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Synthetic Data Augmentation for Functional Data: Wasserstein Risk Bounds and Diffusion-Based Conditional Generation

Yuhao Qi, Wenlin Dai, Xuehu Zhu

arXiv:2610.34120v1stat.MEcs.LGstat.ML

TL;DR

Functional samples are costly and often small, while existing synthetic-data theory uses total variation, which is uninformative for curves. This paper develops a class-conditional functional diffusion generator and Wasserstein-1 guarantees; applications show gains across regression and classification, with the largest imbalanced-class gain from minority-targeted generation.

  • Problem

    Functional samples are often small, and existing synthetic-data guarantees rely on total variation, which is uninformative for function-space distributions.

    Source: §1 · p. 2

  • Method

    The paper uses a class-conditional functional diffusion generator and derives Wasserstein-1 risk and allocation guarantees for synthetic augmentation.

    Source: §1 · p. 3

  • Results

    Across three real datasets, augmentation improved function-to-scalar regression and minority-targeted classification, with consistent gains for penalized function-to-function regression.

    Source: §1 · p. 3, §5 · p. 21

  • Takeaways & Limitations

    Under class imbalance, directing synthetic curves to the minority class gave the largest gain.

    Source: §5 · p. 21

    Future work is needed to derive a statistical rate for conditional score matching and a data-driven allocation rule with guarantees.

    Source: §5 · p. 21

Abstract

from arXiv · show

Functional data are costly to collect; samples of tens to hundreds of curves are common. Synthetic data augmentation can help, but its theory is developed for finite-dimensional data and relies on total variation, which is uninformative for functions. We study augmentation with a class-conditional functional diffusion generator and state guarantees in the Wasserstein-1 distance. For stable downstream estimators, we bound the risk of training on synthetic curves, from above and below, by the risk of training on as many real curves plus a generation penalty, and derive the best number of synthetic curves and a condition under which synthetic data outperform real data. For allocating synthetic curves under class imbalance, we bound the excess risk of training on real and synthetic curves together, without sample splitting. In applications to temperature, hyperspectral and gait curves, augmentation improved classification, function-to-scalar regression and function-to-function regression, and directing synthetic curves to the minority class gave the largest gain under class imbalance.

1 Introduction

Functional samples are often small, while existing synthetic-data guarantees rely on total variation, which is uninformative for curves. This paper develops a conditional functional diffusion generator and Wasserstein-1 risk and allocation theory, then evaluates augmentation on three real datasets.

1 Introduction

  • Functional samples are often small, and prior synthetic-data bounds use total variation, which can be uninformative for function-space distributions.Prior conditional-generation guarantees also use Euclidean latent diffusion and split real data between generator training and downstream fitting.Source: §1 · p. 2
  • The method conditions a functional diffusion generator on class labels or continuous responses, using FiLM to preserve curve structure while generating curves.The score network is trained from scratch on the real sample.Source: §1 · p. 3
  • The Wasserstein-1 volume theory bounds synthetic-training risk above and below by real-training risk plus a generation penalty that does not grow with synthetic volume for stable estimators.It also derives an optimal synthetic volume and a sufficient condition for synthetic data to match or outperform real data.Source: §1 · p. 3
  • The allocation theory bounds excess risk for mixed real-and-synthetic training without sample splitting and gives explicit imbalanced-class rates involving label imbalance and per-class generation error.The bound accounts for approximation, estimation, domain-adaptation, allocation-mismatch, and generation-error terms.Source: §1 · p. 3
  • Across three real datasets, augmentation improved function-to-scalar regression and minority-targeted classification, with consistent gains for penalized function-to-function regression.The applications cover daily temperature, grape hyperspectral reflectance, and gait biomechanics, across classification and both regression settings.Source: §1 · p. 3

2 Methodology

The methodology defines a conditional functional diffusion generator and a pipeline for using its curves in downstream tasks. It specifies functional-space noise, conditioning, data allocation, and evaluation procedures.

2.1 Functional Data Diffusion Models

  • The DDO uses Gaussian-process noise with trace-class covariance, keeping perturbed curves in the function space H.The covariance is selected from a powered-exponential kernel whose parameters control noise amplitude, roughness, and correlation length.Source: §2.1 · p. 4

2.2 Conditional Generation via Feature-wise Linear Modulation

  • FiLM conditions the diffusion network on labels or responses while preserving the functional structure of each curve.Unlike appending a label to a discretized curve, FiLM modulates residual blocks across grid positions.Source: §2.2 · p. 5

2.2.1 FiLM Conditioning

  • For continuous responses, a value-augmented one-hot vector encodes group by position and response by the nonzero value.The response is min–max normalized using training-data extrema, making the embedding vary smoothly within each group.Source: §2.2.1 · p. 5

2.2.2 Network Architecture, Training and Sampling

  • The U-shaped Fourier neural operator is condition-modulated in every residual block to generate curves conditional on Y, and accepts curves of varying lengths.Curves are padded to a required resolution and outputs cropped back to their original grid; training uses standardized curves and group-balanced minibatches.Source: §2.2.2 · p. 6

2.3.1 The Generation-Augmentation Pipeline

  • The Conditional DDO is trained once, generates curves under training-data conditions, and supports real-only, augmented, or synthetic-only downstream fits evaluated on real test data.Generation-quality checks use MMD, fWass, and fFID, which are described as agreement checks rather than estimates of the theorem's generation error.Source: Figure 1 · p. 6, §2.3.1 · p. 7

2.3.2 Generational Effect Analysis

  • Synthetic-volume trajectories pool G=1,…,20 sets of n curves, with five replicates per volume sharing no synthetic curves.Trajectories are smoothed with penalized B-splines and shown with pointwise 95% confidence bands.Source: §2.3.2 · p. 7

2.3.3 Allocation Design

  • For class allocation, the design fixes synthetic volume m and adds the nearest integer to a_km curves from each class k to the real training data.This varies class shares without changing the total synthetic volume.Source: §2.3.3 · p. 7

2.4 Downstream Models

  • Downstream models are evaluated with classification metrics or regression metrics, while the training set is the only deployment component changed by augmentation.Classification uses accuracy, per-class precision, recall, F1, and AUC; regression uses RMSE and R2, including a grid-point-summed functional R2 for function-to-function regression.Source: §2.4 · p. 8

3 Theoretical Results

The theory uses W1 to measure functional generation error, then bounds synthetic-only and mixed real–synthetic risks under stability, generation, and allocation conditions. It characterizes synthetic-volume selection and class allocation, including a guarantee that does not require sample splitting.

3.1 Setting and Choice of Metric

  • TV remains 1 for distinct scaled Gaussian laws in infinite-dimensional function space, whereas W1 tends continuously to zero as the scale approaches equality.The paper therefore measures generation error with W1.Source: §3.1 · pp. 8–9

3.2 Training on Synthetic Data: How Many Curves

  • Under the theorem assumptions, some volume m0 < m* attains minimum synthetic risk, while every m ≥ m* has strictly higher risk; m0 is the reflection point.The finite-optimum condition requires that the generational gap stop shrinking beyond m*; the theorem states its assumptions separately.Source: §3.2 · p. 10
  • For stable estimators, R_syn(m) ≤ Cθm^-α + Lw and R*_syn ≤ Lw; w ≤ (Cθ/L)n^-α suffices for best synthetic risk to match raw-data risk.The transfer bound is two-sided, and stability reduces the generation-error threshold exponent from 1 + α to α.Source: §3.2 · p. 11
  • For synthetic-only training, the two-sided transfer bound limits both loss and gain relative to using the same number of real curves by Lm^(1-κ)w.The direct gap bound is |Gr(m)| ≤ LmW1(eF, F); κ controls how the risk sensitivity decays with sample size.Source: §3.2 · pp. 10–11

3.3 Augmentation: Allocation of Synthetic Curves

  • The mixed estimator’s excess risk has exponential-tail control by the largest of approximation, estimation, domain-adaptation, generation, and allocation-mismatch terms.The bound also includes a tail term for the event that some class-conditional generator misses its rate.Source: §3.3 · p. 13
  • The mixed-data guarantee requires no sample splitting: the generator may use the same real curves that enter the downstream fit.The theorem combines separately bounded real and synthetic blocks with a union bound.Source: §3.3 · p. 13

3.4 Imbalanced Classification

  • For imbalanced classification, allocation trades domain-adaptation against allocation-mismatch costs, while γ ≥ αcls limits generation inflation to at most max{1, Kn}.The certified level is set by the raw-data rate, imbalance cost, and generation term.Source: §3.3 · p. 12, §3.4 · p. 14

4 Real-Data Analyses

The real-data analyses cover temperature, grape reflectance, and gait curves across classification and two regression tasks. The datasets vary in their sampling and conditioning setups.

4 Real-Data Analyses

  • The analyses use three functional datasets from climatology, plant science, and biomechanics to study classification, function-to-scalar regression, and function-to-function regression.The section also asks whether augmentation improves small-sample performance, can replace real training data, and how synthetic curves should be allocated under class imbalance.Source: §4 · p. 14

4.1 Datasets

  • Spanish daily mean temperature curves from 1974–2013 support Madrid-versus-Valladolid classification, with 178 Madrid and 54 Valladolid training curves in a chronological split.Training covers 1974–2001, validation 2002–2007, and testing 2008–2013.Source: §4.1 · pp. 14–15
  • Grape data comprise 274 reflectance spectra spanning 397–1004 nm, paired with sugar measurements and split 60/20/20 stratified by variety and sugar-content tercile.The three varieties are Fer, Mauzac, and Syrah.Source: §4.1 · p. 15
  • The gait dataset analyzes ankle curves for slow and moderate groups, with four channels generated jointly and whole participants assigned to training (25) or test (6).The analyzed axes are vertical for the slow group and anterior–posterior for the moderate group; curves cover loads of 0%, 10%, and 20% of body weight.Source: §4.1 · p. 15

Appendix D.1 gives further details of the three datasets.

Across temperature, grape and gait data, synthetic functional curves improved downstream prediction or classification, with minority-directed allocation producing the strongest class-imbalance gains. The experiments also assess curve fidelity and describe the function-space diffusion model and broader theoretical contribution.

4.3.2 Generation Quality

  • Synthetic city curves closely followed real means, seasonal amplitude and phase, with similar spread; their fWass distances were 6% higher for Madrid and 13% higher for Valladolid than same-city references.The protocol also compares within-class fidelity and cross-class confusion using fWass, with additional metrics in the appendices.Source: §4.2 · p. 15, §4.3.2 · p. 16, Figure 2 · p. 16

4.3.3 Augmentation and Replacement across Synthetic Volume

  • Both augmentation and replacement beat the real-only baseline at every volume but peaked at G = 4, where AUC was 0.756 ± 0.015 and 0.765 ± 0.016, respectively.Performance then declined and plateaued, matching the rise-and-decline pattern described by Theorem 3.1.Source: §4.3.3 · pp. 16–17, Figure 3 · p. 17

4.3.4 Allocation under Class Imbalance

  • Valladolid precision rose from 0.40 to about 0.57, while Madrid recall rose from 0.62 to 0.87; Madrid F1 exceeded its real-only value at every allocation.The majority-class performance was largely maintained, while Valladolid recall declined as its precision improved.Source: §4.3.4 · p. 17, Figure 4 · p. 18
  • AUC reached 0.824 ± 0.007 at m/n = 2.5 when all synthetic curves went to Valladolid, 0.19 above the real-only classifier.Directing synthetic curves to the minority class, with 54 training curves, gave the largest gain.Source: §4.3.4 · pp. 17, 19, Figure 4 · p. 18

4.4.2 Augmentation and Replacement across Synthetic Volume

  • For grape sugar prediction, augmentation raised R2 by G = 4 from 0.658 to 0.747 for M1 and from 0.759 to 0.826 for M2, then remained flat.Because synthetic responses copy training responses, the gain comes from more spectra per response value, stabilizing the REML coefficient-function estimate; it persists with variety in M2.Source: §4.4.2 · p. 19, §4.4.1 · p. 19, Figure 5 · p. 20

4.5.2 Augmentation and Replacement across Synthetic Volume

  • For gait function-to-function regression, augmentation improved both models on both problems by 1.4–6.3% in RMSE and 0.02–0.10 in functional R2; replacement matched augmentation at the best volume.PFFR improved at every volume, while FDboost’s ankle-VT RMSE gain faded by about G = 15–20.Source: §4.5.1 · p. 19, §4.5.2 · p. 20, Figure 6 · p. 21

5 Discussion

  • The paper develops a same-sample conditional diffusion augmentation pipeline and Wasserstein-1 volume and allocation bounds; future work includes deriving conditional score-matching rates and quadratic-penalty theory.Across three datasets, synthetic curves improved on real-only training, replaced real data at moderate volumes, and gave their largest imbalance gains when sent to the minority class.Source: §5 · pp. 20–21

A.1 Diffusion in Function Space

  • The function-space diffusion model uses trace-class Gaussian noise, making the reference measure well defined on H and the noise’s expected squared norm finite.The powered-exponential kernel parameters control noise amplitude, path roughness and length scale.Source: §A.1 · p. 27

A.2 Network Architecture

The architecture combines spectral convolutions in a U-shaped network with condition-aware residual blocks. Its training and sampling procedures generate conditioned curves, while implementation choices handle differing grid lengths and multichannel data.

A.2 Network Architecture

  • The U-shaped network uses spectral convolutions, encoder-decoder resolution changes and skip connections, with a conditional spectral residual block and a final pointwise output convolution.The reported configurations use four resolution levels, with two residual blocks for temperature and spectra and three for gait.Source: §A.2 · pp. 28–29

A.3 Training and Sampling Algorithms

  • The training algorithm standardizes curves, adds trace-class Gaussian noise at sampled timesteps, encodes labels or responses, and trains the network to predict the noise.Sampling starts from a Gaussian reference draw and applies reverse-diffusion steps to return synthetic curves on the original scale.Source: §A.2 · p. 29, §A.3 · p. 30

A.2 Network Architecture

  • Curves are padded at both ends to a grid length divisible by 2^(L−1), then outputs are cropped; multichannel gait curves are stacked and generated jointly.The training loss and generated curves use the original grid points only.Source: §A.2 · pp. 28–29

B.1 Software and Computing Environment

  • The generator and sampler use Python with PyTorch, downstream procedures and generation-quality metrics use R packages, and training runs used single NVIDIA A800 GPUs.The paper says code, configuration files and seeds will be released in a public repository.Source: §B.1 · p. 30, B Implementation Details · p. 30

B.2 Generator Configurations for the Real Data

The appendices document the real-data generator settings, data preparation and experimental replicates, alongside downstream procedures and supplementary theorem remarks. They also state a joint-satisfiability condition for assumptions used in Theorem 3.1.

B.2 Generator Configurations for the Real Data

  • Across real-data generators, training uses group-balanced minibatches, AdamW, gradient clipping and 1,000-step ancestral sampling; dataset-specific settings and preprocessing are documented separately.Standardization constants are computed using training data only, and the gait curves use four stacked channels.Source: §B.2 · p. 30, Table 5 · p. 31

B.3 Generation and Replicates

  • Generational analyses pool independently seeded synthetic sets into volumes with five disjoint replicates, while the temperature allocation sweep varies volume and Valladolid share across 20 replicates.Each generational analysis uses one trained generator; each replicate is a sampler run followed by a downstream fit.Source: §B.3 · p. 32

B.4 Downstream Procedures

  • The documented downstream procedures use FPCA and LDA for classification, penalized functional regression for sugar prediction, and FDboost or PFFR for gait regression.The appendix specifies fitting choices and evaluation metrics for each task.Source: §B.4 · p. 32

C Supplement to the Theoretical Results

  • The supplement organizes remarks and proofs for the theoretical results, including assumptions, estimation-error indices and dependence conditions for the allocation theorems.It identifies the nonshrinking component of generation error as zero for generators trained from scratch.Source: C Supplement to the Theoretical Results · p. 32

C.1.1 Remarks on Theorem 3.1

  • The assumptions of Theorem 3.1 are jointly satisfiable only when R*syn < U.Source: §C.1.1 · p. 33

C.1.2 Remarks on Theorem 3.2

Theorem 3.2’s remarks make its gain conditions and rate requirements explicit, while the allocation-theorem remarks explain how estimation error is handled across real and synthetic blocks. The supplement then records the proof framework.

C.1.2 Remarks on Theorem 3.2

  • A gain can be certified at large n only if the generator’s rate γ is at least eκ, or at least α when κ = 1.The probability-version condition requires the generation error bound to fall below the stated n-dependent threshold; the power-law raw-risk assumption gives explicit optimal-volume and gain-threshold formulas.Source: §C.1.2 · pp. 33–34

C.1.3 Remarks on Theorems 3.3 and 3.4

  • The allocation-theorem remarks treat synthetic and real estimation errors as separate blocks, with conditional analysis for synthetic draws and a union bound to combine the bounds.The generation error component that does not shrink with sample size is zero for generators trained from scratch.Source: §C.1.3 · pp. 34–35

C.2 Proofs

  • The supplement assigns outer probability to events not known to be measurable and uses its monotonicity and countable subadditivity.Source: §C.2 · p. 35

C.2.1 Auxiliary Results

The auxiliary results provide tools for bounding Wasserstein discrepancies between mixtures and for controlling risks and function classes. They also establish the risk comparison used to identify an optimal synthetic volume.

Mixture bounds

  • The mixture comparison separates class-conditional Wasserstein error from error in the class proportions.The corollary applies this decomposition to bound the aggregate discrepancy using classwise bounds and allocation error.Source: §C.2.1 · pp. 35–36

Lipschitz variance bound

  • For bounded Lipschitz losses, differences in variances are controlled by Wasserstein distance between the underlying distributions.The proof applies Kantorovich–Rubinstein duality to the squared loss and then compares the variance identities.Source: §C.2.1 · p. 36

Moment bound

  • A bounded random variable’s centered moments of order k are bounded using its variance and a factor of (2U)^(k−2).The proof establishes the needed factorial inequality for every integer k ≥ 2.Source: §C.2.1 · p. 36

Entropy bound

  • The bracketing entropy of pairwise loss differences is controlled by that of the original loss class.The stated bound uses brackets of half the width, and the associated entropy integral follows by substitution.Source: §C.2.1 · pp. 36–37

Optimal synthetic volume

  • Under the stated risk assumptions, synthetic training has a finite optimal volume, and volumes at or above the threshold cannot attain the minimum risk.The proof locates a minimizer below the threshold and bounds the generation penalty there using the Wasserstein discrepancy.Source: §C.2.1 · pp. 37–38

Penalty bounds

  • The optimal volume’s generation penalty is bounded by the estimator’s risk range, with the stated discrepancy-dependent bounds under the additional conditions.The proof uses the lower bound m*ψ(w) on the generation penalty and the risk upper bound U.Source: §C.2.1 · p. 38

C.2.3 Proof of Theorem 3.2, Lemma 3.1 and Remark C.6

The appendix proves Wasserstein-based risk bounds for synthetic training, gives conditions for matching the real-sample rate, and supplies proof details and dataset descriptions. Its arguments include a joint real-and-synthetic analysis that does not require unconditional independence between the real sample and synthetic draws.

C.2.3 Proof of Theorem 3.2, Lemma 3.1 and Remark C.6

  • The synthetic-training risk differs from the raw-data risk by a generation term bounded by min{L m^(1−κ)w, U}.Here w is the Wasserstein-1 distance between generated and true distributions; the bound follows from Lipschitz stability and bounded risk.Source: §C.2.3 · p. 38
  • When the generator discrepancy decays at a sufficient rate, the optimal synthetic risk is bounded by Cθ n^(−α) on the generation event.The appendix states that the condition holds eventually when the discrepancy rate exponent γ exceeds the required exponent e.Source: §C.2.3 · pp. 38–39

Notation

  • The joint-risk proof conditions on the real data for synthetic draws and does not require independence beyond their conditional law.The final proof step explicitly says no other independence between the real observations and synthetic draws is used.Source: Notation · pp. 40, 42

C.2.5 Proofs for Section 3.4

  • For Section 3.4, the proof reduces the binary allocation bounds to the class-proportion discrepancies and the stated approximation terms.The theorem application uses K = 2 and takes ελ to be ε.Source: §C.2.5 · p. 43

D Real-Data Analyses: Supplementary Details

  • The real-data appendix documents dataset construction, generation-quality metrics, comparisons, and dataset-specific results.It identifies these materials as supplementary details for the Section 4 analyses.Source: D Real-Data Analyses: Supplementary… · p. 43

D.1.1 Spanish Station Temperature

  • The temperature analysis uses daily station curves from 1974–2013 and a chronological split of 232 training, 50 validation, and 50 test curves.The two labels are Madrid and Valladolid, represented by seven and two stations, respectively.Source: §D.1.1 · p. 44

D.1.2 Grape Hyperspectral Spectra

  • The grape analysis uses spectra measured at 204 bands, interpolated to 104 wavelengths, with 165 training, 54 validation, and 55 test samples.The data are stratified by variety and sugar-content tercile, and variety supplies the group encoding.Source: §D.1.2 · p. 44

D.1.3 Gait Biomechanics

  • The gait analyses use 75 training and 18 test curves per problem, with four jointly generated channels and no held-out validation participants.The channels are not constrained to make velocity and acceleration derivatives of angle.Source: §D.1.3 · p. 44

D.2 Generation-Quality Metrics

The appendix defines three generation-quality metrics and explains how to interpret their comparisons. It distinguishes expected sampling discrepancy, within-class fidelity, and cross-class confusion, with an additional response-alignment check for grape spectra.

D.2 Generation-Quality Metrics

  • MMD uses an RBF kernel and the square root of its biased estimate, while fWass compares Gaussian approximations through means and covariances.The biased MMD statistic remains positive in finite samples even for identical laws; fWass’s terms are summed over grid points and channels.Source: §D.2 · p. 45
  • fWass depends on grid and curve units, so the appendix compares it only within a dataset.fFID applies the fWass formula to functional principal component scores fitted on real reference curves.Source: §D.2 · p. 45

D.3 Comparison Scheme

  • The comparison scheme treats real-versus-reference discrepancy as a sampling reference, then checks within-class fidelity and whether synthetic classes remain closer to their matching real class.The reference level is not a lower bound because synthetic curves may lie closer to a reference sample than another real sample.Source: §D.3 · p. 45
  • For grape spectra, overlaid mean and ±1 standard-deviation envelopes within sugar-content terciles check whether generation captures the spectrum–response relation.Misaligned envelopes can reveal a mismatch even when MMD and fWass appear acceptable.Source: §D.3 · p. 45

D.4 Temperature: Further Results

The temperature results report distinguishable synthetic city classes and compare selected synthetic volumes and allocation shares. The reported allocation-grid changes are modest, with the best small-volume share close to all-Madrid augmentation.

Generation quality

  • By MMD, Valladolid validation curves lie slightly closer to real than synthetic Madrid curves (0.477 against 0.510), while both cities are closer to their matching synthetic class.Madrid validation curves are closer to synthetic Madrid than Valladolid (5879 vs. 9840), and Valladolid validation curves are closer to synthetic Valladolid than Madrid (7021 vs. 7827).Source: §D.4 · pp. 45–46, Table 6 · p. 46

Generational design

  • Table 7 reports test accuracy and AUC at selected synthetic volumes, including the trajectory peak at G = 4 pooled sets of 232 curves.The endpoints are G = 1 and G = 20; results are means ± standard errors over five replicates with disjoint synthetic data.Source: §D.4 · p. 46, Table 7 · p. 46

Allocation

  • Across neighboring allocation shares, AUC decreases by at most 0.04 and increases by up to 0.06; adding only Madrid curves changes AUC by at most 0.026.At small volumes, share 0.9 is best rather than 1, but the difference is within one standard error.Source: §D.4 · p. 46

D.5 Grape: Further Results

Grape-spectrum results show variety-dependent fidelity: synthetic spectra are close to validation spectra within varieties and remain closest to their own variety across varieties. Additional analyses examine sugar-content conditioning and downstream regression across synthetic volumes.

Generation quality

  • For Fer, synthetic spectra are closer to validation spectra than real training spectra by fWass (0.8 against 1.1); for Mauzac, 4.1 against 6.4; for Syrah, they are farther (7.8 against 2.8).Values are summed over 104 wavelengths on the reflectance scale.Source: §D.5 · p. 46, Table 9 · p. 47

Class allocation

  • Table 8 reports test AUC over the full synthetic-volume and Valladolid-allocation grid, with real-only AUC 0.636.Rows encode synthetic volume relative to n; columns encode the share allocated to Valladolid.Source: Table 8 · p. 47

Generation quality

  • Synthetic Fer, Mauzac and Syrah spectra are closest to validation spectra of their own variety: 0.8 against at least 7.1, 4.1 against at least 470.8, and 7.8 against at least 8.8, respectively.Figure 7 pairs real validation spectra with synthetic spectra of closest sugar content; Figure 8 overlays real and synthetic spectra by sugar-content tercile within each variety.Source: §D.5 · p. 47, Figure 7 · p. 48, Figure 8 · p. 48
  • For Fer and Mauzac, synthetic spectra are closer than training spectra by fFID, and for Mauzac also by MMD; for Syrah, they are farther by both metrics.These metrics do not assess the relationship between spectrum and sugar content, which Figure 8 examines by tercile.Source: §D.5 · p. 47, Table 10 · p. 47

Generational design

  • The grape generational-design analysis reports that gain is essentially complete at volume G = 4.The passage identifies this as an endpoint of trajectories in Figure 5.Source: §D.5 · p. 48

Grape sugar-content regression

  • Table 11 reports test RMSE and R2 for penalized functional regression models using the spectrum alone or the spectrum plus variety, across synthetic volumes.The table summarizes five replicates with disjoint synthetic data; the real-only fit is deterministic.Source: Table 11 · p. 48

D.6 Gait: Further Results

Gait generation quality varies by problem and metric: ankle VT synthetic curves are as close as training curves by fWass, while ankle AP has weaker within-condition fidelity. Downstream moment-prediction results and performance across synthetic volumes are also reported.

Generation quality

  • For ankle VT, synthetic curves are as close to test curves as training curves by fWass (9.4–12.7 against 11.6–12.7); for ankle AP, synthetic 0% and 20% curves are 2.4 and 1.6 times as far as training curves.Source: §D.6 · p. 49
  • Across ankle VT load conditions, test curves are only weakly separated even in real data, and synthetic curves reproduce that weak separation.For example, 0% test curves are closer to 10% than 0% training curves (10.7 against 11.6).Source: §D.6 · p. 49
  • By MMD, synthetic curves are farther from test curves than training curves in five of six load-condition cases, including two ankle VT conditions where fWass finds them as close.Table 14 compares synthetic or real training curves with test curves of the same condition.Source: §D.6 · p. 49, Table 14 · p. 52

Gait moment prediction

  • Table 15 reports best-over-volume test RMSE and functional R2 for gait moment prediction under real-only, augmentation and replacement fits.The best augmented or replacement result is selected over G = 1–20 and averaged over five replicates; m = 75G.Source: §D.6 · p. 49, Table 15 · p. 52

Curve displays

  • Figure 9 displays ankle VT curves by channel and load condition, with condition-wise means ±1 SD above and individual curves below; Figure 10 presents ankle AP curves in the same layout.The horizontal axis is percentage of the gait cycle.Source: Figure 9 · p. 50, Figure 10 · p. 51

Performance against synthetic volume

  • Figure 11 tracks ankle AP moderate performance against synthetic volume; Table 15 gives the best replicate-mean performance across the 20 volumes.The table compares those results with real-only performance.Source: §D.6 · p. 49, Figure 11 · p. 53

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