LINEAR MODEL

Every statistic study begins with a linear model, here:
Y = a₀ + a₁X₁ + a₂X₂ + a₃X₃ 
In neural language, the variables X₁, X₂, X₃ are input neurons, the variable Y is the output neuron, the coefficients a₀, a₁, a₂, a₃ are the synaptic values, they are adjusted during the learning.

SECOND ORDER POLYNOMIAL MODEL

The second order polynomial model allows characterizing the influence of squares and cross products, here:
Y = a₀ + a₁X₁ + a₂X₂ + a₃X₃ + a₄X₁² + a₅X₁X₂ + a₆X₂ ² + a₇X₁X₃ + a₈X₂X₃ + a₉X₃² 
It is a non-linear model towards the inputs variables, but linear towards the coefficients. The unique solution to this ten-coefficients model requires at least ten examples in the learning base.

THIRD ORDER POLYNOMIAL MODEL

The third order polynomial model add another difficulty, but it is still a linear model towards its coefficients. Here, the unique solution to this twenty-coefficients model requires at least twenty examples in the learning base.

STATIC NETWORKS

Static networks are the most common among neural networks. They are non-linear models towards their entries and towards their coefficients, since the neurons in the hidden layer, here neurons 6,7 and 8, have a non-linear activation function, in a sigmoid type. They are used when every example in the learning base are independent from one another, and when time is not a functional parameter.

MULTI-OUTPUT STATIC NETWORKS

The opposite graphic shows the first type of multi-output network where every output is a linear combination of the neurons from the hidden layer.

MULTI-LAYER STATIC NETWORKS

Multi-layer networks (and here, also multi-output) are useless in regression, but they can be useful for classification.
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NEURAL NETWORKS WITH CONNECTION BETWEEN HIDDEN NEURONS

These networks are characterized by their hidden neurons that depend from one another. They are seldom used.

PROBABILITY NETWORKS

Neural networks are excellent tools to characterize the probability density function of any probability law. The results are very good for thick-tailed laws and for the estimation of the extreme values.

TEMPORAL STATIC NETWORKS

For these networks that are still static networks, time is an important functional parameter, They are called sequences of data separated by constant time intervals. The terms Y*(t-1) and Y*(t-2) are the output values that have been observed (or measured), while Y is the output value that is calculated by the model.

DYNAMIC INPUT-OUTPUT NETWORKS

These networks are dynamic, since they contain loops. Time is an important functional parameter. The terms Y (t-1) and Y (t) are the output values calculated by the model.

MULTI-LOOP DYNAMIC NETWORKS

Neuro One allows using several lateness, two here, but this number is not limited.

ARIMA/GARCH NETWORKS, etc...

Neuro One allows building complex networks, particularly ARIMA or GARCH networks which compare the observed values and the calculated values.
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STATES DYNAMIC NETWORKS

The second type of dynamic networks. The states models characterize the models whose internal states are not measurable.

MULTI-STATE DYNAMIC NETWORKS

The number of states is not limited in Neuro One