ioanalysis (version 0.3.4)

feedback.loop: Feedback Loop Analysis

Description

Calculates the complete hierarchical feedback loop as described in Sonis et al. (1995). A feed back loop is complete if it contains all region-sector pairs. Much like a sudoku puzzle, there may only be one identified cell in each row and one identified cell in each column per loop. The loops are hierarchical in the sense that first loop maximizes the intermediate transactions given the aforementioned constraints.

There are TWO functions for RAM concerns. A singular function storing all feedback loop matrices grows at rate n^3. Alternatively, constructing feedback loop matrices one at a time translates to the output of feedback.loop growth rate of roughly 2n^2.

Note: A feedback loop solves the Linear Programming Assignment problem.

Warning: Computation time depends on size of the system. A progress bar is printed.

Usage

feedback.loop(io, agg.sectors, agg.regions, n.loops)
feedback.loop.matrix(fl, loop)

Arguments

io

An object of class InputOutput calculated from as.inputoutput

agg.sectors

An option to aggregate the sectors to compare regions only. Default is FALSE.

agg.regions

An option to aggregate the regions to compare sectors only. Default is FALSE.

n.loops

The number of loops you wish to calculate. The default is "all". Must either be an integer or "all"

fl

An object of class FeedbackLoop created from feedback.loop

loop

The loop from which you want the selector matrix.

Value

Produces a nested list: fl

fl

Contains "value", "loop_1", "loop_2", ..., and "loop_n"

value

Contains a vector of the total value of intermediate transactions for each loop.

loop_i

Contains a list over each loop's subloops. Retrieve by calling fl$loop_i$subloop_j. Note each loop will likely have a different number of subloops.

Details

The feedback loop solves the following optimization problem:

$$max_S vec(Z)'vec(S)$$ such that: $$i) A_{col}vec(S) = vec(1)$$ $$ii) A_{row}vec(S) = vec(1)$$ $$iii) vec(0) \le vec(S) \le vec(1)$$

where \(Z\) is the intermediate transaction matrix from io, \(S\) is a selctor matrix of the cells in \(Z\), \(A_{col}\) is a constraint matrix to ensure only one cell per column is selected, \(A_{row}\) is a constraint matrix to ensure only one cell per row is selected, and constraint \(iii)\) ensures the values in the selector matrix are either one or zero.

After each loop, the selected cells are set to an extremely negative number to prevent selection in the next loop.

See the documentation on http://www.real.illinois.edu/ for more details and interpretation of the loops.

References

Sonis, M., Hewings, G. J., & Gazel, R (1995). The structure of multi-regional trade flows: hierarchy, feedbacks and spatial linkages. The Annals of Regional Science, 29(4) 409-430.

See Also

as.inputoutput

Examples

# NOT RUN {
##########################
# The base feedback loop #
##########################
data(toy.IO)
class(toy.IO)

fbl = feedback.loop(toy.IO)
fbl$loop_1

fl_3 = feedback.loop.matrix(fbl, 3)
heatmap.io(fl_3, RS_label = toy.IO$RS_label)

fbl$value
fbl$per = fbl$value / sum(fbl$value) * 100

obj = data.frame(x = 1:length(fbl$per), y = fbl$per)

ggplot(obj, aes(x = x, y = y)) + 
  geom_line() + geom_point() +
  labs(x = 'Loop', y = 'Percent', title = 'Proportion of Total Intermediate Transactions per Loop')

###############################
# An aggregated feedback loop #
###############################
fbl_agg = feedback.loop(toy.IO, agg.regions = TRUE)
io_agg  = agg.region(toy.IO, regions = 'all', newname = 'magic')

fl_agg_1 = feedback.loop.matrix(fbl_agg, loop = 1)

heatmap.io(fl_agg_1, RS_label = io_agg$RS_label)
# }