Graph-based covariance for GP

Questions
1

Hi everyone. I have a spatial dataset with a set of measurements for each point in a grid layed over a map, something like the following:

image
image1558×862 313 KB

Per grid point, I have many samples, each multi-dimensional.

I would like to find a function that maps each point in the grid to its measurement vector using gaussian processes. For the covariates I could simply use the (x, y) coordinates, however this would not take into account the shape of the map. (The region outside of the map cannot be “crossed”) Instead my idea was to calculate a kNN-graph on the coordinates of these points, and somehow use the shortest path between two nodes as a way to quantify covariance between two points.

The idea then would be to assign an index to each point, which I can model as a categorical distribution. Then I would use a pre-computed matrix of shortest paths between every pair of points in a custom covariance function. I have got the following at the moment:

X = # n x 1 index vector for each sample
P = # k x k matrix of shortest paths between any two points in grid
Y = # n x p matrix of measurements 

class ShortestPath(pm.gp.cov.Stationary):
    def __init__(self, P, ls=None, ls_inv=None, active_dims=None):
        super().__init__(1, ls, ls_inv, active_dims)
        self.P = tt.as_tensor_variable(P)

    def full(self, X, Xs=None):
        X, Xs = self._slice(X, Xs)
        index = tt.cast(X[:, 0], 'int32')
        if Xs is None:
            index2 = index.T
        else:
            index2 = tt.cast(Xs[:, 0], 'int32').T
        return self.P[index, index2]

 with pm.Model() as model:
    p = pm.Dirichlet('p', 0.1 * np.ones((n, k)), shape=(n, k))
    x = pm.Categorical('x', p, shape=n, observed=X)
    xs = tt.reshape(x, (n, 1))
    
    for i in range(p):
        cov_func = ShortestPath(P, ls=1)
        gp = pm.gp.Marginal(cov_func=cov_func)
        y = gp.marginal_likelihood(f'y{i}', X=xs, y=Y[:, i], noise=0.01)

Do I have the right idea here?

2

If you have a graph, you can use a conditional autoregression to allow for covariance between connected nodes. The CAR will not allow correlation to cross the missing regions of the spatial area (e.g. the concavities in the boundary of the shape you’ve shown) unlike a standard GP and will force correlations to travel along the graph’s edges. Even though it is specified between pairs of sites, it allows for long range correlations between nodes.

However, I’m a little unsure if this is precisely what you were asking for. You are assigning an index to each point which is latent and therefore must be modeled. Are you directly interested in the value of that index or do you just regard it as a device to inform a graph covariance structure?

2 Likes
3

Thanks for the reply. I am actually interested in the value of the index (or rather the probabilities over the indices). Actually later on I would like to include data points with missing locations to infer their spatial location using their measurements. So a kind of semi supervised approach.