Tree ring analysis - coupling measurement series taken on opposite sides of the pith with one ring difference
[bozza non finita]
Aim
Yearly measurements have been taken on almost 200 stem sections in a ‘simplified’ stem analysis study. (Simplified in the sense that sections have been cut adapting to foresters needs, at tree base and at the top of the long commercial logs he defined, not at researcher defind positions). Measurement procedure and data imput method didn’t oblige to take care of readings internal coherence and, acually, of the more than 150 measurements seriese that could effectively proceed from one side of the stem till the opposite side, 140 series present a common incoherence problem: the number of rings on the two sides of the pith differ for one year.
To proceed with the original stem analysis work, ecah of these series has to be corrected either,
(I) inserting a missing mesurement in the short side,
(D) deliting a measurement in excess in the long one.
Aim of the work performed here is to identify the locations within such series where (D) or (I) would best improve the matching between the series taken from opposite sides of the stem.
Measurements have been taken as progressive values. Assuming that total width section is correctly mesured, in order to preserve all other readings not affected by the correction, (I) and (D) respectvely imply that:
(I) the width of a given ring is divided in two (let us assume, equal) parts
(D) the width of two consecutive rings are summed to form a single one
Evaluating matching indicators
Dynamic Time Warping (DTW) is a well known approach to time series matching, see
https://en.wikipedia.org/wiki/Dynamic_time_warping
or http://www.cs.ucr.edu/~eamonn/DTW_myths.pdf for a more critical appraisal.
Although the general approach can be applyed in this context, the problem at hand requires has specific constraints that, as far as I could appreciate, can not be implemented using available packages or algorithms:
1) in this case a single correcion (I or D) is allowed and required,
2) what has to be evaluated is the ‘match gain’ after correction, i.e. after having modified (by summation or halving) the single series values.
Moreover in DTW the focus is on ‘point to point’ distances (however the metric is defined), and ‘warping’ is led by ‘distance minimization’, with no concern for local matching. To overcome this limitation a ‘shape DTW’ approach has been developed but the procedures are available only as code for Matlab (paper: https://arxiv.org/pdf/1606.01601.pdf, code: https://github.com/jiapingz/shapeDTW). Other approaches have been proposed but I couldn’t find any implementation (https://pdfs.semanticscholar.org/98cc/ff83a8c815eed2f3978a47a331373baf332a.pdf). In this work local correlation is considered as viable approach to take ‘local shapes’ into consideration.
Compute Sections synthesis (final code of ‘FromGS2at.R’) and Ring Widths
load("RinaldiniV2.0.Rdata")
library(sqldf)## Loading required package: gsubfn## Loading required package: proto## Loading required package: RSQLiteSects_sy <- sqldf("select TreeId, SectId
, sum(side='a' and bark) abark, sum(side='a' and not bark) nam
, sum(side='b' and not bark) nbm, sum(side='b' and bark) bbark
, count(*) as nm
from Diams_at group by TreeId, SectId order by TreeId, SectId")## Loading required package: tcltk## Warning: Quoted identifiers should have class SQL, use DBI::SQL() if the
## caller performs the quoting.Sects_sy <- sqldf("
select A.*, B.comment
from Sects_sy A natural left join Sects_at B")
Sects_sy$complete <- Sects_sy$bbark | is.na(Sects_sy$comment)
Sects_sy <- sqldf("
select * from Sects_sy natural join
(select TreeId, SectId, max(year) max_b_year
from (select * from Diams_at where side='b')
group by TreeId, SectId)")
pith <- sqldf("
select A.TreeId, A.SectId, year-1 year, avg(value) value
from Diams_at A natural join (
select TreeId, SectId, side, min(year) year
from Diams_at group by TreeId, SectId, side)
group by TreeId, SectId")
missing_bark <- sqldf("
select TreeId, SectId, 'a' side, 2017 year, 0 value
from Sects_sy where abark=0")
tmp <- sqldf("
select TreeId, SectId, side, year, value
from Diams_at where not (side='b' and bark)
UNION select TreeId, SectId, 'a' side, year, value from pith
UNION select TreeId, SectId, 'b' side, year, value from pith
UNION select TreeId, SectId, side, year, value from missing_bark")
rw_all <- sqldf("
select A.TreeId, A.SectId, A.side, A.year gyear
, case when A.side='a' then B.value-A.value else A.value-B.value end rw
from tmp A join tmp B
on A.TreeId=B.TreeId and A.SectId=B.SectId and A.side=B.side
and A.year=B.year+1")
# dubbio: la corteccia b non è inclusa come rw vero?
# verifica: NO (year è messo ad NA e quindi non aggancia nell'accoppiamento)Evaluating similarity measures
library(latticeExtra)## Warning: package 'latticeExtra' was built under R version 3.3.3## Loading required package: lattice## Loading required package: RColorBrewer# library(TTR)
# tested but abanoned because:
# 1 - numeric instability (NaN sometimes when x or y constant)
# 2 - lagged result (result[n] corresponds to x_y[n/2])
# substituted with a custom 'localCor()' function
source('F_localCor.R')
tmp <- sqldf("select A.* from rw_all A natural join Sects_sy where max_b_year = 2017")
rw_ok <- sqldf("select A.*, A.rw/series_width rrw from tmp A natural join (select TreeId, SectId, side, sum(rw) series_width from tmp group by TreeId, SectId, side)")
rw_cor <- sqldf("select TreeId, SectId, gyear
, A.rrw rrw_a, B.rrw rrw_b
from (select * from rw_ok where side='a') A
join (select * from rw_ok where side='b') B
using(TreeId, SectId, gyear)")
xyplot(rw ~ gyear | paste(TreeId,SectId,sep="-"), rw_ok, group = side, t='l', main="'a' and 'b' Series of sections where lengths are equal")
ts <- with(rw_cor, paste(TreeId, SectId, sep="-"))
for (i in unique(ts)) {
# print(i)
rw_cor$rw_ab_cor[ts==i] <- with(rw_cor[ts==i,]
, localCor(rrw_a, rrw_b, 3))
}
# rrw_a+rrw_b+
obj1 <- xyplot(I(rrw_a - rrw_b)^2 ~ gyear| ts, rw_cor, type='l'
, scales=list(y=list(log="e"))
, panel=function(...) {panel.xyplot(...)
panel.grid(h=-1, v=-1, ...)}
, main="Comparing similarity measures")
obj2 <- xyplot(I((rw_ab_cor+1)/2) ~ gyear| ts, rw_cor, type='l')
doubleYScale(obj1, obj2, text = c("E.dist. relativeRW", "scaled_localCor"), add.ylab2 = TRUE, main="Comparing similarity measures")
Sequential evaluation of insertions and delitions on sections with ‘one year difference’ sides
library(splus2R)
tmp <- sqldf("select A.* from rw_all A natural join Sects_sy where max_b_year = 2016")
rw_1yd <- sqldf("select A.*, A.rw/series_width rrw from tmp A natural join (select TreeId, SectId, side, sum(rw) series_width from tmp group by TreeId, SectId, side)")
start_time <- Sys.time()
pcsim <- data.frame()
ts <- with(rw_1yd, paste(TreeId, SectId, sep="-"))
for (i in unique(ts)) {
print(i)
ls <- rw_1yd[ts==i & rw_1yd$side=='a',] # LongSeries
N <- nrow(ls)
ss <- rw_1yd[ts==i & rw_1yd$side=='b',] # ShortSeries (len=N-1)
csim <- data.frame()
# (partial) cumulated differences/correlations squared
# for (y in min(ls$gyear):(min(ls$gyear))) { # compute cumulative 'similarity' I and D in position j
for (y in min(ls$gyear):(max(ls$gyear)-1)) { # compute cumulative 'similarity' I and D in position j
# Insert (after y, splitting rw[y] in two, shifting +1 following years) - in ss
ssI <- ss[ss$gyear <= y, ]
rw <- ss$rrw[ss$gyear == y]
ssI <- rbind(ssI, ss[ss$gyear == y, ])
ssI$rrw[ssI$gyear == y] <- rw/2
p <- nrow(ssI)
ssI <- rbind(ssI, ss[ss$gyear > y,])
ssI$gyear[p:nrow(ssI)] <- 1 + ssI$gyear[p:nrow(ssI)]
# Delete (ring y: sum j and j+1, till N-1) from long series
if (y < max(ls$gyear)) {
lsD <- ls[ls$gyear <= y,]
lsD$rrw[lsD$gyear == y] <- lsD$rrw[lsD$gyear == y] + ls$rrw[ls$gyear == y+1]
p <- nrow(lsD)
lsD <- rbind(lsD, ls[ls$gyear > y+1,])
lsD$gyear[p:nrow(lsD)] <- lsD$gyear[p:nrow(lsD)] -1
} else {lsD = NULL}
n <- 3
csim <- rbind(csim,
data.frame(TeeeSector = i
, modify_y=y
, cdI = sum((ls$rrw-ssI$rrw)^2) / length(ls)
, cdD = sum((lsD$rrw-ss$rrw)^2) / length(lsD)
, crI = sum((localCor(ls$rrw, ssI$rrw, n)+1)/2) / length(ls)
, crD = sum((localCor(lsD$rrw, ss$rrw, n)+1)/2) / length(lsD)
))
}
pcsim_i <- data.frame()
for (c in names(csim)[2:5]) {
sel <- which(peaks(csim[,c],span=7))
pcsim_i <- rbind(pcsim_i, data.frame(c=c, o=c(1,2)
, r=sel[order(csim[,c][sel], decreasing = T)][1:2]))
}
csim$how <- ''
for (i in 1:nrow(pcsim_i)) {
csim$how[pcsim_i$r[i]] <-
paste(csim$how[pcsim_i$r[i]],
paste(pcsim_i$c[i], pcsim_i$o[i], sep=""), sep=":")
}
pcsim <- rbind(pcsim, csim[csim$how != '',])
}## [1] "1-1"
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## [1] "46-4"end_time <- Sys.time()
print(paste("start:", start_time, "- end:", end_time, "- Time elapsed:", end_time - start_time))## [1] "start: 2017-08-17 05:08:31 - end: 2017-08-17 05:09:12 - Time elapsed: 41.2033178806305"pcsim[,c("TeeeSector","modify_y","how")]## TeeeSector modify_y how
## 4 1-1 1971 :cdI1:cdD2
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## 31 19-1 1998 :crI2
## 421 19-1 2009 :crI1
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## 1312 32-6 2003 :crI1
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## 1811 32-7 2011 :cdI2
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## 2211 36-1 1990 :crI1
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## 312 36-1 1999 :cdI2
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## 2013 36-3 2009 :crI1
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## 1521 36-4 2007 :cdI1:cdD1
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## 1115 36-5 2006 :crI1
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## 1522 36-5 2010 :crI2
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## 715 37-1 1973 :cdD1
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## 451 37-1 2011 :crI1
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## 1025 37-2 1988 :crI2
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## 423 38-5 1997 :crI2
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## 1316 38-5 2006 :crI1
## 1710 38-5 2010 :cdI2:cdD2
## 917 38-6 2004 :crI2
## 1118 38-6 2006 :cdD2
## 1523 38-6 2010 :cdI1:cdD1
## 1711 38-6 2012 :crI1
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## 1422 39-1 1982 :crI2
## 1616 39-1 1984 :cdI2
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## 2015 39-4 2006 :crI1
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## 825 40-2 1982 :crI1
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## 918 40-3 1992 :crI2
## 1119 40-3 1994 :cdI1:cdD1
## 1318 40-3 1996 :crI1
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## 1916 43-2 2005 :cdI1:cdD1
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## 287 44-2 2010 :crI1
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## 1812 44-5 2010 :crI1
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## 1425 45-2 1998 :crI2
## 2217 45-2 2006 :crI1
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## 1322 46-1 1982 :cdI2
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## 313 46-1 2000 :crI2
## 4111 46-1 2010 :cdD2
## 426 46-2 1982 :cdI1:cdD1
## 828 46-2 1986 :cdI2:cdD2
## 2710 46-2 2005 :crI2
## 352 46-2 2013 :crI1
## 523 46-3 1990 :cdD2
## 920 46-3 1994 :cdI2
## 1426 46-3 1999 :cdI1:cdD1
## 1813 46-3 2003 :crI2
## 2312 46-3 2008 :crI1
## 427 46-4 1997 :cdI2:cdD2
## 1029 46-4 2003 :crI1
## 1715 46-4 2010 :cdI1:cdD1library(googlesheets)
suppressMessages(library(dplyr))
gsn <- "RotelleVer3"
gsurl <- "https://docs.google.com/spreadsheets/d/19QBPUmwnREaTlZ5Zbw46askcnmPSwc6wW17UELbhmBg/edit#gid=2034395174"
gs_ls(gsn)## # A tibble: 1 × 10
## sheet_title author perm version updated
## <chr> <chr> <chr> <chr> <dttm>
## 1 RotelleVer3 scotti rw old 2017-08-17 02:48:39
## # ... with 5 more variables: sheet_key <chr>, ws_feed <chr>,
## # alternate <chr>, self <chr>, alt_key <chr>Rv3 <- gs_url(gsurl)## Sheet-identifying info appears to be a browser URL.
## googlesheets will attempt to extract sheet key from the URL.## Putative key: 19QBPUmwnREaTlZ5Zbw46askcnmPSwc6wW17UELbhmBg## Sheet successfully identified: "RotelleVer3"# Rv3 <- Rv3 %>%
# gs_ws_new(ws_title = "CorrezioniProposte", input = pcsim)Istruzioni per l’uso
Questa tabella espone il risultato dell’analisi effettuata su tutte le rotelle teoricamente complete sui due lati, che però presentano, sul lato ‘b’, un anello in meno che sul lato ‘a’ (oppure, equivalentemente, un anello di troppo sul lato ‘a’). Per ogni rotella (individuata con la sigla “TreeSection”) vengono presentate diverse possibili opzioni di correzione della sequenza delle misurazioni effettuate, tra loro alternative. La colonna ‘modify_y’ individua su quale anello intervenire, la colonna ‘how’ specifica il tipo di correzzione proposto. I codici utilizzati nella colonna ‘how’ hanno i seguenti significati. I due punti “:” separano possibili interventi diversi relativi allo stesso anello (anno) Le lettere maiuscole “I” e “D” indicano rispettivamento ‘Insertion’ e ‘Delition’ Le due lettere minuscole iniziali indicano il criterio in base al quale di è ottenuta l’indicazione presentata: “cd” è quello analogo al DTW (pasato sulle distanze punto-punto), “cr” è un criterio originale basato sulla ‘correlazione locale tra segmenti di serie’. Infine, il numero in chiusura indica la priorità: 1 = prima opzione da considerare (per quella correzione in base a quel criterio), 2 = seconda opzione da considerare.
In pratica
Delle rotelle considerate in questa tabella prenderne alcune, casualmente, fra quelle che si possono rileggere almeno sulla foto, se non dal vivo. (Se è possibile arrivare a 30 rotelle sarebbe ideale) Per ciscuna di queste, aiutandosi con la tabella di registarazione delle misurazioni(1), individuare gli anelli corrispondenti agli anni selezionati dall’analisi (‘modify_y’). Se ‘how’ contiene una ‘I’, esaminare il lato ‘b’ e valutare se, in effetti, l’anello che segue è stato saltato nelle misurazioni registrate. Se ‘how’ contiene una ‘D’, esaminare il lato ‘a’ e valutare se, in effetti, la misurazione relativa all’anno individuato non corrisponde ad un anello effettivo e sarebbe quindi da eliminare. Valutata la situazione per tutti gli anni selezionati dall’analisi scegliere quale delle correzioni proposte è la più convincente e registrare la scelta in una colonna aggiuntiva, replicando la sigla dell’intervento. (2)
(*1) La tabella da utilizzare quella nel foglio ‘DatabaseV2’ del tabellone Google ‘RotelleVer2’ raggiungibile all’indirizzo https://docs.google.com/spreadsheets/d/15ZLnI780oKr4QKuc4k4UAwZ6w6YmEE8tj4c2sU44xLQ/edit#gid=1996889235 In questa tabella puoi inserire commenti ma non puoi modificare i dati. Se ti è più comodo puoi utilizzare una copia in formato Excel di quel tabellone (ne inivio una io).
(*2) Ho preparato un tabellone con la colonna dove replicare il codice della correzione che hai valutato migliore per ciascuna delle sezioni che potrai esaminare. La trovi all’indirizzo https://docs.google.com/spreadsheets/d/19QBPUmwnREaTlZ5Zbw46askcnmPSwc6wW17UELbhmBg/edit#gid=2034395174 (Se non è possibile inserire i risultati della tua analisi su questo tabellone in rete, invio copia in Excel)