Lowess smoothed as defined for STATA 13.

# The lowess Package

This package provides a function to perform a LOWESS on Pandas Series objects. LOWESS (locally weighted scatterplot smoothing) [1, 2] as defined by STATA [3]. The regressions utilises some of the methods in [4].

## Description

### Methods and Formula

Let x and y be the two variables each of length N, and assume that the data are ordered so that xi = < xi+1 for i = 1,...,N-1. For each yi, a smoothed value yis is calculated. The subset of points used in calculating yis is i- = max(1, i-k) through i+ = min(i+k, N), where

k = Floor((N × bandwidth - 0.5) / 2).

The weights for each of the observations between j = i-,...,i+ are the tricube

wj = [1 - (|xj - xi| / ∆)3]3,

where ∆ = 1.0001 max(xi+-xi, xi-xi-). The smoothed value yis is then the weighted polynomial regression prediction at xi.

NB: In this implemtation x and y should be Pandas Series objects. The series need not be sorted and x and y can be in different orders, so long as their indexes have the same elements.

### Usage

Once the package has been installed it can be imported into a python script
`import lowess`
The package provides a single module `lowess` with a single function `lowess.lowess`. This function has the signiture:
`lowess.lowess(x, y, bandwidth=0.2, polynomialDegree=1)`
where the arguments are:

1. x (pandas.core.series.Series): a Pandas Series containing the x (independent/covariat) values. The indices must be unique.
2. y (pandas.core.series.Series): a Pandas Series containing the y (dependent) values. It must have the same index as x (although not necessarily in the same order.)
3. bandwidth (float, optional): the bandwidth for smoothing. It must be between 0 and 1. Default is 0.2
4. polynomialDegree (int, optional): The degree of polynomial to use in the regression. It must be >= 0. Default is 1.

It returns a Pandas Series containing the smoothed y values, with the same index as y.

If the input is not valid or an error occurs, a `LowessError` exception is raised.

## Examples

Some examples are given in the directory `examples`.

## Installation

### Via the PyPI package manager

The package can be installed with `pip` via the command:
`\$ pip install lowess`

### Via GitHub

The package can be installed from source via GitHub. First download the repository, either via SSH
`\$ git clone git@github.com:CCGE-Cambridge/lowess.git`
or via HTTPS
`\$ git clone https://github.com/CCGE-Cambridge/lowess.git`
Then install the package via

``````\$ cd lowess
\$ pip install .
``````

## Documentaion

Documentaion of the API is provided via Sphinx. To make the documentaion

``````\$ cd docs
\$ make html
\$ open build/html/index.html
``````

This may require installation of the package `sphinx`.

## Testing

Unit tests are implemented via `unittest` and are in the file `tests/test_lowess.py`. To run the tests first download the source code and then run the command:
`\$ python tests/test_lowess.py`

## Requirements

numpy==1.18.2
pandas==1.0.3
python-dateutil==2.8.1
pytz==2019.3
six==1.14.0

Copyright (c) 2020 Andrew Lee

This software is provided as is without any warranty whatsoever. Permission to use, for non-commercial purposes is granted. Permission to modify for personal or internal use is granted, provided this copyright and disclaimer are included in all copies of the software. All other rights are reserved. In particular, redistribution of the code is not allowed.

## References

1. Cleveland, W. S. 1979. Robust locally weighted regression and smoothing scatterplots. Journal of the American Statistical Association 74: 829–836. [https://www.jstor.org/stable/2286407]
2. Wikipedia: Local Regression - [https://en.wikipedia.org/wiki/Local_regression] (accessed 2020-04-20)
3. STATA: Lowess - [https://www.stata.com/manuals13/rlowess.pdf] (accessed 2020-04-20)
4. Cappellari et al. 2013 The ATLAS3D project - XX. Mass-size and mass-σ distributions of early-type galaxies: bulge fraction drives kinematics, mass-to-light ratio, molecular gas fraction and stellar initial mass function Monthly Notices of the Royal Astronomical Society 432: 1862-1893 [https://doi.org/10.1093/mnras/stt644]