How do you justify a PCA-based credit score to a regulator asking which ratio drove it?

## What is actually lost Compress 60 correlated financial ratios - leverage measures, coverage measures, liquidity measures, all moving together - into 8 principal components and you gain a compact, uncorrelated, well-conditioned input set. What you lose is per-ratio traceability. Every component is a weighted blend of all 60 ratios, with a loading on each. "Component 3 pushed this application below the cutoff" is not a reason an applicant or a supervisor can act on, and in regimes that require adverse-action reasons or model explainability it is not a compliant answer either. This is the standard cost of any feature *extraction*: the shipped model's inputs are constructed quantities with no independent meaning, so any explanation naturally lands on constructed quantities too. Attribution methods such as SHAP or LIME do not rescue you here on their own - run them over a model whose features are components and they faithfully tell you that component 3 mattered, which is exactly the sentence nobody can use. ## The technical escape: linearity composes There is a genuine and often-missed nuance. PCA is a **linear** map: each component score is the centred ratios multiplied by that component's loadings and summed. If the model on top is also linear - a scorecard, a logistic model - then the score is a linear function of the components, and a linear function of a linear function is a linear function. Multiply the model's component coefficients through the loading matrix and you obtain a single exact weight per original ratio. The shipped model *is* a weighted sum of the 60 ratios; the components were only the route by which those weights were chosen. Two caveats keep this honest. First, truncation to 8 components does change the weights relative to a model fitted on the raw ratios - it constrains them to a subspace - but the composed weights are exactly correct *for the model as shipped*, which is what a regulator is asking about. Second, the composition disappears the moment a nonlinear model sits on top of the components, because there is no longer a single weight per ratio to recover. So if per-ratio explanation is a hard requirement, keeping the stack linear end to end is the design that satisfies it. ## When a component genuinely does mean something Components are not always opaque. On a 40-item employee engagement survey, it is common to find that every item loads on the first component with the same sign and similar magnitude - people who answer positively answer positively everywhere - so reading PC1 as an overall-satisfaction axis is reasonable shorthand. The second component often contrasts groups of items: pay and benefits items loading positive while management and recognition items load negative, which reads as a pay-versus-management tension in the workforce. Use that, but state it correctly. Naming a component is an **interpretation of a loading pattern**, not a measured fact. Before it goes into a report, check that the pattern is stable: refit on a different sample or a later wave and confirm the same items still group the same way. And remember that the sign of a component is arbitrary, so "positive PC2 means pay-focused" is a labelling convention you must fix and document, not a property of the data. ## The decision a lead actually owns This is a scoping call, made before any modelling: - **Is an individual decision explained to a person?** Credit, hiring, insurance pricing, benefits eligibility - here per-feature reasons are usually a duty, and feature extraction fights that duty. Prefer domain-grouped features and supervised selection, so the shipped inputs are quantities a human already names. - **If PCA is worth keeping anyway**, keep the model on top linear so the weights compose back, and document the composition as part of the model's governance package. - **Is nobody owed an explanation?** Internal ranking, anomaly scoring, latency-driven compression, de-correlating inputs for a model whose explanations are not customer-facing - the interpretability cost is close to free, and the conditioning and compactness gains are real. - **Use PCA as a diagnostic even when you do not ship it.** The variance profile of 60 ratios tells you how much genuine independent information the feature set carries. If 8 components hold 95% of the variance, you have learned that your 60 ratios are roughly 8 ideas - which is an argument for pruning to 8 interpretable *ratios*, not necessarily for shipping 8 components. ## Governance details that bite later Components are refit when the data is refreshed, and loadings drift. A story told about PC2 last year may not survive this year's refit, and the sign can flip on a rerun even when nothing substantive changed. If a component's meaning has been documented to a supervisor, refitting silently changes what was documented. Either freeze the loadings between formal model reviews and treat a refit as a model change, or do not build explanations on component identity at all. ## The answer that lands Acknowledge the loss plainly, then show the two ways out: compose the linear weights back to per-ratio contributions if the stack allows it, or decide up front that a regulated, individually-explained decision is the wrong place for feature extraction. Confidence that PCA is simply "a black box, sorry" is as weak an answer as pretending the components are self-explanatory.