Estudio de localización

In this thesis, a domain adaptation method is proposed in the atemporal domain adaptation phase. However, Hephaestus is flexible enough to work with any other domain adaptation technique. The goal here is to align the domains of all atemporal features and labels to enable them to be handled together. Figure 4.3 illustrates how this works.

For a single feature f, the values Xf from each source domain and the target domainDk,

are subjected to aglobal normalization(if the relationship betweenY andXf are absolute) or

local normalization (if the relationship between the label Y and the feature Xf are relative).

These two normalization methods are described next.

Local Normalization

A relationship betweenY and Xf is relative if the value ofY relies on a proportional value of

Xf. In other words, the conditional distributions P(Y|Ψ(Xf)) of each source or target are the

same, whereΨ() is a normalization function. Local normalizationis illustrated in Figure 4.3, which shows all the resulting similar marginal distributions centered in the same position for

Local normalization

Source 1 Source n Target

μXf T μXf Sn μXf S1 ...

Is the relationship between y and feature xrelative or absolute?

Normalized Source 1 Normalized Source n Normalized Target

0 ... transfer normalization 0 0 Global normalization

Normalized Source 1 Normalized Source n Normalized Target

0 0 0 μ`Xf T μ`Xf Sn μ`Xf S1 ab so lu te re la ti ve ... P(Xf) P(Xf) P(Xf) P(Xf) P(Xf) P(Xf) P(Xf) P(Xf) P(Xf)

all source and target domains.

To illustrate a relative relationship, assume that energy consumption and external temper- ature are correlated and that buildings have different structures adapted to the climate where they were built. Therefore the air conditioner will not be set to an exact outdoor temperature, but turned on when the indoor temperature becomes warm, a decision which depends on the building’s heat-exchange characteristics with the outside environment. In this situation, the relationship between the outside temperature and energy consumption is relative. The idea of normalizing the temperature for each building is to align the concept of warm temperature onto the same scale for all buildings.

Because most features in real-world regression problems follow a normal distribution, the linear transformation proposed for this method is the z-score normalization. The z-score nor- malization maintains all the feature distributions by aligning their means on the center and keeping outliers out of bounds. Figure 4.3 uses z-score normalization as an example.

Z-score normalization can be calculated using Eq. 2.2. For local normalization, only the feature valuesXDk

f from the current domainDk are used.

Similarly in time series domain adaptation, depending on dataset size, it is difficult or sta- tistically irrelevant to achieve proper normalization using only the local dataset. This can be solved by using the normalization parameters from one of the source domains or a composite of all of them, assuming that they have similar distributions.

Labels from source and target datasets can be in different domains, therefore they must also be normalized. For example, energy consumption depends on building size: the energy consumption peak in a bigger building can be expected to be higher than in a smaller building. In this case,local normalizationshould be used for the label.

Global Normalization

A relationship between Y and Xf is absolute if the value ofY relies directly on the absolute

value ofXf. Therefore the Xf of each dataset is a subset of a superset F that contains all the

subsets from all the datasets. In other words, the conditional distributionsP(Y|Ψ(Xf,F)) for all

source or target datasets are the same in a global context. In this case, the featureXf is consid-

is illustrated in Figure 4.3, where the dashed lines represent the assumed marginal distribution of the supersetF and the continuous lines represents the marginal distribution of Xf from the

source and target domains in relation toF.

For example, suppose that all the buildings are outfitted with the same type of equipment (e.g. computers, lighting) with equal consumption characteristics and that the number of pieces of equipment is available in the datasets. Because each piece of equipment consumes the same amount of energy for every building, its relationship with energy consumption would be considered absolute because it depends on the absolute number of pieces of equipment that are turned on.

However, even if features with a global relationship are already in the same global do- main, they should also be normalized in the same way the features with a local relationship to maintain all the features at the same scale and consequently ensure prediction quality.

When using z-score normalization (Eq. 2.2) forglobal normalization, all the feature values Xf from all datasets are used at once.