Spatial Science and Urban Systems, UC Irvine Spatial Science and Urban Systems, UC Irvine

Department of Urban Planning and Public Policy · UC Irvine

We build open-source methods, models, and software for understanding sociospatial structure.

SSUS is the research lab of Elijah Knaap at the University of California, Irvine. We study who lives where and why: how households sort across neighborhoods, how housing and land markets price location, and how the resulting patterns of segregation and opportunity change over time.

Outcomes in one place depend on its neighbors, so these questions need spatial models: we develop spatial econometric and Bayesian models of sorting and spillovers, and the sparse numerical tools that make them feasible for whole metropolitan regions.

The questions come from policy. Our models have informed a federal fair-housing evaluation and a regional plan, and show where harm-reduction services would reach the most overdose risk. The methods behind them ship as open-source software, so others can check the analysis and apply it to their own decisions.

Research areas

Urban spatial structure

Fields

  • Urban economics
  • Residential sorting
  • Housing and land markets
  • Neighborhood dynamics

Methods

  • Discrete choice models
  • Land value models
  • Agglomeration measurement
  • Geodemographic clustering

Spatial science

Fields

  • Spatial econometrics
  • Spatial statistics
  • Spatial data science
  • Urban analytics
  • Computational social science

Methods

  • Spatial autoregressive models
  • Bayesian estimation at scale
  • Areal interpolation
  • Regionalization
  • Network accessibility

Inequality and policy

Fields

  • Segregation
  • Neighborhood effects
  • Access to opportunity
  • Housing policy
  • Public health

Methods

  • Multiscalar segregation profiles
  • Opportunity mapping
  • Facility location models
  • Space-time analysis
Nine small line charts, one per metropolitan area, each showing a segregation index falling as the distance defining the neighborhood grows, with one line per census year from 1980 to 2010.

From the work

Segregation depends on the scale you measure it at

A multiscalar profile recomputes segregation as the neighborhood around each location widens. Here, nine U.S. metropolitan areas from 1980 to 2010: the curves fall with distance, and the shape of the fall differs from one metro to the next.

The segregation package computes these profiles

Projects

All projects

Current

Spatial Concordance Indices of Income Segregation

A new spatial segregation index that locates where in a region's income distribution its geography is most organized.

Two line charts of bias in the posterior mean of rho against the true value of rho, negative values at left and positive at right. Six log-determinant methods lie on zero through the interior; near rho of 0.9 and above, the Barry–Pace line climbs to about 1.4 posterior standard deviations, and two others rise modestly.

Current

Fast and Unbiased Log-Determinants for Large Spatial Autoregressive Models

Log-determinant methods that make Bayesian spatial autoregressive models fast and unbiased at the scale of every U.S. census tract.

Eight histograms, one per Bayesian Lagrange-multiplier statistic on the Columbus crime data, each marked with the 5 percent chi-squared critical value as a dashed line and the posterior mean as a solid red line.

Current

A Bayesian Lagrange-Multiplier Framework for Spatial Cross-Section, Panel, and Flow Models

Bayesian counterparts of the Lagrange-multiplier tests that spatial econometrics uses to choose a specification.

Spatial Economic Analysis, forthcoming

The PySAL logo: a central node branching into colored clusters of packages.

2024–2027 · Current

An Open-Source Ecosystem for Spatial and Urban Data Science

Long-term viability for an open-source ecosystem for spatial data science, with PySAL at its core.

NSF POSE Phase II, award 2345820

Line chart of mean neighborhood opportunity scores from 2003 to 2013 for three types of housing assistance; the Baltimore Housing Mobility Program line rises well above housing choice vouchers and public housing.

Completed

Modeling the Geography of Opportunity for Fair Housing Policy

Opportunity models behind HUD's evaluation of the Baltimore Housing Mobility Program and the Baltimore regional plan.

Map of Riverside County showing age-adjusted opioid overdose death rates by area in 2020, from the project dashboard.

2021–2025 · Completed

Opioid Overdose Disparities in Inland Southern California: Ethnography Meets Advanced Spatial Analytics

Ethnography and spatial data science on where and why fatal overdoses occur in Riverside County.

NIH / NIDA, award R21DA054611

Selected publications

All publications
  1. 2026
    segregation: Segregation Analysis, Inference, and Decomposition in Python

    Renan X. Cortes, Elijah Knaap, Sergio J. Rey · Journal of Open Source Software

  2. 2025
    Exploring Space-Time Interactions in Fatal Opioid Overdoses

    Sergio J. Rey, Elijah Knaap, Alejandra Cabral, Jennifer Syvertsen · Geographical Analysis

  3. 2024
    Segregated by design? Street network topological structure and the measurement of urban segregation

    Elijah Knaap and Sergio J. Rey · Environment and Planning B: Urban Analytics and City Science

  4. 2024
    Measuring two decades of urban spatial structure: The evolution of agglomeration economies in American metros

    Elijah Knaap and Sergio J. Rey · Computers, Environment and Urban Systems

  5. 2024
    The Legacy of Redlining: A Spatial Dynamics Perspective

    Sergio J. Rey and Elijah Knaap · International Regional Science Review

  6. 2017

Software

All software

SSUS Numerical and statistical tools developed in the lab.

sparsax Lead

Sparse direct solvers for JAX, backed by SuiteSparse: factorizations, solves, and log-determinants that run natively inside compiled code.

pip install sparsax Source
pgjax Lead

Pólya-Gamma sampling on device for JAX, inside compiled, scanned, and parallel programs.

pip install pgjax Source
bayesplain Lead

The Bayesian version of the tests an introductory statistics course teaches, printed beside the conventional test.

pip install bayesplain Source
neighbayes Pre-release Lead

Bayesian estimation of spatial econometric models in Python.

pip install neighbayes Source
locpick Pre-release Lead

Discrete choice models of location decisions: households, individuals, or firms choosing among places.

pip install locpick Source

oturns Open tools for urban, regional, and neighborhood science, housed at SSUS.

geosnap Lead

The Geospatial Neighborhood Analysis Package: exploring, modeling, and visualizing the social context and spatial extent of neighborhoods and regions over time.

pip install geosnap Source
pandarm Lead

The pandas routing machine: travel network analysis on contraction hierarchies, with a pandas interface.

pip install pandarm Source
netweave Experimental Lead

Transit and pedestrian networks woven into one routable multimodal graph for accessibility analysis.

Source

PySAL The Python Spatial Analysis Library. The lab leads the packages below; Knaap is a core developer and a member of the steering committee.

segregation Lead

Segregation measurement: single-group, multigroup, and local indices, spatial and aspatial, with inference and decomposition.

pip install segregation Source
tobler Lead

Areal interpolation, dasymetric mapping, change of support, and small area estimation.

pip install tobler Source

People

Elijah Knaap Director Assistant Professor of Urban Planning and Public Policy, UC Irvine.