Bird Game

The goal of the bird game is to provide a probabilistic forecast to track the future location of a dove.

Your model must process a sequence of records that will be received in real time. Each record provides:

• The current dove location

• The current time

• A location of one falcon

• The falcon wingspan

• The falcon identity

The falcon locations are shown as colored dots on the figure above. Some falcons may provide useful information as to (track) the future location of the dove, or the uncertainty of the same, whereas others may not. Their utility or otherwise is for you to determine.

Probabilistic forecasting provides a distribution of possible future values rather than a single point estimate, allowing for uncertainty quantification. Instead of predicting only the most likely outcome, it estimates a range of potential outcomes along with their probabilities by outputting a probability distribution.

A probabilistic forecast models the conditional probability distribution of a future value given past observations . This can be expressed as:

where represents the historical data up to time . Instead of a single prediction , the model estimates a full probability distribution , which can take different parametric forms, such as a Gaussian:

where is the predicted mean and represents the uncertainty in the forecast.

Probabilistic forecasting can be handled through various approaches, including variance forecasters, quantile forecasters, interval forecasters or distribution forecasters, each capturing uncertainty differently.

For example, you can try to forecast the target location by a gaussian density function (or a mixture), thus the model output follows the form:

   { 
       "density": { 
           "name": "normal", 
           "params": { "loc": y_mean, "scale": y_var } 
       }, 
       "weight": weight 
   } 

A mixture density, such as the gaussion mixture allows for capturing multi-modal distributions and approximate more complex distributions.

Mathematical Definition

The informal meaning of probabilistic forecast is a mixture of parametric univariate density functions where each is taken from a standard family (such as exponential, or gaussian).

Engineering Definition

The meaning of probabilistic forecast is made more precise by means of the Python density package which provides a dict specification of continuous univariate density function mixtures using the pydantic Python package. The function validatedensitydict will tell you whether or not your specification is valid.

A tracker is a framework that processes real-time data to track the dove’s movement and predict its future location. It considers inputs like the dove’s position and falcon locations to generate a probabilistic forecast.

To create your tracker, you need to define a class that implements the TrackerBase interface. Specifically, your class must implement the following methods:

• tick(self, payload: dict) -> None
  This method is called at every time step to process new payloads. Use this method to update your internal state or logic as needed.

• predict(self) -> dict
  This method should return your prediction of the dove's location at a future time step. Ensure that the return format complies with the density_pdf specification.

To compare your Tracker's performance against the benchmark Tracker, use the test_run method provided in the TrackerBase class. This method evaluates your Tracker's efficiency over a series of time steps using density_pdf scoring. A higher score reflects more accurate predictions.

• Literature

• Useful Python packages