Compute the following metrics:

`FirstGermTime`

Time of first germination or Germination time lag (\(t_{0}\)) (Edwards 1932; Czabator 1962; Goloff and Bazzaz 1975; Labouriau 1983; Ranal 1999; Quintanilla et al. 2000) .

`LastGermTime`

Time of last germination (\(t_{g}\)) (Edwards 1932; Labouriau 1983; Ranal and de Santana 2006) .

`TimeSpreadGerm`

Time spread of germination (Al-Mudaris 1998; Kader 2005) or Germination distribution (Schrader and Graves 2000) .

`PeakGermTime`

Peak time of germination or Modal time of germination (\(t_{peak}\)) (Ranal and de Santana 2006) .

```
FirstGermTime(germ.counts, intervals, partial = TRUE)
LastGermTime(germ.counts, intervals, partial = TRUE)
PeakGermTime(germ.counts, intervals, partial = TRUE)
TimeSpreadGerm(germ.counts, intervals, partial = TRUE)
```

- germ.counts
Germination counts at each time interval. Can be partial or cumulative as specified in the argument

`partial`

.- intervals
The time intervals.

- partial
logical. If

`TRUE`

,`germ.counts`

is considered as partial and if`FALSE`

, it is considered as cumulative. Default is`TRUE`

.

For `FirstGermTime`

, the time of first germination value in the
same unit of time as specified in the argument `intervals`

.

For `LastGermTime`

, the time of last germination value in the same
unit of time as specified in the argument `intervals`

.

For `TimeSpreadGerm`

, the time spread of germination value in the same
unit of time as specified in the argument `intervals`

.

For `PeakGermTime`

, the time(s) of peak germination value(s) as a
numeric vector in the same unit of time as specified in the argument

`intervals`

.

Time of first germination indicates time of germination of the faster seeds in a seedlot.

Lower value of time of first germination indicates faster initiation of germination and lower value of time of last germination indicates faster termination of germination.

\[t_{0} = \min \lbrace T_{i} : N_{i} \neq 0 \rbrace\]

\[t_{g} = \max \lbrace T_{i} : N_{i} \neq 0 \rbrace\]

Where, \(T_{i}\) is the time from the start of the experiment to the \(i\)th interval and \(N_{i}\) is the number of seeds germinated in the \(i\)th time interval (not the accumulated number, but the number corresponding to the \(i\)th interval).

Time spread of germination (\(t_{g}-t_{0}\)) indicates difference between faster and slower germinating members of a sample.

Peak time of germination is the time in which highest frequency of germinated seeds are observed. Multiple peak times of germination are possible and if detected are indicated by a warning message.

\[t_{peak} = \lbrace T_{i} : N_{i} = N_{max} \rbrace\]

Where, \(N_{max}\) is the maximum number of seeds germinated per interval.

Al-Mudaris MA (1998).
“Notes on various parameters recording the speed of seed germination.”
*Der Tropenlandwirt - Journal of Agriculture in the Tropics and Subtropics*, **99**(2), 147--154.

Czabator FJ (1962).
“Germination value: An index combining speed and completeness of pine seed germination.”
*Forest Science*, **8**(4), 386--396.

Edwards TI (1932).
“Temperature relations of seed germination.”
*The Quarterly Review of Biology*, **7**(4), 428--443.

Goloff AA, Bazzaz FA (1975).
“A germination model for natural seed populations.”
*Journal of Theoretical Biology*, **52**(2), 259--283.

Kader MA (2005).
“A comparison of seed germination calculation formulae and the associated interpretation of resulting data.”
*Journal and Proceedings of the Royal Society of New South Wales*, **138**, 65--75.

Labouriau LG (1983).
*A Germinacao Das Sementes*.
Organizacao dos Estados Americanos. Programa Regional de Desenvolvimento Cientifico e Tecnologico. Serie de Biologia. Monografia 24.

Quintanilla LG, Pajaron S, Pangua E, Amigo J (2000).
“Effect of temperature on germination in northernmost populations of *Culcita macrocarpa* and *Woodwardia radicans*.”
*Plant Biology*, **2**(6), 612--617.

Ranal MA (1999).
“Effects of temperature on spore germination in some fern species from semideciduous mesophytic forest.”
*American Fern Journal*, **89**(2), 149.

Ranal MA, de Santana DG (2006).
“How and why to measure the germination process?”
*Brazilian Journal of Botany*, **29**(1), 1--11.

Schrader JA, Graves WR (2000).
“Seed germination and seedling growth of *Alnus maritima* from its three disjunct populations.”
*Journal of the American Society for Horticultural Science*, **125**(1), 128--134.

```
x <- c(0, 0, 0, 0, 4, 17, 10, 7, 1, 0, 1, 0, 0, 0)
y <- c(0, 0, 0, 0, 4, 21, 31, 38, 39, 39, 40, 40, 40, 40)
z <- c(0, 0, 0, 0, 11, 11, 9, 7, 1, 0, 1, 0, 0, 0)
int <- 1:length(x)
# From partial germination counts
#----------------------------------------------------------------------------
FirstGermTime(germ.counts = x, intervals = int)
#> [1] 5
LastGermTime(germ.counts = x, intervals = int)
#> [1] 11
TimeSpreadGerm(germ.counts = x, intervals = int)
#> [1] 6
PeakGermTime(germ.counts = x, intervals = int)
#> [1] 6
# For multiple peak germination times
PeakGermTime(germ.counts = z, intervals = int)
#> Warning: Multiple peak germination times exist.
#> [1] 5 6
# From cumulative germination counts
#----------------------------------------------------------------------------
FirstGermTime(germ.counts = y, intervals = int, partial = FALSE)
#> [1] 5
LastGermTime(germ.counts = y, intervals = int, partial = FALSE)
#> [1] 11
TimeSpreadGerm(germ.counts = y, intervals = int, partial = FALSE)
#> [1] 6
PeakGermTime(germ.counts = y, intervals = int, partial = FALSE)
#> [1] 6
# For multiple peak germination time
PeakGermTime(germ.counts = cumsum(z), intervals = int, partial = FALSE)
#> Warning: Multiple peak germination times exist.
#> [1] 5 6
```