README.Rmd

---
output: github_document
---

<!-- README.md is generated from README.Rmd. Please edit that file -->

```{r setup, include = FALSE}
knitr::opts_chunk$set(
  collapse = TRUE,
  comment = "#>",
  fig.path = "man/figures/README-",
  out.width = "100%"
)

pas3 <- 'proc_align_slicewise3'
pa3 <- 'proc_align3'
```

# FASE

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The goal of fase is to fit a "Functional Adjacency Spectral Embedding" to an indexed collection of network snapshots. The behavior of each node is summarized by a latent process, which is fit in a prespecifed spline basis. The fase package has options to fit either B-spline (with equally spaced knots) or smoothing spline latent processes. For more details, see [MacDonald et al., (2022+)](https://arxiv.org/abs/2210.07491). 

Additional functions are included in this package to do Procrustes alignments of latent processes stored as 3-dimensional arrays.

## Installation

You can install fase from CRAN with:

``` r
install.packages("fase")
```

## Example

This is a basic example which shows you how to generate functional network data and fit a functional adjacency spectral embedding.

First, we generate $50$ snapshots of a functional network on $100$ nodes, with Gaussian edges (standard deviation $\sigma=4$) and two-dimensional sinusoidal latent processes.

```{r simulating}
# load fase package
library(fase)

# set a seed
set.seed(1)

# generate functional network data
data <- gaussian_snapshot_ss(n=100,d=2,
                             x_vec=seq(0,1,length.out=50),
                             self_loops=FALSE,
                             sigma_edge=4)
```

Then we fit a two-dimensional functional adjacency spectral embedding to this data, using a $B$-spline basis design
with dimension $9$.
This choice of basis dimension is chosen to minimize the network generalized cross validation criterion.

As a post-processing step, and due to the intrinsic non-identifiability of the latent processes, we align our embedding to the original processes used to fit the data. Note that by using `r pa3` (rather than `r pas3`), we apply the same orthogonal transformation to all the evaluations of the latent process snapshots. The difference between slicewise and common identifying transformations is discussed in more detail in [MacDonald et al., (2022+)](https://arxiv.org/abs/2210.07491), Appendix C.

```{r embedding}
# embed with fase (B-spline design)
fit <- fase(data$A,d=2,self_loops=FALSE,
            spline_design=list(type='bs',q=9,x_vec=data$spline_design$x_vec))

# align fitted and true latent processes
Z_align <- proc_align3(fit$Z,data$Z)
```

Finally, we extract a couple of example estimated (and aligned) latent processes: the first dimension for node 34, and the second dimension for node 54. We plot them against the true values over the entire index space.

```{r plotting, echo = FALSE}
par(mfrow=c(1,2))
plot(data$spline_design$x_vec,data$Z[34,1,],type='l',
     xlab='x',ylab='Z_{34,1}(x)',ylim=c(-3,3))
lines(data$spline_design$x_vec,Z_align[34,1,],type='l',col='blue')
legend(x=0.35,y=-1.5,legend=c('True','Estimated'),
       lty=1,pch=NA,col=c('black','blue'),cex=0.6)
plot(data$spline_design$x_vec,data$Z[54,2,],type='l',
     xlab='x',ylab='Z_{54,2}(x)',ylim=c(-3,3))
lines(data$spline_design$x_vec,Z_align[54,2,],type='l',col='blue')
```