matsbynameMatrices 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 |
| Energy | energy carriers | energy conversion steps |
The power of matrices lies in their ability to organize network-wide calculations, thereby simplifying the work of analysts who study entire systems. However, three problems arise when performing matrix operations in R and other languages.
Although built-in matrix functions ensure size conformity of matrix operands, they do not respect the names of rows and columns (known as dimnames in R). In the following example, U represents a use matrix that contains the quantity of each product used by each industry, and Y represents a final demand matrix that contains the quantity of each product consumed by final demand industries. If the rows and columns are not in the same order, the sum of the matrices is nonsensical.
productnames <- c("p1", "p2")
industrynames <- c("i1", "i2")
U <- matrix(1:4, ncol = 2, dimnames = list(productnames, industrynames))
U
#> i1 i2
#> p1 1 3
#> p2 2 4
Y <- matrix(1:4, ncol = 2, dimnames = list(rev(productnames), rev(industrynames)))
Y
#> i2 i1
#> p2 1 3
#> p1 2 4
# This sum is nonsensical. Neither row nor column names are respected.
U + Y
#> i1 i2
#> p1 2 6
#> p2 4 8As a result, analysts performing matrix operations must maintain strict order of rows and columns across all calculations.
# Make a new version of Y (Y2), this time with dimnames in same order as U
Y2 <- matrix(4:1, ncol = 2, dimnames = list(productnames, industrynames))
Y2
#> i1 i2
#> p1 4 2
#> p2 3 1
# Now the sum is sensible. Both row and column names are respected.
U + Y2
#> i1 i2
#> p1 5 5
#> p2 5 5In many cases, operand matrices may have different numbers or different names of rows or columns. This situation can occur when, for example, products or industries changes across time periods. When performing matrix operations, rows or columns of zeros must be added to ensure name conformity.
Y3 <- matrix(5:8, ncol = 2, dimnames = list(c("p1", "p3"), c("i1", "i3")))
Y3
#> i1 i3
#> p1 5 7
#> p3 6 8
# Nonsensical because neither row names nor column names are respected.
# The "p3" rows and "i3" columns of Y3 have been added to
# "p2" rows and "i2" columns of U.
# Row and column names for the sum are taken from the first operand (U).
U + Y3
#> i1 i2
#> p1 6 10
#> p2 8 12
# Rather, need to insert missing rows in both U and Y before summing.
U_2000 <- matrix(c(1, 3, 0,
2, 4, 0,
0, 0, 0),
ncol = 3, byrow = TRUE,
dimnames = list(c("p1", "p2", "p3"), c("i1", "i2", "i3")))
Y_2000 <- matrix(c(5, 0, 7,
0, 0, 0,
6, 0, 8),
ncol = 3, byrow = TRUE,
dimnames = list(c("p1", "p2", "p3"), c("i1", "i2", "i3")))
U_2000
#> i1 i2 i3
#> p1 1 3 0
#> p2 2 4 0
#> p3 0 0 0
Y_2000
#> i1 i2 i3
#> p1 5 0 7
#> p2 0 0 0
#> p3 6 0 8
U_2000 + Y_2000
#> i1 i2 i3
#> p1 6 3 7
#> p2 2 4 0
#> p3 6 0 8The analyst’s burden is cumbersome. But worse problems await.
Respecting names (and adding rows and columns of zeroes) can lead to an inability to invert matrices downstream, as shown in the following example.
# The original U matrix is invertible.
solve(U)
#> p1 p2
#> i1 -2 1.5
#> i2 1 -0.5
# The version of U that contains zero rows and columns (U_2000)
# is singular and cannot be inverted.
tryCatch(solve(U_2000), error = function(err){print(err)})
#> <simpleError in solve.default(U_2000): Lapack routine dgesv: system is exactly singular: U[3,3] = 0>Matrix functions provided by R and other languages do not ensure type conformity for matrix operands to matrix algebra functions. In the example of matrix multiplication, columns of the multiplicand must contain the same type of information as the as the rows of the multiplier. If the columns of A are countries, then the rows of B must also be countries (and in the same order) if A %*% B is to make sense.
The matsbyname package automatically addresses all three problems above. It performs smart matrix operations that
These features are available without analyst intervention, as shown in the following example.
# Same as U + Y2, without needing to create Y2.
sum_byname(U, Y)
#> i1 i2
#> p1 5 5
#> p2 5 5
# Same as U_2000 + Y_2000, but U and Y3 are unmodified.
sum_byname(U, Y3)
#> i1 i2 i3
#> p1 6 3 7
#> p2 2 4 0
#> p3 6 0 8
# Eliminate zero-filled rows and columns. Same result as solve(U).
U_2000 %>% clean_byname(margin = c(1,2), clean_value = 0) %>% solve()
#> p1 p2
#> i1 -2 1.5
#> i2 1 -0.5In addition to sum_byname and clean_byname, the matsbyname package contains many additional matrix algebra functions that respect the names of rows and columns. Commonly-used functions are:
sum_byname()difference_byname()hadamardproduct_byname()matrixproduct_byname()quotient_byname()rowsums_byname()colsums_byname()invert_byname(), andtranspose_byname().The full list of functions can be found with ?matsbyname and clicking the Index link.
Furthermore, matsbyname works well with its sister package, matsindf. matsindf creates data frames whose entries are not numbers but entire matrices, thereby enabling the use of matsbyname functions in tidyverse functional programming.
When used together, matsbyname and matsindf allow analysts to wield simultaneously the power of both matrix mathematics and tidyverse functional programming.
This vignette demonstrates the power of matrix mathematics performed by name.
The matsbyname package has several features that both simplify analyses and ensure their correctness.
In the preceding examples, row and column names were provided by the dimnames argument to the matrix function. However, matsbyname provides the setcolnames_byname and setrownames_byname functions to perform the same tasks using the pipe operator (%>%).
U_2 <- matrix(1:4, ncol = 2) %>%
setrownames_byname(productnames) %>% setcolnames_byname(industrynames)
U_2
#> i1 i2
#> p1 1 3
#> p2 2 4Row and column types can be understood by analogy: row and column types are to matrices in matrix algebra as units are to scalars in scalar algebra. Just as careful tracking of units is necessary in scalar calculations, careful tracking of row and column types is necessary in matrix operations. Because matsbyname keeps track of row and column types automatically, much of the burden of dealing with row and column types is removed from the analyst.
Row and column types are character strings stored as attributes of the matrix object, and matsbyname functions ensure correctness of matrix operations by checking row and column types, throwing errors if needed. Row and column types can be set by the functions setrowtype and setcoltype and retrieved by the functions rowtype and coltype. Consider matrices A, B, and C:
A <- matrix(1:4, ncol = 2) %>%
setrownames_byname(productnames) %>% setcolnames_byname(industrynames) %>%
setrowtype("Products") %>% setcoltype("Industries")
A
#> i1 i2
#> p1 1 3
#> p2 2 4
#> attr(,"rowtype")
#> [1] "Products"
#> attr(,"coltype")
#> [1] "Industries"
B <- matrix(8:5, ncol = 2) %>%
setrownames_byname(productnames) %>% setcolnames_byname(industrynames) %>%
setrowtype("Products") %>% setcoltype("Industries")
B
#> i1 i2
#> p1 8 6
#> p2 7 5
#> attr(,"rowtype")
#> [1] "Products"
#> attr(,"coltype")
#> [1] "Industries"
C <- matrix(1:4, ncol = 2) %>%
setrownames_byname(industrynames) %>% setcolnames_byname(productnames) %>%
setrowtype("Industries") %>% setcoltype("Products")
C
#> p1 p2
#> i1 1 3
#> i2 2 4
#> attr(,"rowtype")
#> [1] "Industries"
#> attr(,"coltype")
#> [1] "Products"B can be added to A, because row and column types are identical.
sum_byname(A, B)
#> i1 i2
#> p1 9 9
#> p2 9 9
#> attr(,"rowtype")
#> [1] "Products"
#> attr(,"coltype")
#> [1] "Industries"However, C cannot be added to A (or B), because row and column types disagree.
tryCatch(sum_byname(A, C), error = function(err){print(err)})
#> <simpleError in organize_args(a, b, fill = 0, match_type = match_type): rowtype(a) (Products) != rowtype(b) (Industries).>In this case, a sum is possible if C is transposed prior to adding to A, because row and column types of A and CT agree.
sum_byname(A, transpose_byname(C))
#> i1 i2
#> p1 2 5
#> p2 5 8
#> attr(,"rowtype")
#> [1] "Products"
#> attr(,"coltype")
#> [1] "Industries"Matrices A and B can be element-multiplied and element-divided for the same reason they can be summed: row and column types agree.
hadamardproduct_byname(A, B)
#> i1 i2
#> p1 8 18
#> p2 14 20
#> attr(,"rowtype")
#> [1] "Products"
#> attr(,"coltype")
#> [1] "Industries"
quotient_byname(A, B)
#> i1 i2
#> p1 0.1250000 0.5
#> p2 0.2857143 0.8
#> attr(,"rowtype")
#> [1] "Products"
#> attr(,"coltype")
#> [1] "Industries"Note that A and C can be matrix-multiplied, because the column type of A and the row type of C are identical (Industries). The result is a Products-by-Products matrix.
matrixproduct_byname(A, C)
#> p1 p2
#> p1 7 15
#> p2 10 22
#> attr(,"rowtype")
#> [1] "Products"
#> attr(,"coltype")
#> [1] "Products"However, A and B cannot be matrix-multiplied, because the column type of A (Industries) and the row type of B (Products) are different.
tryCatch(matrixproduct_byname(A, B), error = function(err){print(err)})
#> <simpleError in organize_args(a, b, fill = 0, match_type = match_type): coltype(a) != rowtype(b): Industries != Products.>Analysts are encouraged to set row and column types on matrices, thereby taking advantage of matsbyname’s type-tracking feature to improve their matrix-based analyses.
*_byname functions work well with listsAnother feature of the matsbyname package is that it works when arguments to functions are lists of matrices, returning lists as appropriate.
sum_byname(A, list(B, B))
#> [[1]]
#> i1 i2
#> p1 9 9
#> p2 9 9
#> attr(,"rowtype")
#> [1] "Products"
#> attr(,"coltype")
#> [1] "Industries"
#>
#> [[2]]
#> i1 i2
#> p1 9 9
#> p2 9 9
#> attr(,"rowtype")
#> [1] "Products"
#> attr(,"coltype")
#> [1] "Industries"
hadamardproduct_byname(list(A, A), B)
#> [[1]]
#> i1 i2
#> p1 8 18
#> p2 14 20
#> attr(,"rowtype")
#> [1] "Products"
#> attr(,"coltype")
#> [1] "Industries"
#>
#> [[2]]
#> i1 i2
#> p1 8 18
#> p2 14 20
#> attr(,"rowtype")
#> [1] "Products"
#> attr(,"coltype")
#> [1] "Industries"
matrixproduct_byname(list(A, A), list(C, C))
#> [[1]]
#> p1 p2
#> p1 7 15
#> p2 10 22
#> attr(,"rowtype")
#> [1] "Products"
#> attr(,"coltype")
#> [1] "Products"
#>
#> [[2]]
#> p1 p2
#> p1 7 15
#> p2 10 22
#> attr(,"rowtype")
#> [1] "Products"
#> attr(,"coltype")
#> [1] "Products"The matsindf package provides functions that collapse tidy data frames of matrix elements into data frames of matrices. Data frames of matrices, such as those created by matsindf, are like magic spreadsheets in which single cells contain entire matrices.
The following example demonstrates an approach to creating a data frame of matrices.
tidy <- data.frame(
matrix = c("A", "A", "A", "A", "B", "B", "B", "B"),
row = c("p1", "p1", "p2", "p2", "p1", "p1", "p2", "p2"),
col = c("i1", "i2", "i1", "i2", "i1", "i2", "i1", "i2"),
vals = c(1, 3, 2, 4, 8, 6, 7, 5)
) %>%
mutate(
rowtype = "Industries",
coltype = "Products"
)
tidy
#> matrix row col vals rowtype coltype
#> 1 A p1 i1 1 Industries Products
#> 2 A p1 i2 3 Industries Products
#> 3 A p2 i1 2 Industries Products
#> 4 A p2 i2 4 Industries Products
#> 5 B p1 i1 8 Industries Products
#> 6 B p1 i2 6 Industries Products
#> 7 B p2 i1 7 Industries Products
#> 8 B p2 i2 5 Industries Products
mats <- tidy %>%
group_by(matrix) %>%
matsindf::collapse_to_matrices(matnames = "matrix", matvals = "vals",
rownames = "row", colnames = "col",
rowtypes = "rowtype", coltypes = "coltype") %>%
rename(
matrix.name = matrix,
matrix = vals
)
mats
#> matrix.name matrix
#> 1 A 1, 2, 3, 4
#> 2 B 8, 7, 6, 5
mats$matrix[[1]]
#> i1 i2
#> p1 1 3
#> p2 2 4
#> attr(,"rowtype")
#> [1] "Industries"
#> attr(,"coltype")
#> [1] "Products"
mats$matrix[[2]]
#> i1 i2
#> p1 8 6
#> p2 7 5
#> attr(,"rowtype")
#> [1] "Industries"
#> attr(,"coltype")
#> [1] "Products"matsbyname with matsindfBecause
matsbyname works well with lists, andmatsbyname functions can be used with tidyr and dplyr functions (such as spread and mutate) to perform matrix algebra within data frames of matrices. A single matsbyname instruction performs the same operation on all rows of a matsindf data frame. Loops begone!
result <- mats %>%
tidyr::spread(key = matrix.name, value = matrix) %>%
# Duplicate the row to demonstrate byname operating simultaneously
# on all rows of the data frame.
dplyr::bind_rows(., .) %>%
dplyr::mutate(
# Create a column of constants.
c = make_list(x = 1:2, n = 2, lenx = 2),
# Sum all rows of the data frame with a single instruction.
sum = sum_byname(A, B),
# Multiply matrices in the sum column by corresponding constants in the c column.
product = hadamardproduct_byname(c, sum)
)
result
#> A B c sum product
#> 1 1, 2, 3, 4 8, 7, 6, 5 1 9, 9, 9, 9 9, 9, 9, 9
#> 2 1, 2, 3, 4 8, 7, 6, 5 2 9, 9, 9, 9 18, 18, 18, 18
result$sum[[1]]
#> i1 i2
#> p1 9 9
#> p2 9 9
#> attr(,"rowtype")
#> [1] "Industries"
#> attr(,"coltype")
#> [1] "Products"
result$sum[[2]]
#> i1 i2
#> p1 9 9
#> p2 9 9
#> attr(,"rowtype")
#> [1] "Industries"
#> attr(,"coltype")
#> [1] "Products"
result$product[[1]]
#> i1 i2
#> p1 9 9
#> p2 9 9
#> attr(,"rowtype")
#> [1] "Industries"
#> attr(,"coltype")
#> [1] "Products"
result$product[[2]]
#> i1 i2
#> p1 18 18
#> p2 18 18
#> attr(,"rowtype")
#> [1] "Industries"
#> attr(,"coltype")
#> [1] "Products"A suggested analysis workflow is as follows:
For more information and a detailed example of this workflow, see the vignette for the matsindf package.
The matsbyname package simplifies analyses in which row and column names ought to be respected. It provides optional row and column types, thereby ensuring that only valid matrix operations are performed. Finally, matsbyname functions work equally well with lists to allow use of *_byname functions with tidyr and dplyr approaches to manipulating data.