Simulation of Datasets Sample Clauses

Simulation of Datasets. = ( ) = ( ) 3 = − {− ( )} ∶= [ ] ≤ ≤ To simulate the different datasets, we first generate the different latent functions. We simulate three different latent functions on an ordering (underlying time) t. Time is used for visualization purposes to give an order to the latent function. It is not needed to learn the latent representation. The first latent function is a cosine A1 cos t of the underlying time t. The second latent input is a sine A2 sin t of the underlying time ordering t. Finally, the last latent input is a negative exponen- tial negative cosine A exp cos 2t with doubled frequency. Note, here A are generating the input matrix X Ai 1 i 4 for the simulation. = [ 1 2] ⋅ We simulate three different datasets Yi from mixtures of the above defined functions A: [ ] = • The first simulated dataset Y1 A , A As1 , is generated using the first two input functions A1, A2 . It has α1 0.3 noise variance added to it. • The second simulated dataset Y2 contains only the first simulated latent input A1 with an added noise variance of α2 = 0.2. αs3 = 0.25. • And Y3 is generated from all three latent inputs combined [A1, A2, A3] with See Table 2.2 for a clear representation of which combination of input functions Ai generates which dataset Yi. Additionally, we show an illustration of the three simulated datasets in Figure 2.7. Input Function Y2 ✓ 0.2 Y1 ✓ ✓ 0.3 Dataset A1 = cos(t) A2 = sin(t) A3 = − exp{− cos(2t)} α ✓ ✓ ✓ Y3 0.25 Table 2.2: Simulation of three datasets Yi to explain MRD dependence structure deduction. The table shows which latent inputs Ai where used to generate each dataset and how much noise variance αi was added. Y1 Y2 Y3