Let’s talk about hulls and bounds for data-based models

Is one of your regular tasks to build data based models of e.g., a combustion engine or a similar technical system? If yes, it is essential to think about the range in which the model is able to predict results in good quality. But what is the difference between all the given possibilities and when to use what?

The following text refers to the tool-suite ETAS ASCMO which is a reference solution for advanced data based modelling and calibration that is used worldwide – mainly in in the automotive industry, but also in some other environments.

A word at the beginning. Using all the below mentioned options at the same time is typically not the best choice for daily use due to calculation time and too strict limitations.

Option 1: Pairwise 2D-Hull

By using the N-dimensional pairwise hull function, a convex hull of all possible pairs of the input dimensions is calculated. But what does that mean? In the screenshot below you can see an example data set where the hull is already calculated and some of the pairs are visualized.


If you just look at the first plot in the upper row, you can see a projection of all measured points on the first two dimensions (speed and load) as black dots. Enveloping these plots there is a black line representing the convex hull. Everything in this convex hull is marked in turquoise.

In addition, all other pairs of input channels can be calculated, visualized and used. The next plot to the right of the first is the combination of the first and third input (speed and injection). The same logic is used here as well to calculate and visualize the information. The same goes for all the other possible pairs and plots.

When to use: If the other options (see Options 2 and 3) are not meaningful or possible, this option is always a good starting point to have a basic set of bounds. But it needs relevant calculation time, so do not activate it if the other options make more sense.

Option 2: Map Bounds

One of the main use cases of ETAS ASCMO is building data-based models of combustion engines in a wide range or even the whole speed/load range of the engine. Here we are typically talking about a “Global Model” where a speed – and a load -channel defines the x- and y-axis of calibration maps. For these global models a special type of hull is used, the so called “Map Bounds over Operating Points”.


The screenshot above shows the map bounds of the input “injection” as a third dimension besides speed and load. There are two colored maps defining the upper as well as lower bound hovering above/below a set of measurement points visualized with small blue dots. Depending on where you are in speed and load, the minimum and maximum allowed value for injection is different. For high speed and low load values (right corner or the map) the range is between ~25 and 45, while for low speed and load the range is between ~10 and 30. So the allowed values for injection depend on the speed/load value (we call it “Operating Point”).

When to use: This option is automatically activated for all global models in ETAS ASCMO. It is a fast end easy way to come up with a basic set of bounds for a successful use and also the optimization of parameters. There is typically no reason for deactivating it.

Option 3: 2D-, 3D- and 4D-Hull

In some cases the “Pairwise 2D-Hull” as well as “Map Bounds” are not enough, for example, if the useful range of an input not only depends on speed and load, but also on other channels. An example for a gasoline engine is speed, load, inlet- and exhaust-cam.

Excessive calculation times make it unreasonable or even unfeasible to always calculate all possible high-dimensional hulls. For this scenario ETAS ASCMO enables a detailed definition of the hull in a configuration dialog where the needed 2D- up to 4D- convex hulls can be defined and activated.


In the first section the pairwise hulls can be activated as mentioned in Option 1. In the second section in this example a 3D-hull is defined using speed, load and EGR. 3D-hulls are visualized in the way you can see in the screenshot below. All others are similar to the pairwise hull plots.


When to use: For global models where the default “Map Bounds” are not sufficient and a very detailed configuration is necessary. Depending on the system there are often some special combinations of 3D- or 4D- hulls that need to be considered. For Diesel engines an example is speed, load, EGR and boost pressure.

Show all hulls/bounds in the “Intersection Plot”

After choosing the right options, the result can also be visualized in the main view of ETAS ASCMO – the so called “Intersection Plot”. Using green and yellow dotted vertical lines, the limits are shown for each input. The remaining range where neither the hulls nor the bounds are limiting stays in white background color, everything that is excluded is in light grey color.


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