ggplot(tp_wq, aes(year_frac, result)) +
geom_point(alpha = 0.5) +
geom_smooth(se = FALSE) +
labs(title = "Total Phosphorus — raw scale", y = "Phosphorus (mg/L)") +
nlt_theme
LOESS Smoothing
Total Phosphorus — the same rise-then-plateau, a smoother local fit
1 A local regression instead of a local average
This is the same Total Phosphorus record used for k-Nearest Neighbors Regression — a rise through the middle years that settles onto a plateau — fit here with LOESS instead, for a direct comparison of two local methods on identical data. Where k-NN predicts each point as the flat average of its \(k\) nearest neighbors, LOESS fits a separate weighted regression at every query point \(x_0\):
\[\hat\beta(x_0) = \arg\min_\beta \sum_{i=1}^n w_i(x_0)\left(y_i - x_i^\top \beta\right)^2, \qquad w_i(x_0) = \left(1 - \left|\frac{x_i - x_0}{h}\right|^3\right)^3_+,\]
using only the nearby observations — the tricube weight \(w_i(x_0)\) is largest for points right at \(x_0\), decays smoothly to zero at the edge of the neighborhood, and is exactly zero beyond it, where \(h\) is set so the neighborhood always contains the proportion of the data given by span. stats::loess() is base R — no additional package is required. The response is modeled on the log scale, for the same reasons covered in Transformations.
degree sets the order of each local fit, not the shape of the whole curve. degree = 0 is a local weighted average — closer in spirit to k-NN’s flat average than to a regression at all. degree = 1 fits a local straight line at every point; degree = 2 fits a local parabola. It’s easy to misread degree = 1 as meaning the fitted curve is a single straight line end to end, but that’s not what’s happening: LOESS fits a different local line at every target point, so degree = 1 still produces a curve that bends over the full record, made up of many local linear pieces stitched together. lm(y ~ x) fits one slope for the entire dataset; loess(y ~ x, degree = 1) fits many, each valid only in its own neighborhood.
2 The shape in the record
Same shape as in the k-NN chapter: a low, roughly flat start, a rise through the middle years, and a plateau at a higher level for the rest of the record.
3 Fitting the shape
grid <- data.frame(year_frac = seq(min(tp_wq$year_frac), max(tp_wq$year_frac), length.out = 300))
span_degree_grid <- expand.grid(span = c(0.3, 0.5, 0.75), degree = c(1, 2))
resid_sd <- mapply(function(sp, deg) {
m <- loess(log(result) ~ year_frac, data = tp_wq, span = sp, degree = deg)
sd(resid(m))
}, span_degree_grid$span, span_degree_grid$degree)
data.frame(span_degree_grid, resid_sd = round(resid_sd, 4)) span degree resid_sd
1 0.30 1 0.1352
2 0.50 1 0.1413
3 0.75 1 0.1596
4 0.30 2 0.1315
5 0.50 2 0.1359
6 0.75 2 0.1400Residual spread drops as span narrows and rises as degree increases within a fixed span — both expected, since a narrower neighborhood and a higher local order each add flexibility. Taken to an extreme, though, a narrow span starts fitting individual points rather than the underlying shape, the same overfitting risk k-NN showed at small \(k\). span = 0.5, degree = 1 is used below: local windows wide enough to average over the record’s monthly noise, without narrowing so far that the fit reduces to interpolating between points.
fit_tp_loess <- loess(log(result) ~ year_frac, data = tp_wq, span = 0.5, degree = 1)
grid <- grid |> mutate(fit = exp(predict(fit_tp_loess, newdata = grid)))4 Comparing the two local fits
knn_fit <- caret::knnreg(log(result) ~ year_frac, data = tp_wq, k = 20)
grid <- grid |> mutate(knn_fit = exp(predict(knn_fit, newdata = grid)))
compare_df <- grid |>
select(year_frac, LOESS = fit, `k-NN` = knn_fit) |>
tidyr::pivot_longer(c(LOESS, `k-NN`), names_to = "method", values_to = "fit")
ggplot() +
geom_point(data = tp_wq, aes(year_frac, result), alpha = 0.3, color = "grey40") +
geom_line(data = compare_df, aes(year_frac, fit, color = method), linewidth = 1) +
labs(title = "Total Phosphorus — LOESS vs. k-NN", y = "Phosphorus (mg/L)", color = NULL) +
nlt_theme +
theme(legend.position = "right")
The two curves track the same rise-and-plateau shape closely, but LOESS’s is visibly smoother through the transition — a consequence of fitting a local line rather than a flat local average, so the curve doesn’t have to jump between step-like neighborhood means as the window slides along the record.
5 No inferential story here either
LOESS doesn’t recover what k-NN gave up. There’s no global coefficient, no standard error, and no significance test attached to this fit — loess() doesn’t return an object structured to support one the way lm() or mgcv::gam() do. This isn’t a limitation specific to the implementation; it’s a structural consequence of fitting a separate regression at every point rather than a single set of parameters over the whole record. LOESS is a genuinely useful tool for prediction, but it comes at the cost of the inferential interpretability a parametric or semi-parametric model provides. Its role in this progression is the smooth-curve bridge to the next chapter, not an answer to “is this trend statistically real.”
6 Interpretation
LOESS reproduces the same rise-then-plateau shape as k-NN, more smoothly, and confirms the two local methods agree on where the record actually bends. Neither, though, can say whether the apparent plateau is a real leveling-off or just where the local windows happen to average out — that requires a model with an actual significance test attached. Generalized Additive Models, next, keep the same flexible-curve idea but restore that inferential machinery: a fitted smooth term with a p-value, and a confidence band that means something.