Machine Learning in Materials Processing & Characterization
Unit 9: ML for Characterization Signals

Prof. Dr. Philipp Pelz

FAU Erlangen-Nürnberg

§1 · Signals & their physics

01. Unit 9 — The Signal Application & Domain Unit

This unit is not a methods unit. The methods were derived elsewhere; here we ask what the signal physics demands of them.

Method	Owned by (derived in)
PCA / SVD	MFML u02 · ML-PC u02
Clustering, autoencoders	MFML u05 · ML-PC u05
NMF	MFML u02 · ML-PC u05
t-SNE / UMAP, latent spaces	MFML u09
MAE / SSL (DINOv2, I-JEPA)	MFML u09 · ML-PC u09b

Important

We reference these methods. We do not re-derive them. Unit 9 is about background subtraction, calibration transfer, quantification, rotational ambiguity, operando streaming — the things the signal physics forces on you.

Learning outcomes. After 90 minutes you can:

1. Explain how the probe physics dictates the structure of XRD / EELS / EDX / XPS / Raman signals.
2. Build a defensible spectral preprocessing pipeline (background → calibrate/align → normalize).
3. Apply MCR-ALS and reason about rotational ambiguity.
4. Turn a latent/peak into a quantified concentration with an error bar (Cliff-Lorimer / ζ-factor).
5. Deploy SSL/MAE-pretrained spectral encoders and operando novelty detection.

Why this slide exists. The old version of this unit re-derived PCA and autoencoders that the students had just seen in two parallel tracks. That is wasted lecture time and it teaches students that the same idea has two unrelated names. Open by killing that confusion explicitly: the table is the contract for the next 90 minutes.

Say this out loud. “If at any point today you think ‘this is just PCA’ or ‘this is just an autoencoder’ — you are right, and that is the point. The method is a black box you already own. What you do not own yet is the judgement about what the EELS edge, the Bragg peak, or the bremsstrahlung background does to that black box. That judgement is today.”

Materials hook. Frame the stakes: a metallurgist does not want a latent vector, they want “this region is 4.2 ± 0.3 at.% Cr, Fe³⁺-rich, and there is an unindexed phase at the interface.” Every method we reference is only a means to that sentence.

Pre-empt the student question. “Will the exam test PCA math?” — No, that is MFML u02 / ML-PC u02 weight. This unit’s exam weight is the pipeline reasoning: which normalization, why a wrong background biases everything, what rotational ambiguity is, how a k-factor becomes a number with an uncertainty.

Forward link. Outcomes 2 and 4 are the spine of the exercise; outcome 5 gestures toward closed-loop / self-driving-microscope ideas (an outlook beyond this term’s units).

Pacing. 3 minutes max. The table is a reference the students will photograph; do not read it line by line.

02. Beyond Images: 1-D Signals

• Most of the course so far: spatial data (micrographs, microstructure maps).

• But many instruments produce 1-D spectral signals — intensity vs. energy / angle / wavenumber:
  • XRD — crystal structure via Bragg peaks
  • EELS — bonding and electronic structure (edges, fine structure)
  • EDX/EDS — elemental composition (characteristic X-ray lines)
  • XPS — surface chemistry and oxidation states (chemical shifts)
  • Raman — molecular / phonon fingerprints

• Modern instruments collect millions of spectra per experiment — every pixel of a STEM scan is a spectrum (spectrum imaging).

Note

A spectrum with \(N\) channels is a vector \(\mathbf{x} \in \mathbb{R}^N\). Every linear-algebra / ML tool applies directly — but each “dimension” is a physical energy channel, and that is what makes this unit different from generic vector ML.

Mechanism in one sentence. A spectrum is a histogram of counts versus a physical axis; the axis carries meaning (an eV, a 2θ, a cm⁻¹), so unlike a generic feature vector you cannot permute the dimensions.

Materials hook. Drive home scale: a single 4D-STEM-EDS map is routinely 512×512 pixels × 2048 channels ≈ half a billion numbers. Nobody hand-fits that. This is the reason ML enters — not because ML is fashionable, but because the data volume crossed a line around 2015.

Anti-pattern to flag now. Treating the energy axis as exchangeable features (e.g., shuffling channels before PCA “to be safe”). The ordering is the physics; convolutional and aligned methods exploit it, channel-permuting methods throw it away.

Forward link. “Spectra are vectors” is the bridge to §2: every preprocessing step is a transformation of that vector chosen to make the chemistry (not the dose or the drift) the dominant source of variance.

Pacing. Light slide, ~2 minutes. Kept material from the old deck — keep it brisk.

03. The Nature of Characterization Signals

• High dimensionality — 1024–4096 channels; each spectrum a point in \(\mathbb{R}^{N}\).

• Sparse peaks — most channels are background; the science lives in a handful of channels.

• Continuous backgrounds — bremsstrahlung (EDX), plasmon tails (EELS), fluorescence (Raman).

• Noise — Poisson (shot) noise dominates at low dose; Gaussian detector/readout noise adds on top.

• Variability — peak positions shift with composition; peak shapes change with bonding/oxidation.

• Low intrinsic dimensionality — a sample with \(P\) phases spans \(\approx P\) + a few directions, \(\ll N\).

Important

The signal-to-noise is Poisson: variance equals the mean. A peak with 100 counts has \(\pm 10\) noise; a background of 10 000 counts has \(\pm 100\). This is why dose, normalization, and background all couple — you cannot treat them independently.

Mechanism. Counting experiments obey Poisson statistics: \(\mathrm{Var}=\mathrm{mean}\). This single fact propagates through the entire unit — it is why total-count normalization changes the noise model, why background subtraction must precede variance-stabilizing transforms, and why MSE losses are technically wrong for raw counts (Poisson NLL is right).

Materials hook / quantitative anchor. EELS core-loss edge on a power-law plasmon tail: the edge might be 5% of the local intensity. At realistic dose the Poisson noise on the background can exceed the entire signal. That is the daily reality of valence-state mapping — not a corner case.

War story. A student “discovers” a new phase in PCA component 4. It is the detector’s per-channel gain pattern beating against Poisson noise. Diagnostic symptom: the “phase” has no spatial coherence and its spectrum is high-frequency hash with no physical edges. Always check spatial coherence and physical plausibility before celebrating a component.

Pre-empt the question. “Why not just acquire longer to beat the noise?” — Beam damage and instrument time. Doubling SNR needs 4× dose (Poisson), and the specimen is often destroyed first. This is exactly why denoising/SSL matter — but see slide 20 for the trap.

Forward link. Low intrinsic dimensionality is why PCA/AE/NMF work on spectra — referenced, not re-derived. Today’s consequence: the preprocessing pipeline (§2) must not inject artificial dimensions (a bad baseline adds a spurious component).

Pacing. ~3 minutes. The Poisson callout is exam-weight; say it twice.

04. Signal Formation: The Physics of the Probe

The signal’s structure is dictated by the physics of the probe — every method in §2–§4 must respect it.

• EELS — ionization edges (sawtooth/hydrogenic) on a steep plasmon / power-law background (\(\sim AE^{-r}\)); near-edge fine structure (ELNES) encodes valence.

• EDX — characteristic lines (Gaussian-ish, detector-broadened) on a bremsstrahlung continuum; absorption + fluorescence distort intensities.

• XRD — Bragg peaks at \(2\theta\) from \(d\)-spacings; instrumental + size/strain broadening (Caglioti, Scherrer); preferred orientation reweights intensities.

• XPS — core-level lines whose binding-energy shift (chemical shift, ~0.1–3 eV) encodes oxidation state; inelastic background (Shirley/Tougaard).

• Raman — sharp phonon lines on a broad, sample-dependent fluorescence background that can dwarf the signal.

Note

Different physics → different background model, different noise, different invariances. There is no universal preprocessing. The pipeline must be chosen per modality.

[FIGURE: five mini-panels, one per modality, each showing the characteristic peak/edge shape sitting on its characteristic background, annotated with the physical origin of each component]

Why this slide is new and essential. The old deck jumped straight to “spectrum = vector, apply PCA,” hiding the most important teaching point: the shape of the background and the invariances of the signal are physics, and they tell you which method is legal.

Mechanism, one line each. EELS: a core electron is ejected, cross-section gives the hydrogenic edge; background is the tail of lower-energy losses (plasmons). EDX: an inner-shell hole is filled emitting a characteristic photon; continuum is decelerated beam electrons (bremsstrahlung). XRD: constructive interference at the Bragg condition. XPS: photoemission; binding energy depends on chemical environment (the chemical shift is the measurement). Raman: inelastic scattering off phonons; fluorescence is a competing radiative process orders of magnitude stronger.

Materials hook / exam trap. A student SNV-normalizes EELS before removing the power-law background. SNV assumes per-spectrum mean and std are meaningful; with a steep \(E^{-r}\) background dominating the integral, SNV encodes the background slope, not chemistry. Order matters and is dictated by the physics on this slide.

Quantitative anchor. XPS chemical shift: metallic Fe 2p₃/₂ ≈ 707 eV, Fe³⁺ in Fe₂O₃ ≈ 711 eV — a ~3–4 eV shift that is the oxidation-state signal. A 1 eV calibration drift (slide 08) throws away a quarter of the discriminating signal.

Pre-empt the question. “Can a foundation model just learn the background?” — In principle (slide 19), but only if the corpus spans the background variability; physical background models remain more sample-efficient and auditable. Revisited on slide 20.

Forward link. Every §2 slide is “given this physics, here is the right operation”: 07 = these background models; 08 = the calibration the XPS shift demands; 09 = why each normalization assumes a particular physics.

Pacing. 4–5 minutes — the conceptual keystone of §1. Two sentences per modality; do not turn it into a spectroscopy lecture.

05. Why ML — and Where Manual Fitting Breaks

• Manual peak fitting is slow, subjective, non-reproducible, and does not scale.
  • 10 peaks in 1 spectrum: a Friday afternoon. 10 peaks in \(10^6\) spectra: impossible.

• Overlapping peaks make decomposition ambiguous.
  • Fe-L and Mn-L (EELS) overlap near ~640 eV; Ti-Kα and Ba-Lα (EDX) overlap near ~4.5 keV.

• Subtle spectral changes encode the science.
  • The Fe-L₂,₃ white-line ratio distinguishes Fe²⁺ from Fe³⁺; a 0.3 eV onset shift = an oxidation-state change.

• Batch effects — calibration drift, detector aging, beam damage, contamination — break naïve models.

Important

Pointer. The methods that solve these — PCA/SVD (MFML u02, ML-PC u02), autoencoders / VAE / conv-AE (MFML u05, ML-PC u05), t-SNE/UMAP & latent spaces & MAE/DINOv2/I-JEPA (MFML u09) — were all derived there. From here we apply them to what the signal physics demands.

Purpose. The hinge slide. Justify ML on materials-domain grounds (scale, overlap, subtlety, drift) and then explicitly hand off the method derivations so §2 onward is unambiguously application.

Materials hook / war story. “Monday model fails on Friday”: a CNN trained on one detector session, deployed a week later, silently mispredicts because the energy axis drifted 0.4 eV and bremsstrahlung shape changed with new contamination. Diagnostic symptom: training-week hold-out fine, new sessions collapse; residual spectra show a coherent shift, not random error. Motivates slides 08 and 11.

Pre-empt the question. “If 0.3 eV shifts matter, can ML even resolve that under Poisson noise?” — Yes, with correct preprocessing and enough averaging / SSL priors; the through-line of §2 and §4. Plant the question now, pay it off later.

Cross-reference discipline. Read the callout verbatim — it is the contract. Whenever a student later asks “how does the autoencoder train?”, point here: “MFML u05 — we do not re-derive; today we ask what the spectrum forces on it.”

Pacing. ~3 minutes. End §1; the divider to §2 is the natural breath.

§2 · The spectral preprocessing pipeline

06. The Pipeline — Garbage In Dominates Everything

flowchart LR
    A["Raw spectra<br>(N × D)"] --> B["Background /<br>baseline subtraction"]
    B --> C["Energy / 2θ<br>calibration & alignment"]
    C --> D["Normalization"]
    D --> E["(optional)<br>reduction / decomposition"]
    E --> F["Downstream:<br>quant · phase ID · anomaly"]
    style A fill:#2d5016,stroke:#4a8c2a,color:#fff
    style D fill:#1a3a5c,stroke:#2a6a9c,color:#fff
    style F fill:#5c1a1a,stroke:#9c2a2a,color:#fff

• Every box is a physics-informed choice, not a default. Order matters (slide 04).

• The reduction/decomposition box is the referenced methods (PCA/AE/NMF) — it is one box, not the unit.

Important

The dominant error term in any spectral-ML result is almost never the model — it is the preprocessing. A 2% baseline error swamps the difference between PCA and a transformer. §2 is the unit’s real content because almost none of it exists in the methods units.

Purpose. Reframe the whole unit: the methods units gave them the reduction box; this unit is the other five boxes, which nobody else teaches and which dominate real results.

War story / quantitative anchor. In real EELS quant, a 2–3% systematic error in the power-law background shifts the extracted Fe²⁺/Fe³⁺ ratio by more than the Fe²⁺/Fe³⁺ ELNES difference itself. The entire downstream ML is then fitting a background artifact. Lesson: spend effort on boxes 1–3, not on swapping PCA for a transformer.

Anti-pattern. “We improved phase-ID accuracy by switching to a deep model.” Viva question: did you hold preprocessing fixed? Nine times in ten the gain came from a bundled preprocessing change; the model contribution is noise.

Pre-empt the question. “Is the reduction box ever skippable?” — Yes; for quantification (slide 15) you go straight from a clean, calibrated, background-subtracted spectrum to a physical model. Reduction is for compression/discovery/denoising, not mandatory.

Forward link. Next four slides walk boxes 1→3 in order: 07 background, 08 calibration/alignment, 09 normalization, 10 deconvolution, 11 cross-instrument transfer, 12 spatial-spectral.

Pacing. ~2 minutes; this is the map. Energy on the callout.

07. Baseline / Background Subtraction

Algorithmic (model-free)

• SNIP — Statistics-sensitive Nonlinear Iterative Peak-clipping (Ryan et al. 1988): iteratively clip each channel to the min of itself and the mean of its \(\pm m\) neighbours; peaks survive, the smooth continuum is estimated.

• Asymmetric Least Squares (AsLS) (Eilers and Boelens 2005): penalized smoother with asymmetric weights — points above the baseline are down-weighted. Two knobs: smoothness \(\lambda\), asymmetry \(p\).

Physical (model-based)

• EELS: power-law \(A E^{-r}\) fitted in a pre-edge window; or plasmon/Drude model.
• EDX: bremsstrahlung continuum (Kramers + detector response + absorption).
• XPS: Shirley / Tougaard inelastic background.

Important

A wrong background biases every downstream feature — peak areas, ratios, latent coordinates. It is a systematic, not random, error: averaging more spectra does not remove it.

[FIGURE: a Raman spectrum with a strong fluorescence background; overlay of SNIP vs AsLS vs a too-stiff polynomial fit, with the resulting baseline-subtracted spectra below]

Mechanism. SNIP = morphological erosion: narrow tall peaks get clipped over iterations, the broad continuum survives; iteration count sets the maximum peak width treated as “signal.” AsLS = minimize \(\|y-b\|_W^2 + \lambda\|D^2 b\|^2\) with \(w_i=p\) if \(y_i>b_i\) else \(1-p\); asymmetry lets the baseline hug the bottom of the signal.

Quantitative recipe anchor. AsLS for Raman: \(\lambda \in [10^5,10^8]\) (more = stiffer), \(p \in [0.001,0.01]\). SNIP for EDX: window \(m\) ≈ FWHM of the broadest line; ~24 iterations. Day-one numbers.

War story / diagnostic. AsLS with \(\lambda\) too small chews into broad-peak bases — symptom = negative lobes flanking strong peaks. \(\lambda\) too large and a sloping fluorescence leaks through — symptom = a residual ramp that PCA/t-SNE then “discovers” as a fake component.

Materials hook. Raman fluorescence can be 10–100× the phonon signal and varies pixel-to-pixel with heterogeneity — the reason Raman mapping is hard; background subtraction is the core of the measurement, not housekeeping.

Pre-empt the question. “Model-free or physical?” — Physical when you trust the physics and want auditable, transferable parameters (EELS power-law for quant); model-free when no clean parametric form (Raman fluorescence). Hybrid is common.

Cross-reference. Systematic-vs-random ties back to slide 03 (Poisson): random → average down; background bias → identical in every spectrum, cannot. Forward to slide 14 (ambiguity) and 16 (uncertainty budget).

Pacing. ~5 minutes — one of the highest-value slides. Exam-weight: the systematic-vs-random distinction.

08. Energy / 2θ Calibration & Spectral Alignment

• The problem. The physical axis is not stable: EELS energy offset drifts (zero-loss wander), XRD \(2\theta\) has zero/sample-height errors, Raman wavenumber depends on laser/grating. A 0.3 eV / 0.05° shift destroys cross-spectrum ML.

Peak-referenced calibration. Anchor to known features:

• EELS: zero-loss peak (0 eV), a known edge onset.
• XRD: a silicon / LaB₆ standard, known reflections.
• Raman: a Si line at 520.7 cm⁻¹.

Warping for non-rigid misalignment.

• DTW — dynamic time warping: optimal monotone alignment of two spectra.
• COW — correlation-optimized warping (Nielsen et al. 1998): piecewise stretch maximizing segment correlation; the chemometrics workhorse.

Important

Misalignment is the silent killer of cross-instrument ML: PCA “discovers” a component that is just the shift; a classifier learns the instrument, not the chemistry. Always align before reduction.

Mechanism. Calibration = an affine (or low-order polynomial) map of the axis fixed by reference peaks. Warping = a non-rigid monotone reparametrization; DTW finds it by dynamic programming under monotonicity+continuity; COW restricts to piecewise-linear segments with a slack parameter, maximizing correlation — cheaper and less pathological than raw DTW.

War story / diagnostic. Two-instrument study: a phase classifier hits 98% on instrument A’s hold-out, 61% on B. The “discriminative” PCA loading is a derivative-of-a-peak shape — the textbook signature of a rigid energy shift (a shift adds a multiple of \(dx/dE\)). Fix calibration, the gap collapses. Teach them to recognize the derivative-shaped loading.

Quantitative anchor. EELS valence needs the axis stable to ≤0.1 eV (the chemical signal is ~0.3 eV, slide 04). XRD: a 0.05° zero-error shifts a refined lattice parameter enough to misassign a phase. Tighter than students expect — say the numbers.

Anti-pattern. Over-warping: unconstrained DTW warps peak A onto peak B of a different compound and reports great alignment. Symptom: alignment “succeeds” but peak areas become physically impossible. Constrain warp flexibility; sanity-check against reference peaks.

Pre-empt the question. “Isn’t shift augmentation enough?” — Buys robustness to small shifts but is a band-aid; cannot recover signal when test shift exceeds the augmentation range, and dilutes the model. Augmentation + explicit calibration is the robust combination (revisited slide 11).

Cross-reference. Same domain-shift phenomenon ML-PC u11 treats from the uncertainty side; here we fix the cause. Forward to slide 11 (full response-function mismatch).

Pacing. ~4 minutes. The derivative-shaped-loading diagnostic is a memorable examable nugget.

09. Normalization Strategies — and What Each Assumes

Strategy	Formula	Physical assumption
Total count / area	\(\mathbf{x}' = \mathbf{x}/\sum_j x_j\)	All variation in total intensity is dose/thickness, not chemistry
Max-peak height	\(\mathbf{x}' = \mathbf{x}/\max_j x_j\)	A reference peak is composition-invariant
SNV	\(\mathbf{x}' = (\mathbf{x}-\bar x)/s_x\)	Per-spectrum mean & std are nuisance scatter/offset (chemometrics)
Reference-peak ratio	\(\mathbf{x}/x_{\text{ref}}\)	An internal-standard line is constant (e.g. matrix element)

• Normalization is not cosmetic: without it PCA/AE learn dose and thickness, not chemistry (variance is dominated by the largest, least interesting effect).

Important

Every normalization changes the noise model. Total-count normalization correlates channels and breaks the clean Poisson assumption — do it after any variance-stabilizing / Poisson-aware step, not before.

Mechanism / the trap. Each row is an assumption false in some regime. Max-peak assumes the reference peak is composition-invariant — but in EDX the biggest peak is often the matrix line, which does change with composition, so you normalize away signal. SNV assumes per-spectrum mean/std are nuisance — true for diffuse-reflectance NIR (its origin), often false for sparse-peak EELS where the mean is background-dominated.

Why it’s not cosmetic (core point). PCA/AE maximize variance. The largest variance in raw spectra is overall intensity (dose × thickness), scientifically boring. Skip normalization and PC1 is “thickness,” chemistry is buried in PC5+ under noise. Normalization makes the interesting variance dominant so referenced methods point at chemistry.

War story. A student’s beautiful UMAP “phase map” was a thickness map — wedge-polished foil. Diagnostic: clusters formed a smooth gradient correlating 0.95 with sample position, not composition. Total-count normalization collapsed the gradient, revealing the actual two phases.

Quantitative recipe. EDX/EELS SI: total-count (or post-background characteristic-counts) is the safe default. Raman/XPS: area or reference-line. SNV: for vibrational reflectance; be suspicious elsewhere.

Pre-empt the question. “Which should I always use?” — None always; the third column is the answer. Exam-weight: given modality + question, justify normalization from physics.

Cross-reference. Noise-model callout ties to slide 03 (Poisson) and slide 19 (Poisson NLL must see counts).

Pacing. ~3 minutes. Kept-and-tightened; the callout is the new examable part.

10. Peak Deconvolution & Physically-Constrained Fitting

• Real peaks have physical line shapes: Gaussian (instrumental/Doppler) ⊗ Lorentzian (lifetime) = Voigt / pseudo-Voigt; EELS edges have hydrogenic/ELNES shape.

• Overlap separation = constrained non-linear least squares: shared widths, fixed multiplet ratios, non-negative amplitudes, positions from physics.

• Differentiable forward models \(f_\theta(\text{physics}) \to\) spectrum: fit by gradient descent, drop into autograd → the bridge to physics-informed fitting and to slide 27 (inverse problems).

Note

Applied callout — DAE-assisted EELS. At low dose the Fe-L₂,₃ fine structure separating Fe²⁺/Fe³⁺ is buried in Poisson noise. A denoising autoencoder trained on simulated Fe-L edges (the method is ML-PC u05 §D) is used here purely as a preprocessing denoiser feeding the constrained fit — not as the analysis. Result: clean white-line ratios at ~10× lower dose. The AE is a tool inside box 1–4, not the deliverable.

[FIGURE: two overlapping EELS L₃/L₂ white lines, raw noisy data, the constrained Voigt-multiplet fit with shared width and fixed branching ratio, and residuals]

Mechanism. A Voigt is Lorentzian (natural lifetime, physical) ⊗ Gaussian (instrument + thermal). Pseudo-Voigt is a cheap linear-combination approximation (true Voigt has no closed form). Constrained NLLS = Levenberg-Marquardt / trust-region with constraints encoding physics (L₃:L₂ branching, fixed spin-orbit splitting, amplitudes ≥ 0).

Why constraints are the whole game. Unconstrained, two overlapping peaks are non-identifiable — infinitely many (position, width, amplitude) triples fit within noise. Physics constraints collapse that to a unique answer. Same identifiability theme as rotational ambiguity in MCR-ALS (slide 14) — flag the parallel.

Differentiable forward model — the modern point. If \(f_\theta\) (peak model, or a FEFF/Rietveld surrogate) is differentiable, fitting is gradient descent and chains with neural components — the seed for physics-informed ML and the inverse problems of slide 27.

Materials hook / war story. Folded-in old slide 22: low-dose Fe valence mapping. Anti-pattern: using the autoencoder’s latent value itself as “the oxidation state.” It is not — it is an uncalibrated coordinate. Defensible pipeline: AE denoises, constrained physical fit extracts the white-line ratio, slide 15 turns it into a number. Diagnostic symptom of the anti-pattern: latent “oxidation state” changes when you retrain with a different seed.

Pre-empt the question. “Why not end-to-end concentration?” — You can, but uncertainty and audit trail vanish; a constrained fit gives parameters a referee can check. ML for the hard representation step, physics for the accountable readout.

Cross-reference. Denoising method = ML-PC u05 §D (do not re-derive). Differentiable model = forward link to slide 27 and ML-PC u08.

Pacing. ~5 minutes. “Constraints resolve non-identifiability” is the spine connecting this slide to 14.

11. Calibration Transfer Between Instruments

• Industrial reality. A model trained on instrument A fails on B: different detector response, resolution, geometry — even after axis calibration. Re-labelling per instrument is unaffordable.

Piecewise Direct Standardization (PDS) (Wang et al. 1991)

• Measure a few transfer standards on both A and B.
• Learn a banded linear map \(\mathbf{X}_B \approx \mathbf{X}_A \mathbf{P}\) (each B-channel from a local window of A-channels).
• Apply \(\mathbf{P}^{-1}\) to bring B into A’s space → reuse A’s model.

Simpler & more general

• Slope/bias correction — affine fix when only gain+offset differ.
• Domain adaptation — align feature distributions A↔︎B (the DA idea is MFML u09; reference, do not re-derive).

Important

PDS needs only a handful of standards measured on both instruments — orders of magnitude cheaper than re-labelling. The difference between a model that ships to one lab and one that ships to a fleet.

Mechanism. PDS: for each target channel \(j\), regress it on a window of source channels \([j-w,j+w]\) using paired standards (PLS/ridge per window), assembling a banded \(\mathbf{P}\). The band encodes “B’s channel \(j\) is a local blur/shift/rescale of A’s neighbourhood” — the response-function mismatch a scalar calibration (slide 08) cannot fix.

Why slide 08 is not enough. Slide 08 fixes the axis (where peaks sit). PDS fixes the response (width/height, detector tails). Both can be wrong; align first, then standardize.

War story / anchor. NIR chemometrics (PDS’s birthplace) transfers a calibration with 15–30 transfer samples instead of re-measuring a 1000-sample set per spectrometer. Same logic now in multi-microscope EDX/EELS facilities: a shared standard (NIST glass, known oxide) measured on each tool is the transfer set.

Anti-pattern / diagnostic. Shipping with no transfer plan, then retraining from scratch per instrument — multiplies labelling cost by instrument count, yields \(N\) silently divergent models. Symptom: same specimen, different quant on two tools, nobody can say which is right.

Pre-empt the question. “DA or PDS?” — PDS when you can measure paired standards (best, auditable, few samples). DA when you cannot pair (only unlabelled target). State which and why — exam-weight.

Cross-reference. DA idea: MFML u09. Uncertainty consequence of unhandled transfer: ML-PC u11. Expands the old “robustness to instrumental shifts” slide into a named, cheap procedure. Forward: §5 case studies only work because the model is portable.

Pacing. ~4 minutes.

12. Spatial-Spectral Models for Spectrum Images

• A spectrum image is a cube \((x, y, E)\) — STEM-EDS easily \(256{\times}256{\times}2048\), STEM-EELS \(100{\times}100{\times}1024\).

• Naïve: unfold to \(N_\text{pix}\times D\), treat each pixel independently. Throws away that neighbouring pixels are almost the same spectrum.

• Better: exploit spatial correlation — factored 2-D spatial + 1-D spectral convolutions, or a 3-D conv-AE. (The conv-AE architecture is MFML u05 / ML-PC u05 — referenced, not re-derived.)

• Benefit: implicit spatial averaging denoises for free; spatial context separates interface pixels from bulk; learns core-shell / gradient structure.

Important

Trade-off. Spatial smoothing blurs sharp interfaces and can invent mixed spectra at boundaries — choose the receptive field to match the smallest real feature, not the noise.

[FIGURE: (x,y,E) datacube schematic; one slice as a noisy single-pixel spectrum vs the same pixel reconstructed by a spatial-spectral model that borrowed strength from its neighbourhood]

Mechanism. Factored conv: depthwise 2-D conv over (x,y) per energy band + 1-D conv over E — far fewer parameters than full 3-D conv, with the right inductive bias (spatial smoothness AND spectral locality) separated. The denoising is bias-variance: borrowing counts from spatially adjacent, chemically-identical pixels raises effective dose without raising actual dose.

Materials hook / war story. Interface-blur failure: a 5×5 spatial receptive field on 2 nm pixels averages a 10 nm window — wider than the grain-boundary segregation layer you want to image. Symptom: the segregation “disappears” and reviewers think the sample is clean. The model did not lie; the receptive field was wrong for the science.

Anti-pattern. Cranking smoothing until the phase map looks “clean” for the figure. Clean is not the objective; correct at the relevant length scale is. Honest move: show the receptive field next to the smallest feature claimed.

Pre-empt the question. “Why not just bin pixels?” — Binning is uniform non-adaptive smoothing; a learned model can preserve edges where data demand (non-linear). Same fundamental risk if the receptive field is too large — no free lunch, only a better-shaped one.

Cross-reference. Conv-AE / 3-D conv: MFML u05, ML-PC u05. The denoising-vs-discovery tension here is paid off on slide 20 (rare-phase erasure).

Pacing. ~3 minutes. End §2; last preprocessing/representation box before decomposition.

§3 · Decomposition & quantification

13. Reference Recap — Spectral Decomposition (one slide)

Important

Derived elsewhere — results only here. PCA/SVD: MFML u02, ML-PC u02. NMF: MFML u02, ML-PC u05 s28. Autoencoders: MFML u05, ML-PC u05. We do not re-derive any of them.

Decomposition	Factorization	Spectral interpretation
PCA / SVD	\(\mathbf{X}\approx \bar{\mathbf{x}} + \mathbf{C}\mathbf{V}^\top\), \(\mathbf{V}\) orthonormal	Eigenspectra = orthogonal variation directions (can be negative — not physical phases)
NMF	\(\mathbf{X}\approx \mathbf{W}\mathbf{H}\), \(\mathbf{W},\mathbf{H}\ge 0\)	\(\mathbf{H}\) = end-member spectra (≈ pure phases), \(\mathbf{W}\) = abundance maps
AE / conv-AE	\(\mathbf{x}\!\to\!\mathbf{z}\!\to\!\hat{\mathbf{x}}\)	Non-linear latent; handles peak shift, not just mixing

• The only thing this unit adds: the spectral reading — non-negativity is physical because photon counts and concentrations cannot be negative; orthogonality is a math convenience with no physical mandate.

Purpose. One slide, on purpose: the explicit “we are not re-teaching factorization” slide. Spend time on the interpretation column, the only domain content.

Teaching point. Why NMF is “the natural choice for spectra” in one breath: a spectrum is non-negative counts; a mixture of phases is a non-negative combination of non-negative pure spectra; PCA’s orthogonality forces negative lobes with no physical referent (you cannot have −3 counts at 700 eV). NMF’s constraint is the physics — one sentence, not a section.

Pre-empt the question. “Then why ever use PCA?” — Speed, uniqueness, denoising, and as initialization/whitening for everything (including NMF and MCR-ALS init). The fast first look; just not the physical model. (Salvaged content of the old PCA/AE/NMF slides, compressed to one row each.)

Cross-reference discipline. If asked for multiplicative-update NMF or the SVD proof: “MFML u02 — open your notes; today the question is what \(\mathbf{H}\) means.” Hold the line.

Forward link. Sets up slide 14: NMF gives a non-negative factorization, not a unique one — that non-uniqueness (rotational ambiguity) and how constraints fix it is the genuinely new content.

Pacing. ~3 minutes, table-driven. Resist deriving.

14. MCR-ALS & Rotational Ambiguity

MCR-ALS — Multivariate Curve Resolution by Alternating Least Squares (Juan et al. 2014): solve \(\mathbf{X}=\mathbf{C}\mathbf{S}^\top+\mathbf{E}\) by alternating non-negative least squares for concentrations \(\mathbf{C}\) and spectra \(\mathbf{S}\).

Rotational ambiguity. For any invertible \(\mathbf{T}\): \[\mathbf{X} = (\mathbf{C}\mathbf{T})(\mathbf{T}^{-1}\mathbf{S}^\top)\] fits equally well. Non-negativity alone leaves a feasible band of solutions, not a unique answer.

Constraints collapse the band:

• Closure — abundances sum to 1 (mass balance).
• Unimodality — a concentration profile has one maximum.
• Known spectra — fix a reference end-member.
• Selectivity / local rank — zones where only one component exists.

Important

Genuinely not in the methods units. NMF is taught there; MCR-ALS + rotational ambiguity + how physical constraints resolve it is the new content.

[FIGURE: feasible band of resolved spectra under non-negativity only (a fan of curves) collapsing to a single curve as closure + unimodality + a known end-member are added]

Why this is the intellectual heart of §3. Students leave the methods units thinking “NMF gives the pure phases.” It does not — it gives one member of a continuum of equally-good factorizations (rotational/intensity ambiguity). The craft of curve resolution is choosing physically-motivated constraints to shrink that continuum to a unique defensible answer. Same identifiability idea as slide 10, now at dataset level — say it out loud.

Mechanism. ALS: fix \(\mathbf{S}\), solve non-negative LS for \(\mathbf{C}\); fix \(\mathbf{C}\), solve for \(\mathbf{S}\); iterate. The \(\mathbf{T}/\mathbf{T}^{-1}\) argument is the key board derivation: write it, then note non-negativity restricts \(\mathbf{T}\) but does not pin it — hence a “feasible band.”

Materials hook. EELS/EDX of a diffusion couple or battery cathode: MCR-ALS resolves pure phase spectra and abundance maps directly, with abundances non-negative and summing to one (closure = a real material, fractions must add up). Unimodality is natural for a diffusion profile.

War story / diagnostic. Two groups publish “the pure phase spectrum” of the same sample and they differ — both valid points in the feasible band, neither reported their constraints. Symptom of unresolved ambiguity: resolved spectra change qualitatively when you re-run from a different initial guess. The fix is more physics in the constraints, not a better optimizer.

Pre-empt the question. “Just NMF with extra steps?” — NMF is a special case (non-negativity only). MCR-ALS adds the other physical constraints (closure, unimodality, selectivity, known spectra) that make the solution unique. The constraints are the science.

Cross-reference. Non-negativity result: slide 13 / MFML u02. Identifiability parallel: slide 10. Uncertainty on resolved profiles: forward to slide 16.

Pacing. ~5 minutes. Exam-weight: state rotational ambiguity and name two constraints that resolve it.

15. Quantification — From Spectra to Concentrations

• A latent coordinate or a peak area is not a number a metallurgist trusts. Quantification produces at.% with a physical basis.

Cliff-Lorimer (EDX, thin film) (Cliff and Lorimer 1975) \[\frac{C_A}{C_B} = k_{AB}\,\frac{I_A}{I_B}\] Ratio of background-subtracted line intensities × a known \(k\)-factor. Needs the thin-film approximation (no absorption).

ζ-factor method (Watanabe and Williams 2006) \[C_A = \zeta_A \frac{I_A}{\rho t}\cdot(\dots)\] Absolute quantification from first principles; folds in mass-thickness, handles absorption self-consistently — the modern standard.

Important

ML can predict \(I_A\) robustly (denoising, deconvolution); the \(k\)/ζ-factor step is physics. Skipping it and regressing concentration end-to-end discards the physical audit trail a lab/certifier requires.

Mechanism. Cliff-Lorimer: in a thin specimen, generated X-ray intensity ∝ concentration; the proportionality constants cancel into one ratio factor \(k_{AB}\) (element pair, beam energy, detector). ζ-factor: keeps the absolute proportionality (ζ relates counts to mass-thickness), so no reference element needed and an absorption correction is built in iteratively with estimated thickness — why it is the modern standard for light elements / thicker foils where Cliff-Lorimer’s no-absorption assumption fails.

Why this must exist in an ML unit. End-to-end-trained students want to regress composition directly. The domain answer: a referee, an ISO certification, or a failure investigation needs to see which physical model and which factor produced the number. A black box saying “4.2% Cr” with no traceable physics is not a deliverable in a certified lab. ML’s job is clean unbiased \(I_A\) (slides 07–10); physics turns \(I_A\) into \(C_A\).

Quantitative anchor. Cliff-Lorimer breaks exactly where it matters: light elements (O, N, C) and thicker regions, absorption biasing results 10–30%. Precisely why Watanabe-Williams ζ exists; quote the failure regime, not just the formula.

War story / anti-pattern. “Our network predicts at.% directly and beats Cliff-Lorimer on the test set.” On whose labels? Usually labels were themselves Cliff-Lorimer outputs — the network mimics the method it claims to beat, plus its biases, with no error model. Diagnostic: ask for absolute-accuracy validation against a certified reference material, not other software.

Pre-empt the question. “Is ML useless for quant?” — No: best tool for intensity extraction under overlap/noise (07, 10) and for uncertainty propagation (next slide). Wrong tool for replacing the traceable physical conversion.

Cross-reference. Intensity extraction = 07/10. Uncertainty on \(C_A\) = slide 16 + ML-PC u11.

Pacing. ~4 minutes. Exam-weight: why the k/ζ step is not replaceable by end-to-end regression.

16. Uncertainty on the Quantification

• A concentration without an error bar is not a deliverable (recall the ML-PC u11 thesis: a threshold decision needs a distribution, not a number).

• Error budget for \(C_A\): counting statistics on \(I_A\) (Poisson) · background-model error (systematic, slide 07) · \(k\)/ζ-factor uncertainty · absorption/thickness uncertainty · ML denoiser bias.

• Propagation. Analytic error propagation through the Cliff-Lorimer / ζ equation for the physical terms; the UQ machinery for the ML-predicted parts is ML-PC u11 (GP CIs, deep ensembles, MC-dropout) — referenced, not re-derived.

Important

The often-dominant term is not counting statistics — it is the background-model systematic (slide 07). Reporting only \(\sqrt{N}\) Poisson error bars is the most common honest-looking lie in the field.

Purpose. Close the quantification arc honestly. The deliverable is \(C_A \pm \sigma_{C_A}\) with a decomposed error budget, not a bare number with a cosmetic Poisson bar.

Mechanism. Error propagation on \(C_A/C_B = k\,I_A/I_B\): relative variances add — \((\sigma_C/C)^2 \approx (\sigma_k/k)^2 + (\sigma_{I_A}/I_A)^2 + (\sigma_{I_B}/I_B)^2 + \dots\). Each term has a different character: counting is random (shrinks with dose), background and k-factor are systematic (do not shrink — connect to 03/07). The ML denoiser contributes a bias invisible to internal cross-validation; assess against a reference.

The honest-looking lie (say it slowly). Most published spectral quant reports \(\pm\sqrt{N}\) bars. That term is frequently the smallest. A 2% background systematic on a peak with 1% Poisson error means the real uncertainty is dominated by a term they did not report. The single most important integrity message of §3 — it ties slide 07 (background bias is systematic) to slide 15 (quant) to ML-PC u11 (calibrated uncertainty is the actual deliverable).

Materials hook / anchor. Light-element EDX (oxygen in an oxide): k-factor uncertainty alone routinely 5–15%; absorption-correction uncertainty similar in thick regions. Counting error might be 1%. Reporting 1% is off by an order of magnitude — a referee should reject it.

Pre-empt the question. “How do I get the ML part’s uncertainty?” — Do not invent it here; it is ML-PC u11’s content (GP closed-form CI, deep-ensemble disagreement, MC-dropout per-pixel variance). This slide says which terms exist and which dominates, and hands ML-UQ to u11.

Cross-reference. Background systematic: 07. Quant equations: 15. UQ methods: ML-PC u11. MCR-ALS feasible-band width (14) is itself an uncertainty contribution for decomposition-based quant — mention if time.

Pacing. ~3 minutes. High exam-weight and integrity-weight; do not rush the “dominant term” callout.

17. Non-Linear Unmixing — When Linear Mixing Fails

• The comfortable model: a boundary pixel is \(\mathbf{x} = \alpha\,\mathbf{e}_A + (1-\alpha)\,\mathbf{e}_B + \boldsymbol{\eta}\) — linear mixing; estimate \(\alpha\) from a latent coordinate. Often fine.

• When linear mixing breaks:
  • EELS multiple scattering — thick specimen: losses convolve, not add.
  • EDX absorption / fluorescence — emitted X-rays re-absorbed depth-dependently.
  • XRD channelling / extinction, preferred orientation — intensities not additive in phase fraction.

• What to do: physical forward model (deconvolve plural scattering; absorption correction) → then unmix; or a non-linear decoder (AE — method in ML-PC u05) whose non-linearity absorbs the curvature. Validate against a known mixture.

Important

A non-linear method fitting a non-linear mixture does not prove it recovered the physics. Always check against a sample of known fractional composition.

Mechanism. Linear unmixing assumes intensities superpose. They do not when the measurement is non-linear in path: EELS, a thicker region means multiple energy losses, so the spectrum is the Poisson-weighted self-convolution series of the single-scattering distribution — not \(\alpha\mathbf{e}_A+(1-\alpha)\mathbf{e}_B\). EDX, generated photons re-absorbed depth-/matrix-dependently — same reason ζ beats Cliff-Lorimer (slide 15). Tie it back.

Materials hook. Sub-pixel composition at an interface where the probe straddles two phases: linear unmixing gives a clean \(\alpha\) only if thin and absorption-negligible. In a real 80 nm foil it is biased, and the bias looks like a shifted phase boundary — a metallurgically meaningful error from a modelling shortcut.

War story / the punchline anti-pattern. An AE reconstructs mixed spectra beautifully and someone reads \(\alpha\) off the latent axis. Low reconstruction error proves the decoder can represent the data; it says nothing about whether the latent equals the physical phase fraction. Diagnostic: a known 30/70 specimen; if the recovered fraction is not ~0.30 with a sane error bar, the latent is not the physics. Same “latent ≠ physical quantity” warning as slide 10 — repeat it; students forget.

Pre-empt the question. “So non-linear methods are dangerous?” — Not dangerous, just non-self-validating. The discipline is identical across the unit: ML for representation, physics + a known reference for the quantitative claim.

Cross-reference. Decoder/AE method: ML-PC u05. Absorption link: 15. Latent-is-not-physics theme: 10, 20.

Pacing. ~3 minutes. Salvaged from the old “signal separation” slide, reframed entirely around failure modes and validation discipline.

§4 · Representation learning for spectra, applied

18. MAE Pretraining on Unlabelled Spectra

The masking trick. Masked Autoencoder (He et al. 2022) on 1-D spectra:

• Split each spectrum into ~64 patches.
• Mask 75%; encode only the visible 25% with a small transformer.
• Lightweight decoder + mask tokens predicts the masked bins.
• Loss = MSE on masked bins only → forced to model spectral structure, not copy.

Pretrain on the lab’s entire unlabelled archive (EELS+Raman+XRD pooled); fine-tune a tiny head on the ~100 labelled spectra.

Why this is the natural materials move. \(10^4\)–\(10^5\) unlabelled spectra rot on the file server; \(10^2\) labelled (labelling = the expensive expert step). MAE converts the unlabelled bulk into a feature extractor.

Note

Rule of thumb. Need \(\gtrsim 10\times\) more unlabelled than labelled spectra before MAE pays off. 200 k pretrain + 100 fine-tune typically beats a from-scratch CNN by 5–15 points small-data.

Method derived in MFML u09 (SSL family). Here: the spectra-specific deployment.

Why mask 75%. Empirical, confirmed for 1-D spectra: at 50% the network interpolates from immediate neighbours (trivial smoothing); at 75% only a quarter is visible, so it must learn what a spectrum of this class looks like — peak positions, edge shapes, baseline — to inpaint. 90% underfits. Sweet spot 75%.

Connection to the vanilla AE (ML-PC u05). Same MSE, same encoder-decoder. The only change is input-side masking + loss on hidden bins. That single change is why it generalises: full-input reconstruction rewards copying through the bottleneck; masked-bin reconstruction rewards modelling structure.

Materials hook. Every spectroscopy lab has years of archived, never-labelled data — labelling is the expensive expert step. MAE finally gives a reason to pay the storage cost: the archive becomes the pretraining corpus for every future supervised task.

Anti-pattern. Running MAE on a labelled-but-scarce set with no unlabelled bulk (100 spectra, nothing else). The whole value is leveraging unlabelled data; with 100 total a GP classifier or augmented CNN wins. The rule-of-thumb callout is exam-weight.

Compute anchor. 4-layer, 128-dim transformer, 64 patches ≈ 1M params; 200 k spectra trains overnight on a single 1080Ti; fine-tuning the head on 100 labelled spectra takes minutes.

Cross-reference. SSL family derived MFML u09 (masking / contrastive / JEPA). Today = the MAE branch on spectra. Forward link: the frozen encoder is exactly the feature extractor an active-learning / self-driving-microscope loop would reuse (an outlook beyond this term’s units).

Pacing. ~4 minutes. Kept-and-tightened; old long notes condensed but the teaching beats preserved.

19. Self-Supervised & Foundation Models for Spectra

• MAE (slide 18) is reconstruction-based SSL. Two other families transfer to spectra:
  • DINOv2-style (Oquab et al. 2024) — self-distillation / invariance to augmentations (shift, broaden, rescale, add Poisson noise).
  • I-JEPA-style — predict latent representations of masked regions, not bins: less low-level, more semantic.

• Spectral foundation models: one encoder pretrained on a huge cross-modality unlabelled corpus, then probed/fine-tuned per task (phase ID, valence, quant features).

Important

Method = MFML u09 (SSL) + ML-PC u09b (transformer backbone). Reference, do not re-derive. Today’s content: which augmentations are physically valid for spectra, and the evidence that pretrained encoders beat from-scratch on small materials sets.

The genuinely new, domain-specific content. Not the algorithms (MFML u09 / ML-PC u09b) but which augmentations encode true spectral invariances. Energy shift within calibration tolerance — valid (it is just drift). Peak broadening within instrumental range — valid. Poisson resampling at lower dose — valid, physically motivated. But arbitrary intensity flips, channel permutation, large nonphysical warps — these teach false invariances and quietly destroy the chemical signal (recall slide 02: the axis is physics). The augmentation set is the inductive bias; choosing it is a materials decision, not an ML default.

Why I-JEPA is interesting for spectra. Predicting in latent space avoids wasting capacity reconstructing the noisy, scientifically-irrelevant background — capacity goes to peak/edge structure. Pre-empt the obvious question: “better than MAE for spectra?” — Often for very small downstream sets; MAE is the safer default and the natural PCA→AE→MAE progression.

Materials hook / war story. A “spectral foundation model” pretrained with vision-style augmentations (random brightness/contrast): brightness scaling ≈ dose (fine), but contrast stretching distorts peak ratios = the chemistry. Symptom: great pretraining loss, downstream valence prediction no better than from-scratch — the encoder learned invariance to exactly the signal you need.

Cross-reference. SSL families: MFML u09. Transformer backbone: ML-PC u09b. This slide = application + the physics-of-augmentation judgement only.

Forward link. Slide 20 immediately problematises all of this: powerful unsupervised compression can erase the rare science.

Pacing. ~3 minutes.

20. Why Unsupervised Compression Can Destroy the Science

Think About This — the PCA rare-phase trap

Question. PCA denoising lets you cut acquisition dose 10×. Why not always do it?

Hint. Think about a feature present in only one or two pixels of a million-pixel map.

Consider.

• Eigenspectra are computed from the whole dataset — they encode the common.
• A rare phase contributes negligible variance → it falls outside the top-\(K\) subspace.
• Truncated reconstruction projects the rare phase away: denoising literally erases the discovery.
• The same logic indicts MAE/SSL latents, NMF with too-few components, aggressive spatial smoothing (slide 12).

Important

Denoising vs. discovery is a fundamental tension, not a tuning issue. Always inspect the residual \(\mathbf{x}-\hat{\mathbf{x}}\) spatially — a rare phase hides there, not in the reconstruction.

Why this kept slide is reframed. Originally a PCA-specific aside. Here it is the philosophical hinge of §4: every powerful unsupervised compressor in this course — PCA, AE, MAE, SSL, NMF, spatial-spectral — optimises for the common and is structurally blind to the rare. Make that the general lesson, PCA the cleanest illustration.

Mechanism. Variance-maximising / reconstruction-minimising objectives weight a component by how many spectra exhibit it. A precipitate in 2 of 10⁶ pixels contributes ~\(10^{-6}\) of the variance — below the noise eigenvalues — so any top-\(K\) truncation discards it by construction. Not a tunable hyperparameter; it is what the objective is.

Materials hook / war story. Real case: an unexpected intermetallic at a diffusion-couple interface, in a thin pixel band, vanished from the PCA-denoised map and was found only because someone plotted the reconstruction residual and saw a spatially-coherent hot region. The discovery lived entirely in the discarded subspace. Diagnostic recipe: residual must be spatially incoherent (looks like noise) if denoising was safe; any spatial structure in the residual = you threw away physics.

Pre-empt the question. “So never denoise?” — Denoise for known, common features (matrix-phase quant). Do not trust a denoised map for discovery; for discovery work with anomaly/residual maps (slide 25). Different question → different tool. This is the through-line that makes §5’s discovery workflow honest.

Cross-reference. Indicts methods from MFML u02/u05/u09 — as an application caution, not a re-derivation. Forward to slide 25 (anomaly workflow) and 26 (operando novelty).

Pacing. ~4 minutes. Run as a genuine think-pause; let them answer the hint before revealing. High exam-weight: state the tension and the residual-inspection remedy.

21. What the Latent Learns — and Seeding It From Simulation

The latent can rediscover the physics.

• Train a 2-D-latent AE on EDX of a ternary Fe-Cr-Ni alloy; colour latent points by known composition.

• \(c_\text{Fe}+c_\text{Cr}+c_\text{Ni}=1\) → only 2 free composition variables → the latent forms a triangle: the network re-derived the Gibbs ternary with no thermodynamics input.

• Emergent, not programmed — but only if preprocessing (§2) made composition the dominant variance.

Seed it from simulation when labels are scarce.

• Pretrain on simulated spectra, fine-tune on few real ones:
  • XRD: Rietveld / structure-factor simulation
  • EELS: FEFF / DFT
  • EDX: Monte-Carlo (CASINO, DTSA-II)
• Add realistic Poisson + readout noise + drift to the sim.

Important

The sim must capture the right physics (peak shapes, backgrounds, artifacts) or you transfer a simulation accent the real instrument never speaks.

Purpose. Merge two kept insights (ternary→Gibbs latent; synthetic→real transfer) into one coherent “what the latent is and how to bootstrap it” slide. They belong together: the latent only rediscovers physics if (a) preprocessing exposed the chemistry and (b) the training distribution contains the physics — simulation guarantees (b) when real labels are scarce.

Mechanism — the Gibbs point. Three concentrations with sum-to-one live on a 2-simplex (triangle). A 2-D-bottleneck AE trained on normalized composition-dominated spectra must lay the data on that 2-manifold; coloured by composition it is the Gibbs triangle. The caveat is the teaching point: skip normalization (slide 09) and the latent encodes thickness, not the simplex — the “magic” is downstream of §2.

Follow-up to pose. “What if the latent were 1-D?” — Collapse the triangle onto a line; two alloys with different Cr/Ni but similar Fe alias to the same code. 60-second Socratic beat.

Materials hook / war story (the simulation accent). A DAE pretrained on FEFF EELS edges that omitted the experimental point-spread / energy-resolution broadening learned to expect sharper near-edge structure than any real spectrometer delivers; on real data it “sharpened” artifacts into fake fine structure mis-read as a valence change. Diagnostic: synthetic-validation excellent, real-data features that disappear when the microscope’s actual energy resolution is convolved into the sim. Fix: physics in the simulator (instrument response, realistic noise, drift), not more network.

Pre-empt the question. “How much real data to fine-tune?” — Enough to correct the residual sim-to-real gap, typically tens to low hundreds if sim physics is good; the regime where slide 18’s SSL pretraining and this slide’s sim pretraining are complementary (sim = physics priors, SSL = instrument-specific structure).

Cross-reference. Latent/AE method: MFML u05 / ML-PC u05. Transfer-learning idea: MFML u09. Today = the materials reading + simulation-fidelity caution.

Pacing. ~3–4 minutes. The Gibbs reveal is a delight — let it land before the caveat.

§5 · Applications & operando

22. Case: Automatic XRD Phase Identification

Problem. Given a noisy multi-phase XRD pattern, identify the crystallographic phases. Database (ICDD) peak-matching is manual, slow, and brittle for mixtures, preferred orientation, and broadening.

Pipeline (methods referenced, not re-derived).

1. §2 preprocessing: background (SNIP), \(2\theta\) calibration (LaB₆ standard), normalization.
2. Encode with an AE/classifier pretrained on simulated patterns for all candidate phases (slide 21).
3. Nearest-neighbour / clustering in latent space → phase set; high reconstruction error → amorphous / unknown (slide 20 residual logic).

Why it beats database matching. Robust to peak broadening (size/strain) and preferred orientation that defeat rigid peak-position matching; decomposes mixtures in latent space.

[FIGURE: noisy multiphase XRD pattern; below, resolved phase fractions and a flagged unindexed amorphous halo highlighted via reconstruction residual]

Purpose. First case study — show the whole pipeline assembled, every box pointing back at §2–§4. The teaching value is the assembly, not new method content.

Materials hook. Preferred orientation is the killer for classical matching: a rolled or thin-film sample suppresses/enhances reflections so the intensity pattern no longer matches the powder reference even though the phase is correct. A representation trained on simulated patterns with texture variation learns the position structure and tolerates intensity reweighting — the concrete win.

War story / diagnostic. Amorphous content reported as “phase X at 12%” by naive matching because the broad amorphous halo overlapped X’s strongest line. The residual/anomaly route (slide 20) flags it as “unindexed broad feature” — the honest answer. Failure symptom: a phase fraction that depends strongly on the background window chosen (slide 07 — backgrounds bias everything).

Pre-empt the question. “Why not a supervised classifier on labelled patterns?” — Labels are scarce for mixtures and the combinatorial mixture space is huge; the sim-pretrain + latent approach (slide 21) sidesteps the labelling bottleneck and gives an anomaly channel for the unexpected.

Cross-reference. Sim pretrain: 21. Anomaly/residual: 20. Methods: MFML u02/u05.

Pacing. ~3 minutes. Brisk; an integration showcase.

23. Case: EELS Spectrum Imaging — Fe²⁺/Fe³⁺ Mapping

Problem. Map iron oxidation state at nm resolution. The Fe-L₂,₃ white-line ratio (\(L_3/L_2\)) and onset shift (~0.3 eV, slide 04) separate Fe²⁺ from Fe³⁺ — but it is buried in Poisson noise at usable dose.

Pipeline. Power-law background (slide 07) → energy calibration to ZLP (slide 08) → DAE denoise pretrained on simulated Fe-L edges (method ML-PC u05; sim slide 21) → constrained white-line fit (slide 10) → continuous valence via the fitted ratio, not a raw latent (slide 17 caution).

Impact. Oxidation-state mapping at ~5× lower dose; resolves continuous mixed-valence gradients at interfaces; validated against Fe²⁺/Fe³⁺ reference standards (slide 16 discipline).

Purpose. The flagship characterization case — the slide where the entire unit’s discipline is visible in one pipeline: background (07) → calibration (08) → denoise (referenced AE) → constrained physical fit (10) → number not latent (17) → reference-validated (16).

Materials hook / anchor. \(L_3/L_2\) white-line ratio is the textbook valence proxy; it shifts measurably between Fe²⁺ and Fe³⁺, onset moves ~0.3 eV. The 5× dose saving matters because the oxides of interest (battery cathodes, corrosion layers) are beam-sensitive — over-dose and you reduce Fe³⁺→Fe²⁺ with the beam, measuring an artifact of your own probe.

War story / killer anti-pattern. Beam-induced reduction: scan too slow/hot and the act of measuring changes the oxidation state, producing a gorgeous fake valence gradient that “correlates with dose.” Diagnostic symptom: the gradient tracks scan direction / dwell time, not microstructure. Why low-dose + denoising is not a convenience here — it is the difference between measuring the sample and measuring the beam. Forward-link to slide 26 (dose-fractionation / beam-damage detection).

Pre-empt the question. “Why not classify Fe²⁺ vs Fe³⁺ with a CNN?” — The science is the continuous mixed-valence gradient, not a binary label; and a fitted physical ratio is referee-checkable while a softmax is not (slide 15 logic).

Cross-reference. Every arrow points to a §2–§4 slide on purpose — say it: “this case study introduced no new method; it is §2–§4 assembled for one physics question.”

Pacing. ~3 minutes.

24. Case: Large-Scale EDS Maps — the Throughput Story

The scale challenge (the point of the slide). \(512{\times}512{\times}2048 \approx 5\times10^5\) spectra per field; multiple fields → \(10^6\)–\(10^7\) spectra per sample. The bottleneck is engineering, not method novelty.

Why PCA, specifically, here. Linear, one-pass, \(O(\min(N,D)^2)\), deterministic, streamable (incremental SVD). Denoise via top-\(K\) reconstruction → K-means in score space → map clusters back to \((x,y)\) → phase map in minutes on a workstation. (Method: ML-PC u02 — referenced.)

Engineering reality. Memory-mapped datacubes, chunked/out-of-core SVD, GPU only where it pays. The win is throughput at fixed accuracy, not a better model.

Important

And it inherits slide 20’s caveat: trace elements below the noise floor and rare phases can be denoised away. Keep a residual/anomaly pass alongside the phase map.

Purpose. Deliberately not a method slide. The lesson: at \(10^7\) spectra, the right question is “what is fast, streamable, deterministic?” — and the boring linear method wins on engineering grounds. Students over-index on model sophistication; this slide is the antidote.

Materials hook / anchor. A modern silicon-drift-detector EDS map is acquired in minutes; analysis taking hours per map is a throughput crisis in a service lab running dozens of samples a day. PCA+K-means at minutes/map is what makes routine phase mapping operationally viable — why it is the production default despite AEs being “better.”

War story. A group replaced PCA with a deep AE for “better” phase maps; per-sample processing went 4 → 40 minutes, the queue backed up two weeks, and the maps were not measurably better because preprocessing (slide 06!) was the actual limiter. The fix was a better background model, not a bigger network.

Pre-empt the question. “When is the AE worth it at scale?” — When non-linear effects (peak shift, slide 17) genuinely break the linear subspace and you amortise training across many samples. Otherwise PCA. Decision = effect size vs throughput cost, same logic as ML-PC u11’s budget table.

Cross-reference. PCA method: ML-PC u02. Rare-phase caveat: 20. Streaming connects forward to slide 26 (operando).

Pacing. ~2–3 minutes.

25. Discovery via Anomaly — the Workflow

• Train a representation on spectra from known phases only (clean nominal set — the prerequisite, ML-PC u05 §E).

Workflow:

1. Apply to a new sample; most pixels reconstruct well.
2. Map reconstruction error spatially → anomaly map (the slide-20 residual, used deliberately).
3. Extract spectra from high-error regions; inspect physically.
4. Identify the unknown phase; add to training; retrain.

• Example. AE trained on two base metals of a diffusion couple; a high-error band at the interface revealed an unexpected intermetallic invisible in the denoised map.

Important

The AE-anomaly method is ML-PC u05 §E (threshold from nominal validation error, never from anomalies). This slide is the materials discovery workflow wrapped around it.

Purpose. Convert slide 20’s warning into a constructive procedure. Deliberate pairing: slide 20 says “denoising hides the rare,” slide 25 says “so make the residual your primary discovery instrument.” Same mathematics, opposite intent.

Mechanism / non-negotiable rule. Reconstruction error is an anomaly score only if the training set is clean nominal. The threshold comes from the nominal validation-error distribution (e.g., 99th percentile), never from labelled anomalies (you usually have none; if you did it would be supervised). Exactly ML-PC u05 §E — say “method is u05 §E, do not re-derive; today is the workflow and the materials judgement.”

War story / diagnostic. Nominal set quietly contaminated with a few unknown-phase pixels → the AE learns to reconstruct the anomaly → never flags it. Symptom: anomaly map uniformly low even on a sample you know contains a surprise. Curation of the nominal set is the real work; the algorithm is trivial.

Materials hook. The diffusion-couple intermetallic: trained on Cu and Ni, the Cu-Ni interface intermetallic lights up the residual because it is in neither training class — discovery precisely because the model is blind to it (slide 20 inverted). Step 4 (add & retrain) is the human-in-the-loop closure — the seed of an active-learning loop (an outlook beyond this term’s units).

Pre-empt the question. “Different from OOD detection?” — It is OOD detection; the materials specifics are (a) the nominal set is your known phases, (b) the score must be spatially mapped (an isolated hot pixel is noise, a coherent region is a phase), (c) the loop ends in a physical identification, not just a flag.

Cross-reference. Method: ML-PC u05 §E. Residual logic: 20. Streaming version: 26. Active-learning closure: outlook (no dedicated unit this term).

Pacing. ~3 minutes.

26. Operando / Streaming Spectral Monitoring

• Operando: time-resolved XRD / Raman / EELS during synthesis, cycling, heating, catalysis — a spectrum stream, not a static cube.

New problems the stream creates

• Drift over time — the nominal distribution itself moves; a fixed threshold goes stale.
• Novelty detection online — a new phase appearing is the result.
• Adaptive acquisition — spend dose where the spectrum is changing.

Beam-damage / dose-fractionation

• Track spectral change vs accumulated dose; stop or fractionate before the probe alters the sample (recall slide 23: beam-induced Fe³⁺→Fe²⁺).
• Poisson model (slide 03) sets the detectable-change floor per frame.

Important

Methods referenced: AE-anomaly ML-PC u05 §E; Poisson noise ML-PC u02. New here: time — non-stationarity, online thresholds, dose as a budgeted resource.

Why this slide is new and important. Everything before §5 assumed a static dataset. Operando breaks that and creates genuinely new domain problems no methods unit covers: the nominal distribution is non-stationary (it drifts as the reaction proceeds), so a fixed anomaly threshold (slide 25) silently degrades; and dose is a depletable budget spent in real time against an irreversible sample.

Mechanism. Streaming/online anomaly: maintain a running (e.g., exponentially-weighted) estimate of the nominal distribution and distinguish expected drift (reaction progressing) from novelty (a new phase nucleating) — that separation is the hard, materials-specific part. Dose-fractionation: the Poisson floor (slide 03, variance=mean) tells you the minimum counts per frame to detect a change of a given size; below that you integrate noise, above it you waste dose / damage the sample — an explicit information-vs-damage optimisation.

Materials hook / war story. Operando battery-cycling XRD: the cathode phase genuinely evolves every cycle (that is the experiment), so a naive anomaly detector screams every frame. The skill is modelling the expected reaction trajectory and flagging only departures. Diagnostic symptom of getting it wrong: anomaly score perfectly correlated with state-of-charge — you are detecting the experiment, not a surprise (compare slide 20’s “you detected thickness”).

Beam damage — the integrity point. Connect to slide 23: in beam-sensitive operando EELS the probe changes the very quantity measured. Online dose tracking that halts acquisition when cumulative spectral change exceeds a threshold is not optimisation polish — it is the difference between data and artifact.

Pre-empt the question. “Just slide 25 in a loop?” — No: slide 25’s nominal set is fixed; here it moves, the threshold must adapt, and the cost of a query (dose) is itself the thing being optimised. That non-stationarity + budgeted-resource framing is the new content and the seed of self-driving-microscope ideas (an outlook beyond this term’s units).

Cross-reference. Anomaly method: ML-PC u05 §E. Poisson: ML-PC u02. Beam damage: 23. Adaptive acquisition closure: outlook (no dedicated unit this term).

Pacing. ~3–4 minutes.

27. Spectral Inverse Problems — EXAFS → Local Structure (stretch)

• Some spectra are an encoded structure, not a feature list. EXAFS oscillations \(\chi(k)\) are a sum over scattering paths → Fourier-related to a radial distribution (neighbour distances, coordination numbers).

• This is a modality-specific inverse problem: recover the structure that generates the spectrum via a known forward model (FEFF), not a generic decomposition.

• ML role: a differentiable / surrogate forward model (seeded on slide 10’s idea) for fast amortised inversion + uncertainty — the general inverse-problem theory is ML-PC u08; reference it, do not re-derive.

Important

An inverse problem can be ill-posed: many structures fit one spectrum. Same disease as MCR rotational ambiguity (slide 14) — physical constraints (known coordination chemistry, path filtering) are the cure.

Purpose / status. Marked stretch — include if pace allows; it generalises the unit’s recurring identifiability theme to a third instance and links to ML-PC u08. If short, state the headline (some spectra are inverse problems; ill-posedness is the recurring disease; constraints the recurring cure) and move to the wrap.

Mechanism. EXAFS: photoelectron scattering off neighbours modulates the absorption fine structure; \(\chi(k)\) is, to first approximation, a sum of damped sinusoids whose frequencies encode neighbour distances — a Fourier transform gives a (phase-shifted) radial distribution. The forward model (FEFF) is well-understood; inversion is the hard, ill-posed direction. ML contributes an amortised/differentiable surrogate for fast inversion with uncertainty.

The unifying teaching point (why the slide earns its place). Three times now the same disease: constrained peak fitting (slide 10, non-identifiable peaks), MCR-ALS (slide 14, rotational ambiguity), inverse problems (here, ill-posedness). Same cure: physical constraints reduce a continuum of fits to a defensible one. Say it explicitly — the single most transferable idea in the unit and a likely exam synthesis question.

Materials hook. EXAFS routinely over-fitted: add enough shells/paths and you fit noise, reporting meaningless coordination numbers. Diagnostic symptom: fit quality keeps improving with more free paths while parameters become physically absurd (negative Debye-Waller factors, non-physical distances). The constraint discipline (path filtering, known chemistry, bounded parameters) is identical in spirit to slide 10’s shared-width/branching-ratio constraints.

Pre-empt the question. “Why not just slide 10?” — Slide 10 fits peaks in the spectrum; here the unknown is a structure behind a non-trivial physical forward operator. The machinery (differentiable forward model, constraints) is shared; the object recovered differs and ill-posedness is more severe.

Cross-reference. General inverse theory: ML-PC u08. Differentiable forward model: 10. Identifiability/ambiguity family: 10, 14.

Pacing. ~2–3 minutes, or a 30-second headline if compressing.

Wrap

28. Key Takeaways — It Was Never About the Method

1. The signal’s structure is dictated by the probe physics — that, not the algorithm, dictates the pipeline.

2. Preprocessing dominates: a wrong background/calibration is a systematic error no model fixes (slides 06–11).

3. Identifiability is the recurring disease — peak overlap (10), MCR rotational ambiguity (14), inverse ill-posedness (27); physical constraints are the recurring cure.

4. A latent is not a quantity: physics + a reference standard turn it into a number with an honest error budget (15–17).

5. Unsupervised compression erases the rare — denoising vs discovery; live in the residual (20, 25, 26).

6. SSL/MAE turns the unlabelled archive into a feature extractor — if augmentations are physically valid (18, 19).

Important

The methods live in MFML u02/u05/u09 and ML-PC u02/u05. This unit was about what the signal physics demands. Next: Unit 10 — Transformers for materials.

Purpose. Land the thesis the unit opened with (slide 01). The six points are the examable spine; read them as “if you remember six things.” The closing callout is the literal contract from slide 01 — closing the frame deliberately.

Five must-know statements for the exam (state now, they recur). 1. Probe physics → pipeline; there is no universal preprocessing (slide 04). 2. Background/calibration error is systematic — averaging does not remove it; usually the dominant uncertainty term (07, 16). 3. Rotational ambiguity / identifiability: non-uniqueness is the default; constraints encode the physics that resolves it (10, 14, 27). 4. A latent/peak ≠ a concentration; the k/ζ physical step is non-negotiable and gives the audit trail (15). 5. Unsupervised compression is structurally blind to the rare; discovery lives in the residual/anomaly channel (20, 25).

Anti-pattern recap. The single behaviour to leave them fearing: swapping in a fancier model while leaving preprocessing untouched and claiming the gain. Nine times in ten the gain was preprocessing bundled with the swap.

Pre-empt the exam-strategy question. “What will be tested?” — Not PCA/AE/MAE math (u02/u05/u09). Domain judgement: choose & justify a pipeline for a given modality+question; explain a failure (drift, background bias, rare-phase erasure); decompose a quantification error budget.

Forward link. Unit 10 (transformers for materials) supplies the backbone behind slide 19’s foundation models.

Pacing. ~3 minutes. End on the callout; add no new content.

Continue

• ← Previous: Unit 08 — Inverse problems and process maps
• → Next: Unit 10 — Transformers for materials (ViT, Flash Attention, Mamba)
• All courses

29. References

Preprocessing & chemometrics

• Background: Ryan et al. (1988) (SNIP), Eilers and Boelens (2005) (Asymmetric Least Squares)
• Alignment: Nielsen et al. (1998) (correlation-optimized warping)
• Calibration transfer: Wang et al. (1991) (piecewise direct standardization)
• Curve resolution: Juan et al. (2014) (MCR-ALS review)

Quantification

• Cliff and Lorimer (1975) (Cliff-Lorimer \(k\)-factors) · Watanabe and Williams (2006) (ζ-factor method)

Representation learning (referenced — derived in MFML u05/u09)

• He et al. (2022) (Masked Autoencoders) · Oquab et al. (2024) (DINOv2)
• Course texts: Sandfeld et al. (2024), McClarren (2021), Neuer et al. (2024)

Cliff, G., and G. W. Lorimer. 1975. “The Quantitative Analysis of Thin Specimens.” Journal of Microscopy 103 (2): 203–7.

Eilers, Paul H. C., and Hans F. M. Boelens. 2005. “Baseline Correction with Asymmetric Least Squares Smoothing.” Leiden University Medical Centre Report 1 (1): 5.

He, Kaiming, Xinlei Chen, Saining Xie, Yanghao Li, Piotr Dollár, and Ross Girshick. 2022. “Masked Autoencoders Are Scalable Vision Learners.” Proceedings of the IEEE/CVF Conference on Computer Vision and Pattern Recognition (CVPR), 16000–16009.

Juan, Anna de, Joaquim Jaumot, and Romà Tauler. 2014. “Multivariate Curve Resolution (MCR). Solving the Mixture Analysis Problem.” Analytical Methods 6 (14): 4964–76.

McClarren, Ryan G. 2021. Machine Learning for Engineers: Using Data to Solve Problems for Physical Systems. Springer.

Neuer, Michael et al. 2024. Machine Learning for Engineers: Introduction to Physics-Informed, Explainable Learning Methods for AI in Engineering Applications. Springer Nature.

Nielsen, Niels-Peter V., Jens M. Carstensen, and Jørn Smedsgaard. 1998. “Aligning of Single and Multiple Wavelength Chromatographic Profiles for Chemometric Data Analysis Using Correlation Optimised Warping.” Journal of Chromatography A 805 (1-2): 17–35.

Oquab, Maxime, Timothée Darcet, Théo Moutakanni, et al. 2024. “DINOv2: Learning Robust Visual Features Without Supervision.” Transactions on Machine Learning Research.

Ryan, C. G., E. Clayton, W. L. Griffin, S. H. Sie, and D. R. Cousens. 1988. “SNIP, a Statistics-Sensitive Background Treatment for the Quantitative Analysis of PIXE Spectra in Geoscience Applications.” Nuclear Instruments and Methods in Physics Research Section B 34 (3): 396–402.

Sandfeld, Stefan et al. 2024. Materials Data Science. Springer.

Wang, Yongdong, David J. Veltkamp, and Bruce R. Kowalski. 1991. “Multivariate Instrument Standardization.” Analytical Chemistry 63 (23): 2750–56.

Watanabe, Masashi, and David B. Williams. 2006. “The Quantitative Analysis of Thin Specimens: A Review of Progress from the Cliff-Lorimer to the New Zeta-Factor Methods.” Journal of Microscopy 221 (2): 89–109.