Module 6 — Spatio-Temporal Change (Part 2)

PAF 516 | Community Analytics

M6 Overview & Learning Materials

Spatio-Temporal Change (Part 2) — Neighborhood Trajectories

Module Overview and Objectives

Module 5 answered the question: what changed? This module asks the deeper question: how does spatial structure shape change? Two neighborhoods can both show improving hardship scores, but if one is a persistent hot spot surrounded by other high-hardship areas while the other is a spatial outlier that has successfully transitioned, they are on fundamentally different trajectories and call for fundamentally different interventions.

This module introduces spatio-temporal analysis — methods that treat space and time jointly rather than separately. You will classify block groups by their LISA trajectory (how their cluster membership changes between 2010 and 2019), test whether change itself is spatially autocorrelated (does decline spread from neighborhood to neighborhood?), and build a multi-point trend using three time observations to distinguish sustained improvement from noise. The result is a policy-ready map that answers not just where hardship is but where it is heading and whether that trajectory is spreading to neighboring areas.

After completing this module, you will be able to:

  • Classify block groups into LISA trajectory categories by cross-tabulating cluster membership across two time periods
  • Interpret the policy implications of persistent hot spots, emerging hot spots, dissolving hot spots, and spatial outliers across time
  • Test whether change scores are spatially autocorrelated using space-time Moran’s I
  • Interpret a significant space-time Moran’s I as evidence of spatial diffusion of decline or recovery
  • Construct a multi-point hardship trend and classify block groups as consistently improving, consistently worsening, or non-monotone
  • Build a trajectory map communicating neighborhood change pathways for policy audiences

Lecture

The lecture notes cover LISA trajectory analysis, transition matrices, space-time Moran’s I, multi-point trend classification, and policy design for different neighborhood trajectory types.

Download the lecture notes: Spatio-Temporal Analysis — Lecture Notes (PDF)

Section 1: LISA Trajectory Analysis

Module 4 classified each block group by its LISA cluster membership at a single point in time: HH (hot spot), LL (cold spot), HL (spatial outlier), LH (spatial outlier), or NS (not significant). Running LISA separately at two time points generates a transition matrix — a cross-tabulation of where each block group started and where it ended up.

The Transition Matrix

If we run LISA on the 2010 hardship index and again on the 2019 hardship index, each block group belongs to one of 5 × 5 = 25 possible transition cells. Most cells will be empty or near-empty. The policy-relevant categories are:

Trajectory 2013 → 2019 Policy Meaning
Persistent HH HH → HH Deep structural hardship; spatial clustering persists. Needs sustained, place-based investment.
Emerging HH NS/LL → HH Deteriorating neighborhood; cluster is forming. Early intervention window is open.
Dissolving HH HH → NS Recovery underway; cluster has broken up. Study these: what worked?
Persistent LL LL → LL Stably low hardship; little policy urgency from an equity standpoint.
Emerging LL HH/NS → LL Rapid improvement; risk of gentrification-driven displacement.
Persistent NS NS → NS No significant spatial clustering in either period.
HL Outlier HL in either period Pocket of high hardship within a low-hardship area. Isolation compounds vulnerability.

Why Trajectories Matter More Than Snapshots

A LISA map at a single point in time cannot distinguish a neighborhood that has always been a hot spot from one that just became a hot spot last year. The trajectory map makes this distinction explicit. Policy implications differ dramatically:

  • A persistent hot spot likely reflects deep structural conditions — historical disinvestment, infrastructure gaps, school district boundaries — that require multi-year, multi-sector intervention.
  • An emerging hot spot is a neighborhood in transition. The cluster has not yet consolidated; targeted early investment may reverse the trajectory at lower cost.
  • A dissolving hot spot is the most important category for learning: what happened here that broke up the cluster? These neighborhoods are natural experiments.

Section 2: Space-Time Moran’s I

Module 4 tested whether hardship values were spatially autocorrelated at a single time point. A natural extension is to ask: are the change scores themselves spatially autocorrelated? If block groups that worsened tend to be located next to other block groups that also worsened, that is evidence of spatial diffusion — decline spreading across the urban landscape like a contagion. If change is spatially random, that suggests neighborhood-specific idiosyncratic factors rather than spreading processes.

The Test

Space-time Moran’s I applies the same moran.mc() function from Module 4, but with the change score (not the level) as the input variable:

moran.mc(change_df$hardship_change, w, nsim = 999)

The same weights matrix w from Lab 4 (queen contiguity, row-standardized) is used.

Interpreting the Result

  • Significant positive I: Change is spatially clustered. Neighborhoods that improved tend to be near other improving neighborhoods; neighborhoods that worsened tend to cluster. This suggests spatial diffusion — economic recovery or decline propagates spatially, likely through housing markets, business investment, or demographic flows.
  • Non-significant I: Change appears spatially random, driven by neighborhood-specific factors (a new employer, a school closure, a redevelopment project) rather than spatial spread.

A significant space-time Moran’s I has direct policy implications: if decline is spreading spatially, intervening at the boundary of the expanding hot spot (before it fully consolidates) is more cost-effective than intervening only at the core.

Section 3: Multi-Point Trend Analysis

Two time points give us a change score but not a trend. A neighborhood could show apparent improvement from 2010 to 2019 because of a one-time shock in 2010 (a factory closure, a flood) rather than sustained structural change. Adding a middle time point — the 2012–2016 ACS 5-year estimate (year = 2016) — lets us distinguish genuine sustained improvement from noise.

The Trend Slope

For a continuous measure of trajectory, you can fit a simple regression for each block group:

library(purrr)

trend_df <- bind_rows(
  r2013_z %>% mutate(year = 2013),
  r2016_z %>% mutate(year = 2016),
  r2019_z %>% mutate(year = 2019)
) %>%
  group_by(GEOID) %>%
  summarise(
    trend_slope = coef(lm(hardship_index ~ year))[["year"]],
    .groups = "drop"
  )

A positive slope means hardship increased on average across the decade; a negative slope means it decreased. The slope is measured in pooled z-score units per year — a slope of +0.05 means hardship increased by 0.05 standard deviations per year on average.

Section 4: From Trajectories to Policy Design

The trajectory map is not an end in itself — it is a decision support tool. Different trajectory types call for different policy responses:

Persistent Hot Spots

These neighborhoods have been spatially clustered in high hardship for at least a decade. The depth and persistence of clustering suggests structural causes — historical disinvestment, school quality gaps, environmental burden, lack of transit access. Single-program interventions are unlikely to be sufficient. Evidence-based approaches include comprehensive community development (housing + economic development + schools + health simultaneously), place-based funding like Promise Zones or Opportunity Zones, and long-term community wealth-building strategies.

Emerging Hot Spots

These neighborhoods are deteriorating and their spatial clustering is intensifying. The early-intervention window is open. Research on neighborhood tipping points (Schelling, 1969; Galster, 2012) suggests that once a neighborhood crosses certain thresholds of disinvestment, self-reinforcing feedback loops make recovery much costlier. Prevention is cheaper than remediation. Targeted interventions include code enforcement, small business lending, anchor institution engagement, and proactive tenant protection before displacement accelerates.

Dissolving Hot Spots

The cluster has broken up. Hardship is declining and the spatial concentration is dissipating. This does not mean the neighborhood’s work is done — it may be in the early stages of displacement-driven gentrification. Policy priorities shift from stimulating investment to protecting existing residents: anti-displacement measures, community land trusts, affordable housing preservation.

Spatial Outliers (HL across time)

High-hardship islands within low-hardship areas. These communities are spatially isolated from the diffusion of nearby recovery. They may face additional barriers not captured in the hardship index: concentrated public housing, language isolation, lack of transportation connectivity to jobs. They are often systematically undercounted and underserved by place-based programs that target larger contiguous areas.

Readings

Spatio-Temporal Methods

  • Rey, S. J., & Ye, X. (2010). Comparative spatial dynamics of regional systems. In A. Páez et al. (Eds.), Spatial Statistics: Methodological Aspects and Applications (pp. 441–463). Springer. — Introduces the LISA transition matrix framework and its application to regional convergence analysis. The theoretical basis for trajectory classification used in Lab 6.

  • Anselin, L., Syabri, I., & Smirnov, O. (2002). Visualizing multivariate spatial correlation with dynamically linked windows. In W. Getis et al. (Eds.), Spatial Econometrics and Spatial Statistics. Palgrave Macmillan. — Formal treatment of space-time LISA and the interpretation of spatial clustering dynamics.

  • Bivand, R. S., & Wong, D. W. S. (2018). Comparing implementations of global and local indicators of spatial association. TEST, 27(3), 716–748. DOI — Already assigned in Module 4; the LISA implementation details are directly relevant to the two-period analysis in Lab 6.

Neighborhood Change Theory

  • Galster, G. (2012). The mechanism(s) of neighbourhood effects: Theory, evidence, and policy implications. In M. van Ham et al. (Eds.), Neighbourhood Effects Research: New Perspectives (pp. 23–56). Springer. — Systematic review of how neighborhood conditions affect resident outcomes; essential theoretical grounding for interpreting trajectory maps.

  • Sampson, R. J. (2012). Great American City: Chicago and the Enduring Neighborhood Effect. University of Chicago Press. — Chapter 3 (“The Durable Effects of Concentrated Poverty”) — The definitive account of why persistent hot spots are not just correlated with poverty but causally amplify it; motivates the distinction between persistent and emerging hot spots.

  • Ellen, I. G., & Turner, M. A. (1997). Does neighborhood matter? Assessing recent evidence. Housing Policy Debate, 8(4), 833–866. — Classic review establishing that neighborhood context matters for outcomes independently of individual characteristics; justifies place-based policy targeting.

Policy Applications

  • Galster, G., & Tatian, P. (2009). Modeling housing appreciation dynamics in disadvantaged neighborhoods. Journal of Planning Education and Research, 29(1), 7–22. — Demonstrates that housing value trajectories in declining neighborhoods exhibit spatial autocorrelation — the spatial diffusion mechanism that space-time Moran’s I detects.

Lab 6

The Lab 6 materials are on the course lab site.

  • Lab 6 Tutorial — Download the tutorial file, knit it to see the complete analysis, then run chunk by chunk to understand each step.
  • Lab 6 Assignment — Download the assignment file, rename it with your last name, complete the three questions, and submit to Canvas.

Yellowdig Discussion

Spatial trajectory analysis reveals something that neither a static map nor a simple change map can show: the difference between a neighborhood on a sustained recovery path and one that experienced a one-year blip. But the categories themselves — “persistent hot spot,” “dissolving hot spot,” “emerging hot spot” — are statistical labels, not explanations.

Your prompt: Select one specific trajectory category from your Lab 6 map (persistent HH, emerging HH, dissolving HH, or HL outlier) and identify at least two block groups in Maricopa County that fall into that category. Research those areas briefly — what do you know about them? What historical, institutional, or policy factors might explain why they followed that trajectory over the 2013–2019 decade? Drawing on the readings from Modules 5 and 6 (especially Galster 2012 and Sampson 2012), discuss what kinds of interventions would be most appropriate given that trajectory type. Be specific: not “invest in the community” but what type of investment, at what scale, targeting what mechanisms.

Key Terms

Term Definition
LISA Trajectory A block group’s LISA cluster membership at two time points (e.g., HH→HH = persistent hot spot)
Transition Matrix A cross-tabulation of LISA cluster membership at period 1 vs. period 2
Persistent Hot Spot A block group classified HH in both periods — deep structural clustering
Emerging Hot Spot A block group that becomes HH between periods — a deteriorating neighborhood
Dissolving Hot Spot A block group that was HH in period 1 but not in period 2 — recovery underway
Space-Time Moran’s I Moran’s I computed on change scores rather than levels; tests whether change is spatially clustered
Spatial Diffusion The process by which neighborhood decline or recovery spreads to adjacent areas
Multi-Point Trend Using three time observations to classify trajectory as monotone (consistent) or non-monotone
Trend Slope The regression coefficient of hardship on year across three time points; sign indicates improving (-) or worsening (+)
Tipping Point A threshold level of neighborhood disinvestment beyond which self-reinforcing decline becomes difficult to reverse