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# LCA#

## Specifying a functional unit#

The functional unit for any LCA calculation is a dictionary of keys and amounts:

```
{
("a database", "the answer"): 42,
("a database", "pi"): 3.14159265358979
}
```

However, you can also use a `Activity`

proxy:

```
In [1]: from brightway2 import *
In [2]: activity = Database("ecoinvent 3.2 cutoff").random()
In [3]: type(activity), activity
Out[3]:
(bw2data.backends.peewee.proxies.Activity,
'quicklime production, milled, packed' (kilogram, CH, None))
In [4]: lca = LCA({activity: 1})
In [5]: lca.demand
Out[5]: {'quicklime production, milled, packed' (kilogram, CH, None): 1}
```

How does this work? It is quite simple - the `Activity`

proxy knows how
to pretend to be a key tuple:

```
In [7]: activity[0], activity[1]
Out[7]: ('ecoinvent 3.2 cutoff', 'ab2f7a551a06a59de9191065128233e4')
In [8]: activity == ('ecoinvent 3.2 cutoff', 'ab2f7a551a06a59de9191065128233e4')
Out[8]: True
```

This is an instance of duck typing - if it walks like a duck and quacks like a duck, then we can treat it like a duck.

If you are interested in the details, see how
`bw2data.proxies.ActivityProxyBase`

defines `__getitem__`

and other `__`

magic methods.

## Turning processed data arrays in matrices {#building-matrices}#

A parameter array is a NumPy structured or record array, where each column has a label and data type. Here is an sample of the parameter array for the US LCI:

input output row col type amount

9829 9829 4294967295 4294967295 0 1.0 9708 9708 4294967295 4294967295 0 1.0 9633 9633 4294967295 4294967295 0 1.0 9276 9276 4294967295 4294967295 0 3.0999 8778 8778 4294967295 4294967295 0 1.0 9349 9349 4294967295 4294967295 0 1000.0 5685 9349 4294967295 4294967295 2 14.895 9516 9349 4294967295 4294967295 1 1032.7 9433 9349 4294967295 4294967295 1 4.4287 8838 9349 4294967295 4294967295 1 1.5490

There are also some columns for uncertainty information, but these would only be a distraction for now. The complete spec for the uncertainty fields is given in the stats_arrays documentation.

We notice several things:

Both the

`input`

and`output`

columns have numbers, but we donâ€™t know what they mean yetBoth the

`row`

and`col`

columns are filled with a large numberThe

`type`

column has only a few values, but they are also mysteriousThe

`amount`

column is the only one that seems reasonable, and gives the values that should be inserted into the matrix

### Input and Output#

The `input`

and `output`

columns gives values for biosphere flows or
transforming activity data sets. The `mapping`

{.interpreted-text
role=â€ťrefâ€ť} is used to translate keys like `("Douglas Adams", 42)`

into
integer values. So, each mapping number uniquely identifies an activity
dataset.

If the `input`

and `output`

values are the same, then this is a
production exchange - it describes how much product is produced by the
transforming activity dataset.

```
::: title
Warning
```

Integer mapping ids are not transferable from machine to machine or
installation to installation, as the order of insertion (and hence the
integer id) is more or less at random. Always `.process()`

datasets on a
new machine.

```
### Rows and columns
The `row` and `col` columns have the data type *unsigned integer, 32
bit*, and the maximum value is therefore $2^{32} - 1$, i.e. 4294967295.
This is just a dummy value telling Brightway2 to insert better data.
The method `MatrixBuilder.build_dictionary` is used to take `input` and
`output` values, respectively, and figure out which rows and columns
they correspond to. The actual code is succinct - only one line - but
what it does is:
> 1. Get all unique values, as each value will appear multiple times
> 2. Sort these values
> 3. Give them integer indices, starting with zero
For our example parameter array, the dictionary from `input` values to
`row` would be:
``` python
{5685: 0,
8778: 1,
8838: 2,
9276: 3,
9349: 4,
9433: 5,
9516: 6,
9633: 7,
9708: 8,
9829: 9}
```
And the dictionary from `output` to `col` would be:
``` python
{8778: 0,
9276: 1,
9349: 2,
9633: 3,
9708: 4,
9829: 5}
```
The method `MatrixBuilder.add_matrix_indices` would replace the
4294967295 values with dictionary values based on `input` and `output`.
At this point, we have enough to build a sparse matrix using
`MatrixBuilder.build_matrix`:
row col amount
----- ----- --------
9 5 1.0
8 4 1.0
7 3 1.0
3 1 3.0999
1 0 1.0
4 2 1000.0
0 2 14.895
6 2 1032.7
5 2 4.4287
2 2 1.5490
Indeed, the [coordinate (coo)
matrix](http://docs.scipy.org/doc/scipy/reference/generated/scipy.sparse.coo_matrix.html)
takes as inputs exactly the row and column indices, and the values to
insert.
Of course, there are some details for specific matrices - technosphere
matrices need to be square, and should have ones by default on the
diagonal, etc. etc., but this is the general idea.
### Types
The `type` column indicates whether a value should be in the
technosphere or biosphere matrix: `0` is a transforming activity
production amount, `1` is a technosphere exchange, and `2` is a
biosphere exchange.
## Brightway2 LCA Reports
The Brightway2 report data format is evolving, and this section should
not be understood as definitive.
```

LCA reports calculated with `bw2analyzer.report.SerializedLCAReport`

are
written as a JSON file to disk. It has the following data format:

```
{
"monte carlo": {
"statistics": {
"interval": [lower, upper values],
"median": median,
"mean": mean
},
"smoothed": [ ## This is smoothed values for drawing empirical PDF
[x, y],
],
"histogram": [ ## This are point coordinates for each point when drawing histogram bins
[x, y],
]
},
"score": LCA score,
"activity": [
[name, amount, unit],
],
"contribution": {
"hinton": {
"xlabels": [
label,
],
"ylabels": [
label,
],
"total": LCA score,
"results": [
[x index, y index, score], ## See hinton JS implementation in bw2ui source code
],
},
"treemap": {
"size:" LCA score,
"name": "LCA result",
"children": [
{
"name": activity name,
"size": activity LCA score
},
]
}
"herfindahl": herfindahl score,
"concentration": concentration score
},
"method": {
"name": method name,
"unit": method unit
},
"metadata": {
"version": report data format version number (this is 1),
"type": "Brightway2 serialized LCA report",
"uuid": the UUID of this report,
"online": URL where this report can be accessed. Optional.
}
}
```

## Graph traversal#

To generate graphs of impact like supply chain or Sankey diagrams, we
need to traverse the graph of the supply chain. The `GraphTraversal`

class does this in a relatively intelligent way, assessing each
inventory activity only once regardless of how many times it is used,
and prioritizing activities based on their LCA score. It is usually
possible to create a reduced graph of the supply chain, with only the
most relevant pathways and flows included, in a few seconds.

### Illustration of graph traversal#

Itâ€™s easiest to understand how graph traversal is implemented with a simple example. Take this system:

This system has four

**nodes**, which are LCI processes, also called transforming activities. Each**node**has one reference product, and a set of zero or more technosphere inputs. By convention, node`A`

produces one unit of product`A`

.This system has four

**edges**which define the inputs of each node. An edge has a start, an end, and an amount.We consider solving this system for a

*functional unit*of one unit of`A`

.

As we traverse this supply chain, we will keep different data for the nodes and the edges. For nodes, we are interested in the following:

`amount`

: The total amount of this node needed to produce the functional unit.`cum`

: The cumulative LCA impact score attributable to the needed amount of this node,*including its specific supply chain*.`ind`

: The individual LCA impact score directly attributable to one unit of this node, i.e. the score from the direct emissions and resource consumption of this node.

For edges, we want to know:

`to`

: The row index of the node consuming the product.`from`

: The row index of the node producing the product.`amount`

: The total amount of product`from`

needed for the amount of`to`

needed.`exc_amount`

: The amount of`from`

needed for*one unit*of`to`

, i.e. the value given in the technosphere matrix.`impact`

: The total LCA impact score embodied in this edge, i.e. the individual score of`from`

times`amount`

.

Our functional unit is one unit of `A`

. Before starting any
calculations, we need to set up our data structures. First, we have an
empty list of **edges**. We also have a **heap**, a list which is
automatically sorted,
and keeps track of the **nodes** we need to examine. **nodes** are
identified by their row index in the *technosphere matrix*. Finally, we
have a dictionary of **nodes**, which looks up nodes by their row
indices.

```
nodes, edges, heap = {}, [], []
```

We create a special node, the functional unit, and insert it into the nodes dictionary:

```
nodes[-1] = {
'amount': 1,
'cum': total_lca_score,
'ind': 1e-6 * total_lca_score
}
```

The *cumulative LCA impact score* is obviously the total LCA score; we
set the *individual LCA score* to some small but non-zero value so that
it isnâ€™t deleted in graph simplification later on.

We next start building our list of edges. We start with all the inputs
to the *functional unit*, which in this case is only one unit of `A`

.
Note that the functional unit can have multiple inputs.

```
for node_id, amount in functional_unit:
edges.append({
"to": -1, ## Special id of functional unit
"from": node_id,
"amount": amount,
"exc_amount": amount,
"impact": LCA(node_id, amount).score, ## Evaluate LCA impact score for node_id/amount
})
```

Finally, we push each node to the **heap**:

```
for node_id, amount in functional_unit:
heappush(heap, (abs(1 / LCA(node_id, amount).score), node_id))
```

This is not so easy to understand at first glance. What is
`1 / LCA(node_id, amount).score`

? Why the absolute value? What is this
`heappush`

thing?

We want one *divided by* the LCA impact score for node `A`

because our
heap is sorted in ascending order, and we want the highest score to be
first.

We take the absolute value because we are interested in the magnitude of node scores in deciding which node to process next, not the sign of the score - leaving out the absolute value would put all negative scores at the top of the heap (which is sorted in ascending order).

`heappush`

is just a call to push something on to the heap, which is our
automatically sorted list of nodes to examine.

After this first iteration, we have the following nodes and edges in our graph traversal:

```
nodes = {-1: {'amount': 1, 'cum': some number, 'ind': some small number}}
edges = [{
'to': -1,
'from': 0, ## Assuming A is 0
'amount': 1,
'exc_amount': 1,
'impact': some number
}]
heap = [(some number, 0)]
```

After this, it is rather simple: pull off the next node from the *heap*,
add it to the list of nodes, construct its edges, and add its inputs to
the heap. Iterate until no new nodes are found.

Because the heap is automatically sorted, at each iteration we will take the node with the highest impact that hasnâ€™t yet been assessed.

There are two more things to keep in mind:

We use a cutoff criteria to stop traversing the supply chain - any node whose cumulative LCA impact score is too small is not added to the heap.

We only visit each node once. The is functionality in

`bw2analyzer`

to â€śunrollâ€ť the supply chain so that afterwards each process can occur more than once.