1.1 What is the Design of experiments?

Algebraic statistics in Design of Experiments

Maria Piera Rogantin

DIMA – Universit`a di Genova – rogantin@dima.unige.it Torino, September 2004

Table of contents

1. General aspects

(a) What is the Design of experiments?

(b) Design & Ideal of the design (c) The full factorial design

2. Fractions and confounding

(a) Fraction of a full factorial design (b) Space of the responses on the

fraction

(c) Identifiability of a model

(d) Fractions and identifiable linear models

(e) Confounding of subspaces

3. Indicator function and orthogo- nality

(a) Orthogonality

(b) Complex coding for full factorial designs

(c) Indicator function of a fraction (d) Results about orthogonality

(e) Generation of fractions with a given orthogonal structure

(f) Regular fractions

4. Models on a fraction and term- orders

(a) Identifiable sub-models and initial orders

(b) Model curvature

(c) Block term-order and factor screening

PART I:

General aspects

Pistone G., Wynn H. P.(1996). Generalized confounding with Gr¨obner bases, Biometrika, 83(1): 653–666.

Robbiano L. (1998). Gr¨obner Bases and Statistics, Gr¨obner Bases and Applications (Proc. of the Conf. 33 Years of Gr¨obner Bases), Buchberg- er, B. & Winkler, F. ed., Cambridge University Press, 179–204.

Pistone G., Riccomagno E. and Wynn H. P. (2001). Algebraic Statistics:

Computational Commutative Algebra in Statistics, Chapman&Hall.

1.1 What is the Design of experiments?

An example: the factorial design

Push-off force of a spark control valve.

(Wu & Hamada. 2000. Experiments. Whiley & Son, p.247)

Influence of three factors on the response:

- weld time (0.3 - 0.5 - 0.7 seconds) - pressure (15 - 20 - 25 psi)

- moisture (0.8 - 1.2 - 1.8 percent) Each factor has three levels.

We consider them as ordinal levels:

low - medium - high;

The levels are coded by integer numbers.

Each treatment corresponds to a point in Z³. full factorial design

∗Full Design

In particular, the experiment is realized on fewer treatments

(because of problems with costs, times, practical constraints, setting the factors and measuring the responses . . . )

How to choose the treatments?

Which fraction for “the best” study of the responses?

time pressure moisture

-1 -1 -1

-1 0 0

-1 1 1

0 -1 0

0 0 1

0 1 -1

1 -1 1

1 0 -1

1 1 0

fractional factorial design or

fraction

∗Fraction

Design and functions on the design^∗

Response Factors

Push-off force (lb) Time Pressure Moisture

111.1 -1 -1 -1

131.0 -1 0 0

65.4 -1 1 1

125.5 0 -1 0

46.9 0 0 1

113.7 0 1 -1

72.5 1 -1 1

141.1 1 0 -1

134.2 1 1 0

Linear influence of the factors and their interactions:

Force = θ0 + θ1 T + θ2 P + θ3 M + θ12 T · P + θ13 T · M + . . .

the θ’s signify the “importance” of the terms w.r.t. the response.

∗General design

Full factorial designs and fractions: notations

• A_i = {a_ij : j = 1, . . . , n_i} factors

a_ij levels coded by rational numbers Q^m or complex numbers C^m

• D = A₁ × . . . × A_m ⊂ Q^m (or D ⊂ C^m_{) with N =} ^Qm_{j=1 nj} points full factorial design

• A fraction is a subset F ⊂ D;

Responses on a design f : D 7→ R (functions defined on D) ^∗

“Design” indicates either “Full factorial design” or “Fractional factorial design” or ...

• X_i : D ∋ (d₁, . . . , d_m) 7→ d_i projection, frequently called factor

• X^α = X₁^α1 · · · X_m^αm, α_i < n_i, i = 1, . . . , m α = (α₁, . . . , α_m) monomial responses or terms or interactions

The term X^α has order k if in α there are k non-null values:

X^α is an interaction of order k (binary case: order = degree)

Definitions:

• Mean value of f on D, E_D(f ): E_D(f ) = _#D^{1 P}_d∈D f (d)

• A contrast is a response f such that E_D(f ) = 0.

• Two responses f and g are orthogonal on D if E_D(f g) = 0.

∗General design

The polynomial complete regression model

• L = {(α1, . . . , αm) : α_i < n_i, i = 1, . . . , m}

exponents (or logarithms) of all the interactions

• complete regression model:

For all d_i ∈ D and for the observed value yi

y_i = X

α∈L

θ_α X^α(d_i) whit θ_α ∈ R or θ_α ∈ C.

In vector notation, Y = (y1, . . . , yn) response measured on the points of D:

Y = X

α∈L

θα X^α

• Z = [X^α(d)]_d∈D,α∈L matrix of the complete regression model

• θ = (θα)_α∈L vector of the coefficients

Matrix notation of the complete regression model Y = Zθ

∗General design

On the full factorial design:

the complete regression model is identifiable, i.e. there is a unique solution w.r.t. θ:

θ = Zˆ ⁻¹ Y (the matrix Z is a square full rank matrix) On a fraction:

there is not a unique solution w.r.t. θ

(Z = [X^α(d)]_{d∈F ,α∈L} has less rows than columns) θ = (Zˆ ^′Z)⁻Z^′ Y

In general, for linear regression models of the form:

Y = W θ + R

where W is a matrix of “explicative variables” and R is a vector of residuals Even if ˆθ = (W^′W )⁻W^′ Y is not unique, the approximation of the response Y through the “explicative variables” is always unique:

Y = W (Wˆ ^′W )⁻W^′ Y

Yˆ is the orthogonal projec- tion of Y in the linear space

generate by the columns of W . _Wθ

Y

Y= W = W (W'W) W' Y = P Y ^ θ ^ - R

Moreover, if Y is a multivariate random variable with normal distribution Y ∼ N (W θ, σ²I) then ˆθ = (W^′W )⁻W^′ Y is a solution of maximum likelihood.

Example: 2 × 3 full factorial design^∗

A1 = {−1, 1}, n1 = 2

A2 = {−1, 0, 1}, n2 = 3 D =

-1 -1 -1 0 -1 1 1 -1

1 0

1 1

monomial responses: 1, X1, X2, X₂², X1X2, X1X₂² L = {(0, 0), (1, 0), (0, 1), (0, 2), (1, 1), (1, 2)}

complete regression model:

Y = θ00 + θ10X1 + θ01X2 + θ02X₂² + θ11X1X2 + θ12X1X₂²

matrix of the complete regression model Z =

1 X1 X2 X₂² X1X2 X1X₂²

1 −1 −1 1 1 −1

1 −1 0 0 0 0

1 −1 1 1 −1 −1

1 1 −1 1 −1 1

1 1 0 0 0 0

1 1 1 1 1 1

∗Full Design

1.2 Design and design ideal

The application of computational commutative algebra to the study of estimability, confounding on the fractions of factorial designs has been proposed by Pistone & Wynn (Biometrika 1996).

1^st idea Each set of points D ⊆ Q^m is the set of the solutions of a system of polynomial equations; we assume that each solution is simple.

2^nd idea Each real function defined on D is a polynomial function with coefficients into the field of real number R.

It is a restriction to D of a real polynomial.

∗General Design

A design D is a finite set of distinct points in a m-dimensional field k

The defining ideal of the design (or design ideal) I(D) is the set of all polynomials on k[x₁, . . . , x_m] that vanish on D.

• The design D is a variety

• The design ideal I(D) is radical

• I(D) is the intersection of the design ideals of each point of D

∗General design

Operations with designs & Operations with ideals

• Product of designs.

D₁ ⊂ k^m1 D₂ ⊂ k^m2 D₁ × D₂ ⊂ k^m1+m2

I (D₁ × D₂) =< I₁, I₂ >

τ term ordering on k[x₁, . . . , x_m1+m2] τ₁, τ₂ t.o. restricted to

k[x₁, . . . , x_m1] and k[x₁, . . . , x_m2]

G_1,τ1, G_2,τ2, G-bases of I (D₁), I (D₂) G = ⁿg₁, g₂ | g₁ ∈ G_1,τ1 g₂ ∈ G_2,τ2^o is G-basis of I (D₁ × D₂)

∗General design

• Restriction of a design^∗

D ⊂ k^m I = I(D)

J ideal in k[x₁, . . . , x_m]

I + J ideal in k[x₁, . . . , x_m]

I + J = {f + g | f ∈ I g ∈ J}

Variety(I + J) = D ∩ Variety(J)

∗General design

• Union of designs

D₁, D₂ ⊂ k^m D₁ ∪ D₂ ⊂ k^m

τ term ordering on k[x₁, . . . , x_m], G₁, G₂ G-bases of I (D₁), I (D₂) A G-basis of I (D₁ ∪ D₂) is

G = {g₁g₂ | g₁ ∈ G₁ g₂ ∈ G₂₂}

∗General design

1.3 The full factorial design.

Polynomial representation

The full factorial design D corresponds to the set of solutions of the system











(X1 − a11)· · · (X1 − a1n1) = 0 (X2 − a21)· · · (X2 − a2n2) = 0

...

(X_m − am1)· · · (Xm − amnm) = 0

or











X₁ⁿ¹ = P_n1−1

k=0 ψ_1k X₁^k ......

X_m^nm = P_nm−1

k=0 ψ_mk X_m^k

In the previous examples:

 (X1 − 1) (X1 + 1) = 0

X2 (X2 − 1) (X2 + 1) = 0 or

 X₁² = 1 rewriting X₂³ = X2 rules







X1(X1 − 1) (X1 + 1) = 0 X2(X2 − 1) (X2 + 1) = 0 X3(X3 − 1) (X3 + 1) = 0

or







X₁³ = X1 rewriting X₂³ = X2 rules

X₃³ = X3

∗Full Design

Space of the real functions on the full factorial design R(D)

For a full factorial design each function is represented in a unique way by an identified complete regression model (i.e. as a linear combination of constant, simple terms and interactions):

R(D) =





 X α∈L

θ_α X^α , θ_α ∈ R







In general, for full factorial designs or fraction:

• R(D) is a Hilbert vector space (classical results derive from this structure)

The scalar product is: < f, g >= E_D(f g) = _N^{1 P}_d∈D f (d)g(d)

• R(D) is a ring (algebraic statistical approach)

The products are reduced with the rules derived by the polynomial representation of the full factorial design:

X_iⁿⁱ =

n_i−1 X k=0

ψ_ik X_i^k , ψ_ik ∈ R for i = 1, . . . , m

∗Full design

Orthogonal decomposition of the space of responses on the full factorial design^∗

R(D) can be decomposed in orthogonal subspaces corresponding to the constants, to the simple factors, to the interactions of order 2, 3, . . . , m:

R(D) = H₀ ⊕

m X i=1

H_i ⊕

m X i,j=1

H_ij ⊕ · · · ⊕ H_12···m .

H_I = span





 Y i∈I

Y_ij, j = 1, . . . , m







where

– I subset of {1, . . . , m}.

– 1, Y_i1, Y_i2, . . . , Y_ni−1 orthogonalization of 1, X_i, X_i², . . . , X_iⁿⁱ⁻¹

∗Full Design

PART II:

Fractions and confounding

Pistone G., Wynn H. P. (1996). Generalized confounding with Gr¨obner bases, Biometrika, 83(1): 653–666.

Robbiano L. (1998). Gr¨obner Bases and Statistics, Gr¨obner Bases and Applications (Proc. of the Conf. 33 Years of Gr¨obner Bases), Buchberg- er, B. & Winkler, F. ed., Cambridge University Press, 179–204.

Pistone G., Riccomagno E. and Wynn H. P. (2001). Algebraic Statistics:

Computational Commutative Algebra in Statistics, Chapman&Hall.

Galetto F., Pistone G., Rogantin M. P. (2003). Confounding revisited with commutative computational algebra. Journal of Statistical Planning and Inference. 117, p. 345-363.

2.1 Fractions of a full factorial design F ⊂ D

A fraction is a subset of a full factorial design, F ⊂ D.

All the fractions are obtained by adding equations (generating equations) to restrict the set of solutions.

In the first example:











X1 (X1 − 1) (X1 + 1) = 0 X2 (X2 − 1) (X2 + 1) = 0 X3 (X3 − 1) (X3 + 1) = 0

X1X2X3 − X1X2 + X1X3 + X2X3 + ¹₃X1 + ¹₃X2 − ¹₃X3 + ¹₃ = 0

∗Fraction

The use of fractions induces a confounding.

• Most of the terms of the complete regression model are not identifi- able.

• Some simple or interaction terms of the complete regression model computed on the fraction points equals a linear combination of other terms.

In the first example:

X₁² = −2X1X2− X₂²+ 2X1X3+ 2X2X3− X₃²+ X1+ X2− X3+ 2

on each point of the displayed fraction.

∗Fraction

2.2 Space of the responses on the fraction R(F)

(Pistone & Wynn, Biometrika 1996)

Algebraic methods allows to find bases of the vector space of responses on the fraction, namely of the quotient space k[x₁, . . . , x_m]/I(F)

- τ a term-ordering,

- G_τ,F the Gr¨obner basis of I(F)

- LT ^G_τ,F^ the set of the leading terms of G_τ,F

Est_τ(F) = ⁿx^β : x^β is not divisible for any x^α, x^α ∈ LT ^G_τ,F^o

R(F ) =





 X β∈M

θ_β X^β , θ_β ∈ R







dim R(F) = #Est_τ(F) = #F Let M be the set of exponents of the elements of Est

∗Fraction

Est_τ(F) is a hierarchical set of terms:

if x^β ∈ Est_τ(F) and x^α divides x^α, then x^α ∈ Est_τ(F) (or order ideal or standard set of power products)

Each response f on F can be written as an unique linear combination of elements of Est_τ(F):

NF_τ,F(f ) = ^X

β∈M

c_β(f ) X^β remainder of f w.r.t. division by G_τ,F,

representative of the equivalence class of f in k[x₁, . . . , x_m]/I(F)

∗Fraction

2.3 Identifiability of a model

Y = (y₁, . . . , y_n) vector of responses B_τ = ^hX^β(d)ⁱ

d∈F ,β∈M matrix of the model

elements of Est_τ(F) valued on the fraction points

The linear system of equation Y = ^X

β∈M

θ_β X^β or Y = B_τθ has one and only one solution w.r.t. θ:

θ = Bˆ _τ⁻¹Y

we say that the polynomial model is identifiable by the fraction Sub-models of an identifiable model are identifiable

∗Fraction

Decomposition of the response space of the full design R(D) (Galetto, Pistone, Rogantin, JSPI, 2003)

The vector space R(D) of the responses on the full factorial D can be decomposed into two orthogonal sub-spaces:

• the space R(F) of the identifiable responses on F

• the space of the null responses on F

∗Full Design

2.3 Fractions and linear polynomial models

1. Direct problem: given a F, what are the linear polynomial models which can be identified by F?

or

What are the possible hierarchical sets of terms E which are bases of the vector space k[X]/I(F)?

2. Inverse problem: given a hierarchical linear polynomial model Y , what are the minimal fractions F ⊂ D which identify Y ?

E hierarchical set of terms associated to Y , E the set of its exponents:

Y = ^X

β∈E

θ_β X^β or

What are the fractions F ⊂ D such that E is a basis of the vector space k[X]/I(F)?

∗Fraction

1. Direct problem:

• Using G-bases:

given I(F), for every term-order τ there is a unique Est_τ(F).

τ are infinite but the Est_τ(F) are finite.

If a model is identifiable w.r.t. a term-ordering then it is identifiable w.r.t. any term-ordering.

• Using indicator function (see below):

If the cardinality of a list of orthogonal terms equals #F then it is a basis of the vector space k[X]/I(F)

∗Fraction

2. Inverse problem^∗

• Given E there exists a fraction s.t. E is a basis of the vector space k[X]/I(F )

Let Distr(E) be the set of such fractions. It is easily computable, but in general such fractions have not good statistical properties.

• There are hierarchical sets of terms E not derived from G-basis method: ∄ τ s.t. E = Est_τ(F)

• Let Sol(E) be set of all the fractions s.t. E is a basis of the vector space k[X]/I(F)

Distr(E) ⊆ Sol(E, ∀τ ) ⊆ Sol(E, ∃τ ) ⊆ Sol(E) where:

- Sol(E, ∀τ ) is the set of fractions s.t. Est_τ(F) = E

- Sol(E, ∃τ ) is the set of fractions s.t. ∃ τ with Est_τ(F) = E In general the inclusions are strict. (Robbiano,1998)

∗Fraction

2.4 Confounding of interaction subspaces on F

(Galetto, Pistone, Rogantin, JSPI 2000)

This definition of confounding does not relate to the non-identifiability of the single monomial responses but it regards the confounding of the full interaction sub-space

An interaction space has clear meaning only when it is fully identifiable on the fraction.

Problems arise with multilevel factors; in the binary case the dimension of each interaction sub-space is 1.

Example: three-level factors H1: subspace of the first factor

on D: dim(H1) = 2; it is generated by an orthogonalization of X1, X₁² H12: subspace of the interaction between the first two factors

on D: dim(H12) = 4 ; it is generated by an orthogonalization of X1X2, X₁²X2, X1X₂², X₁²X₂²

∗Fraction

Identifiability of subspaces of a fraction^∗

• H_J is identifiable on F if

dim H_J on F equals dim H_J on D = ^ n

#J



• H_J1, . . . , H_Jk are simultaneously identifiable on F if dim

k X i=1

H_Ji on F equals

k X i=1

dim H_Ji on D

∗Fraction

Confounding of an interaction subspace H_I

with a set of subspaces H_J1, . . . , H_Jk simultaneously identifiable^∗

• H_I is confounded on F with the spaces H_J1, . . . , H_Jk (H_J1 + . . . + H_Jk 6= R(F)) if:

a) H_I ⊆ H_J1 + . . . + H_Jk

b) the set {H_J1, . . . , H_Jk} is minimal for the property a)

• H_I is totally confounded with the simultaneously identifiable spaces H_J1, . . . , H_Jk if

it is confounded and if:

c) H_I is identifiable,

d) H_I = H_J1 + . . . + H_Jk.

∗Fraction

An algebraic method for studying the confounding of a interaction sub-space H_I is based on the system of the Normal Forms of the basis of H_I on the design D.^∗

advantage: it is easily implementable within symbolic computation soft- wares

disadvantage: it does not display all possible confounding relations

but it applies only to the controlled subspaces, i.e. spaces having a basis contained in Est_τ(F)

but: different choices of the term-ordering τ allow to find different con- founding relations

∗Fraction

PART III:

Indicator function and orthogonality

Fontana R., Pistone G. and Rogantin M. P. (2000). Classification of two-level factorial fractions, J. Statist. Plann. Inference 87(1), 149–172.

Ye K. Q. (2003). Indicator function and its Application in two-level factorial designs, The Annals of Statistics. 31(3).

Pistone G., Rogantin M. P. (2004). Complex coding for multilevel facto- rial designs. Submitted.

∗Fraction

3.1 Orthogonality

of factors : “all level combinations appear equally often”

of responses in a vector space, based on a scalar or Hermitian product:

< f, g >= E_D(f g) = 0

Two orthogonal responses are not confounded and the estimators of their coefficients in a model are not correlated.

Vector orthogonality is affected by the coding of the levels, while factor orthogonality is not.

If the levels are coded with the complex roots of the unity the two notion of orthogonality are essentially equivalent

∗Fraction

3.2 Complex coding for full factorial designs

Pistone G. and Rogantin M. P. (2004)

We code the n levels of a factor A with the complex solutions of the equation ζⁿ = 1:

ω_k = exp



i 2π n k



for k = 0, . . . , n − 1 ^ω0 ω1

ω2

[k]_n the residue of k mod n; especially, for integer h, (ω_k)^h = ω_[hk]n

The mapping Z_{n ∋ k ↔ exp}^i ^2π_n k^ is a group isomorphism on the multi- plicative group of C.

Recoding is a polynomial of degree n in both directions.

∗Fraction

The full factorial design D, as a subset of C , is defined by the system of equations^∗

ζ_j^nj − 1 = 0 for j = 1, . . . , m

The set of all responses on the full design C(D) is a complex Hilbert space with Hermitian product

< f, g >= E_D(f g)

X^αX^β = X^[α−β], where [·] denotes the modulo operation extended to L.

The set of all the monomial response on the full factorial design:

{X^α, α ∈ L} L = {(α₁, . . . , α_m) : α_i < n_i, i = 1, . . . , m}

is a monomial basis of C(D). ^{In fact:}

1. E_D(X⁰) = 1, and E_D(X^α) = 0 for α 6= 0;

2. E_D(X^αX^β) = E_D(X^[α−β]) =

1 if α = β 0 if α 6= β

∗Full design

Example

Integer coding

















−1 −1 −1

−1 0 0

−1 1 1

0 −1 0

0 0 1

0 1 −1 1 −1 1 1 0 −1

1 1 0

















The fraction is defined by:











X1(X1 − 1) (X1 + 1) = 0 X2(X2 − 1) (X2 + 1) = 0 X3(X3 − 1) (X3 + 1) = 0

X1X2X3 − X1X2 + X1X3 + X2X3 + ¹₃X1 + ¹₃X2 − ¹₃X3 + ¹₃ = 0

Complex coding

















ω0 ω0 ω0

ω0 ω1 ω1

ω0 ω2 ω2

ω1 ω0 ω1

ω1 ω1 ω2

ω1 ω2 ω0

ω2 ω0 ω2

ω2 ω1 ω0

ω2 ω2 ω1

















The fraction is defined by:











ζ₁³ − 1 = 0 ζ₂³ − 1 = 0 ζ₃³ − 1 = 0

ζ₁² ζ₂² ζ3 − 1 = 0

∗Fraction

3.3 Indicator function F of a fraction

It is a response defined on the full factorial design D such that

F (ζ) =

1 if ζ ∈ F

0 if ζ ∈ D r _F

It is represented as the polynomial:

F = ^X

α∈L

b_α X^α

whose terms are orthonormal on the full factorial design.

The coefficients b_α satisfy the following properties:

• b_α = _N^{1 P}_ζ∈F X^α(ζ) and especially b₀ = ^#_N^F

• b_α = b_[−α] because F is real valued.

Important statistical features of the fraction can be read out from the form of the polynomial representation of the indicator function.

∗Fraction

Fractions with replicates and Counting function

A fraction with replicates Frep that we denote by Frep, can be consid- ered a multi-subset of a full factorial design D, that is a list of r points.

The counting function R is a response on the full factorial design showing the number of replicates of a point ζ.

The coefficients of the representation of R as:

R(ζ) = ^X

α∈L

c_α X^α(ζ) . are:

c_α = 1 N

X ζ∈Frep

X^α(ζ) and especially c_∅ = r N

∗Fraction

3.4 Results about orthogonality

1. A simple term or an interaction term X^α is a contrast on F if and only if c_α = c_[−α] = 0.

2. Two simple or interaction terms X^α and X^β are orthogonal on F if and only if c_[α−β] = c_[β−α] = 0;

3. If X^α is a contrast then, for any β and γ such that α = [β − γ] or α = [γ − β], X^β is orthogonal to X^γ.

∗Fraction

Some other results about orthogonality

(following from the structure of the roots of the unity as cyclical group)

Let X^α be a term with level set Ω_s on the full factorial design D.

1. If s is prime, then the term X^α is a contrast if and only if its s levels appear equally often

2. If the vector of replicates is a combination with positive weights of indicators of subgroups or laterals of subgroups of Ω_s, then X^α is a contrast.

∗Fraction

3.5 Generation of fractions with a given orthogonal structure

The coefficients of the indicator function of F are related according to:

b_α = ^X

β∈L

b_β b_[α−β] α ∈ L

Let O be the set of exponents corresponding to the given orthogonal structure

(e.g. all the simple terms mutually orthogonal: O corresponds to all interactions of order two)

The solutions of the system of:







b_α = ^P_β∈L b_β b_[α−β] ∀α ∈ L

b_α = 0 ∀α ∈ O

are the coefficients of the indicator functions of all the fractions with such a structure and then the points of such a fraction can be derived.

∗Fraction

We computed all the fraction with mutually orthogonal simple terms of:

• 2⁴ and 2⁵ designs

• 3⁴ and 2 × 3³ designs

Problems:

• classes of equivalence for permutations of factors and levels

• for multilevel case: symbolic softwares with operations on the complex roots of the unity are not available.

∗Fraction

3.6 Regular fractions

A fraction F is regular if:

• all the factors have the same number of levels n

• their defining equations are of the form

X^α = e(α) e(α) ∈ Ω_n, α ∈ H

where the set of exponents H can be completed to be a subgroup L of L.

In a regular fraction the interactions X^α e X^β are

either orthogonal or (totally) confounded

∀ α, β ∈ L .

∗Fraction

Indicator function and regular fractions (Pistone, Rogantin, 2004) The fraction F is regular with defining equations X^α = e(α), α ∈ L.

if and only if the indicator function of F is

F (ζ) = 1 l

X α∈L

e(α) X^α(ζ) ζ ∈ F

(all the coefficients are of the type ^e(α)_l ) Example

The indicator function of the fraction considered before is:

F = 1 3

1 + X₁²X₂²X₃ + X₁X₂X₃^2 We check that it is a regular fraction.

∗Fraction

PART IV:

Models on a fraction and term-orders

Holliday T., Pistone G., Riccomagno E. and Wynn H. (1999). The ap- plication of computational algebraic geometry to the analysis of designed experiments: a case study, Comput. Statist., 14.2, p.213–231

Bates R., Giglio B., Riccomagno E. and Wynn H. (1998). Gr¨obner basis methods in polynomial modelling. Improceeding of COMPSTAT 98, ed.

R. Payne p. 179–184

∗Fraction

4.1 Identifiable sub-models and initial orders

A strategy to find a parsimonious model (few terms well interpolating the response):

1. Fix the term order type (e.g. DegRevLex)

2. Choose an initial order for the factors and compute Est_τ 3. Repeat (2.) for all initial orders τ

4. Consider the sub-model given by E = ^T_τ Est_τ

(note that {1} ⊆ E)

5. Use statistical methods to choose parsimonious models from E

An example of 4-factor design with 23 points in Holliday, Pistone, Riccomagno, Wynn (1999)

∗Fraction

4.2 Model curvature

Basis of the vector space of the responses on F derived from Gr¨obner basis method:

Est_τ(F) =





 X β∈M

c_β X^β







The corresponding saturated identifiable model is Y = ^P_β∈M θˆ_β X^β and the coefficients are θ = Bˆ _τ⁻¹Y

Define the fitted polynomial model function f (x) = ^P_β∈M θˆ_β x^β Hessian matrix of f (x):

H_f(x) =

(∂²f (x)

∂x_i ∂x_j

)

∗Fraction

A measure of smoothness:

φ² = ^X

d∈F

trace ^H_f(d)^2 = ˆθ^′ Q ˆθ

where Q is a non-negative matrix depending only on B and F.

Choice of a good model:

vary the term-order τ and choose the corresponding model with smallest curvature

Notice that the curvature depend on the response values Y (through ˆθ).

An example: A 3-factor design with 16 points^∗

Y P B A

271.4 500 396 8 268.9 500 403 13 282.8 500 404 3 266.2 800 402 8 297.5 200 402 8 295.1 500 645 8 262.2 500 151 8 269.4 500 405 8 274.8 350 248 5 291.1 350 550 5 266.9 650 248 5 285.4 650 555 5 261.0 350 252 5 276.5 350 551 11 263.1 650 254 11 280.4 650 550 11

Two interpolators’ bases:

Est_Lex =

{1, A, A², A³, A⁴, B, BA, BA², B², B²A, B²A², B³, B³A², B⁴, P } Est_DegRevLex =

{1, A, B, P, A², BA, P A, B², P B, P², A³, BA², P A², B²A, P BA, B³}

Curvatures:

φ_Lex = 17374.9

φ_DegRevLex = 257.6

Reduced model via stepwise regression:

{1, P A², A³, AP }

∗Fraction

4.3 Block term-order and factor screening

Statistical and practical methods can give prior information on the rele- vance of the factors: e.g. A’s factors have less influence on the response than B’s factors

Let M (τ_A) and M (τ_B) be the matrices for the term-order corresponding to the A’s and B’s factors.

The block matrix:

"

M (τ_A) 0 0 M (τ_B)

#

represents the matrix for a term-order corresponding to the A and B’s factors, whit A’s factors less relevant than B’s