vignettes/matsindf.Rmd
matsindf.Rmd
Matrices are important mathematical objects, and they often describe networks of flows among nodes. Example networks are given in the following table.
System type  Flows  Nodes 

Ecological  nutrients  organisms 
Manufacturing  materials  factories 
Economic  money  economic sectors 
The power of matrices lies in their ability to organize networkwide calculations, thereby simplifying the work of analysts who study entire systems.
But wouldn’t it be nice if there were an easy way to create data frames whose entries were not numbers but entire matrices? If that were possible, matrix algebra could be performed on columns of similar matrices.
That’s the reason for matsindf
. It provides functions to convert a suitablyformatted tidy data frame into a data frame containing a column of matrices.
Furthermore, matsbyname
is a sister package that …
dimnames
in R
) to free the analyst from the task of aligning rows and columns of operands (matrices) passed to matrix algebra functions andWhen used together, matsindf
and matsbyname
allow analysts to wield simultaneously the power of both matrix mathematics and tidyverse functional programming.
This vignette demonstrates the use of these packages and suggests a workflow to accomplish sophisticated analyses using matrices in data frames (matsindf
).
UKEnergy2000
To demonstrate the use of matsindf
functions, consider a network of energy flows from the environment, through transformation and distribution processes, and, ultimately, to final demand. Such energy flow networks are called energy conversion chains (ECCs), and this example is based on an approximation to a portion of the UK’s ECC circa 2000. (Note that these data are to be used for demonstration purposes only and have been rounded to 1–2 significant digits.) These example data first appeared in Figures 3 and 4 of Heun, Owen, and Brockway (2018).
head(UKEnergy2000, 2)
#> Country Year Ledger.side Flow.aggregation.point Flow
#> 1 GB 2000 Supply Total primary energy supply Resources  Crude
#> 2 GB 2000 Supply Total primary energy supply Resources  NG
#> Product E.ktoe
#> 1 Crude 50000
#> 2 NG 43000
Country
and Year
contain only one value each, GB
and 2000
respectively. Following conventions of the International Energy Agency’s energy balance tables,
Ledger.side
indicates Supply
or Consumption
;Flow.aggregation.point
indicates how data are to be aggregated;Flow
indicates the industry, machine, or final demand sector for this flow;Product
indicates the energy carrier for this flow; andE.ktoe
gives the magnitude of this flow in units of kilotons of oil equivalent (ktoe).Each flow is its own observation (its own row) in the UKEnergy2000
data frame, making it tidy.
The remainder of this vignette demonstrates an analysis conducted using the UKEnergy2000
data frame as a basis. It:
matsbyname
functions,The EnergyUK2000
data frame is similar to “cleaned” data from an external source: there are no missing entries, and it is tidy. But the data are not organized as matrices, and additional metadata is needed.
The collapse_to_matrices
function converts a tidy data frame into a matsindf
data frame using using information within the tidy data frame. So the first task is to prepare for collapse by adding metadata columns.
collapse_to_matrices
needs the following information:
argument to collapse_to_matrices

identifies 

matnames 
Name of the input column of matrix names 
values 
Name of the input column of matrix entries 
rownames 
Name of the input column of matrix row names 
colnames 
Name of the input column of matrix column name 
rowtypes 
Optional name of the input column of matrix row types 
coltypes 
Optional name of the input column of matrix column types 
The following code gives the approach to adding metadata, appropriate for this application, relying on Ledger.side
, the sign of E.ktoe
, and knowledge about the rows and columns for each matrix. Each type of network will have its own algorithm for identifying row names, column names, row types, and column types in a tidy data frame.
UKEnergy2000_with_metadata < UKEnergy2000 %>%
# Add a column indicating the matrix in which this entry belongs (U, V, or Y).
matsindf:::add_UKEnergy2000_matnames(.) %>%
# Add columns for row names, column names, row types, and column types.
matsindf:::add_UKEnergy2000_row_col_meta(.) %>%
mutate(
# Eliminate columns we no longer need
Ledger.side = NULL,
Flow.aggregation.point = NULL,
Flow = NULL,
Product = NULL,
# Ensure that all energy values are positive, as required for analysis.
E.ktoe = abs(E.ktoe)
)
head(UKEnergy2000_with_metadata, 2)
#> Country Year E.ktoe matname rowname colname rowtype coltype
#> 1 GB 2000 50000 V Resources  Crude Crude Industry Product
#> 2 GB 2000 43000 V Resources  NG NG Industry Product
With the metadata now in place, UKEnergy2000_with_metadata
can be collapsed to a matsindf
data frame by the collapse_to_matrices
function. Much like dplyr::summarise
, collapse_to_matrices
relies on grouping to indicate which rows of the tidy data frame belong to which matrices. The usual approach is to tidyr::group_by
the matnames
column and any other columns to be preserved in the output, in this case Country
and Year
.
EnergyMats_2000 < UKEnergy2000_with_metadata %>%
group_by(Country, Year, matname) %>%
collapse_to_matrices(matnames = "matname", matvals = "E.ktoe",
rownames = "rowname", colnames = "colname",
rowtypes = "rowtype", coltypes = "coltype") %>%
rename(matrix.name = matname, matrix = E.ktoe)
# The remaining columns are Country, Year, matrix.name, and matrix
glimpse(EnergyMats_2000)
#> Rows: 3
#> Columns: 4
#> $ Country <chr> "GB", "GB", "GB"
#> $ Year <int> 2000, 2000, 2000
#> $ matrix.name <chr> "U", "V", "Y"
#> $ matrix <list> <<matrix[11 x 9]>>, <<matrix[11 x 12]>>, <<matrix[4 x 2]>>
# To access one of the matrices, try one of these approaches:
(EnergyMats_2000 %>% filter(matrix.name == "U"))[["matrix"]] # The U matrix
#> [[1]]
#> Crude dist. Diesel dist. Elect. grid Gas wells & proc. NG dist.
#> Crude 0 0 0 0 0
#> Crude  Dist. 0 0 0 0 0
#> Crude  Fields 47500 0 0 0 0
#> Diesel 0 15500 0 0 0
#> Diesel  Dist. 25 0 0 50 25
#> Elect 0 0 6400 0 0
#> Elect  Grid 25 0 0 25 25
#> NG 0 0 0 43000 0
#> NG  Dist. 0 0 0 0 0
#> NG  Wells 0 0 0 0 41000
#> Petrol 0 0 0 0 0
#> Oil fields Oil refineries Petrol dist. Power plants
#> Crude 50000 0 0 0
#> Crude  Dist. 0 47000 0 0
#> Crude  Fields 0 0 0 0
#> Diesel 0 0 0 0
#> Diesel  Dist. 50 0 250 0
#> Elect 0 0 0 0
#> Elect  Grid 25 75 0 100
#> NG 0 0 0 0
#> NG  Dist. 0 0 0 16000
#> NG  Wells 0 0 0 0
#> Petrol 0 0 26500 0
#> attr(,"rowtype")
#> [1] "Product"
#> attr(,"coltype")
#> [1] "Industry"
EnergyMats_2000$matrix[[2]] # The V matrix
#> Crude Crude  Dist. Crude  Fields Diesel Diesel  Dist.
#> Crude dist. 0 47000 0 0 0
#> Diesel dist. 0 0 0 0 15150
#> Elect. grid 0 0 0 0 0
#> Gas wells & proc. 0 0 0 0 0
#> NG dist. 0 0 0 0 0
#> Oil fields 0 0 47500 0 0
#> Oil refineries 0 0 0 15500 0
#> Petrol dist. 0 0 0 0 0
#> Power plants 0 0 0 0 0
#> Resources  Crude 50000 0 0 0 0
#> Resources  NG 0 0 0 0 0
#> Elect Elect  Grid NG NG  Dist. NG  Wells Petrol
#> Crude dist. 0 0 0 0 0 0
#> Diesel dist. 0 0 0 0 0 0
#> Elect. grid 0 6275 0 0 0 0
#> Gas wells & proc. 0 0 0 0 41000 0
#> NG dist. 0 0 0 41000 0 0
#> Oil fields 0 0 0 0 0 0
#> Oil refineries 0 0 0 0 0 26500
#> Petrol dist. 0 0 0 0 0 0
#> Power plants 6400 0 0 0 0 0
#> Resources  Crude 0 0 0 0 0 0
#> Resources  NG 0 0 43000 0 0 0
#> Petrol  Dist.
#> Crude dist. 0
#> Diesel dist. 0
#> Elect. grid 0
#> Gas wells & proc. 0
#> NG dist. 0
#> Oil fields 0
#> Oil refineries 0
#> Petrol dist. 26000
#> Power plants 0
#> Resources  Crude 0
#> Resources  NG 0
#> attr(,"rowtype")
#> [1] "Industry"
#> attr(,"coltype")
#> [1] "Product"
EnergyMats_2000$matrix[[3]] # The Y matrix
#> Residential Transport
#> Diesel  Dist. 0 14750
#> Elect  Grid 6000 0
#> NG  Dist. 25000 0
#> Petrol  Dist. 0 26000
#> attr(,"rowtype")
#> [1] "Product"
#> attr(,"coltype")
#> [1] "Sector"
Larger studies will include data for multiple countries and years. The ECC data from UK in year 2000
can be duplicated for 2001
and for a fictitious country AB
. Although the data are unchanged, the additional rows serve to illustrate the functional programming aspects of the matsindf
and matsbyname
packages.
Energy < EnergyMats_2000 %>%
# Create rows for a fictitious country "AB".
# Although the rows for "AB" are same as the "GB" rows,
# they serve to illustrate functional programming with matsindf.
rbind(EnergyMats_2000 %>% mutate(Country = "AB")) %>%
spread(key = Year, value = matrix) %>%
mutate(
# Create a column for a second year (2001).
`2001` = `2000`
) %>%
gather(key = Year, value = matrix, `2000`, `2001`) %>%
# Now spread to put each matrix in a column.
spread(key = matrix.name, value = matrix)
glimpse(Energy)
#> Rows: 4
#> Columns: 5
#> $ Country <chr> "AB", "AB", "GB", "GB"
#> $ Year <chr> "2000", "2001", "2000", "2001"
#> $ U <list> <<matrix[11 x 9]>>, <<matrix[11 x 9]>>, <<matrix[11 x 9]>>, <<…
#> $ V <list> <<matrix[11 x 12]>>, <<matrix[11 x 12]>>, <<matrix[11 x 12]>>,…
#> $ Y <list> <<matrix[4 x 2]>>, <<matrix[4 x 2]>>, <<matrix[4 x 2]>>, <<ma…
An important step in any analysis is data verification. For an ECC analysis, it is important to verify that energy is conserved (i.e., energy is in balance) across all industries. Equations 1 and 2 in Heun, Owen, and Brockway (2018) show that energy balance is verified by
\[\mathbf{W} = \mathbf{V}^\mathrm{T}  \mathbf{U},\]
and
\[\mathbf{W}\mathbf{i}  \mathbf{Y}\mathbf{i} = \mathbf{0}.\]
Energy balance verification can be implemented with matsbyname
functions and tidyverse
functional programming:
Check < Energy %>%
mutate(
W = difference_byname(transpose_byname(V), U),
# Need to change column name and type on y so it can be subtracted from row sums of W
err = difference_byname(rowsums_byname(W),
rowsums_byname(Y) %>%
setcolnames_byname("Industry") %>% setcoltype("Industry")),
EBalOK = iszero_byname(err)
)
Check %>% select(Country, Year, EBalOK)
#> Country Year EBalOK
#> 1 AB 2000 TRUE
#> 2 AB 2001 TRUE
#> 3 GB 2000 TRUE
#> 4 GB 2001 TRUE
all(Check$EBalOK %>% as.logical())
#> [1] TRUE
This example demonstrates that energy balance can be verified for all combinations of Country and Year with a few lines of code. In fact, the exact same code can be applied to the Energy
data frame, regardless of the number of rows in it.
Secure in the knowledge that energy is conserved across all ECCs in the Energy
data frame, other analyses can proceed.
To further illustrate the power of matsbyname
functions in the context of matsindf
, consider the calculation of the efficiency of every industry in the ECC as column vector \(\eta\) as shown by Equation 11 of Heun, Owen, and Brockway (2018).
\[\mathbf{g} = \mathbf{V}\mathbf{i}\]
\[\mathbf{\eta} = \widehat{\mathbf{U}^\mathrm{T} \mathbf{i}}^{\mathrm{}1} \mathbf{g}\]
Etas < Energy %>%
mutate(
g = rowsums_byname(V),
eta = transpose_byname(U) %>% rowsums_byname() %>%
hatize_byname() %>% invert_byname() %>%
matrixproduct_byname(g) %>%
setcolnames_byname("eta") %>% setcoltype("Efficiency")
) %>%
select(Country, Year, eta)
Etas$eta[[1]]
#> eta
#> Crude dist. 0.9884332
#> Diesel dist. 0.9774194
#> Elect. grid 0.9804688
#> Gas wells & proc. 0.9518282
#> NG dist. 0.9987820
#> Oil fields 0.9485771
#> Oil refineries 0.8921933
#> Petrol dist. 0.9719626
#> Power plants 0.3975155
#> attr(,"rowtype")
#> [1] "Industry"
#> attr(,"coltype")
#> [1] "Efficiency"
Note that only a few lines of code are required to perform the same series of matrix operations on every combination of Country
and Year
. In fact, the same code will be used to calculate the efficiency of any number of industries in any number of countries and years!
Plotting values from a matsindf
data frame can be accomplished by expanding the matrices of the matsindf
data frame (in this example, Etas
) back out to a tidy data frame. Expanding is the reverse of collapseing, and the following information must be supplied to the expand_to_tidy
function:
argument to expand_to_tidy

identifies 

matnames 
Name of the input column of matrix names 
matvals 
Name of the input column of matrices to be expanded 
rownames 
Name of the output column of matrix row names 
colnames 
Name of the output column of matrix column name 
rowtypes 
Optional name of the output column of matrix row types 
coltypes 
Optional name of the output column of matrix column types 
drop 
Optional value to be dropped from output (often 0) 
Prior to expand
ing, it is usually necessary to gather
columns of matrices.
etas_forgraphing < Etas %>%
gather(key = matrix.names, value = matrix, eta) %>%
expand_to_tidy(matnames = "matrix.names", matvals = "matrix",
rownames = "Industry", colnames = "etas",
rowtypes = "rowtype", coltypes = "Efficiencies") %>%
mutate(
# Eliminate columns we no longer need.
matrix.names = NULL,
etas = NULL,
rowtype = NULL,
Efficiencies = NULL
) %>%
rename(
eta = matrix
)
# Compare to Figure 8 of Heun, Owen, and Brockway (2018)
etas_forgraphing %>% filter(Country == "GB", Year == 2000)
#> # A tibble: 9 x 4
#> Country Year Industry eta
#> <chr> <chr> <chr> <dbl>
#> 1 GB 2000 Crude dist. 0.988
#> 2 GB 2000 Diesel dist. 0.977
#> 3 GB 2000 Elect. grid 0.980
#> 4 GB 2000 Gas wells & proc. 0.952
#> 5 GB 2000 NG dist. 0.999
#> 6 GB 2000 Oil fields 0.949
#> 7 GB 2000 Oil refineries 0.892
#> 8 GB 2000 Petrol dist. 0.972
#> 9 GB 2000 Power plants 0.398
etas_forgraphing
is a data frame of efficiencies, one for each Country, Year, and Industry, in a format that is amenable to plotting with packages such as ggplot.
The following code creates a bar graph of efficiency results for the UK in 2000:
etas_UK_2000 < etas_forgraphing %>% filter(Country == "GB", Year == 2000)
etas_UK_2000 %>%
ggplot(mapping = aes_string(x = "Industry", y = "eta",
fill = "Industry", colour = "Industry")) +
geom_bar(stat = "identity") +
labs(x = NULL, y = expression(eta[UK*","*2000]), fill = NULL) +
scale_y_continuous(breaks = seq(0, 1, by = 0.2)) +
scale_fill_manual(values = rep("white", nrow(etas_UK_2000))) +
scale_colour_manual(values = rep("gray20", nrow(etas_UK_2000))) +
guides(fill = FALSE, colour = FALSE) +
theme(axis.text.x = element_text(angle = 90, vjust = 0.4, hjust = 1))
This vignette demonstrated the use of the matsindf
and matsbyname
packages and suggested a workflow to accomplish sophisticated analyses using matrices in data frames (matsindf
).
The workflow is as follows:
UKEnergy2000
above.collapse_to_matrices
to create a data frame of matrices with columns for matrix names and matrices themselves, similar to EnergyMats_2000
above.tidyr::spread
the matrices to obtain a data frame with columns for each matrix, similar to Energy
above.Check
above.matsbyname
functions in a manner similar to the process of generating the Etas
data frame above.tidyr::gather
the columns to obtain a tidy data frame of matrices.expand_to_tidy
to create a tidy data frame of matrix elements, similar to etas_forgraphing
above.Data frames of matrices, such as those created by matsindf
, are like magic spreadsheets in which single cells contain entire matrices. With this data structure, analysts can wield simultaneously the power of both matrix mathematics and tidyverse functional programming.