Orthogonal polynomials

Calculates orthogonal polynomial coefficients using stats::lm

opoly(
  x,
  degree,
  raw,
  drop_coo,
  from_coo = coo,
  to_coe = {     {         coe     } },
  ...
)

opoly_i(x, nb_pts, from_coe, ...)

Arguments

x	coo_single, coo_list or mom_tbl
degree	polynomial degree for the fit (degree+1) coefficients are returned, see Details)
raw	logical whether to return raw and full results
drop_coo	logical whether to drop coo column (default to TRUE)
from_coo, to_coe	column names
...	for generics. Useless here.
nb_pts	number of points to sample and on which to calculate polynomials
from_coe	column name for inverse method

Value

a list with components when applied on a single shape:

• coeff the coefficients (including the intercept)

• ortho whether orthogonal or natural polynomials were fitted

• degree degree of the fit (could be retrieved through coeff though)

• baseline1 the first baseline point (so far the first point)

• baseline2 the second baseline point (so far the last point)

• r2 the r2 from the fit

• mod the raw lm model

Details

Curves must be registered on a bookstein baseline, with the first coordinates on (-0.5, 0) and the last on (0.5, 0). Use coo_bookstein

A polynomial of degree n use this fit:

\(y = a_{0} + a_{1}x^1 + a_{...}x^{...} + a_{n}x^{n}\)

And thus returns n+1 coefficients. The +1 being $a_0$ ie the intercept.

Orthogonal polynomials are also called Legendre's polynomials. They are preferred over natural polynomials since adding a degree does not change lower order coefficients.

In retired Momocs, baseline was free and not necessarily Bookstein. Not sure this was really helpful but I'm sure this overcomplicated coding. If your shapes are not registered on Bookstein coordinates, you will be messaged (not for coo_single though)

Functions

• opoly_i: inverse opoly method

See also

Other polynomials: npoly()

Other morphometrics: dfourier(), efourier(), npoly()

Examples

o <- olea %>% pick(5) %>% coo_bookstein()
o %>% pile()
op <- o %>% opoly(degree=3)
op
#> # A tibble: 1 x 4
#>      a0     a1     a2      a3
#>   <dbl>  <dbl>  <dbl>   <dbl>
#> 1 0.216 0.0154 -0.996 0.00544
#> ❯coe_single with 4 coefficients
op %>% class
#> [1] "opoly_single" "coe_single"   "tbl_df"       "tbl"          "data.frame"  

op %>% opoly_i() %>% draw()

olea %>% dplyr::slice(1:3) %>%
  opoly(drop=FALSE) %>% npoly_i
#> ! opoly: 'degree' not provided and set to 5
#> # A tibble: 3 x 7
#>   coo                var   domes view  ind   coe               coe_i            
#>   <list<coo_single[> <fct> <fct> <fct> <fct> <list<coe_single> <list<coo_single>
#> 1 <tibble [99 × 2]>  Aglan cult  VD    O10   <tibble [1 × 6]>  <tibble [120 × 2…
#> 2 <tibble [96 × 2]>  Aglan cult  VL    O10   <tibble [1 × 6]>  <tibble [120 × 2…
#> 3 <tibble [95 × 2]>  Aglan cult  VD    O11   <tibble [1 × 6]>  <tibble [120 × 2…
#> ❯mom_tbl