Assess how linear a logged temperature trace is

A pre-fitting diagnostic with two uses. As quality control on the ramp, it fits temp ~ time per path and reports the realised start and rate, the fit (r_squared), and, given nominal, the rate_bias against the setpoint, so you can see whether the controller held a linear ramp at the intended rate. And as a check on whether a dense or noisy log can be simplified for fitting: a finely sampled temperature series turns every reading into an at-risk-interval boundary in build_hazard_data, so a linear one can be replaced by a clean two-point ramp from ramp_path (exact under the linear interpolation, free of sensor jitter, and far fewer rows). A faithful series is otherwise best fitted directly; this only judges that faithfulness.

Usage

assess_ramp_linearity(path, nominal = NULL)

Arguments

path

A data frame with path_id, time and temp – the logged temperature trace(s).

nominal

An optional named numeric of setpoint rates (names matching path_id). When given, adds rate_nominal and rate_bias (realised minus nominal), the controller's rate error.

Value

A tibble, one row per path_id, with n (the number of readings), the realised T_start and rate, r_squared, rmse, max_abs_dev, and max_dev_hot (the worst absolute deviation in the hot quartile); plus rate_nominal and rate_bias when nominal is supplied.

Details

As a rule of thumb, a series is linear enough to simplify when its deviations (and especially max_dev_hot, the worst deviation in the upper temperature quartile, where the failure rate M(T) is steepest) are small relative to log(10) / b, the number of degrees for a tenfold change in an exponential M(T): half that distance already multiplies M(T) by sqrt(10), about threefold.

That distance is not always the z that derive_tdt_ttl() reports. z is the number of degrees for a tenfold change in the time to failure, k log(10) / b, with k the shape of a Weibull baseline. The two are the same under the memoryless baseline, where k = 1, and differ by the factor k under a Weibull one. Judge the deviations against log(10) / b, not against a reported z.

Examples

# A logged temperature series close to a 0.3 degree/min ramp, with a small
# mid-run wobble.
path <- data.frame(path_id = "r1", time = 0:20,
                   temp = 20 + 0.3 * (0:20) + c(rep(0, 8), 0.4, 0.5, 0.3, rep(0, 10)))
assess_ramp_linearity(path)
#> # A tibble: 1 × 8
#>   path_id     n T_start  rate r_squared  rmse max_abs_dev max_dev_hot
#>   <chr>   <int>   <dbl> <dbl>     <dbl> <dbl>       <dbl>       <dbl>
#> 1 r1         21    20.1 0.298     0.994 0.143       0.441      0.0487

# With the setpoint rate, to see the controller's rate bias.
assess_ramp_linearity(path, nominal = c(r1 = 0.3))
#> # A tibble: 1 × 10
#>   path_id     n T_start  rate r_squared  rmse max_abs_dev max_dev_hot
#>   <chr>   <int>   <dbl> <dbl>     <dbl> <dbl>       <dbl>       <dbl>
#> 1 r1         21    20.1 0.298     0.994 0.143       0.441      0.0487
#> # ℹ 2 more variables: rate_nominal <dbl>, rate_bias <dbl>