Sensitivity coefficients for matrixbased LCA
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Abstract
Background, aim, and scope
Matrixbased life cycle assessment (LCA) is part of the standard ingredients of modern LCA tools. An important aspect of matrixbased LCA that is straightforward to carry out, but that requires a careful mathematical handling, is the inclusion of sensitivity coefficients based on differentiating the matrixbased formulas.
Materials and methods
We briefly review the basic equations for LCA and the basic theory of sensitivity coefficients.
Results
We present the complete set of sensitivity coefficients from inventory to weighting through characterization and normalization. We show the specific formulas for perturbation analysis, uncertainty analysis, and key issue analysis. We also provide an example using the ecoinvent data.
Discussion
The limitations of the present approach include the restriction to small changes and uncertainties and the ignorance of correlation between input uncertainties. In contrast to common thinking, there is no restriction to normally distributed uncertainties: Every uncertainty distribution for which a variance can be defined can be submitted to the analytical uncertainty analysis.
Conclusions
This paper provides a useful set of tables for a number of purposes related to uncertainty and sensitivity analysis.
Recommendations and perspectives
Although the formulas derived are not simple, they are straightforward to implement in software for LCA. Once this is done, the use of these formulas can become routine practice, enabling a key issue analysis and speeding up perturbation and uncertainty analysis.
Keywords
Derivatives Life cycle interpretation Matrixbased LCA Sensitivity Taylor series expansion Uncertainty1 Introduction
2 Background, aim, and scope
By now, the matrix approach to life cycle assessment (LCA) has received wide recognition. Proposed in the early 1990s (Möller 1992; Heijungs et al. 1992), it took a decade or so before this idea was embraced as a general accepted way of doing LCA (see Heijungs and Suh 2006 for a short history of the matrix approach to LCA).
Overview of symbols representing input data in the LCA
Symbol  Name  Dimension (rows × columns)  Defined in 

f  Final demand vector  Economic flows × 1  Goal and scope definition 
A  Technology matrix  Economic flows × processes  Inventory analysis 
B  Intervention matrix  Environmental flows × processes  Inventory analysis 
Q  Characterisation matrix  Categories × environmental flows  Impact assessment 
\( {\mathbf{\dot{g}}} \)  Intervention totals^{a}  Environmental flows × 1  Impact assessment 
\( {\mathbf{\dot{h}}} \)  Category totals^{a}  Categories × 1  Impact assessment 
w  Weighting factors  1 × categories  Impact assessment 
Overview of symbols representing output results in the LCA and the formulas with which these results are obtained
Symbol  Name  Dimension (rows × columns)  Equation 

s  Scaling factors  processes × 1  \( {\mathbf{s}} = {{\mathbf{A}}^{  1}}{\mathbf{f}} \) 
g  Inventory results  Environmental flows × 1  \( {\mathbf{g}} = {\mathbf{Bs}} \) 
h  Characterization results  Categories × 1  \( {\mathbf{h}} = {\mathbf{Qg}} \) 
Λ  Intensity matrix  Environmental flows × economic flows  \( {\mathbf{\Lambda }} = {\mathbf{B}}{{\mathbf{A}}^{  1}} \) 
\( {\mathbf{\tilde{h}}} \)  Normalization results  Categories × 1  \( \forall k:{\tilde{h}_k} = \frac{{{h_k}}}{{{{\dot{h}}_k}}} \) 
\( {\mathbf{\dot{h}}} \)  Category totals^{a}  Categories × 1  \( {\mathbf{\dot{h}}} = {\mathbf{Q\dot{g}}} \) 
W  Weighted index  1 × 1  \( W = {\mathbf{w\tilde{h}}} \) 

the treatment of allocation and cutoff (Heijungs and Frischknecht 1998);

how to connect a processbased LCI to an input–output table (Suh and Huppes 2005);

how to efficiently compute an answer to the inventory problem (Peters 2007);

how to analyze the feedback structure of the system (Suh and Heijungs 2007);

how to calculate sensitivity coefficients (Heijungs 1994).
Most of these details indeed refer to exclusively inventoryoriented questions. The impact assessment follows the inventory results, so all issues related to cutoff, allocation, IObased LCA, efficient algorithms, and structural path analysis are only interesting from an LCI point of view. For the last one mentioned, the sensitivity coefficients, this is different, however.
Sensitivity coefficients are important for both uncertainty analysis and sensitivity analysis (Heijungs 1994). In the context of uncertainty analysis, they serve to establish essential information for a Taylor series expansion, and for sensitivity analysis, they provide the multipliers that enable one to distinguish sensitive from nonsensitive parameters, the socalled key issues for refined data collection (Heijungs 1996). But uncertainty and sensitivity analyses are not only important in LCI but in impact assessment as well. Moreover, the impact assessment adds additional uncertainty to the already uncertain results of the LCI. Likewise, not only the sensitivity of inventory results is of interest but also (or perhaps even more so) the impact assessment results.
The extension of the sensitivity coefficients from LCI to life cycle impact assessment was “left as an exercise” to the LCA practitioner. For instance, Heijungs and Suh (2002, p. 144) write that “In this way, all equations of LCA may be processed,” but they do not pursue this. It is a task that is not so often carried out, we guess, at least we have never seen the explicit results of such an exercise. This paper therefore aims to carry out this exercise and to make the results available. The formulas obtained can easily be implemented in matrixbased software for LCA. We have done so in CMLCA, a program for doing LCA, and some screenshots are shown at the end of the paper.
In this paper, we first review the basic equations of LCA. Then, we proceed to review the theory of sensitivity coefficients in general and their form in LCI. This finally leads to a derivation and coherent presentation of the sensitivity coefficients for the entire LCA process.
3 Materials and methods
3.1 Basic equations for LCA
The basic equations for LCA have been discussed in a consistent notation by Heijungs and Suh (2002). Below, we present two tables of symbols and equations connecting the fundamental concepts of LCA.

A vector of intervention totals, \( {\mathbf{\dot{g}}} \), can be defined for the reference situation. For instance, one can collect data on the emissions of CO_{2}, SO_{2}, etc., which then represent \( {\dot{g}_1} \), \( {\dot{g}_2} \), etc. These then can be processed by the same characterization model to yield the vector of category totals, \( {\mathbf{\dot{h}}} \). This then forms the basis of the normalization. Changing \( {\mathbf{\dot{g}}} \) will induce a change in \( {\mathbf{\dot{h}}} \), but \( {\mathbf{\dot{h}}} \) itself will not be changed directly by the LCA practitioner. We will refer to this as normalization case 1.

Alternatively, the vector of category totals \( {\mathbf{\dot{h}}} \) can be known without a detailed specification of the underlying interventions. In that case, \( {\mathbf{\dot{h}}} \) is not an output result (in the sense of belonging to Table 2), but input data (in the sense of belonging to Table 1). Thus, \( {\mathbf{\dot{h}}} \) can be changed directly, and it will affect the normalization results and the weighted index. We will refer to this as normalization case 2.
Both approaches in fact appear in practice and are therefore elaborated below as separate cases.
3.2 General theory of sensitivity coefficients
4 Results
4.1 Sensitivity coefficients for LCA
It is clear that the higher one moves in the sequence inventory–characterization–normalization–weighting, the larger the number of sensitivity coefficients there will be. Scaling factors only depend on the technology matrix, but the inventory results depend on the technology matrix and on the intervention matrix.
Overview of the sensitivity coefficients that express how LCA output results (columns) change if LCA input data (rows) data change
 \( {{\partial {s_k}} \mathord{\left/{\vphantom {{\partial {s_k}} \cdots }} \right.} \cdots } \)  \( {{\partial {g_k}} \mathord{\left/{\vphantom {{\partial {g_k}} \cdots }} \right.} \cdots } \)  \( {{\partial {h_k}} \mathord{\left/{\vphantom {{\partial {h_k}} \cdots }} \right.} \cdots } \)  \( {{\partial {{\tilde{h}}_k}} \mathord{\left/{\vphantom {{\partial {{\tilde{h}}_k}} \cdots }} \right.} \cdots } \)  \( {{\partial W} \mathord{\left/{\vphantom {{\partial W} \cdots }} \right.} \cdots } \) 

\( { \cdots \mathord{\left/{\vphantom { \cdots {\partial {a_{ij}}}}} \right.} {\partial {a_{ij}}}} \)  \(  {\left( {{{\mathbf{A}}^{  1}}} \right)_{ki}}{s_j} \)  −λ_{ ki } s _{ j }  \(  {s_j}\sum\limits_l {{q_{kl}}{\lambda_{li}}} \)  \(  \frac{{{s_j}}}{{{{\dot{h}}_k}}}\sum\limits_l {{q_{kl}}{\lambda_{li}}} \)  \(  {s_j}\sum\limits_k {\frac{{{w_k}}}{{{{\dot{h}}_k}}}\sum\limits_l {{q_{kl}}{\lambda_{li}}} } \) 
\( { \cdots \mathord{\left/{\vphantom { \cdots {\partial {b_{ij}}}}} \right.} {\partial {b_{ij}}}} \)  0  s _{ j } δ _{ ik }  q _{ ki } s _{ j }  \( \frac{{{q_{ki}}{s_j}}}{{{{\dot{h}}_k}}} \)  \( {s_j}\sum\limits_k {\frac{{{w_k}}}{{{{\left( {{{\dot{h}}_k}} \right)}^2}}}{q_{ki}}} \) 
\( { \cdots \mathord{\left/{\vphantom { \cdots {\partial {q_{ij}}}}} \right.} {\partial {q_{ij}}}} \)  0  0  g _{ j } δ _{ ik }  \( \left( {\frac{{{g_j}}}{{{{\dot{h}}_k}}}  \frac{{{h_k}{{\dot{g}}_j}}}{{{{\left( {{{\dot{h}}_k}} \right)}^2}}}} \right){\delta_{ik}} \)  \( {w_i}\left( {\frac{{{g_j}}}{{{{\dot{h}}_i}}}  \frac{{{h_i}{{\dot{g}}_j}}}{{{{\left( {{{\dot{h}}_i}} \right)}^2}}}} \right) \) 
\( { \cdots \mathord{\left/{\vphantom { \cdots {\partial {{\dot{g}}_i}}}} \right.} {\partial {{\dot{g}}_i}}} \) (normalization case 1)  0  0  0  \(  \frac{{{h_k}{q_{ki}}}}{{{{\left( {{{\dot{h}}_k}} \right)}^2}}} \)  \(  \sum\limits_k {\frac{{{w_k}{h_k}{q_{ki}}}}{{{{\left( {{{\dot{h}}_k}} \right)}^2}}}} \) 
\( { \cdots \mathord{\left/{\vphantom { \cdots {\partial {{\dot{h}}_i}}}} \right.} {\partial {{\dot{h}}_i}}} \) (normalization case 2)  0  0  0  \(  \frac{{{h_k}}}{{{{\left( {{{\dot{h}}_k}} \right)}^2}}}{\delta_{ik}} \)  \(  \frac{{{w_i}{h_i}}}{{{{\left( {{{\dot{h}}_i}} \right)}^2}}} \) 
\( { \cdots \mathord{\left/{\vphantom { \cdots {\partial {w_i}}}} \right.} {\partial {w_i}}} \)  0  0  0  0  \( {\tilde{h}_i} \) 
One remark on these coefficients. Each of the formulas in Table 3 contains A ^{−1} or a symbol that depends on A ^{−1}, such as s, Λ or h. Thus, we need to go through the process of matrix inversion to evaluate the sensitivity coefficients. Moreover, although Ciroth et al. (2004) argue that one can solve LCAs without calculating a full matrix inverse, Table 3 shows that we need the full matrix inverse when we wish to extend the analysis to include sensitivity and analytical uncertainty studies.
4.2 Perturbation analysis
Overview of the relative sensitivity coefficients (multipliers) that express how small changes in input data propagate into changes in output results
Multiplier  Definition  Formula 

σ_{ k }(a_{ ij })  \( \frac{{\partial {s_k}/{s_k}}}{{\partial {a_{ij}}/{a_{ij}}}} \)  \(  \frac{{{a_{ij}}}}{{{s_k}}}{\left( {{{\mathbf{A}}^{  1}}} \right)_{ki}}{s_j} \) 
γ_{ k }(a_{ ij })  \( \frac{{\partial {g_k}/{g_k}}}{{\partial {a_{ij}}/{a_{ij}}}} \)  \(  \frac{{{a_{ij}}}}{{{g_k}}}{\lambda_{ki}}{s_j} \) 
γ_{ k }(b_{ ij })  \( \frac{{\partial {g_k}/{g_k}}}{{\partial {b_{ij}}/{b_{ij}}}} \)  \( \frac{{{b_{ij}}}}{{{g_k}}}{s_j}{\delta_{ik}} \) 
η_{ k }(a_{ ij })  \( \frac{{\partial {h_k}/{h_k}}}{{\partial {a_{ij}}/{a_{ij}}}} \)  \(  \frac{{{a_{ij}}}}{{{h_k}}}{s_j}\sum\limits_l {{q_{kl}}{\lambda_{li}}} \) 
η_{ k }(b_{ ij })  \( \frac{{\partial {h_k}/{h_k}}}{{\partial {b_{ij}}/{b_{ij}}}} \)  \( \frac{{{b_{ij}}}}{{{h_k}}}{q_{ki}}{s_j} \) 
η_{ k }(q_{ ij })  \( \frac{{\partial {h_k}/{h_k}}}{{\partial {q_{ij}}/{q_{ij}}}} \)  \( \frac{{{q_{ij}}}}{{{h_k}}}{g_j}{\delta_{ik}} \) 
\( {\tilde{\eta }_k}\left( {{a_{ij}}} \right) \)  \( \frac{{\partial {{\tilde{h}}_k}/{{\tilde{h}}_k}}}{{\partial {a_{ij}}/{a_{ij}}}} \)  \(  \frac{{{a_{ij}}}}{{{{\tilde{h}}_k}}}\frac{{{s_j}}}{{{{\dot{h}}_k}}}\sum\limits_l {{q_{kl}}{\lambda_{li}}} \) 
\( {\tilde{\eta }_k}\left( {{b_{ij}}} \right) \)  \( \frac{{\partial {{\tilde{h}}_k}/{{\tilde{h}}_k}}}{{\partial {b_{ij}}/{b_{ij}}}} \)  \( \frac{{{b_{ij}}}}{{{{\tilde{h}}_k}}}\frac{{{q_{ki}}{s_j}}}{{{{\dot{h}}_k}}} \) 
\( {\tilde{\eta }_k}\left( {{q_{ij}}} \right) \)  \( \frac{{\partial \tilde{h}/{{\tilde{h}}_k}}}{{\partial {q_{ij}}/{q_{ij}}}} \)  \( \frac{{{q_{ij}}}}{{{{\tilde{h}}_k}}}\left( {\frac{{{g_j}}}{{{{\dot{h}}_k}}}  \frac{{{h_k}{{\dot{g}}_j}}}{{{{\left( {{{\dot{h}}_k}} \right)}^2}}}} \right){\delta_{ik}} \) 
\( {\tilde{\eta }_k}\left( {{{\dot{g}}_i}} \right) \) (normalization case 1)  \( \frac{{\partial {{\tilde{h}}_k}/{{\tilde{h}}_k}}}{{\partial {{\dot{g}}_i}/{{\dot{g}}_i}}} \)  \(  \frac{{{{\dot{g}}_i}}}{{{{\tilde{h}}_k}}}\frac{{{h_k}{q_{ki}}}}{{{{\left( {{{\dot{h}}_k}} \right)}^2}}} \) 
\( {\tilde{\eta }_k}\left( {{{\dot{h}}_i}} \right) \) (normalization case 2)  \( \frac{{\partial {{\tilde{h}}_k}/{{\tilde{h}}_k}}}{{\partial {{\dot{h}}_i}/{{\dot{h}}_i}}} \)  \(  \frac{{{{\dot{h}}_i}}}{{{{\tilde{h}}_k}}}\frac{{{h_k}}}{{{{\left( {{{\dot{h}}_k}} \right)}^2}}}{\delta_{ik}} \) 
ω(a_{ ij })  \( \frac{{\partial W/W}}{{\partial {a_{ij}}/{a_{ij}}}} \)  \(  \frac{{{a_{ij}}}}{W}{s_j}\sum\limits_k {\frac{{{w_k}}}{{{{\dot{h}}_k}}}\sum\limits_l {{q_{kl}}{\lambda_{li}}} } \) 
ω(b_{ ij })  \( \frac{{\partial W/W}}{{\partial {b_{ij}}/{b_{ij}}}} \)  \( \frac{{{b_{ij}}}}{W}{s_j}\sum\limits_k {\frac{{{w_k}}}{{{{\left( {{{\dot{h}}_k}} \right)}^2}}}{q_{ki}}} \) 
ω(q_{ ij })  \( \frac{{\partial W/W}}{{\partial {q_{ij}}/{q_{ij}}}} \)  \( \frac{{{q_{ij}}}}{W}{w_i}\left( {\frac{{{g_j}}}{{{{\dot{h}}_i}}}  \frac{{{h_i}{{\dot{g}}_j}}}{{{{\left( {{{\dot{h}}_i}} \right)}^2}}}} \right) \) 
\( \omega \left( {{{\dot{g}}_i}} \right) \) (normalization case 1)  \( \frac{{\partial W/W}}{{\partial {{\dot{g}}_i}/{{\dot{g}}_i}}} \)  \(  \frac{{{{\dot{g}}_i}}}{W}\sum\limits_k {\frac{{{w_k}{h_k}{q_{ki}}}}{{{{\left( {{{\dot{h}}_k}} \right)}^2}}}} \) 
\( \omega \left( {{{\dot{h}}_i}} \right) \) (normalization case 2)  \( \frac{{\partial W/W}}{{\partial {{\dot{h}}_i}/{{\dot{h}}_i}}} \)  \(  \frac{{{{\dot{h}}_i}}}{W}\frac{{{w_i}{h_i}}}{{{{\left( {{{\dot{h}}_i}} \right)}^2}}} \) 
ω(w_{ i })  \( \frac{{\partial W/W}}{{\partial {w_i}/{w_i}}} \)  \( \frac{{{w_i}}}{W}{\tilde{h}_i} \) 
4.3 Uncertainty analysis
Overview of the variance of output results as a function of the variance of the input data
Uncertainty  Equation 

var(s _{ k })  \( \sum\limits_{i,j} {{{\left( {{s_j}} \right)}^2}{{\left( {{{\left( {{{\mathbf{A}}^{  1}}} \right)}_{ki}}} \right)}^2}} {\rm var} \left( {{a_{ij}}} \right) \) 
var(g _{ k })  \( \sum\limits_{i,j} {{{\left( {{s_j}{\lambda_{ki}}} \right)}^2}} {\rm var} \left( {{a_{ij}}} \right) + \sum\limits_j {{{\left( {{s_j}} \right)}^2}} {\rm var} \left( {{b_{kj}}} \right) \) 
var(h _{ k })  \( \sum\limits_{i,j} {{{\left( {{s_j}\sum\limits_l {{q_{kl}}{\lambda_{li}}} } \right)}^2}} {\rm var} \left( {{a_{ij}}} \right) + \sum\limits_{i,j} {{{\left( {{s_j}{q_{ki}}} \right)}^2}} {\rm var} \left( {{b_{ij}}} \right) + \sum\limits_j {{{\left( {{g_j}} \right)}^2}} {\rm var} \left( {{q_{kj}}} \right) \) 
\( {\rm var} \left( {{{\tilde{h}}_k}} \right) \)  \( \left\{ {\begin{array}{*{20}{c}} {\frac{1}{{{{\left( {{{\dot{h}}_k}} \right)}^2}}}\left[ {\sum\limits_{i,j} {{{\left( {{s_j}\sum\limits_l {{q_{kl}}{\lambda_{li}}} } \right)}^2}} {\rm var} \left( {{a_{ij}}} \right) + \sum\limits_{i,j} {{{\left( {{s_j}{q_{ki}}} \right)}^2}} {\rm var} \left( {{b_{ij}}} \right) + \sum\limits_j {{{\left( {{g_j}  \frac{{{h_k}{{\dot{g}}_j}}}{{{{\dot{h}}_k}}}} \right)}^2}} {\rm var} \left( {{q_{kj}}} \right) + \sum\limits_i {{{\left( {\frac{{{h_k}{q_{ki}}}}{{{{\dot{h}}_k}}}} \right)}^2}} {\rm var} \left( {{{\dot{g}}_i}} \right)} \right]} & {{\hbox{(normalization case 1)}}} \\{\frac{1}{{{{\left( {{{\dot{h}}_k}} \right)}^2}}}\left[ {\sum\limits_{i,j} {{{\left( {{s_j}\sum\limits_l {{q_{kl}}{\lambda_{li}}} } \right)}^2}} {\rm var} \left( {{a_{ij}}} \right) + \sum\limits_{i,j} {{{\left( {{s_j}{q_{ki}}} \right)}^2}} {\rm var} \left( {{b_{ij}}} \right) + {{\left( {\frac{{{h_k}}}{{{{\dot{h}}_k}}}} \right)}^2}{\rm var} \left( {{{\dot{h}}_k}} \right)} \right]} & {{\hbox{(normalization case 2)}}} \\\end{array} } \right. \) 
var(W)  \( \left\{ {\begin{array}{*{20}{c}} {\frac{1}{{{{\left( {{{\dot{h}}_k}} \right)}^2}}}\left[ {\sum\limits_{i,j} {{{\left( {{s_j}\sum\limits_l {{q_{kl}}{\lambda _{li}}} } \right)}^2}} {\text{var}}\left( {{a_{ij}}} \right) + \sum\limits_{i,j} {{{\left( {{s_j}{q_{ki}}} \right)}^2}} {\text{var}}\left( {{b_{ij}}} \right) + \sum\limits_j {{{\left( {{g_j}  \frac{{{h_k}{{\dot{g}}_j}}}{{{{\dot{h}}_k}}}} \right)}^2}} {\text{var}}\left( {{q_{kj}}} \right) + \sum\limits_i {{{\left( {\frac{{{h_k}{q_{ki}}}}{{{{\dot{h}}_k}}}} \right)}^2}} {\text{var}}\left( {{{\dot{g}}_i}} \right)} \right]} & {\left( {{\text{normalization case 1}}} \right)} \\ {\frac{1}{{{{\left( {{{\dot{h}}_k}} \right)}^2}}}\left[ {\sum\limits_{i,j} {{{\left( {{s_j}\sum\limits_l {{q_{kl}}{\lambda _{li}}} } \right)}^2}} {\text{var}}\left( {{a_{ij}}} \right) + \sum\limits_{i,j} {{{\left( {{s_j}{q_{ki}}} \right)}^2}} {\text{var}}\left( {{b_{ij}}} \right) + {{\left( {\frac{{{h_k}}}{{{{\dot{h}}_k}}}} \right)}^2}{\text{var}}\left( {{{\dot{h}}_k}} \right)} \right]} & {\left( {{\text{normalization case 2}}} \right)} \\ \end{array} } \right. \) 
Again, mind that there are two expressions for \( {\rm var} \left( {{{\tilde{h}}_k}} \right) \) and two expressions for var(W) for both normalization variants.
4.4 Key issue analysis
Overview of the contributions to the variance of output results by the individual variance of the input data
Contribution  Formula 

ζ(s_{ k },a_{ ij })  \( \frac{{{{\left( {{{\left( {{{\mathbf{A}}^{  1}}} \right)}_{ki}}{s_j}} \right)}^2}{\rm var} \left( {{a_{ij}}} \right)}}{{{\rm var} \left( {{s_k}} \right)}} \) 
ζ(g_{ k },a_{ ij })  \( \frac{{{{\left( {{s_j}{\lambda_{ki}}} \right)}^2}{\rm var} \left( {{a_{ij}}} \right)}}{{{\rm var} \left( {{g_k}} \right)}} \) 
ζ(g_{ k },b_{ ij })  \( \frac{{{{\left( {{s_j}{\delta_{ik}}} \right)}^2}{\rm var} \left( {{b_{ij}}} \right)}}{{{\rm var} \left( {{g_k}} \right)}} \) 
ζ(h_{ k },a_{ ij })  \( \frac{{{{\left( {{s_j}\sum\limits_l {{q_{kl}}{\lambda_{li}}} } \right)}^2}{\rm var} \left( {{a_{ij}}} \right)}}{{{\rm var} \left( {{h_k}} \right)}} \) 
ζ(h_{ k },b_{ ij })  \( \frac{{{{\left( {{s_j}{q_{ki}}} \right)}^2}{\rm var} \left( {{b_{ij}}} \right)}}{{{\rm var} \left( {{h_k}} \right)}} \) 
ζ(h_{ k },q_{ ij })  \( \frac{{{{\left( {{g_j}} \right)}^2}{\rm var} \left( {{q_{kj}}} \right)}}{{{\rm var} \left( {{h_k}} \right)}} \) 
\( \zeta \left( {{{\tilde{h}}_k},{a_{ij}}} \right) \)  \( \frac{{{{\left( {{s_j}\sum\limits_l {{q_{kl}}{\lambda _{li}}} } \right)}^2}{\text{var}}\left( {{a_{ij}}} \right)}}{{{{\left( {{{\dot{h}}_k}} \right)}^2}{\text{var}}\left( {{{\widetilde{h}}_k}} \right)}} \) 
\( \zeta \left( {{{\tilde{h}}_k},{b_{ij}}} \right) \)  \( \frac{{{{\left( {{s_j}{q_{ki}}} \right)}^2}{\rm var} \left( {{b_{ij}}} \right)}}{{{{\left( {{{\dot{h}}_k}} \right)}^2}{\rm var} \left( {{{\tilde{h}}_k}} \right)}} \) 
\( \zeta \left( {{{\tilde{h}}_k},{q_{ij}}} \right) \)  \( \frac{{{{\left( {{g_j}  \frac{{{h_k}{{\dot{g}}_j}}}{{{{\dot{h}}_k}}}} \right)}^2}{\rm var} \left( {{q_{kj}}} \right)}}{{{{\left( {{{\dot{h}}_k}} \right)}^2}{\rm var} \left( {{{\tilde{h}}_k}} \right)}} \) 
\( \zeta \left( {{{\tilde{h}}_k},{{\dot{g}}_i}} \right) \) (normalization case 1)  \( \frac{{{{\left( {\frac{{{h_k}{q_{ki}}}}{{{{\dot{h}}_k}}}} \right)}^2}{\rm var} \left( {{{\dot{g}}_i}} \right)}}{{{{\left( {{{\dot{h}}_k}} \right)}^2}{\rm var} \left( {{{\tilde{h}}_k}} \right)}} \) 
\( \zeta \left( {{{\tilde{h}}_k},{{\dot{h}}_i}} \right) \) (normalization case 2)  \( \frac{{{{\left( {\frac{{{h_k}}}{{{{\dot{h}}_k}}}} \right)}^2}{\rm var} \left( {{{\dot{h}}_k}} \right)}}{{{{\left( {{{\dot{h}}_k}} \right)}^2}{\rm var} \left( {{{\tilde{h}}_k}} \right)}} \) 
ζ(W,a_{ ij })  \( \frac{{{{\left( {{s_j}} \right)}^2}{{\left( {\sum\limits_l {\frac{{{w_l}}}{{{{\dot{h}}_l}}}\sum\limits_k {{q_{lk}}{\lambda_{ki}}} } } \right)}^2}{\rm var} \left( {{a_{ij}}} \right)}}{{{\rm var} (W)}} \) 
ζ(W,b_{ ij })  \( \frac{{{{\left( {{s_j}} \right)}^2}{{\left( {\sum\limits_l {\frac{{{w_l}}}{{{{\dot{h}}_l}}}{q_{li}}} } \right)}^2}{\rm var} \left( {{b_{ij}}} \right)}}{{{\rm var} (W)}} \) 
ζ(W,q_{ ij })  \( \frac{{{{\left( {\frac{{{w_i}}}{{{{\dot{h}}_i}}}} \right)}^2}{{\left( {{g_j}  \frac{{{h_i}{{\dot{g}}_j}}}{{{{\dot{h}}_i}}}} \right)}^2}{\rm var} \left( {{q_{ij}}} \right)}}{{{\rm var} (W)}} \) 
\( \zeta \left( {W,{{\dot{g}}_i}} \right) \) (normalization case 1)  \( \frac{{{{\left( {\frac{{{w_i}{h_i}}}{{{{\dot{h}}_i}{{\dot{h}}_i}}}} \right)}^2}{\rm var} \left( {{{\dot{h}}_i}} \right)}}{{{\rm var} (W)}} \) 
\( \zeta \left( {W,{{\dot{h}}_i}} \right) \) (normalization case 2)  \( \frac{{{{\left( {\frac{{{w_i}{h_i}}}{{{{\dot{h}}_i}{{\dot{h}}_i}}}} \right)}^2}{\rm var} \left( {{{\dot{h}}_i}} \right)}}{{{\rm var} (W)}} \) 
ζ(W,w_{ i })  \( \frac{{{{\left( {{{\tilde{h}}_i}} \right)}^2}{\rm var} \left( {{w_i}} \right)}}{{{\rm var} (W)}} \) 
4.4.1 Example
We have programmed these equations into CMLCA (http://www.cmlca.eu/). As an example, we loaded the ecoinvent v1.3 (2,630 unit processes) and calculated the system for a reference flow of 1kWh Swiss electricity, low voltage, at grid. In the example below, we have restricted the analysis to the inventory analysis and to the emission of Carbon dioxide, fossil, emitted to air, low population density.
Here, we see that only four coefficients make up 80% of the uncertainty of the CO_{2} emission. In other words, we can obtain a more reliable, less uncertain result by trying to find more accurate data for these four coefficients.
5 Discussion
Three important restrictions must be born in mind.
First, the formulas are for the “normal” LCI. This means that the equations become more complicated once we incorporate other features and developments, such as allocation and IObased LCA. Heijungs et al. (2006) show how the formula for \( \frac{{\partial {g_k}}}{{\partial {a_{ij}}}} \) changes when the inventory is done using a hybrid method, combining processbased and IObased LCA. This modification may be propagated to the impact assessment level as well.
Second, some of the formulas (namely those in Tables 4, 5, and 6) are based on a firstorder Taylor series approximation. That means that they are correct for small changes, uncertainties, and perturbations, but not necessarily for larger ones. To fix this, one may either include secondorder terms (or even further). For instance, for the perturbations, we may improve by going from
into
This requires the evaluation of not only \( \frac{{\partial z}}{{\partial x}} \) but also of the second derivative \( \frac{{{\partial^2}z}}{{\partial {x^2}}} \) or even beyond. Table 3 might be extended to include expressions for \( \frac{{{\partial^2}{s_k}}}{{\partial {{\left( {{a_{ij}}} \right)}^2}}} \) and similar. Alternatively, the approach by Sherman and Morrison (1950) can be used to provide an exact answer to this question.
Third and finally, the formulas for uncertainty (Tables 5 and 6) are based on the ignorance of the covariance between input variables. As we noted above, the complete firstorder expression includes an additional term for the covariance between input data:
The last three terms requires us to specify cov(a _{ ij } ,a _{ lm }), cov(a _{ kj } ,b _{ kl }), and cov(a _{ ij } ,b _{ kl }) for all combinations of processes, economic flows, and environmental flows. Although some uncertainties will be definitely correlated (for instance, the fuel input of a combustion process and the CO_{2} emission of the same process may have a correlation close to 1), most uncertainties will be uncorrelated, or any information on such correlations is lacking. The infrastructure needed (lots of memory for storing a number of covariance matrices, much more complicated formulas, much more data collection and estimation) will probably not offset the relatively limited gain of having a slightly more accurate computation. In the end, there is always something perverse about knowing the uncertainty with certainty.
A question that always arises in connection to analytical error propagation is whether it only works for normally distributed uncertainties. This is not the case. The theory of analytical error propagation through Taylor series expansions (Morgan and Henrion 1990, p. 183 ff.) nowhere contains the assumption of normally distributed distributions. The only assumption is that for a firstorder approximation, the function should be sufficiently close to a linear function within the range of uncertainty. This point has been addressed above. A practical issue is of course that the formulas require a specification of the variance, whereas most distributions are specified without an explicit variance. For instance, a uniform distribution is often specified in terms of its width and a lognormal distribution in terms of the geometric standard deviation (or its square, as in ecoinvent). Heijungs and Frischknecht (2005) provide formulas to easily calculate a variance from the standard parameters of a normal, lognormal, uniform, and triangular distribution. As we can see in Table 5, the propagated variances are the sum of a large number of terms. Following the central limit theorem (e.g., Morgan and Henrion 1990), such a propagated uncertainty will tend to become normally distributed provided that the input uncertainties are independent.
6 Conclusions
This paper has carried out the “exercise” that was left over by Heijungs and Suh (2002) and related work on deriving the complete set of sensitivity coefficients for matrixbased LCA.
7 Recommendations and perspectives
For some, the formulas in Tables 4, 5 and 6 are intimidating and may offer little insight. But they are straightforward to implement in computer code. Once implemented, doing an uncertainty analysis is as easy as doing a Monte Carlo analysis: click a button and wait for the results. Likewise, doing a key issue analysis is as easy as doing a classical contribution analysis.
We hope that the availability of these equations will stimulate developers of software, commercial or not, to implement analytical, Taylor seriesbased approaches toward uncertainty and sensitivity analysis. In particular for the key issue analysis, no good Monte Carlo approach is available, and the analytical solution using Table 6 provides an extremely powerful way of reducing the uncertainties in LCA. For perturbation and uncertainty analysis, numerical approaches can be used as well, but these are extremely timeconsuming for large LCA systems.
As noted above, Monte Carlo analyses for large LCA systems may be unfeasible. This paper develops sensitivity coefficients that serve to derive the formulas for analytical error propagation based on first (or higher)order Taylor series approximation. This has other limitations, for instance, relating to a more restricted range of validity. Therefore, we welcome the simultaneous development of more sophisticated sampling methods such as Latin hypercube sampling and response surface methods, as well as the development of more efficient algorithms such as those based on a power series expansion. Yet, even when this would overtake the analytical error propagation, we still see a role for the sensitivity coefficients in perturbation analysis and in key issue analysis.
Notes
Open Access
This article is distributed under the terms of the Creative Commons Attribution Noncommercial License which permits any noncommercial use, distribution, and reproduction in any medium, provided the original author(s) and source are credited.
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