orthogonal polynomials

Project.toml

@@ -5,6 +5,9 @@ version = "0.1.1"
 
 [deps]
 Combinatorics = "861a8166-3701-5b0c-9a16-15d98fcdc6aa"
+LinearAlgebra = "37e2e46d-f89d-539d-b4ee-838fcccc9c8e"
+Statistics = "10745b16-79ce-11e8-11f9-7d13ad32a3b2"
+StatsAPI = "82ae8749-77ed-4fe6-ae5f-f523153014b0"
 StatsModels = "3eaba693-59b7-5ba5-a881-562e759f1c8d"
 
 [compat]

src/RegressionFormulae.jl

@@ -1,5 +1,8 @@
 module RegressionFormulae
 
+using LinearAlgebra
+using Statistics
+using StatsAPI
 using StatsModels
 using Combinatorics
 using Base.Iterators
@@ -13,4 +16,7 @@ include("fulldummy.jl")
 include("power.jl")
 include("nesting.jl")
 
+include("poly.jl")
+export poly
+
 end # module

src/poly.jl

@@ -0,0 +1,78 @@
+function poly(x, deg; raw=true)
+    raw == true && return [el^d for el in x, d in 1:deg]
+    length(unique(x)) < deg &&
+        error("Degree of orthogonal polynomial must be lower than the " *
+              "number of unique points")
+    x̄ = mean(x)
+    x = x .- mean(x)
+    mat = qr!([el^d for el in x, d in 0:deg])
+    cols = vcat(Diagonal(mat.R), zeros(length(x) - size(mat.R,1), size(mat.R, 2)))
+    cols = mat.Q * cols
+    colnorms = sum.(abs2, eachcol(cols))
+
+    for (j, cn) in zip(axes(cols, 2), colnorms)
+        cols[:, j] ./= sqrt(cn)
+    end
+
+    return cols[:, (begin+1):end]
+
+end
+
+# type of model where syntax applies: here this applies to any model type
+const POLY_CONTEXT = RegressionModel
+
+# struct for behavior
+struct PolyTerm{T,D} <: AbstractTerm
+    term::T
+    deg::D
+    raw::Bool
+end
+
+Base.show(io::IO, p::PolyTerm) = print(io, "poly($(p.term), $(p.deg))")
+
+# for `poly` use at run-time (outside @formula), return a schema-less PolyTerm
+poly(t::Symbol, d::Int; raw::Bool=true) = PolyTerm(term(t), term(d), raw)
+
+# for `poly` use inside @formula: create a schemaless PolyTerm and apply_schema
+function StatsModels.apply_schema(t::FunctionTerm{typeof(poly)},
+                                  sch::StatsModels.Schema,
+                                  Mod::Type{<:POLY_CONTEXT})
+    apply_schema(PolyTerm(t.args_parsed...), sch, Mod)
+end
+
+# apply_schema to internal Terms and check for proper types
+function StatsModels.apply_schema(t::PolyTerm,
+                                  sch::StatsModels.Schema,
+                                  Mod::Type{<:POLY_CONTEXT})
+    term = apply_schema(t.term, sch, Mod)
+    isa(term, ContinuousTerm) ||
+        throw(ArgumentError("PolyTerm only works with continuous terms (got $term)"))
+    isa(t.deg, ConstantTerm) ||
+        throw(ArgumentError("PolyTerm degree must be a number (got $t.deg)"))
+    PolyTerm(term, t.deg.n)
+end
+
+function StatsModels.modelcols(p::PolyTerm, d::NamedTuple)
+    col = modelcols(p.term, d)
+    p.raw && return reduce(hcat, [col.^n for n in 1:p.deg])
+
+    return mapreduce(hcat, 1:p.deg) do n
+
+
+        # INSERT HERE
+    end
+end
+
+# the basic terms contained within a PolyTerm (for schema extraction)
+StatsModels.terms(p::PolyTerm) = terms(p.term)
+# names variables from the data that a PolyTerm relies on
+StatsModels.termvars(p::PolyTerm) = StatsModels.termvars(p.term)
+# number of columns in the matrix this term produces
+StatsModels.width(p::PolyTerm) = p.deg
+
+# highlight the difference between raw and orthogonal in the output
+function StatsAPI.coefnames(p::PolyTerm)
+    p.raw && return coefnames(p.term) .* "^" .* string.(1:p.deg)
+
+    return "poly(" .* coefnames(p.term) .* ", " .* string.(1:p.deg) .* ")"
+end

test/poly.jl

@@ -0,0 +1,22 @@
+# > poly(1:10, 3)
+#                 1           2          3
+#  [1,] -0.49543369  0.52223297 -0.4534252
+#  [2,] -0.38533732  0.17407766  0.1511417
+#  [3,] -0.27524094 -0.08703883  0.3778543
+#  [4,] -0.16514456 -0.26111648  0.3346710
+#  [5,] -0.05504819 -0.34815531  0.1295501
+#  [6,]  0.05504819 -0.34815531 -0.1295501
+#  [7,]  0.16514456 -0.26111648 -0.3346710
+#  [8,]  0.27524094 -0.08703883 -0.3778543
+#  [9,]  0.38533732  0.17407766 -0.1511417
+# [10,]  0.49543369  0.52223297  0.4534252
+
+# > poly(-3:3, 5)
+#                  1             2             3          4             5
+# [1,] -5.669467e-01  5.455447e-01 -4.082483e-01  0.2417469 -1.091089e-01
+# [2,] -3.779645e-01  9.690821e-17  4.082483e-01 -0.5640761  4.364358e-01
+# [3,] -1.889822e-01 -3.273268e-01  4.082483e-01  0.0805823 -5.455447e-01
+# [4,]  2.098124e-17 -4.364358e-01  4.532467e-17  0.4834938  5.342065e-16
+# [5,]  1.889822e-01 -3.273268e-01 -4.082483e-01  0.0805823  5.455447e-01
+# [6,]  3.779645e-01  0.000000e+00 -4.082483e-01 -0.5640761 -4.364358e-01
+# [7,]  5.669467e-01  5.455447e-01  4.082483e-01  0.2417469  1.091089e-01