Signal identification by local functional ANOVA

remi.mahmoud@institut-agro.fr
JDS 2025
Joint work with D. Causeur

Rémi Mahmoud

2025-06-05

Introduction

What today’s talk is about

Context
Functional ANOVA (fANOVA)
Local fANOVA
Simulation study
Application to real-world data
Discussion and perspectives

Functional ANOVA

Functional linear model

A key question in statistic…

Does X has an effect on Y / Is X linked to Y ?

Swiss Army knife of statistic \(\Rightarrow\) Linear Model

The functional linear model:

\((Y(t))_{t \in {\cal T}}\) functional response variable defined on a time domain \({\cal T}\)
\(x=(x_{1},\ldots,x_{p})'\) stand for a \(p\)-vector of time-independent explanatory variables.

\(t \in {\cal T}\), the following model is assumed:

\[\begin{aligned} Y ( t ) & = & \beta_{0} ( t ) + \beta_{1} (t) x_{1} + \ldots + \beta_{p} ( t ) x_{p} + \varepsilon ( t ) , \label{fANOVAmod} \end{aligned}\]

where:

\(\beta_{0}(t)\) pointwise intercept parameter at time \(t\)
\(\beta(t)=(\beta_{1}(t),\ldots,\beta_{p}(t))'\) \(p\)-vector of pointwise regression parameters at time \(t\).
\(\varepsilon(t) \sim \cal N(0,\sigma^2(t))\) & \(\rho(t,t') = cor(\varepsilon(t),\varepsilon(t'))\) (potential time dependance)

Framework for today’s presentation

One-way design with a two-group covariate
Could be applied to more complicated frameworks

What we want to test

Is the mean function the same for each group ?

Pointwise null hypothesis: \(H_{0t}: \beta(t) = 0, \ \forall t \in \cal T\)
Global null hypothesis: \(H_0 = \{H_{0t}, \ t \in \cal T \}\)

A decomposition to take time dependance into account

A proposal made by Sheu et al. 2016 ¹ and used in Causeur et al. 2020 ²

Idea: Decorrelate pointwise test statistics and sum them across a given number of factor
How: decompose the time-correlation matrix R into a q-factor model \(\Sigma = \Lambda \Lambda' + \Psi\)
- \(\Psi\) diagonal \(T \times T\) matrix
- \(\Lambda \text{ is a } T \times q\) matrix containg factor loadings (among of “shared variance”)
Greater power

How to gain accuracy ?

An example of significantly different curves:

We conclude in a difference between mean curves for each conditions…
But another question arises: where does this difference come from ?
Need to find a more accurate approach to conclude

Another point of view: multiple comparison procedure

Compute pointwise F-Statistics (\(F_t\))
Compute p-values and apply a correction (Bonferonni / Benjamini Hochberg for instance)

Some problems with these approaches:
1. May be too conservative
2. May be tricked by spurious time points with higher differences
3. Functional nature of the signal (ex. time dependance) not taken into account

Local FANOVA

A compromise between multiple testing procedures and fANOVA

Idea: for a given set of curves, find the largest intervals at which no significant effect can be detected with the functional ANOVA test

In a nutshell

Screen the whole time frame partitionned into a given number of intervals, and incrementally find the largest union of intervals not significant.

Inputs:

k = window size
m = number of intervals to consider

Local FANOVA: How it works

For each interval i in 1 to m:

Define a time frame centered on interval i with a window size k
Perform a fANOVA on these time frames

Local FANOVA: How it works

For each interval i in 1 to m:

Define a time frame centered on interval i with a window size k
Perform a fANOVA on these time frames

Remove all intervals from the time frame that are significant.

Local FANOVA: How it works

For each interval i in 1 to m:

Define a time frame centered on interval i with a window size k
Perform a fANOVA on these time frames

Remove all intervals from the time frame that are significant.
While the Fanova on the remaining whole time frame is not significant:

Incrementally add intervals to the time frame (decreasing p-values)
Perform a fANOVA on this time frame

Local FANOVA: How it works

For each interval i in 1 to m:

Define a time frame centered on interval i with a window size k
Perform a fANOVA on these time frames

Remove all intervals from the time frame that are significant.
While the Fanova on the remaining whole time frame is not significant:

Incrementally add intervals to the time frame (decreasing p-values)
Perform a fANOVA on this time frame

Local FANOVA: How it works

For each interval i in 1 to m:

Define a time frame centered on interval i with a window size k
Perform a fANOVA on these time frames

Remove all intervals from the time frame that are significant.
While the Fanova on the remaining whole time frame is not significant:

Incrementally add intervals to the time frame (decreasing p-values)
Perform a fANOVA on this time frame

Local FANOVA: How it works

For each interval i in 1 to m:

Define a time frame centered on interval i with a window size k
Perform a fANOVA on these time frames

Remove all intervals from the time frame that are significant.
While the Fanova on the remaining whole time frame is not significant:

Incrementally add intervals to the time frame (decreasing p-values)
Perform a fANOVA on this time frame

Local FANOVA: How it works

For each interval i in 1 to m:

Define a time frame centered on interval i with a window size k
Perform a fANOVA on these time frames

Remove all intervals from the time frame that are significant.
While the Fanova on the remaining whole time frame is not significant:

Incrementally add intervals to the time frame (decreasing p-values)
Perform a fANOVA on this time frame

Simulations

Datasets generated

Underlying observations generated by the model: \[Y_{ij} = z_i^T \beta_j + \varepsilon_{ij}\]

with:

\(Y_{ij}\) observation of curve \(i\) at time \(t_j\), with \(i = 1, ...,n\) and \(j = 1, ...,T\)
\(\beta_j = \beta_i(t_j)\)
\(\varepsilon_{ij} = \varepsilon_i(t_j)\)
\(\cal{T} = [0,1]\)

Parameters:

Temporal resolution \(T \in \{50,500,2000\}\)
Number of individuals \(n \in \{100,1000\}\)
Temporal dependancy of the residuals \(\varepsilon(t) \sim AR(1), \ \rho \in \{0,0.3,0.8\}\)

100 datasets for each parameter combination

Values of \(\beta(t)\)

\[ \beta(t) = \begin{cases} \mathcal{GP}(m(t), K(t,t')) & \text{if } t \in ]0.3 ; 0.4] \ \cup \ ]0.7 ; 1] \\ 0 & \text{else} \end{cases} \]

with covariance function \(K(t,t') = 0.1*e^{-0.3 |t - t'|}\) and

\[ m(t) = \mathbb{1}_{t \in ]0.3, 0.4] } +0.5 \times \mathbb{1}_{t \in ]0.7, 1] }\]

Alternatives methods compared

Lasso regression (Tibshirani):

\(\hat{\beta} = \underset{\beta}{\text{argmin}}(\sum_1^n (y_i - \sum_1^p\beta_j x_{ij})^2 + \lambda \sum_1^p |\beta_k|)\)

Multiple testing procedure (BH correction)

Interval wise testing procedure (Pini et al. 2018¹)

Simulation results

The moment that every statistician waits for / fears

Real world data

Context

Fusarium head blight: Fungal disease \(\Rightarrow\) lot of damages (yield / food safety / added value)
Mycotoxins emitted by Fusarium (ex. Nivalénol (NIV))

How is the overtaking of a given toxin related to a given climate variable ?

Dataset (Arvalis)

\(\approx 1500\) farms: climate timeseries and toxins concentrations
Different agronomic practices
Binarization of the concentrations: above/below of a legal threshold

Example with 320 farms sharing the same agronomic practices

Application

Application of local fANOVA in our case:

\(Y\) is a time serie of a climate variable (ex. water excess) and \(x_1 = \mathbb{1}_{\text{NIV > legal threshold}}\)

Discussion and perspectives

On-going thoughts on the choice of hyperparameters

In our case, two key parameters:
- k = window size
- m = number of intervals to consider

Choice of hyperparameters, different points of view:
- statistical: find the best combination of hyperparameters that optimize a given metric
- practical: find the best compromise between “good results” and acceptable computational costs
- field-related: when possible, set hyperparameters that have a mean

Perspectives and conclusion

Development version available on Github/RemiMahmoud/TrustMe

devtools::install_github("RemiMahmoud/localFaov")

This is an ongoing work

Promising but contrasting simulation results
Convincing application on some real world cases
Some issues still need to be fixed (choice of hyperparameters, situations of bad results etc.).

Functional data

Observations represented by curves or functions
Old but new: original studies by Grenander & Rao (1948 / 1952 resp) but huge works from Ramsay and Silverman 2005¹
Similarly to NN, has arosen in the last years because of higher data availability and computing abilities

Challenges:
- temporal dependency
- specific questions (adaptation of classical methods to FD etc.)

Metrics: how good / bad are the results

Recall of the goal (in our case):

Detect curve segments responsible for a significant difference between modalities
Need for a metric designed for this goal

Let \((I_i)_{i = 1,\cdots,m}\) the collection of intervals linked to this difference
(Reminder) In our simulation study, \(m = 2, \ I_1 = ]0.3,0.4] \text{ and } I_2 = ]0.7,1]\)

Metrics used

Mean Overlap: \(\text{Mean Overlap} = \frac{1}{m} \sum_{i=1}^{m} \frac{\text{# points selected in } I_i}{|I_i|}\)
\(\text{Sensitivity} = \frac{\text{TP}}{\text{TP + FN}}\)
\(F1 = \frac{2 \times \text{Mean Overlap} \times \text{Sensitivity}}{\text{Mean Overlap} + \text{Sensitivity}}\)
Other metrics exist (mean distance to closest interval, FDR etc.)

What we want to test

Is the mean function the same for each group ?

Pointwise null hypothesis: \(H_{0t}: \beta(t) = 0, \ \forall t \in \cal T\)
Global null hypothesis: \(H_0 = \{H_{0t}, \ t \in \cal T \}\)

A common approach:

Compute pointwise F-Statistics (\(F_t\), Ramsay and Silverman 2005)
Aggregate the F-Statistics (\(F_t\))
- \(\int\) (Zhang and Liang 2014¹)
- \(F_{\max} = \sup_{t\in \mathcal{T}} \mathcal{F}(t)\) (Zhang et al. 2019²)
Distribution of F under \(H_0\) known or derived by permutation

But generally time dependance not taken into account (Shen et al. 2016) \(\Rightarrow\) Increase risk of type-I error !