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Table 3 Results for Case Study II

From: Modelling biochemical networks with intrinsic time delays: a hybrid semi-parametric approach

NN Nlag τ BIC    MSE    NN Nlag τ BIC    MSE   
    train valid test train valid test     train valid test train valid test
3 0 0 -18561 -5147 -5000 0.0155 0.0187 0.0157 6 0 0 -18500 -5199 -5030 0.0148 0.0187 0.0154
4 0 0 -18514 -5164 -4994 0.0155 0.0187 0.0150 7 0 0 -18576 -5197 -4990 0.0153 0.0180 0.0144
5 0 0 -18531 -5186 -5011 0.0151 0.0189 0.0153 8 0 0 -18430 -5211 -5053 0.0146 0.0182 0.0158
2 1 2 -17144 -4697 -4475 0.0086 0.0098 0.0065 2 1 2.5 -16343 -4293 -4331 0.0077 0.0076 0.0067
3 1 2 -16981 -4588 -4470 0.0085 0.0108 0.0067 3 1 2.5 -17230 -4725 -4584 0.0186 0.0327 0.0183
4 1 2 -16877 -4581 -4484 0.0087 0.0117 0.0072 4 1 2.5 -16635 -4506 -4690 0.0154 0.0220 0.0170
5 1 2 -16792 -4544 -4490 0.0078 0.0093 0.0064 5 1 2.5 -16927 -4614 -4524 0.0086 0.0111 0.0072
6 1 2 -16911 -4643 -4544 0.0087 0.0119 0.0073 6 1 2.5 -16944 -4523 -4488 0.0082 0.0092 0.0066
7 1 2 -17197 -4632 -4505 0.0097 0.0151 0.0084 7 1 2.5 -17044 -4742 -4608 0.0082 0.0107 0.0075
8 1 2 -17181 -4530 -4496 0.0123 0.0155 0.0105 8 1 2.5 -16976 -4694 -4608 0.0084 0.0110 0.0076
2 2 2 -16466 -4512 -4525 0.0067 0.0079 0.0067 2 2 2.5 -17813 -4833 -4759 0.0130 0.0152 0.0129
3 2 2 -16856 -4656 -4535 0.0081 0.0116 0.0073 3 2 2.5 -16637 -4684 -4595 0.0079 0.0123 0.0079
4 2 2 -16788 -4616 -4525 0.0079 0.0111 0.0072 4 2 2.5 -16703 -4507 -4517 0.0073 0.0083 0.0064
5 2 2 -16734 -4430 -4446 0.0075 0.0088 0.0061 5 2 2.5 -16384 -4327 -4404 0.0068 0.0074 0.0063
6 2 2 -16573 -4271 -4353 0.0077 0.0081 0.0065 6 2 2.5 -16601 -4400 -4432 0.0071 0.0079 0.0062
7 2 2 -16704 -4569 -4541 0.0071 0.0089 0.0065 7 2 2.5 -16569 -4405 -4466 0.0068 0.0072 0.0061
8 2 2 -16921 -4728 -4632 0.0082 0.0132 0.0079 8 2 2.5 -16833 -4790 -4664 0.0080 0.0127 0.0079
2 3 2 -19006 -5136 -5080 0.0181 0.0177 0.0158 2 3 2.5 -16619 -4566 -4514 0.0077 0.0099 0.0071
3 3 2 -16811 -4692 -4549 0.0078 0.0119 0.0073 3 3 2.5 -16037 -4218 -4259 0.0064 0.0072 0.0058
4 3 2 -16737 -4474 -4466 0.0069 0.0089 0.0063 4 3 2.5 -16439 -4224 -4287 0.0068 0.0078 0.0057
5 3 2 -16519 -4357 -4408 0.0066 0.0076 0.0058 5 3 2.5 -16199 -4358 -4295 0.0063 0.0076 0.0056
6 3 2 -16832 -4506 -4415 0.0090 0.0107 0.0078 6 3 2.5 -16604 -4577 -4556 0.0072 0.0094 0.0067
7 3 2 -16565 -4385 -4439 0.0066 0.0072 0.0058 7 3 2.5 -16344 -4475 -4432 0.0064 0.0078 0.0060
8 3 2 -16758 -4672 -4569 0.0079 0.0117 0.0069 8 3 2.5 -16471 -4374 -4505 0.0066 0.0069 0.0061
2 4 2 -16655 -4365 -4504 0.0071 0.0107 0.0077 2 4 2.5 -16562 -4532 -4503 0.0079 0.0097 0.0073
3 4 2 -16377 -4301 -4431 0.0067 0.0078 0.0064 3 4 2.5 -16325 -4471 -4470 0.0072 0.0086 0.0066
4 4 2 -16215 -4183 -4316 0.0062 0.0067 0.0057 4 4 2.5 -16261 -4189 -4281 0.0064 0.0068 0.0058
5 4 2 -16611 -4481 -4484 0.0070 0.0101 0.0066 5 4 2.5 -15954 -4190 -4255 0.0056 0.0060 0.0052
6 4 2 -17503 -4762 -4597 0.0189 0.0374 0.0197 6 4 2.5 -16439 -4334 -4476 0.0064 0.0073 0.0060
7 4 2 -25835 -7171 -7226 10.160 9.2368 11.843 7 4 2.5 -25949 -7288 -7280 13.781 19.040 17.824
8 4 2 -16256 -4328 -4369 0.0058 0.0061 0.0052 8 4 2.5 -16296 -4326 -4381 0.0061 0.0065 0.0054
  1. Results of the performance measures, BIC and MSE, over structure parameters, namely Numbers of Nodes in the hidden layer of the ANN, NN, Number of time lags, Nlag and the time lag, τ, for Pichia pastoris cells with distributed time delays using the structure of Fig. 4B. Integration of material balances along with the equations obtained from the sensitivity method is carried out using the linear approximation integration schema described in the Methods section. The times series were chosen such that one of the delays matched the maximum of the time delay of the weighting function of the simulation case (see Eqs. A 12).
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