.. _`Linear Difference Equation`:

.. _`com.sympathyfordata.timeseriesanalysis.project_linear_difference_equation`:

Linear Difference Equation
``````````````````````````

.. image:: linear_difference_equations.svg
   :width: 48


Performs a forward projection of a linear difference equation system




Documentation
:::::::::::::

Performs a forward projection of the following linear difference equation
system:

yᵢ(t) = aᵢ₀ + ∑ aᵢⱼₖ yⱼ(t-k) + ∑ bᵢⱼₖ uⱼ(t-k)

Where aᵢⱼₖ and bᵢⱼₖ are the coefficients giving the contribution
to yᵢ based on the value of yⱼ or uⱼ at time point t-k.

The system with v variables y_1 ... y_v  and control signals u_1 ... u_r
is described by (K+1) x (v+r) coefficients, where
K is the number of previous times samples considered by
this system.

This system takes as input v columns of (v+r)*K+1 rows as coefficients as
follows:

aᵢ₀ = column[i][0],

aᵢⱼₖ = column[i][j*K + k],

bᵢⱼₖ = column[i][V*K + j*K + k]


Given at least K rows of initial condition this node steps the
linear system forwards in time and outputs all variables at their
corresponding time points.

By default the system has no coefficients b and
the node configuration parameter T, defines the length of the system's
response.


To use the node for a system with a control signal, right click the node
and select
Ports>Input>Create: Control Signal.
In this case, the length of the control sequence defines also the length of
the output.


Definition
::::::::::


Input ports
...........

    **init cond**  table
        Initial conditions

    **coefficients**  table
        Coefficients

    **control signal** 0 - 1,  table
        Control signal


Output ports
............

    **out**  table
        Output




Configuration
.............

    **T** (T)
        Number of points generated in outputs including the initial condition points







Examples
........

* :download:`linear time invariant systems.syx </Examples/Time Series/linear time invariant systems.syx>`




Implementation
..............

.. automodule:: node_difference
    :noindex:

.. class:: DifferenceEquation
    :noindex: