Electronic supplementary material 1
For
Estimating Industrial Solid Waste and Municipal Solid Waste data at high resolution using
economic accounts: An Input-Output approach with Australian case study
For Journal of Material Cycles and Waste Management
The type of article: Original article
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1 Proration
Prorating (from the Latin Prorata i.e. proportion) is a series of operations that disaggregate data sets
using a binary concordance matrix according to the ratios defined by predetermined proportion
vectors [1, 2]. These proportion (or proxy) vectors are independent datasets that pertain to the final
disaggregated product and provide approximate proportions for the disaggregation operation.
Simply put, the proration operation below assigns what proportion of a single total figure is
attributable to a number of sectors. These sectors are found in a separate proportion vector. A
concordance matrix is used to indicate into which of these sectors this single total figure is
disaggregated.
In both Lenzen [2] and Geschke [1] the operation of proration or prorating is described in detail.
Proration can be calculated upon either the rows or columns — for simplicity I use only column
proration
𝐩=
1
[∑𝑖 𝐜̂ 𝐱𝑖 ]
𝐜̂ 𝐱
(A1)
where 𝐜 is an 𝑛×1 dimension binary concordance matrix, with rows that sum to one and the 𝑛 of 𝐜
the same as the 𝑛 of the proportion vector.
There are 3 types of waste discussed in the main text; C&I, C&D, and MSW. As I am concerned with
only one of the three waste types at a time, 𝐜 is understood to be a single column vector, meaning
that in this case rows that do not sum to one. The notation of ‘one’ indicating waste generation and
‘zero’ a lack of waste generation. Furthermore, 𝐜̂ is a 𝑛×𝑛 dimensional diagonal matrix that has the
values of 𝐜 in its main diagonal.
In this manuscript, I am predominantly concerned with single column concordance matrices and
offer the following example where Gross Output (𝐱): is used as the proportion vector to find sectoral
waste production (which for the example below is 𝐈𝐒𝐖).
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A worked example
Consider an economy with 4 industries, with a Gross Output (the total economic output for each
industry a cell) represented by the vector 𝐱.
100
14
𝐱=(
)
26
76
For this example all industries produce ISW, so there is a 1 in every cell of 𝐜 (𝐜 Τ = (1
this example the Total ISW produced (∑4𝑖=1 ISW𝑖 ) is 3000 tonnes.
1 1 1)). In
This example’s 𝐩col can be read as
𝐩col =
1
0
∑ [(
0
0
0
1
0
0
1
0
0
1
0
0
100
0
14
)(
)]
0
26
1
76
1
0
(
0
0
0
1
0
0
0
0
1
0
0
100
100
0.46
1
0
14
14
0.06
)(
)=
(
)=(
)
0
26
26
0.12
216
1
76
76
0.35
Thus,
(3000)
(0.46 0.06 0.12 0.35) = ⏟
(1388.88 194.44 361.11 1055.55).
⏟
⏟
∑ 𝐈SW
𝐈𝐒𝐖 T
𝐏col T
If the third industry did not produce ISW, then 𝐜 Τ = (1 1
𝐩col =
1
0
∑ [(
0
0
0
1
0
0
1
0
0
0
0
0
100
0
14
)(
)]
0
26
1
76
1
0
(
0
0
0
1
0
0
0
0
0
0
0 1)
0
100
100
0.52
1
0
14
14
0.07
)(
)=
(
)=(
)
0
26
0
190 0
1
76
76
0.4
Thus,
(3000)
(0.52 0.07 0 0.4) = ⏟
(1578.94 221.05 0
⏟
⏟
∑ ISW
𝐩col T
1200)
𝐈𝐒𝐖 T
The key assumption is that each sector produces waste with the proportion given by their
proportion of production (gross output) or relative employment, with gross output being the
external proportion vector chosen for this example.
Proration in the estimation method
The proration featured in this papers estimation method utilises three proportion vectors, 𝐱, total
sectoral gross output per sector, 𝐄, employment per sector, and ∑𝑖 𝑇𝑖𝑗 , the amount of inputs of
production per intermediate sector, and two concordance matrices, 𝐜̂ C&I , for commercial and
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industrial (C&I) waste generating sectors, and 𝐜̂ C&D , for construction and demolition (C&D) waste
generating sectors.
The various resulting proportion vectors each represent a different factors interpretation of waste
flow. These proportion vectors are written as variants of Equation (A1)
𝐩x = [∑
1
𝑖 𝐜̂ 𝐱 𝑖 ]
𝐜̂ 𝐱
𝐩T =
1
[∑𝑗 𝑐̂𝑗 ∑𝑖 𝑇𝑖𝑗 ]
𝐜̂ 𝐓ij 𝐓ij 𝟏ij
𝐩E = [∑
1
𝑖 𝐜̂ 𝐄𝑖 ]
𝐜̂ 𝐄
A weight could be added to each of the proportion vectors to represent the level of influence that
the different factors have upon waste generation. This weight has not been added as I assume that
the each factor has equal impact on the generation of waste (as discussed in Section 2.3).
The individual elements (i) from the 3 proportion vectors are then summed and divided by the sum
total of the 3 proportion vectors, to bring together the differing interpretations of the C&I and C&D
waste flow, 𝐩C&𝐼 and 𝐩C&𝐷 respectively.
These can be written as
𝐩C&I = 𝑎T 𝐩T C&I + 𝑎x 𝐩x C&I + 𝑎E 𝐩E C&I
𝐩C&D = 𝑎T 𝐩T C&D + 𝑎x 𝐩x C&D + 𝑎E 𝐩E C&D
(2)
(3)
1
Where 𝑎T + 𝑎X + 𝑎E = 1 , which in this case of equal weighting means 𝑎T = 𝑎X = 𝑎E = 3.
The proportion vectors shown in Equations (2) and (3), are multiplied by the total waste produced by
̃ C&I and 𝐰
̃ C&D, vectors of total waste produced
each waste stream, ∑i wC&I ij and ∑i wC&𝐷 ij, to give 𝐰
by each sector as shown in Equations (4) and (5). Note that the inclusion of the symbol ̃ above 𝐰,
denotes that this is no longer a single value (the total amount of waste generated of that waste
type),rather it is disaggregated to all the sectors (𝑖) of 𝐩.
𝐰
̃ C&I = 𝐩C&I ∑i wC&I ij
(4)
𝐰
̃ C&D = 𝐩C&D ∑i wC&𝐷 ij
(5)
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2 Using concordances in the Direct Estimates waste estimation
method.
This section provides a basic numerical example of the use of various concordances as discussed in
Section 3.3 of the main manuscript.
In the direct inputs waste estimation method it is assumed that that waste is generated by the
manufacture and consumption of products and/or services. Each product/service has one or more
primary waste types associated with its production. The relative volumes or monetary worth of
products and services that are supplied into a sector can determine the hypothetically dominant
types of waste that are generated as a result of production in that sector.
This assumed industry→product→waste relationship was replicated in the Direct Inputs estimation
method through the use of a series of concordance matrices – binary matrices (i.e. containing only
0s and 1s) that describe sector correspondence in order to be suitable for disaggregation. These
matrices have to be row or column normalised.
A worked example
There are three concordance matrices, First a concordance matrix that lists the primary products or
service from each sector, CStoP.
𝐂StoP .
Sector 1
S2
S3
S4
S5
S6
S7
Product 1 P2
1
0
0
0
0
0
0
P3
0
1
0
0
0
0
0
P4
0
0
1
1
0
0
0
0
0
0
0
1
1
1
Each of these products is associated with a waste type as listed in 𝐂PtoW,
𝐂PtoW
Product 1
P2
P3
P4
Waste 1 Waste 2
1
0
1
0
0
1
0
1
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multiplying these matrices gives a concordance of waste produced per sector 𝐂StoW.
𝐂StoW
Sector 1
S2
S3
S4
S5
S6
S7
Waste 1 Waste 2
1
0
1
0
0
1
0
1
0
1
0
1
0
1
The concordance of waste produced per sector, 𝐂StoW, is then transposed and multiplied on the
right by the direct requirements matrix 𝐀
S1
0.075
0.016
0.016
0.006
0.013
0.642
0.009
𝐀
S1
S2
S3
S4
S5
S6
S7
S2
0.067
0.451
0.048
0.002
0.015
0.012
0.209
S3
0.030
0.004
0.223
0.058
0.071
0.030
0.055
S4
0.336
0.035
0.271
0.860
0.589
0.165
0.512
S5
0.119
0.052
0.231
0.060
0.219
0.075
0.199
S6
0.343
0.438
0.191
0.008
0.010
0.075
0.012
S7
0.030
0.003
0.020
0.006
0.082
0.001
0.005
S6
S7
to give the estimated waste production of each sector 𝐌C&I
𝐌C&I
w1
w2
S1
S2
0.091
0.686
S3
0.519
0.286
S4
0.034
0.436
S5
0.371
2.397
0.172
0.785
I normalise the matrix 𝐌C&I by dividing each cell by its column sum
0.781
0.296
1
[∑𝑖 𝑀𝐶&𝐼𝑖𝑗 ]
0.033
0.114
𝑀𝐶&I𝑖𝑗 . This gives the
̅ C&I .
relative waste produced per industry for C&I waste, 𝐌
̅ C&I
𝐌
w1
w2
S1
0.12
0.88
S2
0.64
0.36
S3
0.07
0.93
S4
0.13
0.87
S5
0.18
0.82
S6
0.73
0.27
S7
0.22
0.78
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̅ C&I by 𝐰
̃ C&I , the waste stream produced by each sector gives an
Multiplying as a dot product 𝐌
expanded listing of waste generation of each sector (𝑖, tonnes) sorted by waste type (𝑗), 𝐖C&I as
shown in Equation (9).
̅ C&I 𝐰
̃ C&I
𝐖C&I = 𝐌
̃ C&I
𝐰
Total Waste
generated per
sector
𝐖C&I
w1
w2
1.
2.
(9)
S1
S2
S3
S4
S5
S6
S7
300
400
50
90
600
500
300
S1
35.01
264.99
S2
257.87
142.13
S3
3.60
46.40
S4
12.07
77.93
S5
107.63
492.37
S6
362.61
137.39
S7
67.41
232.59
Geschke, A., Manual for routine prorate, 2012: University of Sydney.
Lenzen, M., Aggregation Versus Disaggregation In Input–Output Analysis Of The
Environment. Economic Systems Research, 2011. 23(1): p. 73-89.
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