In MatthewHeun/LMDIR: Log Mean Divisia Index Decomposition in R
knitr::opts_chunk$set(
collapse = TRUE,
comment = "#>"
)
library(dplyr)
library(ggplot2)
library(LMDIR)
library(magrittr)
library(matsbyname)
library(matsindf)
library(tidyr)
\newcommand{\transpose}[1]{#1^\mathrm{T}}
\newcommand{\inverse}[1]{#1^{\mathrm{-}1}}
\newcommand{\mat}[1]{\mathbf{#1}}
\newcommand{\colvec}[1]{\mathbf{#1}}
\newcommand{\rowvec}[1]{\transpose{\colvec{#1}}}
\newcommand{\inversetranspose}[1]{\transpose{\left( \inverse{\mat{#1}} \right)}}
\newcommand{\transposeinverse}[1]{\inverse{\left( \transpose{\mat{#1}} \right)}}
\newcommand{\hatinv}[1]{\inverse{\widehat{#1}}}
Introduction
Log-mean divisia index (LMDI) decomposition analysis is an important technique
for understanding causes of changes in an energy composite over time.
The technique was pioneered by B.W. Ang and is described in detail in
Ang [-@Ang:2005].
LMDI decomposes an energy aggregate by factors and categories.
Heun and Brockway [-@Heun:2019] developed a matrix representation
of the LMDI mathematics that works well with the
Physical Supply Use Table (PSUT) framework
developed by Heun et al. [-@Heun:2018].
This vignette describes
and provides examples for
the matrix method for LMDI decomposition analysis.
This vignette is a version of Section S3 of the Supplemental Information to
Heun and Brockway [-@Heun:2019].
Description of the matrix method for LMDI decomposition analysis
The LMDI approach pioneered by Ang et al. [-@Ang:1998]
and described in detail by Liu and Ang [-@Liu:2003]
decomposes an energy aggregate ($V$) by factors ($x$)
that cause $V$ to change over time.
(In this vignette, we adopt the notation of Ang [-@Ang:2005].)
Noting that at a given time every combination of energy category ($i$) and decomposition factor ($j$)
has an associated numerical value for the decomposition factor ($x_{ij}$),
the matrix approach to LMDI decomposition analysis begins with formation of an $\mat{X}$ matrix
for each time of interest.
Rows of $\mat{X}$ contain energy categories (indexed by $i$) of the aggregate ($V$) and
columns of $\mat{X}$ contain decomposition factors (indexed by $j$).
Column vector $\colvec{v}$ contains
the value of each energy category ($v_i$)
and is formed from row products of $\mat{X}$.
\begin{equation} \label{eq:v}
v_i = \prod_j x_{ij}
\end{equation}
At any given time, the energy aggregate ($V$) is obtained by
the column sum of elements in the $\colvec{v}$ vector
\begin{equation} \label{eq:V}
V = \rowvec{i} \colvec{v} \; ,
\end{equation}
where $\colvec{i}$ is a column vector of 1's (an identity vector) of same size as $\colvec{v}$.
Additive ($\Delta V$) and multiplicative ($D$) changes in the energy aggregate ($V$)
between two adjacent times,
an initial time (superscript $0$) and a later time (superscript $T$),
are calculated by
\begin{equation} \label{eq:deltaV_def}
\Delta V = V^T - V^0 \; ,
\end{equation}
and
\begin{equation} \label{eq:D_def}
D = \frac{V^T}{V^0} \; .
\end{equation}
A $\mat{Z}$ matrix is formed for two adjacent times by
\begin{equation} \label{eq:Z_eqn_element_notation}
Z_{ij} = \frac{v_i^T - v_i^0}{\ln \left( \frac{v_i^T}{v_i^0}\right)}
\ln \left( \frac{x_{ij}^T}{x_{ij}^0} \right) ,
\end{equation}
with care taken to address the zero-value problem noted by
Ang et al. [-@Ang:1998]
in Table 2
and discussed at length by
Wood and Lenzen [-@Wood:2006].
Given the $\mat{Z}$ matrix defined by the equation for $Z_{ij}$ above,
a $\Delta \mat{v}$ vector is calculated by column sums of $\mat{Z}$
\begin{equation} \label{eq:deltaV_eqn}
\Delta \mat{V} = \rowvec{i} \mat{Z} \; ,
\end{equation}
where entries in $\Delta \mat{V}$ give the additive contribution
of the $j^\mathrm{th}$ decomposition factor
to changes in $V$ from time $0$ to time $T$.
Multiplicative decomposition is obtained with the $\mat{D}$ row vector calculated by
\begin{equation} \label{eq:D_eqn}
D_j = \exp \left( \frac{\Delta V_j}{\frac{V^T - V^0}{\ln \frac{V^T}{V^0}}} \right) \; ,
\end{equation}
where entries in $\mat{D}$ give the multiplicative contribution
of the $j^\mathrm{th}$ decomposition factor
to changes in $V$ from time $0$ to time $T$.
Verification can be performed by
\begin{equation} \label{eq:deltaV_check}
\Delta \mat{V} \colvec{i} \overset{?}{=} V^T - V^0 \; ,
\end{equation}
and
\begin{equation} \label{eq:D_check}
\prod_j D_j \overset{?}{=} \frac{V^T}{V^0}\; .
\end{equation}
Cumulative changes in $V$ over many time periods can be calculated by
cumulative sums of the $\Delta \mat{V}$ row vector for additive decomposition and by
cumulative products of the $\mat{D}$ row vector for multiplicative decomposition.
Application of the matrix method to energy conversion chain (ECC) analysis
An energy conversion chain (ECC) is a set of
energy carriers,
energy transformation devices, and
energy services
within spatial and temporal boundaries of interest.
For the purposes of this vignette, an ECC comprises
primary, final, and useful energy and exergy flowing through society.
Following
Brockway et al. [-@Brockway:2015aa],
one energy aggregate from an ECC is total useful exergy supplied to the economy ($X_u$).
$X_u$ is comprised of
subcategories ($sc$, Heat, Electric, Mechanical Drive (Petroleum), and Muscle Work)
and
subsubcategories ($ssc$, such as KE--Fans and MD--Tractors).
The decomposition factors are given in the following table.
| factor | description
|-:|:----------------
| $X_p$ | Total primary exergy supplied to the economy
| $\phi_{p,sc}$ | Allocation ratio of primary exergy ($p$) to subcategory ($sc$) (Heat, Electric, Mechanical drive, Muscle work)
| $\phi_{sc,ssc}$ | Allocation ratio of exergy from subcategory ($sc)$ to subsubcategory ($ssc$) (MD - Diesel trains, Electric lights, etc.)
| $\eta_{ssc}$ | Primary-to-useful thermodynamic efficiency of subsubcategory ($ssc$)
In equation form, the energy aggregate ($X_u$) is calculated by
\begin{equation} \label{eq:X_u_eqn}
X_u = \sum_{ssc} X_{u,ssc} \; ,
\end{equation}
and
\begin{equation} \label{eq:X_u_i_eqn}
X_{u,ssc} = X_{p,tot} \phi_{p,sc} \phi_{sc,ssc} \eta_{ssc}
= X_{p,tot} \frac{X_{sc}}{X_{p,tot}} \frac{X_{ssc}}{X_{sc}} \frac{X_{u,ssc}}{X_{p,ssc}} \; .
\end{equation}
Note that $X_{p,ssc}$ is the embodied primary exergy
of the useful exergy produced by the subsubcategory ($X_{u,ssc}$).
Simple example
To demonstrate the matrix approach to LMDI decomposition analysis,
we begin with a simple example that is
small enough to allow
replication of the results by hand calculations
if desired
but sufficient detail to illustrate features of the calculations.
A data frame of $\mat{X}$ matrices is the starting point for a simple example.
The example has two fictitious countries (AB and YZ) and
covers four years (1971--1974).
Countries AB and YZ are identical for the purposes of illustration.
library(dplyr)
library(ggplot2)
library(LMDIR)
library(magrittr)
library(matsbyname)
library(matsindf)
library(tidyr)
DF <- create_simple_LMDI()
DF
For the remainder of these examples,
only the first four rows will be shown.
(Rows 5--8 for country YZ are the same as rows 1--4 for country AB.)
The X column of the data frame contains $\mat{X}$ matrices.
The $\mat{X}$ matrices contain all the information about energy categories and factors.
The $\mat{X}$ matrices are all that is needed to perform the LMDI analysis.
DF[1:4, ]$X
To see how the factors affect the aggregate over time,
we can perform the LMDI decomposition analysis.
But we first group by Country to ensure that each Country will be
treated separately.
(lmdi() respects grouping.)
res <- DF %>%
group_by(Country) %>%
lmdi()
The result of the call to lmdi() is DF with additional columns.
glimpse(res)
Each additional column provides results of the LMDI analysis.
Both additive and multiplicative results are provided.
The V column contains aggregates ($V$),
with the first year assigned a value of 0.
res$V %>% unlist()
The Z column contains $\mat{Z}$ matrices,
with the first year assigned a value of 1.
res[1:4, ]$Z
Note that calculating the entries in the $\mat{Z}$ matrices requires
evaluating 8 degenerate cases as shown in Table 2, p. 492 of
Ang et al. [-@Ang:1998].
Evaluation of the 8 degenerate cases is performed by the Zij() function.
Zij
The dV_agg column contains year-to-year differences in the aggregate ($V$).
res[1:4, ]$dV_agg %>% unlist()
The D_agg column contains year-to-year ratios of the aggregate ($V$).
res$D_agg %>% unlist()
The dV column contains $\colvec{\Delta V}$ vectors.
res[1:4, ]$dV
The D column contains $\colvec{D}$ vectors.
res[1:4, ]$D
The remaining columns have the _cum suffix and represent cumulative values.
The dV_agg_cum column gives cumulative differences in the aggregate ($V$) relative to the first time.
res$dV_agg_cum %>% unlist()
The D_agg_cum column gives cumulative ratios of the aggregate ($V$) relative to the first time.
res$D_agg_cum %>% unlist()
The dV_cum column gives the cumulative differences in the $\colvec{\Delta V}$ vectors
relative to the first time.
res[1:4, ]$dV_cum
The D_cum column gives the cumulative ratios of the $\colvec{D}$ vectors
relative to the first time.
res[1:4, ]$D_cum
Realistic example (Ghana)
The objective of this analysis is to discern the drivers of changes
to useful exergy supplied to the Ghanaian economy over time,
expressed in multiplicative form.
To achieve this objective, we will perform and report results of an
LMDI analysis of the Ghanaian energy conversion chain.
This analysis also serves to demonstrate the use of the LMDIR package with data
from a real energy conversion chain.
We'll use data from Heun and Brockway [-@Heun:2019]
for exergy flows through Ghanaian society.
The data frame consists of annual $\mat{X}$ matrices for Ghana over the period 1971--2013.
This data frame is available as LMDIR::XGH.
glimpse(XGH)
Each $\mat{X}$ matrix contains
useful exergy subsubcategories (in rows)
and factors (in columns)
as shown below for Ghana for 2003 and 2004.
prettifyXZ <- function(XZ){
out <- XZ %>%
sort_rows_cols(roworder = c(
# Heat categories (phi_i = 0.622)
"HTH.600.C - Electric heaters", "MTH.100.C - Charcoal stoves",
"MTH.100.C - Kerosene stoves", "MTH.100.C - LPG stoves",
"MTH.100.C - Wood stoves",
# Electricity categories (phi_i = 0.020)
"KE - Fans", "Light - Electric lights",
"Light - Televisions", "LTH.-10.C - Refrigerators",
"MD - Electric motors", "MD - Other appliances",
"MTH.100.C - Electric heaters", "MTH.200.C - Irons",
"MTH.200.C - Electric heaters",
# Liquid petroleum categories (phi_i = 0.131)
"MD - Boat engines", "MD - Diesel cars",
"MD - Diesel trains", "MD - Industry static diesel engines",
"MD - Petrol cars", "MD - Tractors",
# Food and feed categories (phi_i = 0.226)
"MD - Draught animals", "MD - Manual laborers"),
colorder = c("EXp.ktoe", "phi_i", "phi_ij", "eta_ij"))
out[ , "EXp.ktoe"] <- round(out[ , "EXp.ktoe"], 1)
out[ , "phi_i"] <- round(out[ , "phi_i"], 3)
out[ , "phi_ij"] <- round(out[ , "phi_ij"], 3)
out[ , "eta_ij"] <- round(out[ , "eta_ij"], 3)
return(out)
}
XGH %>%
filter(Year == 2003) %>%
extract2("X") %>%
extract2(1) %>%
prettifyXZ()
XGH %>%
filter(Year == 2004) %>%
extract2("X") %>%
extract2(1) %>%
prettifyXZ()
Note that row and column ordering is provided for readability.
The ordered rows show identical values of $\phi_i$
for allocation to the subcategories of
Heat, Electricity, Mechanical drive (fueled by petroleum), and
Muscle work (fueled by food and feed).
Column order follows flows of energy and exergy through the economy,
from primary stage to allocation to subcategories
through allocation to subsubcategories and, finally, to
a primary-to-useful efficiency for each task.
All unique $\phi_i$ values sum to 1.
(Specifically for 2003, 0.519 + 0.081 + 0.188 + 0.212 = 1.)
Within each subcategory, $\phi_{ij}$ values sum to 1.
(E.g., for Mechanical drive fueled by food and feed, 0.789 + 0.211 = 1)
Perform the LMDI decomposition analysis
To perform LMDI decomposition analysis, we use the lmdi() function after grouping on metadata variables
(in this case, only Country).
fillrowX <- matrix(c(1, 1, 0, 1), byrow = TRUE, nrow = 1, ncol = 4,
dimnames = list("row",
c("EXp.ktoe", "phi_i", "phi_ij", "eta_ij")))
GHlmdi <- XGH %>%
group_by(Country) %>%
lmdi(fillrow = list(fillrowX))
glimpse(GHlmdi)
(The need for and values in fillrowX are described below.)
A note about special cases for calculating entires in $\mat{Z}$
When a category of useful exergy is present in one year
but absent in an adjacent year,
the corresponding row in $\mat{X}^0$ or $\mat{X}^T$ is missing compared to the other.
That happened in Ghana between 2003 and 2004;
in 2003, VALCO's aluminum smelters were operating, but in 2004 they were not.
Thus, we treat the HTH.600.C - Electric heaters row specially in 2004.
For 2004, we insert a row for HTH.600.C - Electric Heaters and fill it
with positive numbers for
primary exergy ($X_{p,tot}$ or EXp,ktoe),
allocation from primary exergy to subcategory ($\phi_{p,sc}$ or phi_i), and
thermodynamic efficiency ($\eta_{ssc}$ or eta_ij).
(The fill values are irrelevant to the calculated result,
so long as they are positive numbers.)
We set the allocation from subcategory to subsubcategory ($\phi_{sc,ssc}$ or phi_ij) to 0.
This approach reflects the reality that
despite the fact that there is no useful exergy of this type in 2004,
we still have total primary exergy ($X_{p,tot}$),
there is still an allocation of primary exergy
to the subcategory Heat ($\phi_{p,sc} \neq 0$), and
if there were electric heaters making HTH.600.C - Electric heaters in this year,
they would have done so with a non-zero thermodynamic efficiency ($\eta_{ssc}$).
XGH %>%
filter(Year == 2004) %>%
extract2("X") %>%
extract2(1) %>%
sum_byname(fillrowX) %>%
prettifyXZ()
fillrowX triggers special calculation of $\mat{Z}$ values for 2004.
When
$X_{p,tot} > 0$,
$\phi_{p,sc} > 0$,
$\eta_{ssc} > 0$,
and $\phi_{sc,ssc} = 0$,
we employ Table 2, Row 2
of Ang et al. [-@Ang:1998]
to calculate Z[HTH.600.C - Electric heaters, phi_ij] = -6.416,
as shown below.
The other values on the row are 0.0.
(If this example were switched in time such that the smelters
were not used in 2003 but turned \emph{on} in 2004,
we would employ Table 2, Row 1
of Ang et al. [-@Ang:1998].)
GHlmdi %>%
filter(Year == 2004) %>%
extract2("Z") %>%
extract2(1) %>%
prettifyXZ()
Presentation of results
The GHlmdi object contains everything needed to present the results.
The focus here is on the ratioed aggregate (D_agg_cum) and disaggregate (D_cum)
factors that affect changes in useful exergy delivered to the economy.
So we first need to expand the data from matrices to tidy format and
select the desired columns.
GHlmdi_tidy <- GHlmdi %>%
select(Country, Year, D_agg_cum, D_cum) %>%
gather(key = matnames, value = matvals, -Country, -Year) %>%
expand_to_tidy() %>%
select(-colnames, -rowtypes, -coltypes)
head(GHlmdi_tidy, 10)
Next, we map the values of matnames and rownames
to variable names appropriate for graphing, according to the table below.
| graph variable | factor | description
|-:|:-:|:---------------
| D~tot~ | | Total change in useful exergy
| D~ex~ | $X_p$ | Primary exergy supplied to the economy
| D~str~ | $\phi_{p,sc}$ | Allocation of primary exergy ($p$) to subcategory ($sc$) (Heat, Electric, Mechanical drive, Muscle work) or main class structural change
| D~eff~ | $\eta_{ssc}$ | Primary-to-useful thermodynamic efficiency of subsubcategory ($ssc$)
| D~dil~ | $\phi_{sc,ssc}$ | Allocation ratio of exergy from subcategory ($sc)$ to subsubcategory ($ssc$) (MD - Diesel trains, Electric lights, etc.) or subclass structural change
We also set the levels of the Factors,
and thereby the order in which they will appear in the legend of the graph.
We choose an order that reflects the vertical positioning of the lines on the graph.
GHlmdi_forgraphing <- GHlmdi_tidy %>%
mutate(
Factor = case_when(
matnames == "D_agg_cum" ~ "D_tot",
rownames == "EXp.ktoe" ~ "D_ex",
rownames == "eta_ij" ~ "D_eff",
rownames == "phi_i" ~ "D_str",
rownames == "phi_ij" ~ "D_dil"
),
Factor = factor(Factor, levels = c("D_tot", "D_ex", "D_str", "D_eff", "D_dil"))
) %>%
rename(
value = matvals
) %>%
select(-rownames, -matnames) %>%
group_by(Country, Factor)
head(GHlmdi_forgraphing, 10)
Finally, we create a graph showing both the total changes ($D_{tot}$) and
the multiplicative factors that comprise the total change
($D_{ex}$, $D_{str}$, $D_{eff}$, and $D_{dil}$).
GHlmdi_forgraphing %>%
ggplot(mapping = aes(x = Year, y = value, linetype = Factor)) +
geom_line() +
geom_hline(yintercept = 1, colour = "gray50", linewidth = 0.1) +
labs(x = NULL, y = "Cumulative changes relative to 1971 [-]", linetype = NULL, colour = NULL) +
scale_linetype_manual(values = c("D_tot" = 1, "D_ex" = 5, "D_str" = 2, "D_dil" = 6, "D_eff" = 3),
labels = c("D_tot" = expression(D[tot]),
"D_ex" = expression(D[ex]),
"D_str" = expression(D[str]),
"D_dil" = expression(D[dil]),
"D_eff" = expression(D[eff]))) +
theme_classic()
Note that the graph above is the same as Figure 7 of
Heun and Brockway [-@Heun:2019].
Conclusion
The LMDIR package provides a convenient and powerful capability
for log-mean divisia index
decomposition analysis of energy conversion chains
expressed as Physical Supply Use Table (PSUT) matrices.
References
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knitr::opts_chunk$set( collapse = TRUE, comment = "#>" ) library(dplyr) library(ggplot2) library(LMDIR) library(magrittr) library(matsbyname) library(matsindf) library(tidyr)
\newcommand{\transpose}[1]{#1^\mathrm{T}} \newcommand{\inverse}[1]{#1^{\mathrm{-}1}} \newcommand{\mat}[1]{\mathbf{#1}} \newcommand{\colvec}[1]{\mathbf{#1}} \newcommand{\rowvec}[1]{\transpose{\colvec{#1}}} \newcommand{\inversetranspose}[1]{\transpose{\left( \inverse{\mat{#1}} \right)}} \newcommand{\transposeinverse}[1]{\inverse{\left( \transpose{\mat{#1}} \right)}} \newcommand{\hatinv}[1]{\inverse{\widehat{#1}}}
Introduction
Log-mean divisia index (LMDI) decomposition analysis is an important technique for understanding causes of changes in an energy composite over time. The technique was pioneered by B.W. Ang and is described in detail in Ang [-@Ang:2005]. LMDI decomposes an energy aggregate by factors and categories.
Heun and Brockway [-@Heun:2019] developed a matrix representation of the LMDI mathematics that works well with the Physical Supply Use Table (PSUT) framework developed by Heun et al. [-@Heun:2018]. This vignette describes and provides examples for the matrix method for LMDI decomposition analysis. This vignette is a version of Section S3 of the Supplemental Information to Heun and Brockway [-@Heun:2019].
Description of the matrix method for LMDI decomposition analysis
The LMDI approach pioneered by Ang et al. [-@Ang:1998] and described in detail by Liu and Ang [-@Liu:2003] decomposes an energy aggregate ($V$) by factors ($x$) that cause $V$ to change over time. (In this vignette, we adopt the notation of Ang [-@Ang:2005].)
Noting that at a given time every combination of energy category ($i$) and decomposition factor ($j$) has an associated numerical value for the decomposition factor ($x_{ij}$), the matrix approach to LMDI decomposition analysis begins with formation of an $\mat{X}$ matrix for each time of interest. Rows of $\mat{X}$ contain energy categories (indexed by $i$) of the aggregate ($V$) and columns of $\mat{X}$ contain decomposition factors (indexed by $j$).
Column vector $\colvec{v}$ contains the value of each energy category ($v_i$) and is formed from row products of $\mat{X}$.
\begin{equation} \label{eq:v} v_i = \prod_j x_{ij} \end{equation}
At any given time, the energy aggregate ($V$) is obtained by the column sum of elements in the $\colvec{v}$ vector
\begin{equation} \label{eq:V} V = \rowvec{i} \colvec{v} \; , \end{equation}
where $\colvec{i}$ is a column vector of 1's (an identity vector) of same size as $\colvec{v}$.
Additive ($\Delta V$) and multiplicative ($D$) changes in the energy aggregate ($V$) between two adjacent times, an initial time (superscript $0$) and a later time (superscript $T$), are calculated by
\begin{equation} \label{eq:deltaV_def} \Delta V = V^T - V^0 \; , \end{equation}
and
\begin{equation} \label{eq:D_def} D = \frac{V^T}{V^0} \; . \end{equation}
A $\mat{Z}$ matrix is formed for two adjacent times by
\begin{equation} \label{eq:Z_eqn_element_notation} Z_{ij} = \frac{v_i^T - v_i^0}{\ln \left( \frac{v_i^T}{v_i^0}\right)} \ln \left( \frac{x_{ij}^T}{x_{ij}^0} \right) , \end{equation}
with care taken to address the zero-value problem noted by Ang et al. [-@Ang:1998] in Table 2 and discussed at length by Wood and Lenzen [-@Wood:2006].
Given the $\mat{Z}$ matrix defined by the equation for $Z_{ij}$ above, a $\Delta \mat{v}$ vector is calculated by column sums of $\mat{Z}$
\begin{equation} \label{eq:deltaV_eqn} \Delta \mat{V} = \rowvec{i} \mat{Z} \; , \end{equation}
where entries in $\Delta \mat{V}$ give the additive contribution of the $j^\mathrm{th}$ decomposition factor to changes in $V$ from time $0$ to time $T$.
Multiplicative decomposition is obtained with the $\mat{D}$ row vector calculated by
\begin{equation} \label{eq:D_eqn} D_j = \exp \left( \frac{\Delta V_j}{\frac{V^T - V^0}{\ln \frac{V^T}{V^0}}} \right) \; , \end{equation}
where entries in $\mat{D}$ give the multiplicative contribution of the $j^\mathrm{th}$ decomposition factor to changes in $V$ from time $0$ to time $T$.
Verification can be performed by
\begin{equation} \label{eq:deltaV_check} \Delta \mat{V} \colvec{i} \overset{?}{=} V^T - V^0 \; , \end{equation}
and
\begin{equation} \label{eq:D_check} \prod_j D_j \overset{?}{=} \frac{V^T}{V^0}\; . \end{equation}
Cumulative changes in $V$ over many time periods can be calculated by cumulative sums of the $\Delta \mat{V}$ row vector for additive decomposition and by cumulative products of the $\mat{D}$ row vector for multiplicative decomposition.
Application of the matrix method to energy conversion chain (ECC) analysis
An energy conversion chain (ECC) is a set of energy carriers, energy transformation devices, and energy services within spatial and temporal boundaries of interest. For the purposes of this vignette, an ECC comprises primary, final, and useful energy and exergy flowing through society.
Following Brockway et al. [-@Brockway:2015aa], one energy aggregate from an ECC is total useful exergy supplied to the economy ($X_u$). $X_u$ is comprised of subcategories ($sc$, Heat, Electric, Mechanical Drive (Petroleum), and Muscle Work) and subsubcategories ($ssc$, such as KE--Fans and MD--Tractors). The decomposition factors are given in the following table.
| factor | description |-:|:---------------- | $X_p$ | Total primary exergy supplied to the economy | $\phi_{p,sc}$ | Allocation ratio of primary exergy ($p$) to subcategory ($sc$) (Heat, Electric, Mechanical drive, Muscle work) | $\phi_{sc,ssc}$ | Allocation ratio of exergy from subcategory ($sc)$ to subsubcategory ($ssc$) (MD - Diesel trains, Electric lights, etc.) | $\eta_{ssc}$ | Primary-to-useful thermodynamic efficiency of subsubcategory ($ssc$)
In equation form, the energy aggregate ($X_u$) is calculated by
\begin{equation} \label{eq:X_u_eqn} X_u = \sum_{ssc} X_{u,ssc} \; , \end{equation}
and
\begin{equation} \label{eq:X_u_i_eqn} X_{u,ssc} = X_{p,tot} \phi_{p,sc} \phi_{sc,ssc} \eta_{ssc} = X_{p,tot} \frac{X_{sc}}{X_{p,tot}} \frac{X_{ssc}}{X_{sc}} \frac{X_{u,ssc}}{X_{p,ssc}} \; . \end{equation}
Note that $X_{p,ssc}$ is the embodied primary exergy of the useful exergy produced by the subsubcategory ($X_{u,ssc}$).
Simple example
To demonstrate the matrix approach to LMDI decomposition analysis,
we begin with a simple example that is
small enough to allow
replication of the results by hand calculations
if desired
but sufficient detail to illustrate features of the calculations.
A data frame of $\mat{X}$ matrices is the starting point for a simple example.
The example has two fictitious countries (AB and YZ) and
covers four years (1971--1974).
Countries AB and YZ are identical for the purposes of illustration.
library(dplyr) library(ggplot2) library(LMDIR) library(magrittr) library(matsbyname) library(matsindf) library(tidyr) DF <- create_simple_LMDI() DF
For the remainder of these examples,
only the first four rows will be shown.
(Rows 5--8 for country YZ are the same as rows 1--4 for country AB.)
The X column of the data frame contains $\mat{X}$ matrices.
The $\mat{X}$ matrices contain all the information about energy categories and factors.
The $\mat{X}$ matrices are all that is needed to perform the LMDI analysis.
DF[1:4, ]$X
To see how the factors affect the aggregate over time,
we can perform the LMDI decomposition analysis.
But we first group by Country to ensure that each Country will be
treated separately.
(lmdi() respects grouping.)
res <- DF %>% group_by(Country) %>% lmdi()
The result of the call to lmdi() is DF with additional columns.
glimpse(res)
Each additional column provides results of the LMDI analysis. Both additive and multiplicative results are provided.
The V column contains aggregates ($V$),
with the first year assigned a value of 0.
res$V %>% unlist()
The Z column contains $\mat{Z}$ matrices,
with the first year assigned a value of 1.
res[1:4, ]$Z
Note that calculating the entries in the $\mat{Z}$ matrices requires
evaluating 8 degenerate cases as shown in Table 2, p. 492 of
Ang et al. [-@Ang:1998].
Evaluation of the 8 degenerate cases is performed by the Zij() function.
Zij
The dV_agg column contains year-to-year differences in the aggregate ($V$).
res[1:4, ]$dV_agg %>% unlist()
The D_agg column contains year-to-year ratios of the aggregate ($V$).
res$D_agg %>% unlist()
The dV column contains $\colvec{\Delta V}$ vectors.
res[1:4, ]$dV
The D column contains $\colvec{D}$ vectors.
res[1:4, ]$D
The remaining columns have the _cum suffix and represent cumulative values.
The dV_agg_cum column gives cumulative differences in the aggregate ($V$) relative to the first time.
res$dV_agg_cum %>% unlist()
The D_agg_cum column gives cumulative ratios of the aggregate ($V$) relative to the first time.
res$D_agg_cum %>% unlist()
The dV_cum column gives the cumulative differences in the $\colvec{\Delta V}$ vectors
relative to the first time.
res[1:4, ]$dV_cum
The D_cum column gives the cumulative ratios of the $\colvec{D}$ vectors
relative to the first time.
res[1:4, ]$D_cum
Realistic example (Ghana)
The objective of this analysis is to discern the drivers of changes to useful exergy supplied to the Ghanaian economy over time, expressed in multiplicative form. To achieve this objective, we will perform and report results of an LMDI analysis of the Ghanaian energy conversion chain. This analysis also serves to demonstrate the use of the LMDIR package with data from a real energy conversion chain.
We'll use data from Heun and Brockway [-@Heun:2019]
for exergy flows through Ghanaian society.
The data frame consists of annual $\mat{X}$ matrices for Ghana over the period 1971--2013.
This data frame is available as LMDIR::XGH.
glimpse(XGH)
Each $\mat{X}$ matrix contains useful exergy subsubcategories (in rows) and factors (in columns) as shown below for Ghana for 2003 and 2004.
prettifyXZ <- function(XZ){ out <- XZ %>% sort_rows_cols(roworder = c( # Heat categories (phi_i = 0.622) "HTH.600.C - Electric heaters", "MTH.100.C - Charcoal stoves", "MTH.100.C - Kerosene stoves", "MTH.100.C - LPG stoves", "MTH.100.C - Wood stoves", # Electricity categories (phi_i = 0.020) "KE - Fans", "Light - Electric lights", "Light - Televisions", "LTH.-10.C - Refrigerators", "MD - Electric motors", "MD - Other appliances", "MTH.100.C - Electric heaters", "MTH.200.C - Irons", "MTH.200.C - Electric heaters", # Liquid petroleum categories (phi_i = 0.131) "MD - Boat engines", "MD - Diesel cars", "MD - Diesel trains", "MD - Industry static diesel engines", "MD - Petrol cars", "MD - Tractors", # Food and feed categories (phi_i = 0.226) "MD - Draught animals", "MD - Manual laborers"), colorder = c("EXp.ktoe", "phi_i", "phi_ij", "eta_ij")) out[ , "EXp.ktoe"] <- round(out[ , "EXp.ktoe"], 1) out[ , "phi_i"] <- round(out[ , "phi_i"], 3) out[ , "phi_ij"] <- round(out[ , "phi_ij"], 3) out[ , "eta_ij"] <- round(out[ , "eta_ij"], 3) return(out) } XGH %>% filter(Year == 2003) %>% extract2("X") %>% extract2(1) %>% prettifyXZ() XGH %>% filter(Year == 2004) %>% extract2("X") %>% extract2(1) %>% prettifyXZ()
Note that row and column ordering is provided for readability. The ordered rows show identical values of $\phi_i$ for allocation to the subcategories of Heat, Electricity, Mechanical drive (fueled by petroleum), and Muscle work (fueled by food and feed). Column order follows flows of energy and exergy through the economy, from primary stage to allocation to subcategories through allocation to subsubcategories and, finally, to a primary-to-useful efficiency for each task. All unique $\phi_i$ values sum to 1. (Specifically for 2003, 0.519 + 0.081 + 0.188 + 0.212 = 1.) Within each subcategory, $\phi_{ij}$ values sum to 1. (E.g., for Mechanical drive fueled by food and feed, 0.789 + 0.211 = 1)
Perform the LMDI decomposition analysis
To perform LMDI decomposition analysis, we use the lmdi() function after grouping on metadata variables
(in this case, only Country).
fillrowX <- matrix(c(1, 1, 0, 1), byrow = TRUE, nrow = 1, ncol = 4, dimnames = list("row", c("EXp.ktoe", "phi_i", "phi_ij", "eta_ij"))) GHlmdi <- XGH %>% group_by(Country) %>% lmdi(fillrow = list(fillrowX)) glimpse(GHlmdi)
(The need for and values in fillrowX are described below.)
A note about special cases for calculating entires in $\mat{Z}$
When a category of useful exergy is present in one year
but absent in an adjacent year,
the corresponding row in $\mat{X}^0$ or $\mat{X}^T$ is missing compared to the other.
That happened in Ghana between 2003 and 2004;
in 2003, VALCO's aluminum smelters were operating, but in 2004 they were not.
Thus, we treat the HTH.600.C - Electric heaters row specially in 2004.
For 2004, we insert a row for HTH.600.C - Electric Heaters and fill it
with positive numbers for
primary exergy ($X_{p,tot}$ or EXp,ktoe),
allocation from primary exergy to subcategory ($\phi_{p,sc}$ or phi_i), and
thermodynamic efficiency ($\eta_{ssc}$ or eta_ij).
(The fill values are irrelevant to the calculated result,
so long as they are positive numbers.)
We set the allocation from subcategory to subsubcategory ($\phi_{sc,ssc}$ or phi_ij) to 0.
This approach reflects the reality that
despite the fact that there is no useful exergy of this type in 2004,
we still have total primary exergy ($X_{p,tot}$),
there is still an allocation of primary exergy
to the subcategory Heat ($\phi_{p,sc} \neq 0$), and
if there were electric heaters making HTH.600.C - Electric heaters in this year,
they would have done so with a non-zero thermodynamic efficiency ($\eta_{ssc}$).
XGH %>% filter(Year == 2004) %>% extract2("X") %>% extract2(1) %>% sum_byname(fillrowX) %>% prettifyXZ()
fillrowX triggers special calculation of $\mat{Z}$ values for 2004.
When
$X_{p,tot} > 0$,
$\phi_{p,sc} > 0$,
$\eta_{ssc} > 0$,
and $\phi_{sc,ssc} = 0$,
we employ Table 2, Row 2
of Ang et al. [-@Ang:1998]
to calculate Z[HTH.600.C - Electric heaters, phi_ij] = -6.416,
as shown below.
The other values on the row are 0.0.
(If this example were switched in time such that the smelters
were not used in 2003 but turned \emph{on} in 2004,
we would employ Table 2, Row 1
of Ang et al. [-@Ang:1998].)
GHlmdi %>% filter(Year == 2004) %>% extract2("Z") %>% extract2(1) %>% prettifyXZ()
Presentation of results
The GHlmdi object contains everything needed to present the results.
The focus here is on the ratioed aggregate (D_agg_cum) and disaggregate (D_cum)
factors that affect changes in useful exergy delivered to the economy.
So we first need to expand the data from matrices to tidy format and
select the desired columns.
GHlmdi_tidy <- GHlmdi %>% select(Country, Year, D_agg_cum, D_cum) %>% gather(key = matnames, value = matvals, -Country, -Year) %>% expand_to_tidy() %>% select(-colnames, -rowtypes, -coltypes) head(GHlmdi_tidy, 10)
Next, we map the values of matnames and rownames
to variable names appropriate for graphing, according to the table below.
| graph variable | factor | description |-:|:-:|:--------------- | D~tot~ | | Total change in useful exergy | D~ex~ | $X_p$ | Primary exergy supplied to the economy | D~str~ | $\phi_{p,sc}$ | Allocation of primary exergy ($p$) to subcategory ($sc$) (Heat, Electric, Mechanical drive, Muscle work) or main class structural change | D~eff~ | $\eta_{ssc}$ | Primary-to-useful thermodynamic efficiency of subsubcategory ($ssc$) | D~dil~ | $\phi_{sc,ssc}$ | Allocation ratio of exergy from subcategory ($sc)$ to subsubcategory ($ssc$) (MD - Diesel trains, Electric lights, etc.) or subclass structural change
We also set the levels of the Factors,
and thereby the order in which they will appear in the legend of the graph.
We choose an order that reflects the vertical positioning of the lines on the graph.
GHlmdi_forgraphing <- GHlmdi_tidy %>% mutate( Factor = case_when( matnames == "D_agg_cum" ~ "D_tot", rownames == "EXp.ktoe" ~ "D_ex", rownames == "eta_ij" ~ "D_eff", rownames == "phi_i" ~ "D_str", rownames == "phi_ij" ~ "D_dil" ), Factor = factor(Factor, levels = c("D_tot", "D_ex", "D_str", "D_eff", "D_dil")) ) %>% rename( value = matvals ) %>% select(-rownames, -matnames) %>% group_by(Country, Factor) head(GHlmdi_forgraphing, 10)
Finally, we create a graph showing both the total changes ($D_{tot}$) and the multiplicative factors that comprise the total change ($D_{ex}$, $D_{str}$, $D_{eff}$, and $D_{dil}$).
GHlmdi_forgraphing %>% ggplot(mapping = aes(x = Year, y = value, linetype = Factor)) + geom_line() + geom_hline(yintercept = 1, colour = "gray50", linewidth = 0.1) + labs(x = NULL, y = "Cumulative changes relative to 1971 [-]", linetype = NULL, colour = NULL) + scale_linetype_manual(values = c("D_tot" = 1, "D_ex" = 5, "D_str" = 2, "D_dil" = 6, "D_eff" = 3), labels = c("D_tot" = expression(D[tot]), "D_ex" = expression(D[ex]), "D_str" = expression(D[str]), "D_dil" = expression(D[dil]), "D_eff" = expression(D[eff]))) + theme_classic()
Note that the graph above is the same as Figure 7 of Heun and Brockway [-@Heun:2019].
Conclusion
The LMDIR package provides a convenient and powerful capability
for log-mean divisia index
decomposition analysis of energy conversion chains
expressed as Physical Supply Use Table (PSUT) matrices.
References
R Package Documentation
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