Generally, if we denote the activation of a parent node by a p and the activations of its child node by a ci, with i=1,…,z and ac 1>ac 2>…>a cz, then the following activation spreading rule applies for an exclusive relation: a p =median({a ci |a ci ≥a cU }).
Let x k, s, t, k=1,…, K, t=1,…, T, s=1,…, S denote the activation level of the intermediate of the k-th cluster at the t-th time point under the s-th stimulation and x s, t = x1, s, t,…, x K, s, t )′.
f j ( j = 1, 2, …, m ) and g i ( i = 1, 2, …, n ) denote the activation function of the j th neurons of Y-layer on the i th neurons of X-layer and the i th neurons of X-layer on the j th neurons of Y-layer at time t and in space x, respectively.
The function f denotes the activation function of each node, and it will be discussed in the following section.
b j i, b ˜ j i, b ¯ j i, d i j, d ¯ i j and d ˜ i j are known constants denoting the synaptic connection strengths between the neurons in the two layers, respectively; f j, f ˜ j, f ¯ j, g i, g ˜ i and g ¯ i denote the activation functions of the neurons and the signal propagation functions, respectively.
More specifically, ({{widehat rho _{v}} = frac {1}{M}sum limits _{m = 1}^{M} {{a_{v}}left ({{mathbf {u}}_{(m)}}right)}}) is the average activation of unit v with the m-th input sample u (m), where ({a_{v}} = text {sigm}left ({mathbf {w}}_{v}^{(1)}{mathbf {u}} + b_{v}^{(1)}right)) denotes the activation value of the hidden layer unit v in the autoencoder and M is the number of training samples.
where t ≥ 0 is the time, x ( t ) = ( x 1 ( t ), x 2 ( t ), …, x n ( t ) ) T ∈ R n is the state vector associated with n neurons, f ( x ( t ) ) = ( f 1 ( x 1 ( t ) ), f 2 ( x 2 ( t ) ), …, f n ( x n ( t ) ) ) T ∈ R n denote the activation functions of the neurons, τ ( t ) is the transmission delay satisfying that 0 < τ ( t ) ≤ τ ¯ and τ ˙ ( t ) ≤ τ ˆ < 1, where τ ¯, τ ˆ are constants.
The rest of the used symbols are as follows: -, are weights set by the user, - is the predicted value of species j in the experiment k, - is the measured value of species j in experiment k, - denotes the activation or not of reaction i in experiment k. - y i, is a variable denoting whether a reaction is possible or not y i = 0 ∨ y i = 1.
These characterisation tests denoted the significant effect of acid activation in both preparation procedures to obtain CMD samples with high % of Mn(IV oxides.
F j ( j = 1, 2, …, n ) denotes the fuzzy activation function of the j th neurons on the i th neurons inside of X-layer.
