P2-E003
=======

You are encouraged to familiarise yourself with a few topics prior to the exercise. The background section will provide you with some basic information and pointers. The exercise itself is designed to give you some practice in working with these concepts to give you a more intuitive grasp of them. The teacher will be able to supervise your during the exercise and explain the underlying theory if uncertainties remained during your preparation for the exercise.

Background
----------

The Bayes Theorem allows you to from new data and update your knowlege about a system/model/hypothesis. This exercise is designed to introduce you to a different use for the same theorem in order to give you a more intuitive grasp of it. If you have completed previous exercises, you will have sufficient skill to work with provided datasets. To prepare for this exercise, you will need to familiarise yourself with the **Bayes Theorem**:

.. math::

    \begin{equation}
      P(A|B)=\frac {P(B/A)·P(A)}{P(B)}
    \end{equation}

Make sure you understand each term. Reading about the basics of **Naive Bayes Classifiers** prior to the exercise will be an advantage.


Exercise
-------- 

Information
...........

+----------------------+--------------------------------------------------------+
| Learning goals                                                                |
+======================+========================================================+
|**Skills**            |                                                        |    
|                      |   * working with multiple datasets                     |
|                      |   * Bayesian categorisation and selection              |
+----------------------+--------------------------------------------------------+


.. topic:: What to Submit

      Submit your script(s) and 1-2 sentence answers to the questions for the tasks of this exercise. You may include the answers as comments in your script(s). Make sure to also comment each of your calculations (using the terminology you were introduced to). The script should be named *[your surname]_e103.[ext]*, where *[ext]* is the file extension. In case of Python, this would be *.py*, for Matlab it is *.m*, etc.
      
Bayesian Categorisation - Assigning Data to a Location     
......................................................

.. figure:: img/categorisation_ccby_KirstinUebernickel.jpg  
   :align: center
   
   Bayes Theorem can be used for categorisation. In this exercise, we categorise data of unknown origin into records from 3 different locations. [photo: Weather stations at Santa Gracia, La Campana and Nahuelbuta; cc-by Kirstin Übernickel]

You are provided with ERA-Interim re-analysis data extracted from three different (southern hemisphere) locations (near weather stations) with different latitudinal and similar longitudinal coordinates: 

Central Chile:
:download:`Antofagasta.cvs <Antofagasta.csv>`   

Central-South Chile: 
:download:`Quintero.cvs <Quintero.csv>`   

South Chile:
:download:`Puerto_Montt_el_Tepual.cvs <Puerto_Montt_El_Tepual.csv>`   

.. figure:: img/winds.jpg  

.. note:: It is common practice in climatology to separate wind vectors into their meridional (along longitudes) and zonal (along latitudes) components. Those are denoted as v-winds and u-winds respectively.  

For this exercise, you can assume that the wind directions at the locations of the weather stations represent fairly typical wind directions at these latitudes and are not affected much by local near-surface geographical disparities. Consequently, the station data is suitable for constructing models for wind directions at the three different latitudes. 

Additionally, you are given a few different data points (of unknown origin) for June wind directions: 
11.4°, -30.8°, 60.2°

You are interested in the latitudes the winds were probably measured at. Use Bayes theorem to update the probabilities for the wind direction models (see above) based on the wind direction data you are given.

**HINT**

* Think carefully about how to take seasons into consideration when constructing your wind direction model.
* Does it make sense to use wind direction in degrees for this exercise? Feel free to discuss with students and teachers.

.. warning::

    Late submissions won't be accepted!