MADS Notebook: Blind Source Separation

MADS is an integrated high-performance computational framework for data/model/decision analyses.

MADS can be applied to perform:

• Sensitivity Analysis
• Parameter Estimation
• Model Inversion and Calibration
• Uncertainty Quantification
• Model Selection and Model Averaging
• Model Reduction and Surrogate Modeling
• Machine Learning (e.g., Blind Source Separation, Source Identification, Feature Extraction, Matrix / Tensor Factorization, etc.)
• Decision Analysis and Support

Here, it is demonstrated how MADS can be applied to solve a general blind Source Separation contamination problem.

Problem setup

Import Mads (if MADS is not installed, first execute in the Julia REPL: import Pkg; Pkg.add("Mads")):

Pkg.resolve()
import Revise
import Mads

┌ Info: Precompiling Mads [d6bdc55b-bd94-5012-933c-1f73fc2ee992]
└ @ Base loading.jl:1317

Mads: Model Analysis & Decision Support
====

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MADS is an integrated high-performance computational framework for data- and model-based analyses.
MADS can perform: Sensitivity Analysis, Parameter Estimation, Model Inversion and Calibration, Uncertainty Quantification, Model Selection and Model Averaging, Model Reduction and Surrogate Modeling, Machine Learning, Decision Analysis and Support.

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import Random
Random.seed!(2015)
nk = 3
s1 = (sin.(0.05:0.05:5) .+1) ./ 2
s2 = (sin.(0.3:0.3:30) .+ 1) ./ 2
s3 = rand(100);

Source matrix (assumed unknown)

S = [s1 s2 s3]

100×3 Matrix{Float64}:
 0.52499      0.64776     0.518466
 0.549917     0.782321    0.249446
 0.574719     0.891663    0.244758
 0.599335     0.96602     0.304535
 0.623702     0.998747    0.0407869
 0.64776      0.986924    0.258012
 0.671449     0.931605    0.0868265
 0.694709     0.837732    0.882364
 0.717483     0.71369     0.850562
 0.739713     0.57056     0.216851
 0.761344     0.421127    0.333606
 0.782321     0.27874     0.285782
 0.802593     0.156117    0.889261
 ⋮                        
 0.0171135    0.999997    0.577248
 0.0112349    0.978188    0.956294
 0.00657807   0.913664    0.62366
 0.0031545    0.812189    0.914383
 0.000972781  0.682826    0.578899
 3.83712e-5   0.537133    0.515968
 0.000353606  0.388122    0.415338
 0.0019177    0.249105    0.961519
 0.00472673   0.1325      0.795982
 0.00877369   0.0487227   0.73677
 0.0140485    0.00525646  0.653462
 0.0205379    0.00598419  0.607898

Mads.plotseries(S; title="Original sources", name="Source", quiet=true)

Mixing matrix (assumed unknown)

H = [[1,1,1] [0,2,1] [1,0,2] [1,2,0]]

3×4 Matrix{Int64}:
 1  0  1  1
 1  2  0  2
 1  1  2  0

Data matix (known)

X = S * H

100×4 Matrix{Float64}:
 1.69122   1.81399   1.56192   1.82051
 1.58168   1.81409   1.04881   2.11456
 1.71114   2.02808   1.06423   2.35805
 1.86989   2.23657   1.2084    2.53137
 1.66324   2.03828   0.705276  2.6212
 1.8927    2.23186   1.16378   2.62161
 1.68988   1.95004   0.845102  2.53466
 2.4148    2.55783   2.45944   2.37017
 2.28173   2.27794   2.41861   2.14486
 1.52712   1.35797   1.17341   1.88083
 1.51608   1.17586   1.42856   1.6036
 1.34684   0.843262  1.35389   1.3398
 1.84797   1.20149   2.58111   1.11483
 ⋮                             
 1.59436   2.57724   1.17161   2.01711
 1.94572   2.91267   1.92382   1.96761
 1.5439    2.45099   1.2539    1.83391
 1.72973   2.53876   1.83192   1.62753
 1.2627    1.94455   1.15877   1.36663
 1.05314   1.59023   1.03198   1.0743
 0.803814  1.19158   0.83103   0.776598
 1.21254   1.45973   1.92496   0.500128
 0.933209  1.06098   1.59669   0.269727
 0.794267  0.834216  1.48231   0.106219
 0.672767  0.663975  1.32097   0.0245614
 0.63442   0.619866  1.23633   0.0325062

Mads.plotseries(X, title="Mixed signals", name="Signal", quiet=true)

Blind signal reconstruction

Wipopt, Hipopt, pipopt = Mads.NMFipopt(X, nk);

OF = 3.5187622911849673e-13

Reconstructed sources

Mads.plotseries(Wipopt, title="Reconstructed sources", name="Source", quiet=true)

Reproduced signals

Mads.plotseries(Wipopt * Hipopt, title="Reproduced signals", name="Signal", quiet=true)