.. Created by Adam Cunnningham on Fri June 3 2016.

**Mauna Loa CO2 Levels**
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Report Description
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**Exercise 1: Load and Plot the Data.**

- Download the Mauna Load CO\ :sub:`2` data from the "National Oceanic and Atmospheric Administration" website and save it to a local file named "co2_mm_mlo.txt".
- Load the data into a NumPy array using NumPy's **loadtxt** function and plot the monthly CO\ :sub:`2` levels against time in years.
- Make some observations about your results.

**Exercise 2: Simple Linear Regression.**

Given a set of data points :math:`\{(x_1, y_1), (x_2, y_2) \ldots, (x_n, y_n)\}`, the parameters :math:`w_0` and :math:`w_1` of the straight line :math:`f(x) = w_0 + w_1 x` that best fits the data are given by the solution to the matrix equation:

.. math::
   \begin{bmatrix} 1 & \overline{x} \\ \overline{x} & \overline{x^2} \end{bmatrix}
   \begin{bmatrix} w_0 \\ w_1 \end{bmatrix} =
   \begin{bmatrix} \overline{y} \\ \overline{xy} \end{bmatrix}

- Find the "best fit" of a straight line to the data using the solutions :math:`w_0` and :math:`w_1` as the parameters for the straight line. Plot this straight line along with the original data on the same graph.
- Predict CO\ :sub:`2` levels in the years 2050 and 2100 on the basis of this model.
- Plot the residual difference against time between the original data and this straight line.
- Make some observations about your results.

.. hint::
    - The matrix equation **Aw** = **b** can be solved using the NumPy function **linalg.solve(A, b)**.
    - The NumPy function **mean** can be used to find the mean value of the elements of an array.


**Exercise 3: Fitting a Quadratic Function.**

The parameters :math:`w_0`, :math:`w_1` and :math:`w_2` of the quadratic function :math:`f(x) = w_0 + w_1 x + w_2 x^2` that best fits the data are given by the solution to the matrix equation:

.. math::
   \begin{bmatrix} 1 & \overline{x} & \overline{x^2} \\ \overline{x} & \overline{x^2} & \overline{x^3} \\ \overline{x^2} & \overline{x^3} & \overline{x^4} \end{bmatrix}
   \begin{bmatrix} w_0 \\ w_1 \\ w_2 \end{bmatrix} =
   \begin{bmatrix} \overline{y} \\ \overline{xy} \\ \overline{x^2y} \end{bmatrix}


- Find the "best fit" of a quadratic function to the data using the solutions :math:`w_0`, :math:`w_1` and :math:`w_2` as the parameters for the function. Plot this curve along with the original data on the same graph.
- Predict CO\ :sub:`2` levels in the years 2050 and 2100 on the basis of this model.
- Plot the residual difference against time between the original data and this curve.
- Make some observations about your results.

**Exercise 4: Seasonal Variation.**

- Plot the residual difference against month of the quadratic residuals.
- Plot the mean monthly residual difference of the quadratic residuals on the same graph.
- Plot the remaining residuals after the mean monthly residuals have also been removed.
- Make some observations about your results.

.. note:: For each of these exercises, say how you have used NumPy arrays, indexing and slicing, and array operations to perform these tasks.