# geoFourierFDA

In this paper, we have observsed *n* curves *χ*_{s1}(*t*), …, *χ*_{sn}(*t*) in a region, where **s**_{i} = (*θ*_{i}, *η*_{i}), *i* = 1, …, *n*, and *θ*_{i} is the latitude and *η*_{i} is the longitude where the curve *χ*_{si} was sampled. The goal of this package is to estimate an unsampled curve *χ*_{s0}(*t*) at **s**_{0} ∉ {**s**_{1}, …, **s**_{n}}. The Ideia proposed by Giraldo (2011) was simple: the curve *χ*_{s0}(*t*) is a linear combination of all curves *χ*_{s1}(*t*), …, *χ*_{sn}(*t*), i.e., \(\\widehat{\\chi\_{\\mathbf{s}\_0}}(t) = \\lambda\_1 \\chi\_{\\mathbf{s}\_1}(t) + \\lambda\_2 \\chi\_{\\mathbf{s}\_2}(t) + \\dots + \\lambda\_n \\chi\_{\\mathbf{s}\_1}(t)\) where *λ*_{1}, …, *λ*_{n} is solution of the linear system given by

where *μ* is an constant from the method of Lagrange’s multipliers and the function *γ*(*h*) = ∫*γ*(*h*; *t*)*d**t* is called the trace-variogram, where, for each ***t*, *γ*(*h*; *t*) is the semivariogram for the process *χ*_{}s_{1}(*t*), …, *χ*_{}s_{n}(*t*). More precisely, for each *t*, a weakly and isotropic spatial process is assumed for *χ*_{}s_{1}(*t*), …, *χ*_{}s**_{n}(*t*) and the integration of the semivariogram is carried out. Usually, the integration in the equation (1) is approximated using a modified version of the empirical semivariogram. In this pcackage, we have used the Legendre-Gauss quadrature, which is simple and it explicitly used the definition of the semivariogram.

## Installation

This package can be installed using the `devtools`

package.

`devtools::install_github("gilberto-sassi/geoFourierFDA")`

## Examples

In this package, we have used the temperature dataset present in the package `fda`

and in the package `geofd`

. This dataset has temperature measurements from 35 weather stations from Canada. This data can be downloaded at weather.gov.ca. For illustration, we have separated the time series at *The Pas* station and used all others stations to estimate the curve temperature at *The Pas*.

### How to interpolate a curve at an unmonitored location

```
# interpolating curve at Halifax using all remaining curves in the functional dataset
data(canada)
# Estimating the temperature at The Pas
geo_fda(canada$m_data, canada$m_coord, canada$ThePas_coord)
```

### Coefficients of smoothing using Fourier series polynomial

```
# Coefficients of smoothing using Fourier series polynomial
# Coefficients of smoothing at The Pas
data(canada)
coefs <- coef_fourier(canada$ThePas_ts)
```

### Smoothed curve using Fourier series

```
# Coefficients of smoothing using Fourier series polynomial
# Coefficients of smoothing at The Pas
data(canada)
# coefficients of Fourier series polynomial
coefs <- coef_fourier(canada$ThePas_ts, m)
# points to evaluate curve at interval [-pi, pi]
x <- seq(from = -pi, to = pi, by = 0.01)
# smoothed curve at some points x
y_est <- fourier_b(coefs, x)
```

## References

Giraldo, R, P Delicado, and J Mateu. 2011. “Ordinary Kriging for Function-Valued Spatial Data.” *Environmental and Ecological Statistics* 18 (3): 411–26.