---
author: Claudio Maggioni
title: Information Modelling & Analysis -- Project 2
geometry: margin=2cm
---

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# Code Repository

The code and result files, part of this submission, can be found at:

- Repository: [https://github.com/infoMA2023/project-02-bug-prediction-maggicl](https://github.com/infoMA2023/project-02-bug-prediction-maggicl)
- Commit ID: **TBD** 

# Data Pre-Processing

I use the sources of the [Closure]{.smallcaps} repository that were already downloaded using the command:

```shell
defects4j checkout -p Closure -v 1f -w ./resources
```

and used the code in the following subfolder for the project:

```
./resources/defects4j-checkout-closure-1f/src/com/google/javascript/jscomp/
```

relative to the root folder of the repository. The resulting CSV of extracted, labelled feature vectors can be found in
the repository at the path:

```
./metrics/feature_vectors_labeled.csv
```

relative to the root folder of the repository. 

Unlabeled feature vectors can be computed by running the script `./extract_feature_vectors.py`.
The resulting CSV of unlabeled feature vectors is located in `./metrics/feature_vectors.csv`.

Labels for feature vectors can be computed by running the script `./label_feature_vectors.py`.

## Feature Vector Extraction

I extracted **291** feature vectors in total. Aggregate metrics
about the extracted feature vectors, i.e. the distribution of the values of each
code metric, can be found in Table [1](#tab:metrics){reference-type="ref"
reference="tab:metrics"}. 

::: {#tab:metrics}
| **Metric** | **Minimum** | **Average** | **Maximum** |
|:------|-:|---------:|-----:|
| `BCM` |0|  13.4124  |  221 |
| `CPX` |0|   5.8247  |   96 |
| `DCM` |0|   4.8652  |176.2 |
| `EX`  |0|   0.1134  |    2 |
| `FLD` |0|   6.5773  |  167 |
| `INT` |0|   0.6667  |    3 |
| `MTH` |0|  11.6529  |  209 |
| `NML` |0|  13.5622  |   28 |
| `RET` |0|   3.6735  |   86 |
| `RFC` |0| 107.2710  |  882 |
| `SZ`  |0|  18.9966  |  347 |
| `WRD` |0| 314.4740  | 3133 |

  : Distribution of values for each extracted code metric.
:::


## Feature Vector Labelling

After feature vectors are labeled, I determine that the dataset contains
**75** buggy classes and **216** non-buggy classes.

# Classifiers

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In this section I explain how I define and perform training for each classifier.

Since the dataset has an unbalanced number of feature vectors of each class, in order
to increase classification performance I upsample the dataset by performing sampling with
replacement over the least frequent class until the number of feature vectors matches the
most frequest class[^1].

[^1]: Upsampling due to unbalanced classes was suggested by *Michele Cattaneo*, who is attending this class.

Other than for the `GaussianNB` (Naive Bayes) classifier, the classifiers chosen for the 
project offer to select hyperparameter values. In order to choose them, I perform a grid
search over each classifier. The hyperparameter values I have considered in the grid search
for each classifier are the following:

- For *DecisionTreeClassifier*:

  | **Parameter** | **Values** |
  |:------------|:--------------|
  | criterion   | gini, entropy |
  | splitter    | best, random  |

- For *SVC*:

  | **Parameter** | **Values** |
  |:------------|:---------------------------|
  | kernel      | linear, poly, rbf, sigmoid |
  | gamma       | scale, auto                |

- For *MLPClassifier*:

  | **Parameter** | **Values** |
  |:------------|:---------------------------|
  | max_iter           | 500000 |
  | hidden_layer_sizes | $[5, 10, 15, ..., 100]$, $[15, 30, 45, 60, 75, 90]^2$, $[20, 40, 60, 80, 100]^3$ |
  | activation         | identity, logistic, tanh, relu |
  | solver             | lbfgs, sgd, adam |
  | learning_rate      | constant, invscaling, adaptive |

  Note that the $[...]^2$ denotes a cartesian product of the array with itself, and $[...]^3$
  denotes the cartesian product of $[...]^2$ with the array (i.e. $[...]^3 = [...]^2 \times [...] = ([...] \times [...]) \times [...]$). 

  Note also the high upper bound on iterations (500000). This is to allow convergence of the less optimal hyperparameter configurations and avoid `ConvergenceWarning` errors.

- For *RandomForestClassifier*:

  | **Parameter** | **Values** |
  |:-------------|:-----------------------------|
  | criterion    | gini, entropy                |
  | max_features | sqrt, log2                   |
  | class_weight | balanced, balanced_subsample |

The script `./train_classifiers.py`, according to the random seed $3735924759$, performs upscaling of the dataset and the grid search training, by recording precision, accuracy, recall and the F1 score of each configuration of hyperparameters. These metrics are then collected and stored in `./models/models.csv`.

The metrics for each classifier and each hyperparameter configuration in decreasing order of
accuracy are reported in the following sections.

For each classifier, I then choose the hyperparameter configuration with highest accuracy. Namely, these configurations are:

| **Classifier** | **Hyper-parameter configuration** | **Precision** | **Accuracy** | **Recall** | **F1 Score** |
|:----|:--------|-:|-:|-:|--:|
| DecisionTreeClassifier | `criterion`: gini, `splitter`: best | 0.7885 | 0.8506 | 0.9535 | 0.8632 |
| GaussianNB | -- | 0.8 | 0.6782 | 0.4651 | 0.5882 |
| MLPClassifier | `activation`: logistic, `hidden_layer_sizes`: (60, 80, 100), `learning_rate`: constant, `max_iter`: 500000, `solver`: lbfgs | 0.8958 | 0.9425 | 1 | 0.9451 |
| RandomForestClassifier | `class_weight`: balanced, `criterion`: gini, `max_features`: sqrt | 0.8367 | 0.8851 | 0.9535 | 0.8913 |
| SVC | `gamma`: scale, `kernel`: rbf | 0.7174 | 0.7356 | 0.7674 | 0.7416 |

## Decision Tree (DT)

| criterion   | splitter   |   precision |   accuracy |   recall |       f1 |
|:------------|:-----------|------------:|-----------:|---------:|---------:|
| gini        | best       |    0.788462 |   0.850575 | 0.953488 | 0.863158 |
| gini        | random     |    0.784314 |   0.83908  | 0.930233 | 0.851064 |
| entropy     | random     |    0.736842 |   0.816092 | 0.976744 | 0.84     |
| entropy     | best       |    0.745455 |   0.816092 | 0.953488 | 0.836735 |

## Naive Bayes (NB)

|   precision |   accuracy |   recall |       f1 |
|------------:|-----------:|---------:|---------:|
|         0.8 |   0.678161 | 0.465116 | 0.588235 |

## Support Vector Machine (SVP)

| gamma   | kernel   |   precision |   accuracy |   recall |       f1 |
|:--------|:---------|------------:|-----------:|---------:|---------:|
| scale   | rbf      |    0.717391 |   0.735632 | 0.767442 | 0.741573 |
| scale   | linear   |    0.75     |   0.735632 | 0.697674 | 0.722892 |
| auto    | linear   |    0.75     |   0.735632 | 0.697674 | 0.722892 |
| auto    | rbf      |    0.702128 |   0.724138 | 0.767442 | 0.733333 |
| scale   | sigmoid  |    0.647059 |   0.678161 | 0.767442 | 0.702128 |
| auto    | sigmoid  |    0.647059 |   0.678161 | 0.767442 | 0.702128 |
| auto    | poly     |    0.772727 |   0.643678 | 0.395349 | 0.523077 |
| scale   | poly     |    0.833333 |   0.597701 | 0.232558 | 0.363636 |

## Multi-Layer Perceptron (MLP)

For sake of brevity, only the top 100 results by accuracy are shown.

| activation   | hidden_layer_sizes   | learning_rate   |   max_iter | solver   |   precision |   accuracy |   recall |       f1 |
|:-------------|:---------------------|:----------------|-----------:|:---------|------------:|-----------:|---------:|---------:|
| logistic     | (60, 80, 100)        | constant        |     500000 | lbfgs    |    0.895833 |   0.942529 | 1        | 0.945055 |
| logistic     | (40, 80, 100)        | adaptive        |     500000 | lbfgs    |    0.86     |   0.91954  | 1        | 0.924731 |
| tanh         | (40, 80, 100)        | invscaling      |     500000 | adam     |    0.86     |   0.91954  | 1        | 0.924731 |
| tanh         | (60, 100, 80)        | adaptive        |     500000 | lbfgs    |    0.86     |   0.91954  | 1        | 0.924731 |
| tanh         | (100, 60, 20)        | constant        |     500000 | adam     |    0.86     |   0.91954  | 1        | 0.924731 |
| tanh         | (100, 80, 80)        | constant        |     500000 | adam     |    0.86     |   0.91954  | 1        | 0.924731 |
| relu         | (75, 30)             | adaptive        |     500000 | lbfgs    |    0.86     |   0.91954  | 1        | 0.924731 |
| logistic     | (20, 40, 60)         | adaptive        |     500000 | lbfgs    |    0.875    |   0.91954  | 0.976744 | 0.923077 |
| logistic     | (40, 60, 80)         | adaptive        |     500000 | lbfgs    |    0.843137 |   0.908046 | 1        | 0.914894 |
| logistic     | (80, 40, 20)         | constant        |     500000 | lbfgs    |    0.843137 |   0.908046 | 1        | 0.914894 |
| tanh         | 30                   | invscaling      |     500000 | lbfgs    |    0.843137 |   0.908046 | 1        | 0.914894 |
| tanh         | 60                   | constant        |     500000 | lbfgs    |    0.843137 |   0.908046 | 1        | 0.914894 |
| tanh         | 85                   | adaptive        |     500000 | lbfgs    |    0.843137 |   0.908046 | 1        | 0.914894 |
| tanh         | (30, 30)             | constant        |     500000 | adam     |    0.843137 |   0.908046 | 1        | 0.914894 |
| tanh         | (45, 45)             | adaptive        |     500000 | lbfgs    |    0.843137 |   0.908046 | 1        | 0.914894 |
| tanh         | (60, 60)             | invscaling      |     500000 | lbfgs    |    0.843137 |   0.908046 | 1        | 0.914894 |
| tanh         | (75, 45)             | invscaling      |     500000 | lbfgs    |    0.843137 |   0.908046 | 1        | 0.914894 |
| tanh         | (75, 75)             | adaptive        |     500000 | lbfgs    |    0.843137 |   0.908046 | 1        | 0.914894 |
| tanh         | (90, 90)             | invscaling      |     500000 | adam     |    0.843137 |   0.908046 | 1        | 0.914894 |
| tanh         | (20, 40, 60)         | invscaling      |     500000 | adam     |    0.843137 |   0.908046 | 1        | 0.914894 |
| tanh         | (20, 100, 20)        | invscaling      |     500000 | lbfgs    |    0.843137 |   0.908046 | 1        | 0.914894 |
| tanh         | (40, 20, 100)        | constant        |     500000 | adam     |    0.843137 |   0.908046 | 1        | 0.914894 |
| tanh         | (40, 80, 60)         | invscaling      |     500000 | adam     |    0.843137 |   0.908046 | 1        | 0.914894 |
| tanh         | (40, 80, 100)        | adaptive        |     500000 | adam     |    0.843137 |   0.908046 | 1        | 0.914894 |
| tanh         | (60, 20, 40)         | adaptive        |     500000 | lbfgs    |    0.843137 |   0.908046 | 1        | 0.914894 |
| tanh         | (60, 60, 80)         | constant        |     500000 | adam     |    0.843137 |   0.908046 | 1        | 0.914894 |
| tanh         | (60, 80, 80)         | adaptive        |     500000 | lbfgs    |    0.843137 |   0.908046 | 1        | 0.914894 |
| tanh         | (80, 20, 40)         | adaptive        |     500000 | lbfgs    |    0.843137 |   0.908046 | 1        | 0.914894 |
| tanh         | (80, 40, 80)         | constant        |     500000 | adam     |    0.843137 |   0.908046 | 1        | 0.914894 |
| tanh         | (80, 60, 60)         | constant        |     500000 | lbfgs    |    0.843137 |   0.908046 | 1        | 0.914894 |
| relu         | (20, 20, 80)         | constant        |     500000 | adam     |    0.843137 |   0.908046 | 1        | 0.914894 |
| relu         | (20, 40, 100)        | constant        |     500000 | lbfgs    |    0.843137 |   0.908046 | 1        | 0.914894 |
| relu         | (20, 60, 20)         | adaptive        |     500000 | adam     |    0.843137 |   0.908046 | 1        | 0.914894 |
| relu         | (20, 60, 100)        | adaptive        |     500000 | adam     |    0.843137 |   0.908046 | 1        | 0.914894 |
| relu         | (20, 100, 20)        | constant        |     500000 | adam     |    0.843137 |   0.908046 | 1        | 0.914894 |
| relu         | (20, 100, 40)        | adaptive        |     500000 | adam     |    0.843137 |   0.908046 | 1        | 0.914894 |
| relu         | (40, 20, 80)         | constant        |     500000 | lbfgs    |    0.843137 |   0.908046 | 1        | 0.914894 |
| relu         | (40, 80, 60)         | constant        |     500000 | lbfgs    |    0.843137 |   0.908046 | 1        | 0.914894 |
| relu         | (60, 20, 100)        | constant        |     500000 | adam     |    0.843137 |   0.908046 | 1        | 0.914894 |
| relu         | (80, 20, 60)         | constant        |     500000 | lbfgs    |    0.843137 |   0.908046 | 1        | 0.914894 |
| relu         | (80, 60, 20)         | adaptive        |     500000 | adam     |    0.843137 |   0.908046 | 1        | 0.914894 |
| relu         | (100, 20, 60)        | invscaling      |     500000 | adam     |    0.843137 |   0.908046 | 1        | 0.914894 |
| logistic     | (20, 60, 80)         | invscaling      |     500000 | lbfgs    |    0.857143 |   0.908046 | 0.976744 | 0.913043 |
| logistic     | (60, 20, 20)         | adaptive        |     500000 | lbfgs    |    0.857143 |   0.908046 | 0.976744 | 0.913043 |
| tanh         | (15, 45)             | constant        |     500000 | lbfgs    |    0.857143 |   0.908046 | 0.976744 | 0.913043 |
| tanh         | (45, 90)             | invscaling      |     500000 | lbfgs    |    0.857143 |   0.908046 | 0.976744 | 0.913043 |
| tanh         | (90, 30)             | constant        |     500000 | lbfgs    |    0.857143 |   0.908046 | 0.976744 | 0.913043 |
| tanh         | (20, 80, 100)        | invscaling      |     500000 | adam     |    0.857143 |   0.908046 | 0.976744 | 0.913043 |
| tanh         | (20, 80, 100)        | adaptive        |     500000 | lbfgs    |    0.857143 |   0.908046 | 0.976744 | 0.913043 |
| tanh         | (40, 40, 40)         | adaptive        |     500000 | lbfgs    |    0.857143 |   0.908046 | 0.976744 | 0.913043 |
| tanh         | (40, 60, 100)        | adaptive        |     500000 | adam     |    0.857143 |   0.908046 | 0.976744 | 0.913043 |
| tanh         | (60, 80, 60)         | constant        |     500000 | lbfgs    |    0.857143 |   0.908046 | 0.976744 | 0.913043 |
| tanh         | (100, 40, 60)        | invscaling      |     500000 | lbfgs    |    0.857143 |   0.908046 | 0.976744 | 0.913043 |
| tanh         | (100, 80, 100)       | adaptive        |     500000 | adam     |    0.857143 |   0.908046 | 0.976744 | 0.913043 |
| relu         | (30, 30)             | adaptive        |     500000 | lbfgs    |    0.857143 |   0.908046 | 0.976744 | 0.913043 |
| relu         | (20, 20, 40)         | adaptive        |     500000 | lbfgs    |    0.857143 |   0.908046 | 0.976744 | 0.913043 |
| relu         | (20, 40, 40)         | adaptive        |     500000 | adam     |    0.857143 |   0.908046 | 0.976744 | 0.913043 |
| relu         | (40, 20, 100)        | adaptive        |     500000 | adam     |    0.857143 |   0.908046 | 0.976744 | 0.913043 |
| relu         | (60, 80, 20)         | invscaling      |     500000 | lbfgs    |    0.857143 |   0.908046 | 0.976744 | 0.913043 |
| logistic     | (40, 80, 60)         | adaptive        |     500000 | lbfgs    |    0.87234  |   0.908046 | 0.953488 | 0.911111 |
| logistic     | 35                   | adaptive        |     500000 | lbfgs    |    0.826923 |   0.896552 | 1        | 0.905263 |
| logistic     | (15, 60)             | invscaling      |     500000 | lbfgs    |    0.826923 |   0.896552 | 1        | 0.905263 |
| logistic     | (45, 45)             | constant        |     500000 | lbfgs    |    0.826923 |   0.896552 | 1        | 0.905263 |
| logistic     | (20, 20, 60)         | adaptive        |     500000 | lbfgs    |    0.826923 |   0.896552 | 1        | 0.905263 |
| logistic     | (60, 60, 80)         | adaptive        |     500000 | lbfgs    |    0.826923 |   0.896552 | 1        | 0.905263 |
| logistic     | (80, 40, 100)        | invscaling      |     500000 | lbfgs    |    0.826923 |   0.896552 | 1        | 0.905263 |
| logistic     | (100, 100, 100)      | constant        |     500000 | lbfgs    |    0.826923 |   0.896552 | 1        | 0.905263 |
| tanh         | 60                   | constant        |     500000 | adam     |    0.826923 |   0.896552 | 1        | 0.905263 |
| tanh         | (15, 15)             | invscaling      |     500000 | adam     |    0.826923 |   0.896552 | 1        | 0.905263 |
| tanh         | (15, 45)             | adaptive        |     500000 | adam     |    0.826923 |   0.896552 | 1        | 0.905263 |
| tanh         | (30, 30)             | invscaling      |     500000 | adam     |    0.826923 |   0.896552 | 1        | 0.905263 |
| tanh         | (30, 60)             | constant        |     500000 | adam     |    0.826923 |   0.896552 | 1        | 0.905263 |
| tanh         | (60, 90)             | invscaling      |     500000 | lbfgs    |    0.826923 |   0.896552 | 1        | 0.905263 |
| tanh         | (75, 15)             | constant        |     500000 | lbfgs    |    0.826923 |   0.896552 | 1        | 0.905263 |
| tanh         | (75, 45)             | constant        |     500000 | lbfgs    |    0.826923 |   0.896552 | 1        | 0.905263 |
| tanh         | (90, 15)             | adaptive        |     500000 | adam     |    0.826923 |   0.896552 | 1        | 0.905263 |
| tanh         | (90, 45)             | invscaling      |     500000 | lbfgs    |    0.826923 |   0.896552 | 1        | 0.905263 |
| tanh         | (20, 40, 20)         | invscaling      |     500000 | adam     |    0.826923 |   0.896552 | 1        | 0.905263 |
| tanh         | (20, 40, 40)         | invscaling      |     500000 | adam     |    0.826923 |   0.896552 | 1        | 0.905263 |
| tanh         | (20, 60, 20)         | adaptive        |     500000 | adam     |    0.826923 |   0.896552 | 1        | 0.905263 |
| tanh         | (20, 80, 60)         | constant        |     500000 | adam     |    0.826923 |   0.896552 | 1        | 0.905263 |
| tanh         | (20, 80, 80)         | constant        |     500000 | adam     |    0.826923 |   0.896552 | 1        | 0.905263 |
| tanh         | (20, 80, 100)        | constant        |     500000 | adam     |    0.826923 |   0.896552 | 1        | 0.905263 |
| tanh         | (40, 20, 60)         | invscaling      |     500000 | lbfgs    |    0.826923 |   0.896552 | 1        | 0.905263 |
| tanh         | (40, 60, 60)         | constant        |     500000 | adam     |    0.826923 |   0.896552 | 1        | 0.905263 |
| tanh         | (40, 60, 60)         | invscaling      |     500000 | adam     |    0.826923 |   0.896552 | 1        | 0.905263 |
| tanh         | (40, 80, 20)         | adaptive        |     500000 | adam     |    0.826923 |   0.896552 | 1        | 0.905263 |
| tanh         | (40, 100, 60)        | constant        |     500000 | adam     |    0.826923 |   0.896552 | 1        | 0.905263 |
| tanh         | (60, 40, 20)         | constant        |     500000 | adam     |    0.826923 |   0.896552 | 1        | 0.905263 |
| tanh         | (60, 40, 40)         | invscaling      |     500000 | adam     |    0.826923 |   0.896552 | 1        | 0.905263 |
| tanh         | (60, 40, 80)         | constant        |     500000 | lbfgs    |    0.826923 |   0.896552 | 1        | 0.905263 |
| tanh         | (60, 60, 20)         | constant        |     500000 | adam     |    0.826923 |   0.896552 | 1        | 0.905263 |
| tanh         | (60, 80, 60)         | constant        |     500000 | adam     |    0.826923 |   0.896552 | 1        | 0.905263 |
| tanh         | (60, 80, 80)         | invscaling      |     500000 | adam     |    0.826923 |   0.896552 | 1        | 0.905263 |
| tanh         | (60, 100, 20)        | invscaling      |     500000 | adam     |    0.826923 |   0.896552 | 1        | 0.905263 |
| tanh         | (60, 100, 40)        | adaptive        |     500000 | adam     |    0.826923 |   0.896552 | 1        | 0.905263 |
| tanh         | (60, 100, 60)        | constant        |     500000 | adam     |    0.826923 |   0.896552 | 1        | 0.905263 |
| tanh         | (60, 100, 60)        | adaptive        |     500000 | adam     |    0.826923 |   0.896552 | 1        | 0.905263 |
| tanh         | (60, 100, 80)        | constant        |     500000 | adam     |    0.826923 |   0.896552 | 1        | 0.905263 |
| tanh         | (80, 40, 40)         | constant        |     500000 | adam     |    0.826923 |   0.896552 | 1        | 0.905263 |

## Random Forest (RF)

| criterion   | class_weight       | max_features   |   precision |   accuracy |   recall |       f1 |
|:------------|:-------------------|:---------------|------------:|-----------:|---------:|---------:|
| gini        | balanced           | sqrt           |    0.836735 |   0.885057 | 0.953488 | 0.891304 |
| entropy     | balanced           | sqrt           |    0.807692 |   0.873563 | 0.976744 | 0.884211 |
| gini        | balanced_subsample | sqrt           |    0.807692 |   0.873563 | 0.976744 | 0.884211 |
| entropy     | balanced_subsample | sqrt           |    0.807692 |   0.873563 | 0.976744 | 0.884211 |
| gini        | balanced           | log2           |    0.82     |   0.873563 | 0.953488 | 0.88172  |
| entropy     | balanced           | log2           |    0.82     |   0.873563 | 0.953488 | 0.88172  |
| gini        | balanced_subsample | log2           |    0.803922 |   0.862069 | 0.953488 | 0.87234  |
| entropy     | balanced_subsample | log2           |    0.803922 |   0.862069 | 0.953488 | 0.87234  |

# Evaluation

To evaluate the performance of each selected classifier, each model has trained 100 times by repeating a 5-fold
cross validation procedure 20 times. A graphical and statistical analysis to compare the distribution of
performance metrics (precision, recall and F1 score) follows.  

Numeric tables, statistical analysis conclusions and the boxplot diagram shown in this section are obtained by
running the script:

```
./evaluate_classifiers.py
```

## Output Distributions

A boxplot chart to show the distribution of each of precision, recall, and F1 score
for all classifiers (including the biased classifier) is shown in figure [1](#fig:boxplot){reference-type="ref" reference="fig:boxplot"}. Table [2](#tab:meanstd){reference-type="ref" reference="tab:meanstd"} is a numeric
table summing up mean and standard deviation of each metric.

![Precision, Recall and F1 score distribution for each classifier for 20-times cross validation.](../models/boxplot.svg){#fig:boxplot}

::: {#tab:meanstd}
| **Classifier**     | **Metric** | **Mean** | **Std. dev.** |
|:-----------------------|:----------|---------:|-----------:|
| BiasedClassifier       | f1        | 0.666238 | 0.00511791 |
| BiasedClassifier       | precision | 0.49954  | 0.00575757 |
| BiasedClassifier       | recall    | 1        | 0          |
| DecisionTreeClassifier | f1        | 0.888104 | 0.0302085  |
| DecisionTreeClassifier | precision | 0.832669 | 0.0425814  |
| DecisionTreeClassifier | recall    | 0.953261 | 0.0358021  |
| GaussianNB             | f1        | 0.54952  | 0.0912106  |
| GaussianNB             | precision | 0.820866 | 0.066323   |
| GaussianNB             | recall    | 0.418885 | 0.0950382  |
| MLPClassifier          | f1        | 0.884807 | 0.0348063  |
| MLPClassifier          | precision | 0.836507 | 0.0489391  |
| MLPClassifier          | recall    | 0.941823 | 0.0454851  |
| RandomForestClassifier | f1        | 0.91079  | 0.0301481  |
| RandomForestClassifier | precision | 0.870685 | 0.0427383  |
| RandomForestClassifier | recall    | 0.956665 | 0.0381288  |
| SVC                    | f1        | 0.752656 | 0.0537908  |
| SVC                    | precision | 0.7557   | 0.0515768  |
| SVC                    | recall    | 0.754709 | 0.0819603  |

  : Mean and standard deviation for each classifier for 20-times cross validation.
:::


## Comparison and Significance

Given the distribution of metrics presented in the previous section, I perform a statistical analysis
using the Wilixcon paired test to determine for each pair of classifiers which one performs better 
according to each performance metric. When the *p-value* is too high, *ANOVA* power analysis (corrected
by *alpha* = $= 0.05$) is performed to determine if the metrics are equally distributed or if the statistical
test is inconclusive.

# F1 Values

- Mean *F1* for *DT*: 0.8881, mean *F1* for *NB*: 0.5495 $\Rightarrow$ *DT* is better than *NB* (*p-value* $= 0.0$)
- Mean *F1* for *DT*: 0.8881, mean *F1* for *MLP*: 0.8848 $\Rightarrow$ statistical test inconclusive (*p-value* $= 0.4711$, *5% corrected ANOVA power* $= 0.2987$)
- Mean *F1* for *DT*: 0.8881, mean *F1* for *RF*: 0.9108 $\Rightarrow$ *RF* is better than *DT* (*p-value* $= 0.0$)
- Mean *F1* for *DT*: 0.8881, mean *F1* for *SVP*: 0.7527 $\Rightarrow$ *DT* is better than *SVP* (*p-value* $= 0.0$)
- Mean *F1* for *NB*: 0.5495, mean *F1* for *MLP*: 0.8848 $\Rightarrow$ *MLP* is better than *NB* (*p-value* $= 0.0$)
- Mean *F1* for *NB*: 0.5495, mean *F1* for *RF*: 0.9108 $\Rightarrow$ *RF* is better than *NB* (*p-value* $= 0.0$)
- Mean *F1* for *NB*: 0.5495, mean *F1* for *SVP*: 0.7527 $\Rightarrow$ *SVP* is better than *NB* (*p-value* $= 0.0$)
- Mean *F1* for *MLP*: 0.8848, mean *F1* for *RF*: 0.9108 $\Rightarrow$ *RF* is better than *MLP* (*p-value* $= 0.0$)
- Mean *F1* for *MLP*: 0.8848, mean *F1* for *SVP*: 0.7527 $\Rightarrow$ *MLP* is better than *SVP* (*p-value* $= 0.0$)
- Mean *F1* for *RF*: 0.9108, mean *F1* for *SVP*: 0.7527 $\Rightarrow$ *RF* is better than *SVP* (*p-value* $= 0.0$)
- Mean *F1* for *Biased*: 0.6662, mean *F1* for *DT*: 0.8881 $\Rightarrow$ *DT* is better than *Biased* (*p-value* $= 0.0$)
- Mean *F1* for *Biased*: 0.6662, mean *F1* for *NB*: 0.5495 $\Rightarrow$ *Biased* is better than *NB* (*p-value* $= 0.0$)
- Mean *F1* for *Biased*: 0.6662, mean *F1* for *MLP*: 0.8848 $\Rightarrow$ *MLP* is better than *Biased* (*p-value* $= 0.0$)
- Mean *F1* for *Biased*: 0.6662, mean *F1* for *RF*: 0.9108 $\Rightarrow$ *RF* is better than *Biased* (*p-value* $= 0.0$)
- Mean *F1* for *Biased*: 0.6662, mean *F1* for *SVP*: 0.7527 $\Rightarrow$ *SVP* is better than *Biased* (*p-value* $= 0.0$)

### Precision

- Mean *precision* for *DT*: 0.8327, mean *precision* for *NB*: 0.8209 $\Rightarrow$ *DT* is as effective as *NB* (*p-value* $= 0.0893$, *5% corrected ANOVA power* $= 0.8498$)
- Mean *precision* for *DT*: 0.8327, mean *precision* for *MLP*: 0.8365 $\Rightarrow$ statistical test inconclusive (*p-value* $= 0.4012$, *5% corrected ANOVA power* $= 0.2196$)
- Mean *precision* for *DT*: 0.8327, mean *precision* for *RF*: 0.8707 $\Rightarrow$ *RF* is better than *DT* (*p-value* $= 0.0$)
- Mean *precision* for *DT*: 0.8327, mean *precision* for *SVP*: 0.7557 $\Rightarrow$ *DT* is better than *SVP* (*p-value* $= 0.0$)
- Mean *precision* for *NB*: 0.8209, mean *precision* for *MLP*: 0.8365 $\Rightarrow$ *MLP* is better than *NB* (*p-value* $= 0.0348$)
- Mean *precision* for *NB*: 0.8209, mean *precision* for *RF*: 0.8707 $\Rightarrow$ *RF* is better than *NB* (*p-value* $= 0.0$)
- Mean *precision* for *NB*: 0.8209, mean *precision* for *SVP*: 0.7557 $\Rightarrow$ *NB* is better than *SVP* (*p-value* $= 0.0$)
- Mean *precision* for *MLP*: 0.8365, mean *precision* for *RF*: 0.8707 $\Rightarrow$ *RF* is better than *MLP* (*p-value* $= 0.0$)
- Mean *precision* for *MLP*: 0.8365, mean *precision* for *SVP*: 0.7557 $\Rightarrow$ *MLP* is better than *SVP* (*p-value* $= 0.0$)
- Mean *precision* for *RF*: 0.8707, mean *precision* for *SVP*: 0.7557 $\Rightarrow$ *RF* is better than *SVP* (*p-value* $= 0.0$)
- Mean *precision* for *Biased*: 0.4995, mean *precision* for *DT*: 0.8327 $\Rightarrow$ *DT* is better than *Biased* (*p-value* $= 0.0$)
- Mean *precision* for *Biased*: 0.4995, mean *precision* for *NB*: 0.8209 $\Rightarrow$ *NB* is better than *Biased* (*p-value* $= 0.0$)
- Mean *precision* for *Biased*: 0.4995, mean *precision* for *MLP*: 0.8365 $\Rightarrow$ *MLP* is better than *Biased* (*p-value* $= 0.0$)
- Mean *precision* for *Biased*: 0.4995, mean *precision* for *RF*: 0.8707 $\Rightarrow$ *RF* is better than *Biased* (*p-value* $= 0.0$)
- Mean *precision* for *Biased*: 0.4995, mean *precision* for *SVP*: 0.7557 $\Rightarrow$ *SVP* is better than *Biased* (*p-value* $= 0.0$)

### Recall

- Mean *recall* for *DT*: 0.9533, mean *recall* for *NB*: 0.4189 $\Rightarrow$ *DT* is better than *NB* (*p-value* $= 0.0$)
- Mean *recall* for *DT*: 0.9533, mean *recall* for *MLP*: 0.9418 $\Rightarrow$ *DT* is better than *MLP* (*p-value* $= 0.0118$)
- Mean *recall* for *DT*: 0.9533, mean *recall* for *RF*: 0.9567 $\Rightarrow$ statistical test inconclusive (*p-value* $= 0.3276$, *5% corrected ANOVA power* $= 0.2558$)
- Mean *recall* for *DT*: 0.9533, mean *recall* for *SVP*: 0.7547 $\Rightarrow$ *DT* is better than *SVP* (*p-value* $= 0.0$)
- Mean *recall* for *NB*: 0.4189, mean *recall* for *MLP*: 0.9418 $\Rightarrow$ *MLP* is better than *NB* (*p-value* $= 0.0$)
- Mean *recall* for *NB*: 0.4189, mean *recall* for *RF*: 0.9567 $\Rightarrow$ *RF* is better than *NB* (*p-value* $= 0.0$)
- Mean *recall* for *NB*: 0.4189, mean *recall* for *SVP*: 0.7547 $\Rightarrow$ *SVP* is better than *NB* (*p-value* $= 0.0$)
- Mean *recall* for *MLP*: 0.9418, mean *recall* for *RF*: 0.9567 $\Rightarrow$ *RF* is better than *MLP* (*p-value* $= 0.0001$)
- Mean *recall* for *MLP*: 0.9418, mean *recall* for *SVP*: 0.7547 $\Rightarrow$ *MLP* is better than *SVP* (*p-value* $= 0.0$)
- Mean *recall* for *RF*: 0.9567, mean *recall* for *SVP*: 0.7547 $\Rightarrow$ *RF* is better than *SVP* (*p-value* $= 0.0$)
- Mean *recall* for *Biased*: 1.0, mean *recall* for *DT*: 0.9533 $\Rightarrow$ *Biased* is better than *DT* (*p-value* $= 0.0$)
- Mean *recall* for *Biased*: 1.0, mean *recall* for *NB*: 0.4189 $\Rightarrow$ *Biased* is better than *NB* (*p-value* $= 0.0$)
- Mean *recall* for *Biased*: 1.0, mean *recall* for *MLP*: 0.9418 $\Rightarrow$ *Biased* is better than *MLP* (*p-value* $= 0.0$)
- Mean *recall* for *Biased*: 1.0, mean *recall* for *RF*: 0.9567 $\Rightarrow$ *Biased* is better than *RF* (*p-value* $= 0.0$)
- Mean *recall* for *Biased*: 1.0, mean *recall* for *SVP*: 0.7547 $\Rightarrow$ *Biased* is better than *SVP* (*p-value* $= 0.0$)

## Practical Usefulness

Discuss the practical usefulness of the obtained classifiers in a
realistic bug prediction scenario (1 paragraph).