9.1 Concepts and Data

Part: 9 — Spatio-Temporal Analysis (Chapter 9.1 of 7)

Dataset: data_1998_2010_long_lbc.csv (3,107 counties x 15 years, 1998-2012); crime_data_2018.csv (14,001 geocoded incidents)

Focus: space-time dependence; the locations x time "cube"; separable vs. non-separable structure; loading both datasets honestly

Library: spatialstats.spacetime is an empty stub — this Part builds everything from esda, cluster, and gwmodel directly

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Table of Contents

1. Overview

2. Python Setup

3. A Note on This Part's Library Support

4. The Space-Time Cube

5. Separable vs. Non-Separable Structure

6. The County-Panel Dataset

7. The Crime Point Dataset: a CRS Gotcha, Verified

8. Summary and Conclusions

9. Resources

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1. Overview

Every earlier Part treated space and (implicitly) a single point in time together. This Part adds time as a genuine second dimension of structure — not just "the same analysis, repeated at different dates," but the question of whether patterns in time are themselves spatially patterned, and vice versa.

Step	What we do
Be honest first	spatialstats.spacetime ships as an empty stub — Section 3 says exactly what that means for this Part
The cube	Locations x time as a 3-dimensional array — and why "flatten it and use Part 2's tools" is not always safe
Separability	When a space-time process factors into a purely spatial part times a purely temporal part, and when it does not
Two datasets	A 15-year, 3,107-county panel (RATE, SMOKING, POVERTY, PM25, NO2, SO2) and 14,001 geocoded 2018 crime incidents

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2. Python Setup

import sys, warnings
from pathlib import Path

import numpy as np
import pandas as pd
import geopandas as gpd
import matplotlib.pyplot as plt

warnings.filterwarnings("ignore", category=FutureWarning)
warnings.filterwarnings("ignore", category=DeprecationWarning)


def find_repo_root(start: Path = Path.cwd()) -> Path:
    for p in [start, *start.parents]:
        if (p / "pyproject.toml").exists() and (p / "data").is_dir():
            return p
    raise FileNotFoundError("Run this notebook from inside the PySpatialStats repository.")


ROOT = find_repo_root()
DATA = ROOT / "data"
try:
    import spatialstats
except ImportError:
    sys.path.insert(0, str(ROOT))
    import spatialstats

plt.rcParams.update({
    "figure.dpi": 100, "savefig.dpi": 100, "font.size": 10, "axes.titlesize": 11,
    "axes.titleweight": "bold", "axes.spines.top": False, "axes.spines.right": False,
    "axes.grid": True, "grid.alpha": 0.25, "legend.frameon": False,
})
COLORS = {"blue": "#2b6cb0", "red": "#c0392b", "green": "#2f855a", "orange": "#dd6b20",
          "grey": "#718096", "purple": "#6b46c1", "teal": "#2c7a7b"}
print(f"spatialstats : version {spatialstats.__version__}")

spatialstats : version 0.2.0

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3. A Note on This Part's Library Support

Checked directly, not assumed:

import spatialstats.spacetime as st

print(f"spatialstats.spacetime exports: {[a for a in dir(st) if not a.startswith('_')]}")
print(f"\n{st.__doc__}")

spatialstats.spacetime exports: []


spatialstats.spacetime
======================
Spatio-temporal methods (planned).

Future API will cover space-time weights and autocorrelation, emerging hot spots,
space-time KDE, ST kriging, and spatial panel regression.

The module is genuinely empty — a stub with a docstring describing planned scope (space-time weights and autocorrelation, emerging hot spots, space-time KDE, ST kriging, spatial panel regression) and zero implemented functions. This is not a gap this Part papers over. Every chapter that follows says explicitly, up front, which of its content is:

• Directly supported by an existing module (esda.Moran/Moran_Diff applied year by year, cluster.getis_ord_gi per year, gwmodel.GTWR) — genuine library functionality, just applied across a time dimension rather than once.
• Built by hand in the notebook, from smaller pieces this library already provides (a Knox test, Mann-Kendall trend categorization, two-way fixed-effects panel regression) — real, working code, but not a single spatialstats function call.
• Not attempted, where the honest answer is that a faithful version needs infrastructure (a true space-time variogram, a Kronecker space-time weights matrix, non-separable ST-KDE) this repository does not yet have, and where hand-building a convincing approximation risks teaching something subtly wrong.

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4. The Space-Time Cube

Data indexed by both location and time naturally form a cube: one axis for space (itself often two-dimensional), one axis for time. Part 2's tools (Moran's I, LISA) were all built for a single 2-D slice of this cube — a snapshot. The central methodological question of this whole Part is: what changes when the cube has a third axis, and when is "just slice it and reuse Part 2's tools" actually a safe simplification?

panel = pd.read_csv(DATA / "data_1998_2010_long_lbc.csv")
print(f"shape: {panel.shape}  ({panel['FIPS'].nunique()} counties x {panel['Year'].nunique()} years)")
print(f"year range: {panel['Year'].min()} to {panel['Year'].max()}  "
     f"({'matches' if panel['Year'].max()-panel['Year'].min()+1==15 else 'does NOT match'} the outline's '15 years')")
print(f"columns: {panel.columns.tolist()}")

cube_shape = (panel["FIPS"].nunique(), panel["Year"].nunique())
print(f"\nthe 'cube' for a single variable (RATE) has shape {cube_shape}: "
     f"{cube_shape[0]:,} counties x {cube_shape[1]} years = {cube_shape[0]*cube_shape[1]:,} cells")
print(f"actual rows in the file: {len(panel):,}  (balanced panel: every county observed every year)")

shape: (46605, 10)  (3107 counties x 15 years)
year range: 1998 to 2012  (matches the outline's '15 years')
columns: ['FIPS', 'X', 'Y', 'Year', 'SMOKING', 'POVERTY', 'PM25', 'NO2', 'SO2', 'RATE']

the 'cube' for a single variable (RATE) has shape (3107, 15): 3,107 counties x 15 years = 46,605 cells
actual rows in the file: 46,605  (balanced panel: every county observed every year)

The file is named ..._2010_... but actually spans 1998 to 2012 (15 years) — a real, checkable discrepancy between the filename and the data worth confirming directly rather than trusting the name, exactly the kind of small provenance mismatch this whole tutorial has repeatedly found worth verifying rather than assuming away. The panel is balanced: every one of the 3,107 counties has exactly one row per year, all 15 years, no gaps — confirmed by the row count matching counties x years exactly.

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5. Separable vs. Non-Separable Structure

A space-time covariance (or, informally, a space-time pattern) is separable if it factors into a purely spatial component times a purely temporal component: $$ C(\mathbf{s}, t) = C_s(\mathbf{s}) \cdot C_t(t) $$ Separability is a strong, convenient assumption — it means the spatial pattern of correlation looks the same at every point in time, just rescaled by a time-varying factor, and a great deal of space-time machinery (including much of what spatialstats.spacetime's stub docstring describes) is built to exploit it when it holds. It is also frequently wrong in practice: a disease outbreak that starts in one region and spreads outward has a spatial correlation structure that genuinely changes shape over time (an expanding cluster), not just in magnitude — exactly the kind of pattern a separable model cannot represent no matter how its parameters are tuned. Chapter 9.3's per-year Gi* categorization (new, intensifying, persistent, diminishing, sporadic, oscillating hot spots) is, in effect, a hand-built way of detecting non-separable behaviour: a hot spot whose spatial footprint genuinely moves or changes shape over the 15 years is showing exactly the kind of time-varying spatial structure a separable model would smooth away.

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6. The County-Panel Dataset

The same national dataset Chapters 6.10–6.12 used for GTWR and performance timing — worth previewing here on its own terms, since this Part uses it far more extensively.

print(panel.describe().round(2))

fig, axes = plt.subplots(1, 2, figsize=(13, 5))
yearly_mean = panel.groupby("Year")["RATE"].mean()
axes[0].plot(yearly_mean.index, yearly_mean.values, "o-", color=COLORS["blue"])
axes[0].set_xlabel("Year"); axes[0].set_ylabel("mean RATE (national)")
axes[0].set_title("National mean RATE, 1998-2012")

snapshot_2012 = panel[panel["Year"] == 2012]
sc = axes[1].scatter(snapshot_2012["X"], snapshot_2012["Y"], c=snapshot_2012["RATE"], s=3, cmap="viridis")
plt.colorbar(sc, ax=axes[1], label="RATE")
axes[1].set_title("A single slice of the cube: RATE, 2012")
axes[1].set_aspect("equal")
plt.tight_layout()
plt.savefig("panel_overview.png", dpi=100, bbox_inches="tight")
plt.show()

           FIPS           X           Y      Year   SMOKING   POVERTY  \
count  46605.00    46605.00    46605.00  46605.00  46605.00  46605.00   
mean   30666.87   387333.70  1761335.86   2005.00     25.78     14.84   
std    14981.37   965741.77   559176.97      4.32      4.04      6.01   
min     1001.00 -2303830.44   343702.67   1998.00      7.10      1.70   
25%    19043.00  -171368.89  1340753.45   2001.00     23.20     10.60   
50%    29211.00   505203.32  1772684.13   2005.00     25.90     13.90   
75%    46007.00  1121743.84  2160297.04   2009.00     28.50     18.10   
max    56045.00  2200344.81  3126214.37   2012.00     44.00     62.00   

           PM25       NO2       SO2      RATE  
count  46605.00  46605.00  46605.00  46605.00  
mean       9.87      1.55      0.04     65.91  
std        3.50      1.39      0.06     16.80  
min        1.04      0.06      0.00     11.18  
25%        7.21      0.65      0.00     54.00  
50%        9.66      1.10      0.02     65.39  
75%       12.24      1.97      0.05     76.83  
max       26.05     21.56      0.63    223.60  

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7. The Crime Point Dataset: a CRS Gotcha, Verified

crime_data_2018.csv carries two coordinate pairs: long/lat and x_m/y_m. The outline warns the projected pair sits in a "mis-zoned CRS" — worth confirming directly, the same way earlier Parts always verified rather than assumed a provenance warning.

crime = pd.read_csv(DATA / "crime_data_2018.csv")
print(f"rows: {len(crime):,}   incident types: {crime['incident_type'].nunique()}")
print(f"long range: {crime['long'].min():.3f} to {crime['long'].max():.3f}  "
     f"(Buffalo, NY area, consistent with a US city)")
print(f"x_m range:  {crime['x_m'].min():,.0f} to {crime['x_m'].max():,.0f}")

crime_gdf = gpd.GeoDataFrame(crime, geometry=gpd.points_from_xy(crime["long"], crime["lat"]), crs="EPSG:4326")
utm17 = crime_gdf.to_crs("EPSG:32617")   # WGS84 UTM zone 17N -- correct zone for -78.9 longitude
print(f"\ncorrectly projected (UTM 17N) x range: {utm17.geometry.x.min():,.0f} to {utm17.geometry.x.max():,.0f}")
print(f"file's own x_m range:                  {crime['x_m'].min():,.0f} to {crime['x_m'].max():,.0f}")
print(f"do these match (within 10km)? {abs(utm17.geometry.x.min() - crime['x_m'].min()) < 10_000}")

rows: 14,001   incident types: 8
long range: -78.912 to -78.798  (Buffalo, NY area, consistent with a US city)
x_m range:  3,607,134 to 3,623,134

correctly projected (UTM 17N) x range: 670,309 to 679,722
file's own x_m range:                  3,607,134 to 3,623,134
do these match (within 10km)? False

Confirmed: x_m/y_m do not match any sensible projection of the same long/lat values — correctly projecting to UTM zone 17N (the right zone for this longitude) gives coordinates on a completely different scale than the file's own x_m/y_m column. Whatever CRS produced x_m/y_m, it is not consistent with the long/lat pair in the same file. This chapter, and every later one in this Part that uses point locations, projects fresh from long/lat rather than trusting x_m/y_m — exactly the outline's own guidance, verified rather than taken on faith.

crime_utm = crime_gdf.to_crs("EPSG:32617")
crime["x_utm"] = crime_utm.geometry.x
crime["y_utm"] = crime_utm.geometry.y

fig, ax = plt.subplots(figsize=(7, 7))
sc = ax.scatter(crime["x_utm"], crime["y_utm"], s=3, alpha=0.3, c=crime["hour_of_day"], cmap="twilight")
plt.colorbar(sc, ax=ax, label="hour_of_day")
ax.set_title(f"{len(crime):,} geocoded 2018 incidents, correctly projected (UTM 17N)")
ax.set_aspect("equal")
plt.tight_layout()
plt.savefig("crime_overview.png", dpi=100, bbox_inches="tight")
plt.show()

print(crime["incident_type"].value_counts())

incident_type
Theft                   6652
Assault                 2770
Breaking & Entering     2377
Theft of Vehicle        1014
Robbery                  834
Other Sexual Offense     158
Sexual Assault           132
Homicide                  64
Name: count, dtype: int64

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8. Summary and Conclusions

This chapter set the conceptual stage for the whole Part, and established both datasets on solid ground before any real analysis begins.

What we did

1. Confirmed directly that spatialstats.spacetime is an empty stub — zero implemented functions — and laid out how each following chapter will be explicit about what is library-supported, hand-built, or not attempted.
2. Introduced the space-time "cube" and confirmed the county panel is genuinely balanced: 3,107 counties x 15 years, no gaps.
3. Found the file's own name (..._2010_...) understates its actual span (1998–2012, 15 years) — a small, checkable provenance mismatch.
4. Explained separability and why real space-time processes (an expanding disease cluster, a moving hot spot) often are not separable — directly motivating Chapter 9.3's per-year hot-spot categorization.
5. Confirmed the crime dataset's x_m/y_m columns do not match any sensible projection of its own long/lat — verified numerically, not just taken on the outline's word — and established long/lat, freshly projected to UTM 17N, as the trustworthy coordinate source for the rest of this Part.

Practical takeaways

• Before trusting a spatialstats.spacetime-shaped claim anywhere in this Part, check Section 3's list: is this a real library call, a hand-built demonstration, or explicitly out of scope?
• A balanced panel (every unit observed every period, no gaps) is a genuine convenience — always check before assuming it, the way Section 6 did here.
• When a dataset ships two coordinate systems, verify they actually agree before picking one — Section 7's check took two lines of code and caught a real, silent inconsistency.

Next steps

Chapter 9.2 (Space-Time Weights and Global Analysis) puts the county panel to work: year-by-year Moran's I, Moran_Diff for testing whether change itself is spatially clustered, and a hand-built Knox test for space-time interaction in the crime data.

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9. Resources

Textbooks

Reference	Notes
Cressie, N. & Wikle, C. K. (2011). Statistics for Spatio-Temporal Data. Wiley.	The separable/non-separable framing in Section 5, and this Part's overall theoretical foundation

In this library

Object	Role
spatialstats.spacetime	The stub this whole Part works around, confirmed empty in Section 3
Part 2	The single-snapshot autocorrelation tools this Part extends across time
Chapter 9.2	The first real analysis, built from esda directly