```
library(BSSoverSpace)
set.seed(16)
```

Blind Source Separation Over Space (BSSS) is a tool for analyzing
spatial multivariate data. Blind Source Separation method assumes that
observed variables are formed by linear combination of underlying
independent latent variables, which cannot be observed directly. The
goal is to estimate the latent variables, which also includes estimating
the mixing matrix. This package `BSSoverSpace`

is an
implementation of the work Zhang, Hao, and Yao
(2022). This manual provides introduction and simple instructions
on how to use the functions within.

`BSSS`

In this package `BSSoverSpace`

, the main function is
`BSSS`

. It implemented the method in Zhang, Hao, and Yao (2022) for estimating the
latent field \(Z(s)\), and the unmixing
matrix. This function takes 5 inputs: `x`

is the data matrix
of observed random field \(X(s)\).
`coord`

is the coordinate of observed random field \(X(s)\). `kernel_type`

and
`kernel_parameter`

are the specifications of kernels for us
to select. For each `kernel_type`

, specification of
`kernel_parameter`

slightly differs. If
`kernel_type`

equals `'ring'`

, there must be an
even number of parameters in `kernel_parameter`

. Check
`spatial_kernel_matrix`

from package `SpatialBSS`

for more details.

Here, we generate a random field and use it to demonstrate the usage of this function. First we generate 500 2-dimensional coordinates:

```
<- 500
sample_size <- runif(sample_size * 2) * 50
coords dim(coords) <- c(sample_size, 2)
```

Next, we generate a 5-variate latent gaussian random field \(Z(s)\) with matern covariance function in the following way:

```
library('BSSoverSpace')
<- 5 # specify the dimensionality of random variable
dim <- runif(dim, 0, 6) # parameter for matern covariance function
nu <- runif(dim, 0, 2) # parameter for matern covariance function
kappa <- gen_matern_gaussian_rf(coords=coords, dim=dim, nu=nu, kappa=kappa) zs
```

Then, we create a mixing matrix \(\Omega\) , and mix our latent field to get the observed random field \(X(s)\):

```
<- diag(dim) # create a diagonal matrix as the mixing matrix
mix_mat <- t(mix_mat %*% t(zs)) xs
```

Now the observed random field \(X(s)\) is created, and we need to choose kernels. Here we choose 3 ring kernels, with parameters \((0, 0.5)\), \((0.5, 1)\) and \((1, 8)\).

`<-BSSS(xs, coord = coords, kernel_type = 'ring', kernel_parameter = c(0, 0.5, 0.5, 1, 1, 8)) example`

The function `BSSS`

returns both the estimated mixing
matrix \(\hat{\Omega}\)
`mix_mat_est`

and the estimated latent field \(\widehat{Z(s)}\). To see how good \(\hat{\Omega}\) is, we can use function
`d_score`

, which gives a numeric value between 0 and 1, with
0 meaning that the estimator is a column permutation of true value:

```
d_score(example$mix_mat_est, mix_mat)
#> [1] 0.123749
```

We can further explore the validity of our estimation, by looking at the eigenvalues of \(\hat{W}\). Larger gap between first few eigenvalues would strengthen the validity of our estimation. One can see the details of \(\hat{W}\) in Zhang, Hao, and Yao (2022).

`plot(example$w_eigenvalue)`

Zhang, Bo, Sixing Hao, and Qiwei Yao. 2022. “Blind
Source Separation over
Space.” arXiv. https://arxiv.org/abs/2201.02023.