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Anticipated and adaptive prediction in functional discriminant analysis

Cristian PREDA

Polytech’Lille, USTL Lille, France

Gilbert SAPORTA

CNAM Paris, France

Mohamed Hedi BEN MBAREK

Institut Sup´ erieur de Gestion, Sousse, Tunisia

Anticipated and adaptive prediction in functional discriminant analysis – p.1/21

The context

Cookie’s quality at Danone and the kneading process A cookie from Danone :

• choose a type of flour (components, density, etc)
• kneading process (≈ 1h)
• Put the dough in form and cook it
• Evaluate quality of the obtained cookies

   > 2h. Idea : predict the cookie’s quality from elements derived from the kneading process : Danone gets time and money !

Anticipated and adaptive prediction in functional discriminant analysis – p.2/21

Dough resistance during the kneading process : X = X(t), t ∈ {0, 2, 4, . . . 480s}

100 200 300 400 200 300 400 500 time X(t)

Anticipated and adaptive prediction in functional discriminant analysis – p.3/21

Flour evaluation The quality of cookies obtained with some type of flour is given by experts. The response (Y ) is : this type of flour is Good or Bad. 90 flours were evaluated : 50 are good and 40 are bad.

Anticipated and adaptive prediction in functional discriminant analysis – p.4/21

90 flours : quality and dough resistance during 480s.

100 200 300 400 100 300 500 700 time X(t)

Good(black) Bad(red)

Anticipated and adaptive prediction in functional discriminant analysis – p.5/21

Functional discriminant analysis

X = {Xt}t∈[0,T], Xt : Ω → R,

• E(X2

t ) < ∞,

• L2–continuous,
• ∀ω ∈ Ω : (Xt(ω))t∈[0,T] ∈ L2([0, T]),
• E(Xt) = 0, ∀t ∈ [0, T].

Y : Ω → {0, 1}. Discriminant score : dT = Φ(X)

Anticipated and adaptive prediction in functional discriminant analysis – p.6/21

Discriminant score estimation {(X1, Y1), (X2, Y2), . . . , (Xn, Yn)}

• Linear discriminant score :

Φ(X) = β, XL2[0,T], β ∈ L2[0, T]. Criterion (Fisher): max

β∈L2[0,T]

V(E(Φ(X)|Y )) V(Φ(X)) Estimation by functional linear regression model (PCR, PLS, etc).

Anticipated and adaptive prediction in functional discriminant analysis – p.7/21

• Nonparametric estimation

ˆ Φ(X) = n

i=1 YiK(u(X, Xi)/h)

n

i=1 K(u(X, Xi)/h)

• RKHS approximation

ˆ Φ(X) =

n

• i

αiK(Xi, X) Criterion (logistic loss) : L(x, y, Φ) = −yΦ(x) + log(1 + eΦ(x))

Anticipated and adaptive prediction in functional discriminant analysis – p.8/21

Kneading data results

100 200 300 400 100 300 500 700 time X(t) 100 200 300 400 100 300 500 700 time X(t)

Good (black) and bad (red) flours. Left : original data. Right : smoothed data Model PLS_FLDA NP PC_FLDA Gaussian(6) LDA Error rate 0.112 0.103 0.142 0.108 0.154 Error rate averaged over 100 test samples.

Anticipated and adaptive prediction in functional discriminant analysis – p.9/21

Anticipated prediction

X is observed on [0, T] Problem : find the smallest T ∗, T ∗ < T, such that the prediction of Y by X observed on [0, T ∗] is "similar" to the prediction obtained with X observed on [0, T].

Discrimination power : ROC curve – d : the discriminant score. – threshold r : Y = 1 if d > r. – "Sensitivity" : P(d > r|Y = 1) – "Specificity" : P(d > r|Y = 0). Measure of discrimination : area under the ROC curve (AUC)

Anticipated and adaptive prediction in functional discriminant analysis – p.10/21

– Estimation of AUC {Y = 1} : X1 = d|Y =1, sample of size n1 {Y = 0} : X0 = d|Y =0, sample of size n2

• AUC = #{X1 > X0}

n1n2 D = {dt}0<t≤T, { AUC(t)}0<t≤T Criterion for T ∗ : compare AUC(t) and AUC(T) for t < T and chose T ∗ as the largest t such that the test H0 : AUC(t) = AUC(T), H1 : AUC(t) < AUC(T) is significant (p-value < 0.05).

Anticipated and adaptive prediction in functional discriminant analysis – p.11/21

Simulation

Class {Y = 0} : Xt = 8 < : W(1 − t), 0 ≤ t ≤ 1 −2 sin(t − 1) + W(t − 1), 1 < t ≤ 2 Class {Y = 1} : Xt = 8 < : W(1 − t), 0 ≤ t ≤ 1 2 sin(t − 1) + W(t − 1), 1 < t ≤ 2

0.0 0.2 0.4 0.6 0.8 1.0 1.2 1.4 1.6 1.8 2.0

• 5
• 4
• 3
• 2
• 1

1 2 3 4 5 Y=1 (black), Y=0 (blue) t X(t)

Figure : Sample of size n = 100 for each class of Y .

Anticipated and adaptive prediction in functional discriminant analysis – p.12/21

M = 50 learning and test samples of size 100. For each t ∈ {0, 2.00, 1.98, . . . , 0} : sample of size M = 50 of

• AUC(t).

– T ∗ = 1.46. – test statistic : S = 1.663

• AUC(t∗) = 0.856,

AUC(T) = 0.872

Anticipated and adaptive prediction in functional discriminant analysis – p.13/21

Kneading data Y ∈ {Bad, Good}. The sample of 90 flours is randomly divided into a learning sample of size 60 and a test sample of size 30. Error test rate (PLS estimation) T = 480 : 0.112, – AUC(T) = 0.746. Anticipated prediction : T ∗ = 186. Error test rate (PLS estimation) T ∗ = 186 : 0.121, – AUC(T ∗) = 0.778. Conclusion : the predictive power of the dough curves for the cookies quality is resumed by the first 186 seconds of the kneading process.

Anticipated and adaptive prediction in functional discriminant analysis – p.14/21

Adaptive prediction

Remark : in anticipated prediction T ∗ is a constant. Let ω ∈ Ω be a new observation for which one wants to predict Y from X. Suppose that X is observed in a sequential way. The problem addressed by the adaptive prediction is : Problem : find the smallest T ∗(ω), T ∗(ω) < T, such that X

• bserved on [0, T ∗(ω)] provides similar prediction as it is
• bserved on [0, T].

Remark :

• here T ∗ is a random variable.
• to observe X(ω) on [T ∗, T] will not change the prediction for

Y (ω) obtained with X on [0, T ∗(ω)].

Anticipated and adaptive prediction in functional discriminant analysis – p.15/21

Conservative index for prediction :

T

1 n

X

x x

t dt [ ]

The discriminant score dt.

Anticipated and adaptive prediction in functional discriminant analysis – p.16/21

Denote by Ωω(t) = {ωi ∈ Ω|ˆ Yt(ω) = ˆ Yt,i} and Ωω(t) = Ω − Ωω(t) the class of elements having the same prediction as ω, respectively its complement with respect to Ω.

T t

ω W W Wω

ω

(t) (t) d d t T p p p p 1 1 Y=0 Y=1 Y=0 Y=1 ^ ^ ^ ^

Conservation rate of the prediction for ω and t.

Anticipated and adaptive prediction in functional discriminant analysis – p.17/21

Let p0|Ωω(t) =

• {ω′ ∈ Ω|ˆ

YT(ω′) = 0} ∩ Ωω(t)}

• |Ωω(t)|

. be the observed rate of elements in Ωω(t) predicted in the class Y = 0 at the time T using the score dT. Similarly, let p1|Ωω(t), p0|Ωω(t)and p1|Ωω(t) Let define by CΩω(t) = max{p0|Ωω(t), p1|Ωω(t)}, respectively by CΩω(t) = max{p0|Ωω(t), p1|Ωω(t)} the conservation rate of the prediction at the time t with respect to the time T for the elements of Ωω(t), respectively of Ωω(t). As a global measure of conservation we consider CΩ(ω, t) = min{CΩω(t), CΩω(t)}.

Anticipated and adaptive prediction in functional discriminant analysis – p.18/21

Remark : For each t ∈ [0, T], CΩ(ω, t) is such that 0.5 ≤ CΩ(ω, t) ≤ 1 and CΩ(ω, T) = 1. Given a confidence conservation threshold γ ∈ (0, 1), e.g. γ = 0.90, we define the following adaptive prediction rule for ω and t :

(1) if CΩ(ω, t) ≥ γ then the observation of X for ω on the time interval [0, t] is sufficient for the prediction of Y (ω). ˆ Y (ω) is then the same as the prediction at time T of the subgroup of Ωω(t) corresponding to CΩω(t). (2) if CΩ(ω, t) < γ then the observation process of X for ω should continue after t. Put t = t + h and repeat the adaptive prediction procedure.

Then, T ∗(ω) is the smallest t such that the condition (1) of the adaptive prediction rule is satisfied.

Anticipated and adaptive prediction in functional discriminant analysis – p.19/21

100 200 300 400 200 300 400 500 time X(t) 100 200 300 400 0.70 0.75 0.80 0.85 0.90 0.95 1.00 time C(w,t)

Left : new flour ω. Right : CΩ(ω, t), t ∈ [100, 480], γ = 0.90.

T ∗(ω) = 220.

Anticipated and adaptive prediction in functional discriminant analysis – p.20/21

For 25 new flours, the adaptive procedure is applied.

150 200 250 300 0.0 0.2 0.4 0.6 0.8 1.0 time Fn(x)

Empirical cumulative distribution function of T ∗ (in red, the time point t=186).

Anticipated and adaptive prediction in functional discriminant analysis – p.21/21