f.influence: Field of Influence
In jjpwade/ioanalysis: Input-Output Analysis

Description Usage Arguments Details Value Author(s) References See Also Examples

Calculates the field of influence. Can handle first to nth order field of influence. Uses the method as Sonis & Hewings 1992. This is a recursive technique, so computation time depends on the size of the data and order of field of influence.

WARNING: Since this is a recursive function, each recursion saves the object for each recursion. This would make it impossible to run for large datasets, even with only a small order of field of influence. Therefore, it is WILDLY more memory efficient to call the object from the global environment (workspace). Thus, for this function you only input the name of the InputOutput object, not the object itself.

This function is primarily intended as a helper function for inverse.important

1	f.influence(ioname, i , j)

`ioname`	Character. The name verbatim of your `InputOutput` from `as.inputoutput` as saved in your workspace.
`i`	Numeric. The row component(s) of the coefficient(s) of interest
`j`	Numeric. The column component(S) of the coefficient(s) of interest

First Order Field of Influence - This is simply the product of the jth column of the Leontief inverse multiplied by the ith row of the Leontief inverse. In matrix notation:

F_1[i, j] = L_{.j} L_{i.}

where F denotes the field of influence, and i and j are scalars

Nth Order Field of Influence - This is a recursive function used to calculate higher order fields of influence. The order cannot exceed the size of the Intermediate Transaction Matrix (Z). I.e. if Z is 20x20, you can only calculate up to the 19th order. The formula is as follows:

F_k[(i_1,...,i_k), (j_1,...,j_k)] = \frac{1}{k-1} ∑_{s=1}^k∑_{r=1}^k (-1)^{s+r+1} l_{i_s,j_r} F_{k-1}[i_{-s}, j_{-r}]

where F is the field of influence, k is order of influence, l_ij is the ith row and jth column element of the Leontief Inverse and -s indicates the sth element has been removed.

Returns a matrix of the Field of Influence

John J. P. Wade, Ignacio Sarmiento-Barbieri

Sonis, Michael & Hewings, Geoffry J.D. (1992), "Coefficient Chang in Input-Output Models: Theory and Applications," Economic Systems Research, 4:2, 143-158 (https://doi.org/10.1080/09535319200000013)

Blair, P.D. and Miller, R.E. (2009). "Input-Output Analysis: Foundations and Extensions". Cambridge University Press

Nazara, Suahasil & Guo, Dong & Hewings, Geoffrey J.D., & Dridi, Chokri, 2003. "PyIO. Input-Output Analysis with Python". REAL Discussion Paper 03-t-23. University of Illinois at Urbana-Champaign. (http://www.real.illinois.edu/d-paper/03/03-t-23.pdf)

inverse.important

data(toy.IO)
class(toy.IO)
# First order field of influence on L[3,2]
i <- 3
j <- 2
f.influence("toy.IO", i, j)

# Second order field of influence on L[3,2], L[4,5], L[6, 3], and L[1,10]
i <- c(3, 4, 6,  1)
j <- c(2, 5, 3, 10)
f.influence("toy.IO", i, j)