Sensor Fusion
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using AbstractGPs
using Plots
using Random
using Stheno
Define and inspect our model
rng = MersenneTwister(123456);
In this example, f
is an unknown real-valued function that we wish to infer. To achieve this, we have access to two sensors. The first returns noisy estimates of f
, where we have been reliably informed by whoever designed the sensor that the mean of the noise is given by sin(x) - 5 + sqrt(abs(x))
, and that it's variance is low (1e-2). How the designer estimated this function, and why a sensor might possibly have such a strange mean error, is beyond the scope of this example. The second returns biased measurements of f
, where the bias is known to be 3.5. The model below specifies a model for this scenario.
model = @gppp let
# Define a smooth latent process that we wish to infer.
f = GP(SEKernel())
# Define the two noise processes described.
noise1 = sqrt(1e-2) * GP(WhiteKernel()) + (x->sin.(x) .- 5.0 .+ sqrt.(abs.(x)))
noise2 = sqrt(1e-1) * GP(3.5, WhiteKernel())
# Define the processes that we get to observe.
y1 = f + noise1
y2 = f + noise2
end;
Generate some toy observations of y₁
and y₂
.
x1 = GPPPInput(:y1, sort(rand(rng, 3) * 10));
x2 = GPPPInput(:y2, sort(rand(rng, 10) * 10));
x = BlockData(x1, x2);
ŷ = rand(rng, model(x));
ŷ1, ŷ2 = split(x, ŷ);
Compute the posterior processes.
model′ = posterior(model(x), ŷ);
Sample jointly from the posterior processes and compute posterior marginals.
xp_ = range(-2.5, stop=12.5, length=500);
xp_f = GPPPInput(:f, xp_);
xp_y1 = GPPPInput(:y1, xp_);
xp_y2 = GPPPInput(:y2, xp_);
xp = BlockData(xp_f, xp_y1, xp_y2);
model′_xp = rand(rng, model′(xp, 1e-9));
f′xp, y1′xp, y2′xp = split(xp, model′_xp);
Plot results
gr();
posterior_plot = plot();
Plot posterior over y1.
plot!(posterior_plot, xp_, model′(xp_y1); color=:red, label="y1");
plot!(posterior_plot, xp_, y1′xp; color=:red, label="", linewidth=1, linealpha=0.2);
Plot posterior over y2.
plot!(posterior_plot, xp_, model′(xp_y2); color=:green, label="y2");
plot!(posterior_plot, xp_, y2′xp; color=:green, label="", linewidth=1, linealpha=0.2);
Plot posterior over f.
plot!(posterior_plot, xp_, model′(xp_f); color=:blue, label="Latent Function");
plot!(posterior_plot, xp_, f′xp; color=:blue, label="", linewidth=1, linealpha=0.2);
Plot samples on which we conditioned.
scatter!(posterior_plot, x1.x, ŷ1;
markercolor=:red,
markershape=:circle,
markerstrokewidth=0.0,
markersize=4,
markeralpha=0.8,
label="Sensor 1",
);
scatter!(posterior_plot, x2.x, ŷ2;
markercolor=:green,
markershape=:circle,
markerstrokewidth=0.0,
markersize=4,
markeralpha=0.8,
label="Sensor 2",
);
posterior_plot
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