Quick start notebook for `HePPCAT.jl`

8.6 μs

Step 1: load the package.

16.2 μs

 
using HePPCAT

5.8 s

Step 2: form data into a Vector with Matrix elements - each matrix is a group of samples (columns) having equal noise variance.

19.5 μs

Array{Float64,2}1

100×800 Array{Float64,2}:
  3.07391    -0.98157    -0.699786   2.6159    …   0.550283   -1.2851     -1.1412
 -4.50313     4.68978    -2.07655    0.665155      0.539257    4.68418    -1.32645
 -1.7521     -0.890531   -2.83606   -2.9092        0.0711276  -2.11742    -2.8706
  2.86943    -4.41902    -0.077702   0.147607     -0.500605    0.412021   -2.69618
 -2.92047    -1.00833    -2.31805   -2.68526      -1.30343     0.0444585  -1.88006
  0.0287089   0.0713829   0.903092  -3.61745   …   4.60365    -1.55906    -0.713535
 -1.73201    -1.44542     0.788838   1.34847      -2.8036     -1.55413     3.3247
  ⋮                                            ⋱                          
 -3.4713      0.830812   -4.29106    0.154452      2.35659     0.647456    1.36104
  6.60686    -1.40712     0.21455   -3.35208   …   2.9285     -0.655483    2.03497
 -2.12712     0.608553   -1.71688   -4.31894       4.6775     -0.917588   -5.22581
 -2.88758    -5.92654     0.172228  -0.556818      2.56659    -3.76559    -1.66873
 -4.67977    -4.4196     -3.76059    0.260593      1.16646    -0.332024   -0.779526
 -0.122026    2.45712     2.13976    1.77679      -4.94248    -0.604571   -1.58459

100×200 Array{Float64,2}:
  0.203052   -0.00674578  -0.119656    …   0.0856759   -0.151344    -0.0163422
 -0.0988053   0.035819     0.159803       -0.00993912  -0.0588247    0.158511
  0.152398   -0.0689631    0.0267519       0.233296     0.0522925   -0.058361
  0.229074   -0.040401    -0.118963        0.157122    -0.0157249    0.00401438
 -0.0746297   0.0808521    0.192419       -0.126884     0.00859645  -0.0592678
  0.0907297  -0.0831365   -0.158995    …   0.0872615    0.0208823   -0.169194
  0.259222    0.0932636   -0.218621        0.165271    -0.227068    -0.0410862
  ⋮                                    ⋱                            
  0.077498   -0.0552509    0.0836566       0.0425496   -0.0678956   -0.107604
 -0.0532653   0.0727037   -0.0164148   …  -0.0887275   -0.131049     0.025677
 -0.300898   -0.0414545    0.00292205     -0.0864283    0.00120629   0.090873
 -0.141627   -0.0181705    0.0522777      -0.135203    -0.136556    -0.112361
  0.283674    0.0192399   -0.158253       -0.0215563   -0.0527849    0.0250238
 -0.219649   -0.061693     0.0747251       0.0243852    0.166031     0.124621

x
 
begin # Example: generating synthetic data with heteroscedastic noise
   d = 100        # number of features (ambient dimension)
   k = 1          # number of factors (latent dimension)
​
   n = [800,200]  # number of samples in each group/block
   v = [10,0.01]  # noise variance for each group/block
   L = length(n)  # number of groups/blocks
​
   F = randn(d,k)/sqrt(d) # synthetic factor matrix
   X = [F*randn(k,n[l]) + sqrt(v[l])*randn(d,n[l]) for l in 1:L]  # synthetic data
end

20.9 ms

This code generates an example synthetic dataset with heteroscedastic noise: the n[1] = 800 samples in X[1] have noise variance v[1] = 10.0, and the n[2] = 200 samples in X[2] have noise variance v[2] = 0.01.

17.3 μs

Step 3: run the heppcat method.

30.7 μs

model

HePPCATModel{Float64,Float64}U

100×1 Array{Float64,2}:
  0.08545307301599947
 -0.04131315824148372
 -0.040369724823846105
  0.14167175171076196
 -0.12541484420891294
  0.04727082503427198
  0.1559739967622669
  ⋮
  0.003190286398210981
  0.0016334942077702972
 -0.09775005299825644
 -0.007695169512153544
  0.08897560084927778
 -0.10818750504322917

λFloat641

1.24644

1×1 Array{Float64,2}:
 1.0

vFloat641

10.0689

0.00997343

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model = heppcat(X,k,100)  # run 100 iterations

33.4 ms

For more info, use ?heppcat to access the docstring.

12.4 μs

Step 4: extract factor matrix and noise variance estimates.

10.2 μs

100×1 Array{Float64,2}:
  0.09540347619500922
 -0.04612378197439529
 -0.045070492438695216
  0.15816842057057826
 -0.14001851170110163
  0.052775176733916136
  0.17413606043592242
  ⋮
  0.0035617725811919786
  0.0018237030017226028
 -0.10913235212189487
 -0.008591217324997152
  0.09933617736569156
 -0.1207851712957901

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model.F

8.4 μs

Float641

10.0689

0.00997343

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model.v

146 ns

For more info about the returned datatype use ?HePPCATModel for the docstring.

17.8 μs

Evaluating recovery: check recovery of the factor covariance and noise variances.

13.1 μs

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using LinearAlgebra

2.5 ms

0.517880090519894

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norm(model.F*model.F' - F*F')/norm(F*F')  # normalized estimation error

86.9 μs

2×2 Array{Float64,2}:
 10.0689      10.0
  0.00997343   0.01

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[model.v v]

2.0 μs

Compared with homoscedastic PPCA:

10.7 μs

ppca

HePPCATModel{Float64,Float64}U

100×1 Array{Float64,2}:
 -0.019637630403928213
 -0.022884783218402385
  0.10709578399944776
 -0.03466448445995213
 -0.07889845223748505
  0.03648735232319211
 -0.04349089258619432
  ⋮
 -0.037032139405059894
 -0.02281947747300657
  0.058604357203842855
 -0.014297849239765969
  0.15139590399091565
  0.006202155963090815

λFloat641

6.45015

1×1 Array{Float64,2}:
 1.0

vFloat641

8.00306

8.00306

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ppca = heppcat(X,k,0)  # initialization is homoscedastic PPCA

15.1 ms

7.769109021285145

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norm(ppca.F*ppca.F' - F*F')/norm(F*F')  # normalized estimation error

79.7 μs

2×2 Array{Float64,2}:
 8.00306  10.0
 8.00306   0.01

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[ppca.v v]

2.0 μs

HePPCAT accounts for heterogeneous quality among the samples and is generally more robust.

9.0 μs

Quick start notebook for HePPCAT.jl

Quick start notebook for `HePPCAT.jl`