek:

print(round(par, 1))

The parameter $r_t$ can be interpreted as the ratio of energy consumed during 1 second with a burst to that consumed during 1 second without a burst. Estimates of this parameter vary only by 5% across tag types, and in monotonic fashion, perhaps due to variation in battery internal resistance. The table provided by Lotek only covers $2 <= BI <= 20$ (the larger value from BI=10s @ 50% duty cycle); curves are extrapolated down to 1s and from 20 to 40s using the fitted model.

xyplot(log10(lifespan)~bi|as.factor(battery), pred,
       main="Reported (+) and Predicted Tag Lifespan\nby Battery Type and Burst Interval",
       xlab="Burst Interval (seconds)",
       ylab="Lifespan (log10(days); 1->10, 2->100, 3->1000)",
       type="l",
       panel = function(x, y, type, subscripts, ...) {
           panel.xyplot(x, y, type, ...)
           panel.points(x[pred$isMeasured[subscripts]], y[pred$isMeasured[subscripts]], pch="+", cex=2, col="black")
           }
       )