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Metalog Distribution

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# metalog
Sergey Kim, Reidar Brumer Bratvold

## Metalog Distribution

The metalog distributions constitute a new system of continuous univariate probability distributions designed for flexibility, simplicity, and ease/speed of use in practice. The system is comprised of unbounded, semi-bounded, and bounded distributions, each of which offers nearly unlimited shape flexibility compared to Pearson, Johnson, and other traditional systems of distributions.

The following [paper](http://www.metalogdistributions.com/images/TheMetalogDistributions.pdf) and [website](http://www.metalogdistributions.com/home.html) provide a full background of the metalog distribution.

## Using the Package

This Python package was transfered from [RMetalog](https://github.com/isaacfab/RMetalog) package by Isaac J. Faber and therefore shares the same R-based structure.

The package requires the following packages: **numpy, pandas, matplotlib and scipy (ver 1.3.1)**.

The [data](https://www.sciencebase.gov/catalog/item/5b45380fe4b060350a140b7b) used for demonstration are body length of salmon and were collected in 2008-2010:

```
import numpy as np
import pandas as pd

salmon = pd.read_csv("Chinook and forage fish lengths.csv")

# Filtered data for eelgrass vegetation and chinook salmon
salmon = salmon[(salmon['Vegetation'] == 'Eelgrass') & (salmon['Species'] == 'Chinook_salmon')]
salmon = np.array(salmon['Length'])
```

To import package with metalog distribution run the code:

```
from metalog import metalog
```

To **fit the data to metalog distribution** one should use function ```metalog.fit()```. It has the following arguments:

- ```x```: data.

- ```bounds```: bounds of metalog distribution. Depending on ```boundedness``` argument can take zero, one or two values.

- ```boundedness```: boundedness of metalog distribution. Can take values ```'u'``` for unbounded, ```'sl'``` for semi-bounded lower, ```'su'``` for semi-bounded upper and ```'b'``` for bounded on both sides.

- ```term_limit```: maximum number of terms to specify the metalog distribution. Can take values from 3 to 30.

- ```term_lower_bound```: the lowest number of terms to specify the metalog distribution. Must be greater or equal to 2 and less than ```term_limit```. The argument is optional. Default value is 2.

- ```step_len```: size of steps to summarize the distribution. The argument is optional. Default value is 0.01.

- ```probs```: probabilities corresponding to data. The argument is optional. Default value is ```numpy.nan```.

- ```fit_method```: fit method ```'OLS'```, ```'LP'``` or ```'any'```. The argument is optional. Default value is ```'any'```.

- ```save_data```: if ```True``` then data will be saved for future update. The argument is optional. Default values is ```False```.

Fit metalog distribution to data and store the result to variable ```salmon_metalog```. The distribution is bounded on both sides: from 0 to 200. Term limit is set to 10:

```
metalog_salmon = metalog.fit(x=salmon, boundedness='b', bounds=[0, 200], term_limit=10)
```

To get **summary of distribution** call the following function with only one argument ```m``` - the variable that stores fitted metalog distribution:

```
metalog.summary(m=metalog_salmon)
```

Output:

```
 -----------------------------------------------
 SUMMARY OF METALOG DISTRIBUTION OBJECT
 -----------------------------------------------

PARAMETERS
 
 Term Limit:  10 
 Term Lower Bound:  2 
 Boundedness:  b 
 Bounds (only used based on boundedness):  [0, 200] 
 Step Length for Distribution Summary:  0.01 
 Method Use for Fitting:  any 
 Number of Data Points Used:  138 
 Original Data Saved:  False

VALIDATION AND FIT METHOD
 
     term valid method
2      2   yes    OLS
3      3   yes    OLS
4      4   yes    OLS
5      5   yes    OLS
6      6   yes    OLS
7      7   yes    OLS
8      8   yes    OLS
9      9   yes    OLS
10    10   yes    OLS
```

It's possible **to plot corresponding PDF and CDF** of metalog distribution:

```
metalog.plot(m=metalog_salmon)
```

Output:

![pdf_cdf](https://raw.githubusercontent.com/kimsergeo/metalog/master/figures/figure_1.png)

**To draw samples** from distribution use ```metalog.r()``` function where ```n``` is number of samples and ```term``` specifies the terms of distribution to sample from:

```
metalog.r(m=metalog_salmon, n=5, term=10)
```

Output:

```
array([73.81897286, 86.74055734, 84.22509619, 83.80426247, 97.79800677])
```

**To get densities** based on quantiles type ```metalog.d()``` function where ```q``` is vector of quantiles:

```
metalog.d(m=metalog_salmon, q=[50, 110, 150], term=10)
```

Output:

```
array([0.00038265, 0.00712032, 0.00373991])
```

**To calculate probabilities** based on quantiles use ```metalog.p()``` function:

```
metalog.p(m=metalog_salmon, q=[50, 110, 150], term=10)
```

Output:

```
array([0.00275336, 0.82349578, 0.98686581])
```

Finally, **to get quantiles** from probabilites input ```metalog.q()```:

```
metalog.q(m=metalog_salmon, y=[0.00275336, 0.82349578, 0.98686581], term=10)
```

Output:

```
array([ 50.02583336, 109.99861143, 149.99737059])
```

metalog

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