Expose PC4 and make the PCA behavioral map configurable

- compute_pca now fits 4 components; PC4 added to PC_LABELS/PC_BLURB
  ("off-peak spread vs headliner pull", ~5% variance, framed as
  diminishing returns).
- "Read an axis" loadings explorer gains PC4.
- The behavioral-map scatter becomes interactive: pick any two
  components for the X/Y axes, and color by category tag, spatial
  area, or any component's loading (continuous RdBu, symmetric).
- Headline caption pinned to evr[:3] so "3 axes = 60%" stays true;
  the by-hour clock stays PC1-3 to preserve the three-axes story.

Co-Authored-By: Claude Opus 4.7 <noreply@anthropic.com>

app.py

@@ -91,6 +91,7 @@ PC_LABELS = {
     "PC1": "Overall busyness",
     "PC2": "Kiddieland ↔ main midway",
     "PC3": "Daytime crowd ↔ evening crowd",
+    "PC4": "Off-peak spread ↔ headliner pull",
 }
 PC_BLURB = {
     "PC1": "How many people are in the park. Every ride loads the same sign — "
@@ -101,6 +102,13 @@ PC_BLURB = {
     "PC3": "What time of day it is. Daytime family flat-rides the school/daycare "
            "groups hit at open (positive) vs. the evening coaster & date-night "
            "crowd (negative).",
+    "PC4": "The faint fourth axis (~5% of variance — diminishing returns). A "
+           "contrast between a few secondary rides (super round-up, tumbling "
+           "timbers, flying tigers — positive) and the headliners (Phoenix, "
+           "Impulse, Flying Turns — negative). Its score drifts with the "
+           "calendar: positive in quiet spring weekday mornings, negative on "
+           "busy summer/fall weekend evenings when demand piles onto the "
+           "marquee rides. Real, but mostly where the clean signal ends.",
 }
 CATEGORY_COLORS = {"Family": "#4C9BE8", "Kiddie": "#E8A33D", "Thrill": "#E0524E"}
 
@@ -161,12 +169,12 @@ def compute_pca():
     )
 
     scaled = StandardScaler().fit_transform(matrix)  # each ride mean 0, var 1
-    pca = PCA(n_components=3)
+    pca = PCA(n_components=4)  # PC4 is faint (~5%) — exposed for exploration only
     sc = pca.fit_transform(scaled)
 
-    scores = pd.DataFrame(sc, index=matrix.index, columns=["PC1", "PC2", "PC3"])
-    loadings = pd.DataFrame(pca.components_.T, index=matrix.columns,
-                            columns=["PC1", "PC2", "PC3"])
+    pc_cols = ["PC1", "PC2", "PC3", "PC4"]
+    scores = pd.DataFrame(sc, index=matrix.index, columns=pc_cols)
+    loadings = pd.DataFrame(pca.components_.T, index=matrix.columns, columns=pc_cols)
     loadings = loadings.join(meta)
 
     # Spatial "areas" from GPS alone — k-means on standardized lat/lon, the same
@@ -406,30 +414,53 @@ with tab4:
             col.metric(f"{pc} · {PC_LABELS[pc]}", f"{evr[int(pc[-1]) - 1]:.1%}",
                        help=PC_BLURB[pc])
         st.caption(f"Just these 3 axes (out of {loadings.shape[0]}) capture "
-                   f"**{evr.sum():.1%}** of all the variation in park wait times.")
+                   f"**{evr[:3].sum():.1%}** of all the variation in park wait times.")
 
         st.markdown("---")
 
-        # --- the behavioral map: PC2 vs PC3, colored by the park's own tag ---
+        # --- the behavioral map: any two components, freely colored ---
         st.markdown("#### The map of the park, drawn from wait times alone")
         st.markdown(
-            "Each ride placed by its **PC2** (left↔right = Kiddieland↔midway) and "
-            "**PC3** (bottom↔top = evening↔daytime) loading — its *behavioral* "
-            "position — and colored by the park's category tag. The position comes "
-            "purely from how each queue moves over time, yet **the picture is the "
-            "park**: Kiddieland huddles left, evening thrill rides sink bottom-right, "
-            "daytime family rides float to the top."
+            "Each ride placed by two of its component loadings — its *behavioral* "
+            "position, drawn purely from how its queue moves over time. The "
+            "default (**PC2 × PC3**, colored by category) is the write-up's view: "
+            "Kiddieland huddles left, evening thrill rides sink bottom-right, "
+            "daytime family rides float to the top. **Swap either axis** to any "
+            "component, or **recolor by a third component's loading**, to hunt for "
+            "structure the default hides."
         )
+        pcs = ["PC1", "PC2", "PC3", "PC4"]
+        axis_labels = {p: f"{p} · {PC_LABELS[p]}" for p in pcs}
+        axis_labels.update({"category": "Park's tag", "area": "Spatial area"})
+        mc1, mc2, mc3 = st.columns(3)
+        with mc1:
+            xpc = st.selectbox("X axis", pcs, index=1, key="map_x",
+                               format_func=lambda p: f"{p} — {PC_LABELS[p]}")
+        with mc2:
+            ypc = st.selectbox("Y axis", pcs, index=2, key="map_y",
+                               format_func=lambda p: f"{p} — {PC_LABELS[p]}")
+        with mc3:
+            color_opt = st.selectbox(
+                "Color by", ["Category tag", "Spatial area"] + [f"{p} loading" for p in pcs],
+                key="map_color")
+
+        if xpc == ypc:
+            st.info("Pick two *different* components for the axes.")
+        else:
             plot_df = loadings.reset_index().copy()
             plot_df["category"] = plot_df["category"].fillna("Untagged")
-        fig_map_pca = px.scatter(
-            plot_df, x="PC2", y="PC3", color="category", text="ride_name",
-            color_discrete_map=CATEGORY_COLORS,
-            labels={"PC2": "PC2  ·  Kiddieland ←→ main midway",
-                    "PC3": "PC3  ·  evening crowd ←→ daytime crowd",
-                    "category": "Park's tag"},
-            hover_name="ride_name", height=620,
-        )
+            common = dict(x=xpc, y=ypc, text="ride_name", hover_name="ride_name",
+                          height=620, labels=axis_labels)
+            if color_opt == "Category tag":
+                fig_map_pca = px.scatter(plot_df, color="category",
+                    color_discrete_map={**CATEGORY_COLORS, "Untagged": "#888"}, **common)
+            elif color_opt == "Spatial area":
+                fig_map_pca = px.scatter(plot_df, color="area", **common)
+            else:
+                cpc = color_opt.split()[0]  # "PC3 loading" -> "PC3"
+                lim = plot_df[cpc].abs().max()  # symmetric scale → white at 0
+                fig_map_pca = px.scatter(plot_df, color=cpc, color_continuous_scale="RdBu",
+                    range_color=[-lim, lim], **common)
             fig_map_pca.update_traces(textposition="top center",
                                       textfont=dict(size=9, color="#9aa0a6"),
                                       marker=dict(size=11, line=dict(width=1, color="#1A1C18")))
@@ -539,7 +570,7 @@ with tab4:
         lcol, rcol = st.columns([1, 2])
         with lcol:
             st.markdown("#### Read an axis")
-            pick = st.radio("Component", ["PC1", "PC2", "PC3"],
+            pick = st.radio("Component", ["PC1", "PC2", "PC3", "PC4"],
                             format_func=lambda p: f"{p} — {PC_LABELS[p]}")
             st.info(PC_BLURB[pick])
         with rcol:
@@ -565,7 +596,7 @@ with tab4:
             "evening as crowds build; **PC3** humps at midday (the school-group "
             "window) then falls — the fingerprint of *time of day*."
         )
-        by_hour = scores.groupby(scores.index.hour).mean()
+        by_hour = scores[["PC1", "PC2", "PC3"]].groupby(scores.index.hour).mean()
         by_hour.index.name = "hour"
         hour_long = by_hour.reset_index().melt("hour", var_name="Component",
                                                value_name="Mean score")