The method enables making predictions of spatio-temporal quantities such as precipitation, air pollution, etc. from a stream of data. It is also equally applicable if the entire training data is available. The method makes use of a localized spatio-temporal basis and uses an online algorithm for learning the predictor in a sequential manner as space-time datapoints are collected. For details see and please cite: Muhammad Osama, Dave Zachariah, Thomas B. Schön, "Learning Localized Spatio-Temporal Models From Streaming Data" [http://proceedings.mlr.press/v80/osama18a.html]
There are two main files implemented in Python: 'basis.py' and 'Learning_from_streaming_data.py'. An 'example.py' file describes what variables to initialize and how to use these former two files for training and then evaluating the predictor at desired spatio-temporal coordinates. Below we provide a short description of each file. It is assumed that the training data is organized in a csv file in a way described in the example.py file. For example for 2D space time, the i:th line in the training.csv represents an observed data point with format [y_i, s1_i, s2_i, t_i], where y is the quantity of interest.
Below we show some example plots from our results on real precipitation data. The first plot shows the predicted precipitation over a spatial region for a specific month (t=53). The red dots denote the training points. The next plot is a comparison of the actual and predicted percipitation over time for spatial point marked by red cross in the contour plot. The black dashed box and dashed line in these plots constitute a contiguous spatio-temporal test region.
This function is used to learn the weights 'w_hat' from training data which are used later for evaluating the predictor.
The function is called in a for loop for each training point in the following way:
[w_hat, Gamma_sparse, rho, kappa] = func_newsample_covlearn(y_n, alpha_sparse_n, ind_alpha_n, Gamma_sparse, rho, kappa, w_hat, n+1, L, U)
Inputs: y_n: the spatio-temporal quantity of interest
alphas_sparse_n, ind_alpha_n: from basis.py
Gamma_sparse, rho, kappa, w_hat: Have to be initialized with zeros outside the for loop before calling the above function. See example.py file
n+1: see example.py file
L: The number of itereations of the algortihm for each training point
y_n : spatio-temporal quantity e.g. precipitation at the nth training point
U: mean parameter set to 1 corresponding to constant spatio-temporal mean
The basis.py evaluates a spatio-temporal basis at a specific spatio-temporal point. The function is called in the following way:
[alpha_sparse_n,ind_alpha_n] = basis.spatio_temporal_basis(x_n,mn,mx,Ns,Nt,sup)
Inputs: x_n: Dx1 array containing spatio-temporal coordinates e.g. for 2D space time: x_n = [s1, s2, t] where [s1,s2] are the spatial coordinates in Euclidean space and 't' is the time coordinates. If spatial coordinates are [longitude, latitude] they must be converted to Euclidean space. We provide a function latlon_conversion.py to do that
mn: Dx1 array containing the lower limit of each dimension
mx: Dx1 array containing the upper limit of each dimension
Ns: the number of basis in space per dimension
Nt: the number of basis in time
sup: the support of local b-spline spatial basis in fraction of range of dimension
alpha_sparse_n: Ns^(D-1)*(Nt+1) x 1 sparse array of spatio-temporal basis for the nth point
ind_alpha_n: array containing indices of the non-zero elements in alpha_sparse_n