Downscaling Global Economic Activity Data : A Spatial and Sectorial perspective using Bayesian Hierarchical Modelling
Where we left off
- Pycnophylactic condition Tobler (1979)
- Rescaling, using Overture POIs and their categories as source for density distribution of activity on the ground.
Pros
- (too ?) Simple
- Efficient
- Global
Cons
- Did not use additional available information : sector level data, regional accounts from local sources, other global layers such as population, Non-Residential built infrastructure etc…
- Univariate model
- Hard constraint
- No flexibility
Can we do better ?
BHM
This, in practice, takes the form \(p(x,z)\). And in such cases, Bayes rule tells us that
\[ p(x,z) = p(z) * p(x|z) \\ p(x,z) = p(x) * p(z|x) \]
Which can be combined into:
\[ p(x) = \frac{p(z) \cdot p(x|z)}{p(z|x)} = \frac{p(x|z)}{p(z|x)}\cdot p(z) \]
The first term is called the likelihood. If we can estimate the RHS using \(z\), then we can estimate the marginal of \(x\).
Now the most important point in BM, is that we apply such reasoning to the parameters of our model. In other words we are asking the question :
What are the chances of observing certain data from my model under some parameters \(\mathbf{\theta}\) ?
And we refine the parameter if we have some observations \(\{X_i\}\).
Simple example
Let’s say we have some data \(\{ X_i \}\) and we model it as \(X \sim \mathcal{N(\mu,\sigma)}\). Usually, we would use the data to make an estimate on the values of \((\mu,\sigma)\), with both parameters fixed and computed from the data. But what if we set one parameter to have variability? In this case, the model becomes parametrised as \(p_X \equiv p_X(.|\mu)\). Formally, let’s model the parameters as RVs as well. The values we will draw from our original model will be conditioned on the parameter, which will itself be sampled. This is where Bayesian thinking. We defined an additional probability density, associated to our belief on the values of the mean \(\mu \sim \mathcal{N(\tau,\gamma)}\). We have a model defined not only for the data, but for the parameters of our model. The distribution on \(\mu\) as well as the one for \(X\) form our prior, encompassing the best of our knowledge about the observed phenomenon. The parameters \((\tau,\gamma)\) are referred to as hyper-parameters and can be used to fine tune our belief on the mean of the data we observe and sample down the way.
What about observed data ?
Let’s say now we have some observed sample data \(\{X_i\}\). Having laid a general behavior for the model, we can now turn to it and ask the question :
“What are the chances of observing the sample \(\{X_i\}\) in our model, conditioned to the parameter \(\mu\) ?”
In other words we are looking at \(p(\{X_i\}|\mu)\), recall Bayes rule from an earlier slide.
Ex : Height distribution in human population
Spatial Economic Activity
How do we develop a downscaling model with this ?
After this rushed summary, we get in the specific details of our problem. The idea is to embed an econometrics model predicting the fine scale values into this Bayesian framework. On the one hand informing the behavior at the fine spatial scale, dictated by the econometrics model, and controlling that this behavior aligns with our prior knowledge and constraints at the coarse spatial and sector scale through the BHM.
Spatial Hierarchy
Economic Hierarchy
All together
Data
Input1
Current
POIs, GHSL NRES, DOSE-WDI, Copernicus, GHSL pop, Bureau of Economic Analysis (BEA), ILOSTAT, UK Business Value Added, EU IO tables
Validation
- Kummu
- Bea
- EU IO
- Null model (OLD)
Subsectors
GEM, CGFI, Climatrace, Edgar, MAPSPAM
Catalogue
Is up on the cluster with access and manipulation facilitated by the scalenav2 package
Selected Regions
FRA.13_1
Selected Regions
Selected Regions
Selected Regions
Japan
Selected Regions
Selected Regions
Modelling Diagnostics
The BHM side of things 
Validation 1
We use existing available downscaled data sets, in which one of the 2 dimensions (spatial, sectorial) is coarse, and reduce our data to validate at the resolution of the available data. Example with (Kummu, Taka, and Guillaume 2018b), where the data is total GDP at a fine spatial scale. We reduce the BHM model data by aggregating each locations output across sectors.
FRA.13_1
Validation 1
JPN
Validation 2
Florida
Validation 2
Validation 2
Validation 2
Using national reports on per sector productivity at coarse geographic resolution.
Reversing the question
The flexibility of BHM allows us to reverse the use of the finer resolution economic output data and use it as a prior knowledge. This is a work in progress feature that is almost implemented.