Iterative Principal-Curve Fitting

From Projections to a Fitted Curve

A principal curve is not specified by a single straight direction: its shape is estimated from the observations themselves. Fitting it therefore requires two linked operations. Given a current curve, project each observation onto it to obtain a curve position; given those positions, smooth the observations to update the curve. Repeating these operations produces an iterative fit.

Assume observations \(x_1,\ldots,x_n\) lie in a Euclidean feature space, such as a chosen low-dimensional representation of cells. The distance used for projection is the Euclidean distance in that space. The feature coordinates and their scaling must be meaningful for that distance: a coordinate measured on a much larger numerical scale can otherwise dominate the projection. Let \(f(t)\) denote a curve parameterized by arc length, so differences in \(t\) measure distance traveled along the curve. For observation \(x_i\), its projection parameter on the current curve \(f^{(r)}\) is a value minimizing its squared distance to the curve.

There may be more than one minimizing value, for example when an observation is equally close to two parts of a curve. The fitting procedure must use a consistent rule for choosing among ties. In computations, the curve is represented by a finite set of points or a smooth basis, so projection and smoothing are implemented numerically rather than over every possible curve.

Why the Smoothing Step Is Natural

Once projection parameters have been assigned, the curve is treated as a function of position along itself. At a given parameter value, the target curve location is the conditional mean of observations having that projection position. The mean arises because it minimizes expected squared Euclidean error. To see this, let \(m=E[X\mid\lambda=t]\) and consider any candidate vector \(a\). Expanding the squared distance and using \(E[X-m\mid\lambda=t]=0\) gives:

The second term is nonnegative and is zero only when \(a=m\). Thus the conditional mean is the least-squares target at each curve position. With finite observations, there are usually too few observations at any exact value of \(t\) to estimate that conditional mean directly. Instead, smooth each coordinate against the projected parameters. A local regression, spline smoother, or another suitable regression smoother can be used. The choice controls how much the fitted curve can bend.

After smoothing, the resulting path may need to be reparameterized by arc length before the next projection step. This keeps the parameter's interpretation consistent: equal increments represent equal distances along the fitted path. A practical implementation may also approximate the smoothed path by a polyline for projection.

The Alternating Updates

This is an alternating procedure, not a guarantee that every choice of smoother will decrease one single objective at every update. Projection reduces distances to a fixed curve, but smoothing may trade local fit for smoothness. Even when updates stabilize, that only shows that the chosen numerical procedure has settled; it does not establish that the curve is globally best among all possible curves.

Assumptions, Diagnostics, and Alternatives

The interpretation of these updates depends on several choices. The observations must be represented in a space where Euclidean distance is meaningful for the intended analysis. The smoother must be appropriate for the number and distribution of projected observations: a very flexible smoother can chase noise, while an overly rigid one can flatten genuine curvature. Projection assignments should be stable enough that smoothing against them is informative. Finally, a single smooth path is a modeling choice. It may be unsuitable when the cell states form distinct paths or when the geometry is not well represented by one curve.

Useful diagnostics include plotting observations and the curve in the fitted feature space, checking the distribution of projection parameters, tracking changes in the curve and the fit criterion across iterations, and inspecting whether a few observations have unusually large distances. Repeated large jumps in projection positions or a curve that oscillates between updates can indicate that the smoothing and projection steps are not settling reliably. A flat-looking curve despite visibly curved data may indicate excessive smoothing; sharp bends that follow individual observations may indicate too little smoothing.

A principal component line is a simpler alternative: it summarizes variation along a straight direction and does not adapt its shape to curved structure. A spline fitted against a fixed external ordering is another alternative, but it assumes that the ordering is supplied independently rather than recalculated as closest positions on the curve. Principal-curve fitting combines a flexible path with data-dependent projection positions, at the cost of iterative computation and sensitivity to the representation and smoothing choices.

• Mistake: Treating a small change in the distance criterion as proof that the inferred trajectory is correct. Correction: Small change indicates numerical stabilization under the chosen representation and smoother; it does not validate the model or the biological interpretation.
• Mistake: Smoothing once and keeping the original projection parameters for all later calculations. Correction: Each updated curve changes the closest points, so recompute projections before the next smoothing update.
• Mistake: Assuming the squared-distance criterion must fall at every step. Correction: The projection step minimizes distances for a fixed curve, but a smoother may impose regularization or otherwise trade fit for smoothness. Monitor the actual procedure rather than assuming monotone improvement.