Natural Polynomial Transform

Compute natural polynomial coefficients from open curve coordinates.

Usage

npoly(x, degree = 5, raw = FALSE, ..., .cols = NULL, .name = "_coe")

Arguments

x

A matrix (nx2), list of matrices, or tibble with coo columns.

degree

Integer. Polynomial degree for the fit. Default is 5.

raw

Logical. If TRUE, returns a list with coefficients and fit details (only for single matrix). Default is FALSE.

...

Additional arguments (reserved for future use).

.cols

Column name(s) to process when x is a tibble. If NULL, processes all coo columns automatically.

.name

Character. Name suffix for output coefficient columns when x is a tibble. Default is "_coe". Output columns will be named originalname_coe. If a single custom name is provided and multiple columns are processed, an error is raised.

Value

• If x is a single matrix and raw = FALSE: returns a named numeric vector with classes c("npoly", "numeric") containing degree+1 coefficients (x0, x1, ..., xN)

• If x is a single matrix and raw = TRUE: returns a list with elements coeff, degree, baseline1, baseline2, r2, mod

• If x is a list: returns a list of coefficient vectors (each with class c("npoly", "numeric"))

• If x is a tibble: returns the tibble with new list-column(s) of class c("npoly", "coe", "list") containing coefficient vectors

Details

Natural polynomial fitting uses standard power basis polynomials (1, x, x², x³, ...) to decompose open curves. The method fits a polynomial model to the y-coordinates as a function of x-coordinates, after baseline normalization.

Unlike orthogonal polynomials, natural polynomials have coefficients that change when higher degrees are added. However, they are more interpretable and directly correspond to the polynomial equation.

The curve is automatically baseline-normalized using the first and last points as landmarks, positioned at (-0.5, 0) and (0.5, 0) respectively.

Choosing the polynomial degree

The default degree = 5 works well for most open curves. Higher degrees capture more detail but increase dimensionality and risk overfitting. Use raw = TRUE to examine the R² value and assess fit quality.

See also

npoly_i() for inverse transform, opoly() for orthogonal polynomials, dct() for discrete cosine transform.

Examples

# Single curve
data(olea)
cur <- olea$VD[[1]]
npoly_coefs <- npoly(cur)
npoly_coefs
#>          x0          x1          x2          x3          x4          x5 
#>  0.31027929  0.03654682 -0.64016581 -0.35437960 -2.26592353  0.64569559 
#> attr(,"class")
#> [1] "npoly"   "numeric"

# Get raw output with R² and model details
npoly(cur, raw = TRUE)
#> $coeff
#>          x0          x1          x2          x3          x4          x5 
#>  0.31027929  0.03654682 -0.64016581 -0.35437960 -2.26592353  0.64569559 
#> 
#> $degree
#> [1] 5
#> 
#> $baseline1
#> [1] -0.5  0.0
#> 
#> $baseline2
#> [1] 0.5 0.0
#> 
#> $r2
#> [1] 0.998722
#> 
#> $mod
#> 
#> Call:
#> lm(formula = coo[, 2] ~ x_poly)
#> 
#> Coefficients:
#> (Intercept)      x_poly1      x_poly2      x_poly3      x_poly4      x_poly5  
#>     0.31028      0.03655     -0.64017     -0.35438     -2.26592      0.64570  
#> 
#> 

# With higher degree
npoly(cur, degree = 8)
#>          x0          x1          x2          x3          x4          x5 
#>  0.31045114  0.01056954 -0.64587938  0.53074384 -2.39943119 -6.75387457 
#>          x6          x7          x8 
#>  1.99846549 17.47625125 -5.68546437 
#> attr(,"class")
#> [1] "npoly"   "numeric"

# List of curves
cur_list <- list(cur, cur * 1.5, cur * 2)
npoly(cur_list)
#> [[1]]
#>          x0          x1          x2          x3          x4          x5 
#>  0.31027929  0.03654682 -0.64016581 -0.35437960 -2.26592353  0.64569559 
#> attr(,"class")
#> [1] "npoly"   "numeric"
#> 
#> [[2]]
#>          x0          x1          x2          x3          x4          x5 
#>  0.31027929  0.03654682 -0.64016581 -0.35437960 -2.26592353  0.64569559 
#> attr(,"class")
#> [1] "npoly"   "numeric"
#> 
#> [[3]]
#>          x0          x1          x2          x3          x4          x5 
#>  0.31027929  0.03654682 -0.64016581 -0.35437960 -2.26592353  0.64569559 
#> attr(,"class")
#> [1] "npoly"   "numeric"
#> 
#> attr(,"class")
#> [1] "npoly" "coe"   "list" 

if (FALSE) { # \dontrun{
# Tibble - processes all coo columns by default
library(dplyr)
olea %>%
  npoly(degree = 6)  # Creates VD_coe and VL_coe

# Process specific column only
olea %>%
  npoly(.cols = VD)

# Custom name for single column
olea %>%
  npoly(.cols = VL, .name = "custom_name")
} # }