State-Space Models for Temperature Dynamics and Spatial Forecasting

Time Series Analysis project, Bocconi University, 2025–2026.

This project applies Hidden Markov Models (HMMs) and Dynamic Linear Models (DLMs) to two related problems in temperature data:

1. identifying long-run changes in Northern Hemisphere temperature anomalies; and
2. evaluating whether spatial dependence across weather stations improves short-horizon temperature forecasts.

The analysis combines time-series decomposition, latent-state modelling, maximum-likelihood estimation, Kalman filtering and smoothing, spatial covariance modelling, and out-of-sample forecast evaluation.

Read the final report

Project overview

1. Long-run temperature dynamics — GISTEMP

The first part uses monthly Northern Hemisphere temperature anomalies from NASA GISTEMP.

After exploring and decomposing the series, we compare two approaches to representing long-run temperature dynamics:

• Hidden Markov Models, which approximate the evolution of temperatures using discrete latent regimes;
• Dynamic Linear Models, including random-walk and locally-linear-trend specifications, which represent the underlying temperature level as a continuously evolving latent state.

The analysis illustrates both the usefulness and limitations of discrete regime models for a process dominated by a persistent trend. The DLM specifications provide a more natural description of the continuous evolution of the latent temperature level, while the estimated slope dynamics provide little evidence for a separately identified stochastic acceleration process.

2. Spatial temperature forecasting — GHCN

The second part uses daily maximum temperatures from the Global Historical Climatology Network.

We model three US weather stations:

• Baker City, Oregon
• Paso Robles, California
• San Francisco, California

The three-station setup extends the baseline two-location exercise and provides variation between a relatively nearby pair and a substantially more distant station.

Four multivariate DLM specifications are compared:

1. Independent — station-level latent states evolve independently;
2. Full covariance — latent-state innovations have an unrestricted covariance matrix;
3. Distance-based covariance — dependence decays exponentially with geographic distance;
4. Seasonal distance model — the spatial model is augmented with a stochastic annual seasonal component.

Models are estimated by maximum likelihood on data through 2014 and evaluated using one-step-ahead forecasts from 2015 onward.

Main findings

• The GISTEMP series is dominated by a persistent low-frequency increase in temperature anomalies. HMM regimes provide a useful discrete summary, but largely discretize an underlying continuous trend.
• In the weather-station application, allowing latent temperature innovations to co-move substantially improves the modelling of the joint predictive distribution.
• The unrestricted covariance model provides the best empirical fit and leaves considerably less residual cross-station dependence.
• Spatial dependence produces only very small improvements in individual-station point forecasts: MAE and RMSFE remain similar across specifications.
• The distance-decay model sacrifices some fit relative to an unrestricted covariance matrix but provides a parsimonious and geographically interpretable dependence structure.
• Adding stochastic seasonality produces little improvement over deterministic seasonal adjustment in this application.

Methods

The project uses:

• state-space models and Kalman filtering/smoothing
• Hidden Markov Models
• multivariate Dynamic Linear Models
• maximum-likelihood estimation
• Cholesky parametrization of covariance matrices
• spatial covariance functions
• delta-method uncertainty estimates
• out-of-sample forecast evaluation
• probabilistic forecast diagnostics
• residual autocorrelation and cross-sectional dependence diagnostics

The analysis is implemented in R, primarily using dlm, depmixS4, and the tidyverse ecosystem.

Repository structure

.
├── Final_Project_Group_27.pdf          # Final report
├── Final_Project_Group_27.tex          # Final report source
├── Final_Project_Part_1_Group_27.rmd   # GISTEMP / HMM / univariate DLM analysis
├── Final_Project_Part_2_Group_27.rmd   # Multivariate and spatial DLM analysis
├── data/
│   ├── gistemp.csv
│   └── ghcn.txt
└── models/                             # Cached maximum-likelihood model fits

Reproducibility

The repository uses renv to record the R package environment.

After cloning the repository, restore the package environment with:

renv::restore()

The two R Markdown files contain the full analysis used to generate the figures, tables, and numerical results in the final report.

Some multivariate DLM specifications are relatively expensive to estimate. Pre-estimated model objects are therefore retained under models/ for convenient reproduction. The models can also be re-estimated from the source code.

Data

The project uses two datasets supplied for the course project:

• GISS Surface Temperature Analysis (GISTEMP), version 4, NASA Goddard Institute for Space Studies;
• Global Historical Climatology Network – Daily (GHCN-Daily), NOAA.

The copies used for the analysis are included under data/.

Project context

This repository documents my implementation and empirical analysis for a group assignment completed with E. Artoni, A. Dreyfus, M. van Weerdenburg as part of the Time Series Analysis of Economic-Financial Data course at Bocconi University.