Chapter 2 Caddo bottle morphology

2.1 Load packages + data

# load packages
#devtools::install_github("mlcollyer/RRPP")
#devtools::install_github("geomorphR/geomorph", ref = "Stable", build_vignettes = TRUE)
library(here)
## here() starts at D:/github/bottle.traj/supp
library(geomorph)
## Loading required package: RRPP
## Loading required package: rgl
## Loading required package: Matrix
library(tidyverse)
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## ✔ tidyr   1.2.1      ✔ stringr 1.4.1 
## ✔ readr   2.1.3      ✔ forcats 0.5.2 
## ── Conflicts ──────────────────────────────────────────────────────────── tidyverse_conflicts() ──
## ✖ tidyr::expand() masks Matrix::expand()
## ✖ dplyr::filter() masks stats::filter()
## ✖ dplyr::lag()    masks stats::lag()
## ✖ tidyr::pack()   masks Matrix::pack()
## ✖ tidyr::unpack() masks Matrix::unpack()
library(wesanderson)

# read GM data
source('readmulti.csv.R')
setwd("./data")
filelist <- list.files(pattern = ".csv")
coords <- readmulti.csv(filelist)
setwd("../")

# read qualitative data
qdata <- read.csv("qdata.csv", 
                  header = TRUE, 
                  row.names = 1)
qdata <- qdata[match(dimnames(coords)[[3]],
                     rownames(qdata)),]

2.2 Generalised Procrustes Analysis

Landmark data were aligned to a global coordinate system (Kendall 1981, 1984; Slice 2001), achieved through generalised Procrustes superimposition (Rohlf and Slice 1990) performed in R 4.0.3 (R Core Development Team, 2017) using the geomorph library v. 3.3.2 (Dean C. Adams et al. 2017; Dean C. Adams and Otárola-Castillo 2013). Procrustes superimposition translates, scales, and rotates the coordinate data to allow for comparisons among objects (Gower 1975; Rohlf and Slice 1990). The geomorph package uses a partial Procrustes superimposition that projects the aligned specimens into tangent space subsequent to alignment in preparation for the use of multivariate methods that assume linear space (Rohlf 1999; Slice 2001).

# gpa
Y.gpa <- gpagen(coords, 
                PrinAxes = TRUE, 
                ProcD = TRUE, 
                Proj = TRUE, 
                print.progress = FALSE)

# geomorph data frame
gdf <- geomorph.data.frame(shape = Y.gpa$coords, 
                           size = Y.gpa$Csize,
                           geo = qdata$geo,
                           time = qdata$time,
                           comb = qdata$comb)

# render 3d gpa plot
#plot(Y.gpa)

# gpa plot
knitr::include_graphics('images/gpa3d.png')

# add centroid size to qdata
qdata$csz <- Y.gpa$Csize

# print updated qdata with centroid size
knitr::kable(qdata, 
             align = "lccccc", 
             caption = "Modified attributes included in qdata.")
Table 2.1: Modified attributes included in qdata.
site type geo time comb csz
1-15-3-Pot3 Atkinson Farm Hodges Engraved north Late-Historic north, late historic 336.4793
1054 Belcher Mound Taylor Engraved north Late-Historic north, late historic 221.1933
1073 Belcher Mound Belcher Engraved north Late-Historic north, late historic 396.8419
121 Salt Lick Wilder Engraved south Late-Historic south, late historic 308.8975
132 Paul Mitchell Hickory Fine Engraved north Formative-Early north, formative early 687.3149
142 Allen Plantation Hickory Fine Engraved south Formative-Early south, formative early 810.0585
152 Smithport Landing Smithport Plain south Formative-Early south, formative early 222.3931
157 Smithport Landing Hickory Fine Engraved south Formative-Early south, formative early 635.9682
16 Pohler Coll UID north Late-Historic north, late historic 718.4339
16sa37-121 Eleven Ton Bridge Wilder Engraved south Late-Historic south, late historic 305.4943
18 Pohler Coll Smithport Plain north Formative-Early north, formative early 624.6644
19-361 George C Davis Holly Fine Engraved south Formative-Early south, formative early 652.1150
19-4020 George C Davis Holly Fine Engraved south Formative-Early south, formative early 719.2440
2 Mustang Creek Mound Hickory Fine Engraved north Formative-Early north, formative early 715.1674
20 Pohler Coll Hickory Fine Engraved north Formative-Early north, formative early 487.5317
2007.18.02 Green UID north Late-Historic north, late historic 632.7158
2015-1 George C Davis Hickory Fine Engraved south Formative-Early south, formative early 425.1642
2097-1 George C Davis Holly Fine Engraved south Formative-Early south, formative early 593.4289
23 Pohler Coll Hickory Fine Engraved north Formative-Early north, formative early 669.7663
24 Pohler Coll UID north Late-Historic north, late historic 418.6807
256 Belcher Mound Taylor Engraved north Late-Historic north, late historic 458.8372
267 Belcher Mound Belcher Engraved north Late-Historic north, late historic 463.8621
269 Belcher Mound Belcher Engraved north Late-Historic north, late historic 557.5507
27 Pohler Coll Smithport Plain north Formative-Early north, formative early 639.5040
271 Belcher Mound Taylor Engraved north Late-Historic north, late historic 477.0324
29 Pohler Coll UID north Late-Historic north, late historic 625.1986
30 Pohler Coll Maxey Noded Redware north Formative-Early north, formative early 571.6977
31-1-79 Poole Maxey Noded Redware north Formative-Early north, formative early 571.5949
31 Pohler Coll UID north Late-Historic north, late historic 620.5723
317 Belcher Mound UID north Late-Historic north, late historic 686.0103
325 Belcher Mound Wilder Engraved north Late-Historic north, late historic 550.6656
340 Belcher Mound UID north Late-Historic north, late historic 620.1775
361 Belcher Mound Belcher Engraved north Late-Historic north, late historic 628.7876
363 Belcher Mound Belcher Engraved north Late-Historic north, late historic 585.9939
404 Belcher Mound Hickory Fine Engraved north Formative-Early north, formative early 504.5038
41sy25-2 S. H. Latham Wilder Engraved south Late-Historic south, late historic 413.9977
41sy25-3 S. H. Latham UID south Late-Historic south, late historic 433.8256
42 Pohler Coll UID north Late-Historic north, late historic 641.7042
427 Paul Mitchell Hickory Fine Engraved north Formative-Early north, formative early 609.9637
46 Pohler Coll UID north Late-Historic north, late historic 588.9848
464 Paul Mitchell Hickory Fine Engraved north Formative-Early north, formative early 476.9909
55-16-3 Battle Mound Haley Engraved north Formative-Early north, formative early 617.9615
55-16-39 Battle Mound Belcher Engraved north Late-Historic north, late historic 535.6979
6-6-609 Hatchel Belcher Engraved north Late-Historic north, late historic 459.0304
67 Paul Mitchell Smithport Plain north Formative-Early north, formative early 319.0530
7 Frank Norris Farm Hickory Fine Engraved north Formative-Early north, formative early 673.2136
775 Belcher Mound Belcher Engraved north Late-Historic north, late historic 502.9991
78 Paul Mitchell Smithport Plain north Formative-Early north, formative early 385.3598
784 Belcher Mound Keno Trailed north Late-Historic north, late historic 343.7336
787 Belcher Mound Taylor Engraved north Late-Historic north, late historic 528.2451
788 Belcher Mound Belcher Engraved north Late-Historic north, late historic 546.2135
8-91-Pot2 Atkinson Farm Avery Engraved north Late-Historic north, late historic 648.7273
8 Crenshaw Mound Hickory Fine Engraved north Formative-Early north, formative early 737.8038
803 Belcher Mound Belcher Engraved north Late-Historic north, late historic 433.7740
804 Belcher Mound Hodges Engraved north Late-Historic north, late historic 487.3065
805 Belcher Mound Belcher Engraved north Late-Historic north, late historic 458.4330
845 Belcher Mound Belcher Engraved north Late-Historic north, late historic 516.3754
852 Belcher Mound Keno Trailed north Late-Historic north, late historic 392.7014
897 Belcher Mound Belcher Engraved north Late-Historic north, late historic 667.6909
904 Greer Farm UID north Late-Historic north, late historic 454.3649
929 Lawton Plantation Keno Trailed south Late-Historic south, late historic 252.2038
95 Smithport Landing Smithport Plain south Formative-Early south, formative early 651.5567
955 Gahagan Mound Hickory Fine Engraved south Formative-Early south, formative early 696.8243
956 Gahagan Mound Hickory Fine Engraved south Formative-Early south, formative early 784.2710
96 Smithport Landing Hickory Fine Engraved south Formative-Early south, formative early 750.3802
979 Belcher Mound Hempstead Engraved north Late-Historic north, late historic 910.2240
997 Belcher Mound Belcher Engraved north Late-Historic north, late historic 617.3610
B1V3 Griffin UID south Late-Historic south, late historic 487.6034
BELCH1 Haley Place Belcher Engraved north Late-Historic north, late historic 389.2537
FIN-S18 Vanderpool UID north Late-Historic north, late historic 567.2520
FIN-S3 Vanderpool Hume Engraved north Late-Historic north, late historic 359.3132
FIN-S4 Vanderpool Poynor Engraved north Late-Historic north, late historic 486.6301
FS7 Hatchel Hickory Fine Engraved north Formative-Early north, formative early 489.5597
HBE1 Haley Place Belcher Engraved north Late-Historic north, late historic 389.2819
HCN3 Hancock UID south Late-Historic south, late historic 673.4124
HE3 Haley Place Haley Engraved north Formative-Early north, formative early 643.6377
HEB36 Haley Place Haley Engraved north Formative-Early north, formative early 540.5575
HEB40 Haley Place Haley Engraved north Formative-Early north, formative early 582.1375
HFE1 Haley Place Hickory Fine Engraved north Formative-Early north, formative early 783.8957
HFE2 Haley Place Hickory Fine Engraved north Formative-Early north, formative early 617.2341
HFE3 Haley Place Hickory Fine Engraved north Formative-Early north, formative early 454.2029
HFE4 Haley Place Hickory Fine Engraved north Formative-Early north, formative early 543.2370
HFE5 Haley Place Hickory Fine Engraved north Formative-Early north, formative early 435.9170
MFB Hancock UID south Late-Historic south, late historic 496.5131
SMPRTPLN1 Haley Place Smithport Plain north Formative-Early north, formative early 676.8090
SMPRTPLN2 Haley Place Smithport Plain north Formative-Early north, formative early 680.3718
SMU16 Bison B Taylor Engraved south Late-Historic south, late historic 159.8901
SMU34 Bison B Taylor Engraved south Late-Historic south, late historic 261.5380
SMU67 Bison B Taylor Engraved south Late-Historic south, late historic 361.6921
vessel-E M. W. Burks UID north Late-Historic north, late historic 211.4420
x16sa4-17 Bison B Wilder Engraved south Late-Historic south, late historic 603.9271
x16sa4-2 Bison B Wilder Engraved south Late-Historic south, late historic 628.6545
x16sa4-26 Bison B Wilder Engraved south Late-Historic south, late historic 440.6329
x16sa4-65 Bison B Wilder Engraved south Late-Historic south, late historic 376.5012

2.2.1 Boxplot

# attributes for boxplot
csz <- qdata$csz # centroid size
comb <-  qdata$comb # comb + time

# boxplot of Caddo bottle centroid size by comb (north/south)
csz.comb <- ggplot(qdata, aes(x = comb, y = csz, color = comb)) + 
  geom_boxplot() +
  geom_dotplot(binaxis = 'y', stackdir = 'center', dotsize = 0.3) +
  scale_colour_manual(values = wes_palette("Moonrise2")) +
  theme(legend.position = "none") +
  labs(x = 'North/South + Formative/Early or Late/Historic?', y = 'Centroid Size')

# render plot
csz.comb
## Bin width defaults to 1/30 of the range of the data. Pick better value with `binwidth`.
Boxplot of centroid size by spatial/temporal unit.

Figure 2.1: Boxplot of centroid size by spatial/temporal unit.

2.3 Principal Components Analysis

Principal components analysis (Jolliffe 2002) was used to visualise shape variation among the bottles. The shape changes described by each principal axis are commonly visualised using thin-plate spline warping of a reference 3D mesh (Klingenberg 2013; Sherratt et al. 2014).

# principal components analysis
pca<-gm.prcomp(Y.gpa$coords)
summary(pca)
## 
## Ordination type: Principal Component Analysis 
## Centering by OLS mean
## Orthogonal projection of OLS residuals
## Number of observations: 94 
## Number of vectors 93 
## 
## Importance of Components:
##                              Comp1       Comp2      Comp3        Comp4       Comp5       Comp6
## Eigenvalues            0.007873265 0.004425944 0.00144765 0.0006653946 0.000365315 0.000167238
## Proportion of Variance 0.508905787 0.286080584 0.09357203 0.0430092429 0.023612937 0.010809793
## Cumulative Proportion  0.508905787 0.794986371 0.88855840 0.9315676477 0.955180585 0.965990377
##                               Comp7        Comp8        Comp9       Comp10       Comp11
## Eigenvalues            0.0001312098 8.902624e-05 6.987494e-05 4.385343e-05 3.896804e-05
## Proportion of Variance 0.0084810327 5.754406e-03 4.516520e-03 2.834563e-03 2.518785e-03
## Cumulative Proportion  0.9744714102 9.802258e-01 9.847423e-01 9.875769e-01 9.900957e-01
##                              Comp12       Comp13       Comp14       Comp15       Comp16
## Eigenvalues            3.306192e-05 2.857672e-05 1.884971e-05 1.511738e-05 1.166749e-05
## Proportion of Variance 2.137030e-03 1.847119e-03 1.218392e-03 9.771453e-04 7.541536e-04
## Cumulative Proportion  9.922327e-01 9.940798e-01 9.952982e-01 9.962754e-01 9.970295e-01
##                              Comp17       Comp18       Comp19       Comp20       Comp21
## Eigenvalues            7.340905e-06 6.485190e-06 5.514481e-06 5.038877e-06 3.634505e-06
## Proportion of Variance 4.744955e-04 4.191845e-04 3.564406e-04 3.256989e-04 2.349242e-04
## Cumulative Proportion  9.975040e-01 9.979232e-01 9.982796e-01 9.986053e-01 9.988403e-01
##                              Comp22       Comp23       Comp24       Comp25       Comp26
## Eigenvalues            2.656661e-06 1.977827e-06 1.789915e-06 1.358874e-06 1.085909e-06
## Proportion of Variance 1.717191e-04 1.278412e-04 1.156951e-04 8.783383e-05 7.019010e-05
## Cumulative Proportion  9.990120e-01 9.991398e-01 9.992555e-01 9.993434e-01 9.994135e-01
##                              Comp27       Comp28       Comp29       Comp30       Comp31
## Eigenvalues            1.053445e-06 1.003879e-06 7.939259e-07 7.736694e-07 6.187956e-07
## Proportion of Variance 6.809170e-05 6.488795e-05 5.131715e-05 5.000782e-05 3.999721e-05
## Cumulative Proportion  9.994816e-01 9.995465e-01 9.995978e-01 9.996479e-01 9.996878e-01
##                              Comp32       Comp33       Comp34       Comp35       Comp36
## Eigenvalues            5.799077e-07 4.863319e-07 3.974048e-07 3.823365e-07 3.261221e-07
## Proportion of Variance 3.748361e-05 3.143513e-05 2.568713e-05 2.471316e-05 2.107962e-05
## Cumulative Proportion  9.997253e-01 9.997568e-01 9.997825e-01 9.998072e-01 9.998282e-01
##                              Comp37       Comp38       Comp39       Comp40       Comp41
## Eigenvalues            3.124219e-07 2.554880e-07 2.182827e-07 2.110372e-07 1.823501e-07
## Proportion of Variance 2.019408e-05 1.651403e-05 1.410918e-05 1.364085e-05 1.178660e-05
## Cumulative Proportion  9.998484e-01 9.998650e-01 9.998791e-01 9.998927e-01 9.999045e-01
##                              Comp42       Comp43       Comp44       Comp45       Comp46
## Eigenvalues            1.685742e-07 1.398269e-07 1.323013e-07 1.298645e-07 9.935186e-08
## Proportion of Variance 1.089616e-05 9.038019e-06 8.551585e-06 8.394075e-06 6.421825e-06
## Cumulative Proportion  9.999154e-01 9.999244e-01 9.999330e-01 9.999414e-01 9.999478e-01
##                              Comp47       Comp48       Comp49       Comp50       Comp51
## Eigenvalues            9.258452e-08 8.187882e-08 7.199289e-08 6.583570e-08 5.398235e-08
## Proportion of Variance 5.984404e-06 5.292417e-06 4.653418e-06 4.255435e-06 3.489268e-06
## Cumulative Proportion  9.999538e-01 9.999591e-01 9.999637e-01 9.999680e-01 9.999715e-01
##                              Comp52       Comp53       Comp54       Comp55       Comp56
## Eigenvalues            4.923070e-08 4.416412e-08 4.172011e-08 3.566585e-08 3.361173e-08
## Proportion of Variance 3.182134e-06 2.854645e-06 2.696671e-06 2.305340e-06 2.172568e-06
## Cumulative Proportion  9.999747e-01 9.999775e-01 9.999802e-01 9.999825e-01 9.999847e-01
##                              Comp57       Comp58       Comp59       Comp60       Comp61
## Eigenvalues            2.829523e-08 2.414620e-08 2.015781e-08 1.997507e-08 1.689190e-08
## Proportion of Variance 1.828924e-06 1.560743e-06 1.302944e-06 1.291132e-06 1.091845e-06
## Cumulative Proportion  9.999865e-01 9.999881e-01 9.999894e-01 9.999907e-01 9.999918e-01
##                              Comp62       Comp63       Comp64       Comp65       Comp66
## Eigenvalues            1.617265e-08 1.548022e-08 1.294355e-08 1.164172e-08 8.643788e-09
## Proportion of Variance 1.045354e-06 1.000598e-06 8.366350e-07 7.524882e-07 5.587102e-07
## Cumulative Proportion  9.999928e-01 9.999938e-01 9.999946e-01 9.999954e-01 9.999959e-01
##                              Comp67       Comp68       Comp69       Comp70       Comp71
## Eigenvalues            8.191522e-09 7.429701e-09 7.198853e-09 5.768361e-09 4.702120e-09
## Proportion of Variance 5.294770e-07 4.802351e-07 4.653137e-07 3.728507e-07 3.039318e-07
## Cumulative Proportion  9.999965e-01 9.999970e-01 9.999974e-01 9.999978e-01 9.999981e-01
##                              Comp72       Comp73       Comp74       Comp75       Comp76
## Eigenvalues            4.304749e-09 4.124299e-09 3.917233e-09 3.083503e-09 2.539693e-09
## Proportion of Variance 2.782469e-07 2.665831e-07 2.531989e-07 1.993090e-07 1.641586e-07
## Cumulative Proportion  9.999984e-01 9.999986e-01 9.999989e-01 9.999991e-01 9.999993e-01
##                              Comp77       Comp78       Comp79       Comp80       Comp81
## Eigenvalues            2.263717e-09 1.777849e-09 1.457729e-09 1.279492e-09 9.079646e-10
## Proportion of Variance 1.463203e-07 1.149152e-07 9.422351e-08 8.270274e-08 5.868828e-08
## Cumulative Proportion  9.999994e-01 9.999995e-01 9.999996e-01 9.999997e-01 9.999998e-01
##                              Comp82       Comp83       Comp84       Comp85       Comp86
## Eigenvalues            7.941435e-10 7.255847e-10 5.093597e-10 4.732068e-10 3.669145e-10
## Proportion of Variance 5.133121e-08 4.689976e-08 3.292358e-08 3.058676e-08 2.371633e-08
## Cumulative Proportion  9.999998e-01 9.999999e-01 9.999999e-01 9.999999e-01 9.999999e-01
##                              Comp87       Comp88       Comp89       Comp90       Comp91
## Eigenvalues            2.123219e-10 1.996531e-10 1.987433e-10 1.252079e-10 7.996617e-11
## Proportion of Variance 1.372390e-08 1.290502e-08 1.284621e-08 8.093089e-09 5.168789e-09
## Cumulative Proportion  1.000000e+00 1.000000e+00 1.000000e+00 1.000000e+00 1.000000e+00
##                              Comp92       Comp93
## Eigenvalues            2.609662e-11 9.102545e-12
## Proportion of Variance 1.686812e-09 5.883630e-10
## Cumulative Proportion  1.000000e+00 1.000000e+00
# set plot parameters
pch.gps.comb <- c(15:18)[as.factor(comb)]
col.gps.comb <- wes_palette("Moonrise2")[as.factor(comb)]
col.hull <- c("#C27D38","#29211F","#798E87","#CCC591")

# plot pca by comb
pc.plot <- plot(pca, 
                 asp = 1,
                 pch = pch.gps.comb,
                 col = col.gps.comb)
                    shapeHulls(pc.plot, 
                             groups = comb,
                             group.cols = col.hull)

# pca warp                   
knitr::include_graphics('images/pca-warp.jpg')
Results of PCA summarising shape variation in the aggregated sample; gray, north-formative/early; orange, north-late/historic; brown, south-formative/early; and black, south-late/historic; with mean shapes (gray) contrasted with max/min shapes (black).

(#fig:pca.warp)Results of PCA summarising shape variation in the aggregated sample; gray, north-formative/early; orange, north-late/historic; brown, south-formative/early; and black, south-late/historic; with mean shapes (gray) contrasted with max/min shapes (black).

2.4 Define models

A residual randomisation permutation procedure (RRPP; n = 10,000 permutations) was used for all Procrustes ANOVAs (D. C. Adams and Collyer 2015; Michael L. Collyer and Adams 2018), which has higher statistical power and a greater ability to identify patterns in the data should they be present (Anderson and Ter Braak 2003). To assess whether shape changes differ by group (geography and time), Procrustes ANOVAs (Goodall 1991) were also run that enlist effect-sizes (zscores) computed as standard deviates of the generated sampling distributions (M. L. Collyer, Sekora, and Adams 2015).

2.4.1 Size and shape

# size.geo
fit.size <- procD.lm(size ~ geo + time,
                data = gdf,
                print.progress = FALSE,
                iter = 9999)

# shape.geo
fit.shape <- procD.lm(shape ~ geo + time,
                data = gdf,
                print.progress = FALSE,
                iter = 9999)

2.5 Test Hypothesis

Hypothesis: Formative/Early and Late/Historic Caddo bottles found north and south of the shape boundary express unique patterns of shape change.

2.5.1 Procrustes ANOVA

# ANOVA: do Caddo bottle sizes differ by geography and/or time?
anova(fit.size)
## 
## Analysis of Variance, using Residual Randomization
## Permutation procedure: Randomization of null model residuals 
## Number of permutations: 10000 
## Estimation method: Ordinary Least Squares 
## Sums of Squares and Cross-products: Type I 
## Effect sizes (Z) based on F distributions
## 
##           Df      SS     MS     Rsq       F       Z Pr(>F)    
## geo        1   25282  25282 0.01231  1.3094 0.73333 0.2475    
## time       1  271285 271285 0.13210 14.0506 2.92981 0.0003 ***
## Residuals 91 1756998  19308 0.85558                           
## Total     93 2053565                                          
## ---
## Signif. codes:  0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1
## 
## Call: procD.lm(f1 = size ~ geo + time, iter = 9999, data = gdf, print.progress = FALSE)
# ANOVA: do Caddo bottle shapes differ by geography and/or time?
anova(fit.shape)
## 
## Analysis of Variance, using Residual Randomization
## Permutation procedure: Randomization of null model residuals 
## Number of permutations: 10000 
## Estimation method: Ordinary Least Squares 
## Sums of Squares and Cross-products: Type I 
## Effect sizes (Z) based on F distributions
## 
##           Df      SS       MS     Rsq      F      Z Pr(>F)    
## geo        1 0.15087 0.150868 0.10486 12.527 3.9641  1e-04 ***
## time       1 0.19200 0.192001 0.13345 15.943 4.0840  1e-04 ***
## Residuals 91 1.09593 0.012043 0.76170                         
## Total     93 1.43880                                          
## ---
## Signif. codes:  0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1
## 
## Call: procD.lm(f1 = shape ~ geo + time, iter = 9999, data = gdf, print.progress = FALSE)

2.5.2 Mean Shapes

# subset landmark coordinates to produce mean shapes
new.coords<-coords.subset(A = Y.gpa$coords, 
                          group = qdata$comb)
names(new.coords)
## [1] "north, formative early" "north, late historic"   "south, formative early"
## [4] "south, late historic"
# group shape means
mean <- lapply(new.coords, mshape)

# plot mean shapes
plot(mean$`north, formative early`)
plot(mean$`north, late historic`)
plot(mean$`south, formative early`)
plot(mean$`south, late historic`)

# comparison plots
plotRefToTarget(mean$`north, formative early`, 
                mean$`north, late historic`, 
                method = c("points"),
                mag = 1)

plotRefToTarget(mean$`north, formative early`, 
                mean$`south, formative early`, 
                method = "points",
                mag = 1)

plotRefToTarget(mean$`north, formative early`, 
                mean$`south, late historic`, 
                method = "points",
                mag = 1)

plotRefToTarget(mean$`north, late historic`, 
                mean$`south, formative early`, 
                method = "points",
                mag = 1)

plotRefToTarget(mean$`north, late historic`, 
                mean$`south, late historic`, 
                method = "points",
                mag = 1)

plotRefToTarget(mean$`south, formative early`, 
                mean$`south, late historic`, 
                method = "points",
                mag = 1)

knitr::include_graphics('images/comp.composite.jpg')

2.6 Trajectory analysis

# trajectory analysis::shape
TA <- trajectory.analysis(fit.shape, 
                          groups = qdata$geo, 
                          traj.pts = qdata$time, 
                          print.progress = FALSE)

# magnitude difference
summary(TA, attribute = "MD")
## 
## Trajectory analysis
## 
## 10000 permutations.
## 
## Points projected onto trajectory PCs
## 
## Trajectories:
## Trajectories hidden (use show.trajectories = TRUE to view)
## 
## Observed path distances by group
## 
##      north      south 
## 0.09211959 0.09211959 
## 
## Pairwise absolute differences in path distances, plus statistics
##                        d    UCL (95%)         Z  Pr > d
## north:south 8.326673e-17 1.457168e-16 0.5188623 0.26785
# plot
TP <- plot(TA, 
           pch = as.numeric(qdata$geo), 
           bg = as.numeric(qdata$time),
           cex = 0.9,
           col = "gray")
add.trajectories(TP, traj.pch = c(15, 19),
                 start.bg = 1, 
                 end.bg = 2)

knitr::include_graphics('images/trajectory.jpg')

References

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