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Functional Boxplot

The functional boxplot (López-Pintado & Romo, 2009) is the canonical method for summarising a sample of curves in the same spirit as Tukey's scalar boxplot: a depth-ranked median curve, a shaded 50 % central region, whisker/fence lines, and flagged outlier curves. fdars.depth.functional_boxplot implements this construction using any of the four depth measures shipped in fdars.depth.

Functional Boxplot — median, 50% central region, whiskers/fence, flagged outliers

Theory

Functional depth and ranking

Every curve \(X_i\) in the sample is assigned a functional depth score \(D(X_i)\). A larger score means the curve is more central relative to the sample; a score near zero means the curve is extreme or peripheral. functional_boxplot supports four depth measures via the method parameter:

method Description
"fraiman_muniz" Pointwise CDF-based depth (Fraiman & Muniz, 2001)
"band" Band depth — proportion of curve inside sample bands
"modified_band" Modified band depth (default); faster, avoids degenerate bands
"random_projection" Projection depth via random half-space projections

Median

The median is the observed curve with the highest depth score:

\[ i^* = \operatorname*{argmax}_{i \in \{1,\dots,n\}} D(X_i), \qquad \text{median}(t) = X_{i^*}(t). \]

It is always an observed sample curve (not a synthetic average).

50 % central region

Rank all \(n\) curves by depth and keep the deepest half. The central region \(C_{0.5}\) is the pointwise envelope of those \(\lceil n/2 \rceil\) most-central curves:

\[ C_{0.5}(t) = \bigl[\,\mathrm{central\_lower}(t),\; \mathrm{central\_upper}(t)\,\bigr], \]
\[ \mathrm{central\_lower}(t) = \min_{i \in S_{0.5}} X_i(t), \quad \mathrm{central\_upper}(t) = \max_{i \in S_{0.5}} X_i(t), \]

where \(S_{0.5}\) is the index set of the deepest half. The central region plays the role of the interquartile range in a scalar boxplot.

Whiskers / fence

The fence is formed by inflating the central region by factor times its pointwise width:

\[ w(t) = \mathrm{central\_upper}(t) - \mathrm{central\_lower}(t), \]
\[ \mathrm{whisker\_upper}(t) = \mathrm{central\_upper}(t) + \texttt{factor} \cdot w(t), \qquad \mathrm{whisker\_lower}(t) = \mathrm{central\_lower}(t) - \texttt{factor} \cdot w(t). \]

The default factor = 1.5 follows the Tukey convention. Sun & Genton (2011) recommend factor = 1.5 for detecting genuine outliers while avoiding swamping.

Outlier flagging

A curve is an outlier if it exceeds the fence at any evaluation point:

\[ X_i \text{ is an outlier} \iff \exists\, t :\; X_i(t) > \mathrm{whisker\_upper}(t) \;\text{or}\; X_i(t) < \mathrm{whisker\_lower}(t). \]

Outlier indices are returned as a Python list of int (0-based row indices).

API

from fdars.depth import functional_boxplot

result = functional_boxplot(data, method="modified_band", factor=1.5)

Parameters

Parameter Type Default Description
data ndarray (n, m) — Functional data matrix; rows are observations. Requires \(n \ge 2\).
method str "modified_band" Depth measure: "fraiman_muniz", "band", "modified_band", or "random_projection".
factor float 1.5 Fence inflation factor (Tukey convention). Must be \(\ge 0\).
scale bool True Passed to "fraiman_muniz" depth.
nproj int 50 Number of random projection directions for "random_projection".
seed int \| None None RNG seed for "random_projection" (None resolves to 0).

Returns — a dict with seven keys:

Key Shape Description
"median" (m,) The deepest observed curve's values.
"central_lower" (m,) Pointwise lower bound of the 50 % central region.
"central_upper" (m,) Pointwise upper bound of the 50 % central region.
"whisker_lower" (m,) Lower fence: central_lower − factor × width.
"whisker_upper" (m,) Upper fence: central_upper + factor × width.
"outliers" Python list[int] 0-based row indices of curves outside the fence.
"depths" (n,) Per-curve depth scores (the ranking used to build the boxplot).

Worked example

The fence below loads Canadian daily temperature curves (35 stations × 365 days), downsamples to every other day to keep compute tiny, and runs functional_boxplot with default settings. It then plots the canonical López-Pintado–Romo picture and reports the flagged outlier stations.

import numpy as np
from docs_data import load_canadian_weather
from docs_fig import fig, render
from fdars.depth import functional_boxplot

day, X, meta = load_canadian_weather("temperature")
# X: 35 stations × 365 days  → downsample to every other day (183 pts) for speed
X_sub = X[:, ::2]
day_sub = day[::2]

bp = functional_boxplot(X_sub)

median = np.asarray(bp["median"])
cl     = np.asarray(bp["central_lower"])
cu     = np.asarray(bp["central_upper"])
wl     = np.asarray(bp["whisker_lower"])
wu     = np.asarray(bp["whisker_upper"])
out_idx = bp["outliers"]

f, ax = fig(figsize=(10.0, 4.2))

# All curves faint
for i, xi in enumerate(X_sub):
    color = "#dc3545" if i in out_idx else "#adb5bd"
    lw    = 1.8       if i in out_idx else 0.6
    alpha = 0.9       if i in out_idx else 0.35
    ax.plot(day_sub, xi, color=color, lw=lw, alpha=alpha)

# 50 % central region shaded
ax.fill_between(day_sub, cl, cu, color="#3f51b5", alpha=0.18, label="50 % central region")

# Fence lines
ax.plot(day_sub, wu, color="#e8710a", lw=1.4, ls="--", label=f"whisker (factor={1.5})")
ax.plot(day_sub, wl, color="#e8710a", lw=1.4, ls="--")

# Median curve bold
ax.plot(day_sub, median, color="#3f51b5", lw=2.4, label="median (deepest curve)")

ax.set(title="Functional Boxplot — Canadian Weather Temperature",
       xlabel="day of year", ylabel="temperature (°C)")
ax.legend(fontsize=9)
print(render(f))

n_out = len(out_idx)
if n_out:
    names = [meta["station"].iloc[i] for i in out_idx]
    print(f"Flagged outliers ({n_out}): indices {out_idx} → {names}  FDARS_FENCE_OK")
else:
    print(f"Flagged outliers: none at factor=1.5  FDARS_FENCE_OK")
image/svg+xml Matplotlib v3.11.1, https://matplotlib.org/
Flagged outliers (6): indices [6, 18, 31, 32, 33, 34] → ['Scheffervll', 'Churchill', 'Yellowknife', 'Iqaluit', 'Inuvik', 'Resolute'] FDARS_FENCE_OK

Interpreting the result

Stations flagged as outliers exceed the fence at at least one point in the year. For Canadian temperatures this typically picks out arctic stations (very cold winters) or Pacific-coast stations (unusually mild winters). Increase factor to 2.0 to suppress borderline cases; decrease it to flag more.

Numeric only — no plot helper yet

functional_boxplot returns the numeric dict. fdars.plot does not yet ship a plot_functional_boxplot() convenience function. Build your own figure from the seven keys as shown above.

See also

  • Outlier Detection — LRT-based, outliergram, and magnitude-shape outlier detectors.
  • Depth Functions — the depth measures (fraiman_muniz, band, modified_band, random_projection) used to rank curves in the boxplot.

References

  1. López-Pintado, S., and Romo, J. (2009). "On the concept of depth for functional data." Journal of the American Statistical Association, 104(486), 718–734. — the depth-fence functional boxplot this page implements.
  2. Sun, Y., and Genton, M. G. (2011). "Functional boxplots." Journal of Computational and Graphical Statistics, 20(2), 316–334. — enhanced functional boxplot with the factor = 1.5 fence convention and visualisation refinements.