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
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=1icxi cx (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, c 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).
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.
x=100142676
For this example all industries produce ISW, so there is a 1 in every cell of c (cΤ=1111). In this example the Total ISW produced (i=14ISWi) is 3000 tonnes.
This example’s pcol can be read as
pcol=110000100001000011001426761000010000100001100142676=1216100142676=0.460.060.120.35
Thus,
3000ISW0.460.060.120.35=PcolT1388.88194.44361.111055.55ISWT.
If the third industry did not produce ISW, then cΤ=1101
pcol=110000100000000011001426761000010000000001100142676=119010014076=0.520.0700.4
Thus,
3000ISW0.520.0700.4pcolT=1578.94221.0501200ISWT
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 iTij, the amount of inputs of production per intermediate sector, and two concordance matrices, cCI, for commercial and industrial (C&I) waste generating sectors, and cCD, 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 =1icxicx pT=1jcjiTijcTijTij1ij pE=1icEicE
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
pCI=aTpTCI+axpxCI+aEpECI (2)
pCD=aTpTCD+axpxCD+aEpECD (3)
Where aT+aX+aE=1 , which in this case of equal weighting means aT=aX=aE=13.
The proportion vectors shown in Equations (2) and (3), are multiplied by the total waste produced by each waste stream, iwCIij and iwC&Dij, to give wCI and wCD, vectors of total waste produced 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.
wCI=pCIiwCIij (4)
wCD=pCDiwC&Dij (5)
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. / Product 1 / P2 / P3 / P4Sector 1 / 1 / 0 / 0 / 0
S2 / 0 / 1 / 0 / 0
S3 / 0 / 0 / 1 / 0
S4 / 0 / 0 / 1 / 0
S5 / 0 / 0 / 0 / 1
S6 / 0 / 0 / 0 / 1
S7 / 0 / 0 / 0 / 1
Each of these products is associated with a waste type as listed in CPtoW,
CPtoW / Waste 1 / Waste 2Product 1 / 1 / 0
P2 / 1 / 0
P3 / 0 / 1
P4 / 0 / 1
multiplying these matrices gives a concordance of waste produced per sector CStoW.
CStoW / Waste 1 / Waste 2Sector 1 / 1 / 0
S2 / 1 / 0
S3 / 0 / 1
S4 / 0 / 1
S5 / 0 / 1
S6 / 0 / 1
S7 / 0 / 1
The concordance of waste produced per sector, CStoW, is then transposed and multiplied on the right by the direct requirements matrix A
A / S1 / S2 / S3 / S4 / S5 / S6 / S7S1 / 0.075 / 0.067 / 0.030 / 0.336 / 0.119 / 0.343 / 0.030
S2 / 0.016 / 0.451 / 0.004 / 0.035 / 0.052 / 0.438 / 0.003
S3 / 0.016 / 0.048 / 0.223 / 0.271 / 0.231 / 0.191 / 0.020
S4 / 0.006 / 0.002 / 0.058 / 0.860 / 0.060 / 0.008 / 0.006
S5 / 0.013 / 0.015 / 0.071 / 0.589 / 0.219 / 0.010 / 0.082
S6 / 0.642 / 0.012 / 0.030 / 0.165 / 0.075 / 0.075 / 0.001
S7 / 0.009 / 0.209 / 0.055 / 0.512 / 0.199 / 0.012 / 0.005
to give the estimated waste production of each sector MCI
MCI / S1 / S2 / S3 / S4 / S5 / S6 / S7w1 / 0.091 / 0.519 / 0.034 / 0.371 / 0.172 / 0.781 / 0.033
w2 / 0.686 / 0.286 / 0.436 / 2.397 / 0.785 / 0.296 / 0.114
I normalise the matrix MCI by dividing each cell by its column sum 1iMC&Iij MC&Iij . This gives the relative waste produced per industry for C&I waste, MCI .
MCI / S1 / S2 / S3 / S4 / S5 / S6 / S7w1 / 0.12 / 0.64 / 0.07 / 0.13 / 0.18 / 0.73 / 0.22
w2 / 0.88 / 0.36 / 0.93 / 0.87 / 0.82 / 0.27 / 0.78
Multiplying as a dot product MCI by wCI, the waste stream produced by each sector gives an expanded listing of waste generation of each sector (i, tonnes) sorted by waste type (j), WCI as shown in Equation (9).
WCI=MCIw CI (9)
w CI / S1 / S2 / S3 / S4 / S5 / S6 / S7Total Waste generated per sector / 300 / 400 / 50 / 90 / 600 / 500 / 300
WCI / S1 / S2 / S3 / S4 / S5 / S6 / S7
w1 / 35.01 / 257.87 / 3.60 / 12.07 / 107.63 / 362.61 / 67.41
w2 / 264.99 / 142.13 / 46.40 / 77.93 / 492.37 / 137.39 / 232.59
1. Geschke, A., Manual for routine prorate, 2012: University of Sydney.
2. 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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