## Spatial functional data analysis

Spatial functional data analysis (FDA) concerns the statistical analysis of spatial and spatiotemporal data.

Spatial FDA is used to better analyse, model and predict the complex dependence structure inherent in spatiotemporal processes and thus is able to provide
more accurate predictions at high temporal and spatial resolutions.

Spatial FDA also accounts for attributes of the geometry of the physical problem such as irregular shaped domains, external and internal boundary features and strong concavities.

Spatial FDA can also include a priori information about the spatial structure of the phenomenon described by a partial differential equation (PDE). This facilitates a causal explanation for the drivers and impediments of the underlying spatiotemporal process.

## What is Data2PDE?

Data to PDE estimates the parameters and the solutions of linear PDEs de noted by data that are partially observed noisy measurements distributed over complex geometries.

## Modelling Temperature in Croatia with Data2PDE

Figure (1) shows the temperature measurements on February 2, 2008, at 127 meteorological stations across Croatia.

We model the dynamics of the temperature in Croatia with the diffusion advection reaction equation

$\frac{\partial^2z}{\partial x^2} + \beta_{yy} \frac{\partial^2z}{\partial y^2}+\beta_x\frac{\partial z}{\partial x}+\beta_y\frac{\partial z}{\partial y}+\beta z = 0$

where $\beta_{yy},\beta_{x},\beta_{y},\beta$ are unknown parameters that require estimation from data.

The DAR PDE is used to describe the flow of heat, particles, or other physical quantities in situations where there is both diffusion and advection. Diffusion is the movement of a substance from an area of high concentration to an area of low concentration, resulting after a passage of time in the uniform distribution of the substance. Advection or flow refers to the transport due to linear or gently curvilinear movement of the substance. Reaction occurs when the substance has a baseline to which it is either accumulating or decaying over either space or time.

The parameter $\beta_{yy}$ represent the rate of the spread of temperature from high concentration to low concentration in space; $\beta_x$ and $\beta_y$ denote the movement of the temperature across space; and the reaction multiplier $\beta$ defines the exponential increase or decrease of the temperature.

Let $z$ be the solution of the diffusion advection reaction equation and let the observations of the process $z$ at the locations $\textsf{x}_{i},\textsf{y}_{i}$ be $\textsf{z}_{i}$ for $i=1,\ldots,N.$ We assume that

$\textsf{z}_{i}=z(\textsf{x}_{i},\textsf{y}_{i})+\alpha w_{i} +\epsilon_{i}$

where $w$ is a co-variate representing scaled elevation (elevation/100) at each station; $\alpha$ measures the relationship between scaled elevation and $\textsf{z},$; and $\epsilon_{i}$ is an independent and identically distributed measurement error following a distribution with zero mean and finite variance $\sigma_{\epsilon}^{2}.$ As weather systems originating or crossing over Croatia are strongly influenced by its diverse topography we include the scaled elevation at each station as a covariate in this model.

The estimated relationship between the elevation and the temperature across Croatia with its 95% con dence interval is $\alpha=-0.67 \pm 0.06.$ Indicating that with a 100-meter rise in altitude, temperature decreases by about $0.67$ degrees Celsius.

The estimated $\beta = 0.55\pm 0.43$ indicates an accumulation of temperature.

The estimated rates of advection are $\beta_{y}=2.89 \pm 1.7$ and $\beta_{x}=-0.71\pm2.$ Indicating that the temperature is moving in a north-easterly direction.

The estimated rate of di ffusion in the y-direction is $\beta_{yy}=1.72 \pm 0.96.$ Indicating anisotropic behaviour in the temperature as it is di ffusing at a faster rate from south to north than from west to east.

The estimated temperature is shown in Figure (2), the warmer air is coming in from the Adriatic sea. The Bura, which is a north to north-eastern wind, blows this warm air from the coastline at Rijeka across Croatia to northeast border near Bilogora.