# Introduction

## Why would you do this ?

After all, scikit learn already has the DecisionTreeClassifier and it works really well and is highly optimized!

Well, I can see four reasons to implement it anyway!

• It is a good exercise if you want to learn the inner details of the decision trees
• The DecisionTreeClassifier only supports two criterions:
criterion{“gini”, “entropy”}, default=”gini”


However, I may be willing to play with other criterions if the metric I am working with is not a standard one.

• With a code in python that does not require any compilation, pyx files and what not, you can perform plenty of experimentations of the logic of the training tree (and given the problem, obtain a better accuracy)
• It is fun!

## Starting point

So, we will use numpy and implement the DecisionTree without the knowledge of any penalty function. This one will be provided by the user.

We will also follow the fit and predict interface, as we want to be able to reuse this class without a lot of efforts.

## The algorithm

Quoting Wikipedia:

A tree is built by splitting the source set, constituting the root node of the tree, into subsets—which constitute the successor children. The splitting is based on a set of splitting rules based on classification features. This process is repeated on each derived subset in a recursive manner called recursive partitioning.

Put another way: given a dataset A and labels, find a colum and a threshold, so that the data is partitionned it two datasets. Repeat this until the whole dataset has been splitted in small datasets whose size is lower than the minimum sample size given to the algorithm. The splitting part must be performed so that the split achieves the highest improvement in terms of the chosen criterion.

Parameters can be added: the maximum depth of the tree, the minimum number of elements in a leaf, the minimum gain to achieve to decide to split or not the data…

# Implementation

## Imports

from sklearn.utils.validation import check_X_y
import datetime
import numpy as np

class bcolors:
OKBLUE = '\033[94m'
OKGREEN = '\033[92m'
WARNING = '\033[93m'
FAIL = '\033[91m'
ENDC = '\033[0m'
BOLD = '\033[1m'
UNDERLINE = '\033[4m'
CYAN = '\033[36m'


Ok, I imported check_X_y from scikit learn. It would be super easy to remove it in the following, but it saves a lot of debugging to use it, so I will leave it here for now.

bcolors is just a convenient class to store the colors, before printing them to the terminal.

## The Tree class

class Tree:

def __init__(self):
self.left = None
self.right = None
self.data = None


A tree is just a recursive data structure, it can hold data in a node and has to children, a left and a right leaf.

There are plenty of things to know about trees in computer science, but we will only need it to store data. So this class will be enough for our purposes!

## The CustomDecisionTree

Let’s decompose the work a little bit more in what follows. Our CustomDecisionTree will expose fit() and predict() and will operate on numpy arrays. Making it available for pandas DataFrame could be done as well, but let’s put it aside as it require more work and does not help to understand the algorithm used to train a decision tree.

class CustomDecisionTree:

def __init__(self, penalty_function, max_depth=3, min_sample_size=3, max_thresholds=10,
verbose=False):
self._max_depth = max_depth
self._min_sample_size = min_sample_size
self._max_thresholds = max_thresholds
self._penalty_function = penalty_function
self._verbose = verbose
self._y = None


The constructor will need:

• penalty_function (the criterion we will try to optimize)
• max_depth (the depth of the tree)
• min_sample_size (the minimum size of a sample to split it)
• max_thresholds (the number of splits proposed per numeric value)

Storing y could have been performed later, but I like to have all the variables used by my class in the constructor.

    def fit(self, X, y, indices=None):
check_X_y(X, y)
self._y = y
self._tree = Tree()
splitters = self._build_splitters(X)

if indices is None:
indices = np.arange(X.shape[0])

if self._verbose:
self._print("{} splitters proposed".format(len(splitters)))

self._train(self._tree, indices, 0, splitters, 0, X, y)

return self


Still not much done here. We make sure that X and y have compatible shapes (the check_X_y function does it for us), we store y and build the splitters.

Let’s get rid of the _print() method (it is just a habit of mine to distinguish prints from different classes with colors, I find this helpful for debugging if needed, and to monitor the execution of the algorithms).

    def _print(self, input_str):
time = datetime.datetime.now().strftime("%Y-%m-%d %H:%M:%S")
print(bcolors.CYAN + "[CustomDecisionTree | " +
time + "] " + bcolors.ENDC + str(input_str))


Note that we will work on indices to perform all the splits recursively. It was not necessary to pass indices to fit(), but I plan to implement a ranom forest later with this class, so we will need them!

The splitters, themselves, will be at the core of the algorithm. A splitter will simply be an index (the column index) and a threshold.

A splitter will just say: the elements of column i whose value is larger than threshold must go to the right leaf, the elements which are smaller must go to the left leaf.

    def _build_splitters(self, X):
splitters = []

for i, column in enumerate(X.T):
sorted_unique_values = np.sort(np.unique(column))
thresholds = (
sorted_unique_values[:-1] + sorted_unique_values[1:]) / 2
n_thresholds = len(thresholds)

if len(thresholds) > self._max_thresholds:
thresholds = thresholds[[round(
i*n_thresholds / self._max_thresholds) for i in range(self._max_thresholds)]]

for threshold in thresholds:
splitters.append((i, threshold))

return splitters


The splitters are the average between sorted values for each column, subsampled so that we do not have too many splitters (a large number of splitters slows down the algorithm and provides a very limited accuracy improvement).

So, we have the fit() entry point to our interface, we briefy went throgh the splitter building part, let’s continue:

    def _split(self, splitter, indices, X):
index, threshold = splitter
mask = X[indices, index] > threshold


As I said, a splitter just splits the data in two subsets (represented by their indices). It should be clear enough from this method that this is exactly what is performed (with a slight help from numpy).

Now if we remember the algorithm, we need to find the best splitter at each step of the recursive splits. This is where the user defined penalty will come in:

    def _splitter_score(self, splitter, indices, X, y):
indices_left, indices_right = self._split(splitter, indices, X)
n_left, n_right = len(indices_left), len(indices_right)

if n_left < self._min_sample_size:
return -100000

if n_right < self._min_sample_size:
return -100000

return (n_left * self._penalty(indices_left, y) +
n_right * self._penalty(indices_right, y)) / \
(n_left + n_right)


Note that the weighted mean of the penalty for a splitter is returned. If you wanted to modify it, this could be performed here.

So we have our splitters, we can, for each subset, give a score to a splitter, we are ready to implement the full train method:

    def _train(self, tree, indices, depth, splitters, current_score, X, y):
if depth >= self._max_depth:
tree.data = indices
else:
splitter_and_scores = list(
map(lambda ns: (ns, self._splitter_score(ns, indices, X, y)), splitters))
scores = list(map(lambda sp: sp[1], splitter_and_scores))
if len(scores) == 0:
tree.data = indices
return
max_score = max(scores)
max_index = scores.index(max_score)
non_trival_splitters_and_scores = list(
filter(lambda p: p[1] != -100000, splitter_and_scores))
non_trival_splitters = list(
map(lambda p: p[0], non_trival_splitters_and_scores))

best_splitter, best_score = splitter_and_scores[max_index]
indices_left, indices_right = self._split(
best_splitter, indices, X)

if len(indices_left) < self._min_sample_size or \
len(indices_right) < self._min_sample_size:
tree.data = indices

else:
tree.data = best_splitter

tree.left = Tree()
tree.right = Tree()

self._train(tree.left, indices_left, depth + 1,
non_trival_splitters, best_score, X, y)
self._train(tree.right, indices_right, depth + 1,
non_trival_splitters, best_score, X, y)


If we reach max_depth, the leaf we are currently in will store the indices remaining for this leaf.

Otherwise, we find the best splitter, split the data into indices_left and indices_right (induced from this best splitter) and call _train() (recursively) twice : once on each subset. At this step, the node of the tree holds a splitter.

Note that each call to train updates the children of the tree. Once all the calls to train are executed, the tree attribute of the class contains all the splitters (for intermediate nodes) and the indices for the final nodes (the ones that could not be splitted any more).

## Predictions

We have to add the methods that enable to propose predictions once the tree is trained.

    def _find_indices_for_row(self, row):
return self._traverse_trained_tree(self._tree, row)

def _predict_one(self, row):
indices = self._find_indices_for_row(row)
return np.bincount(self._y[indices]).argmax()

def _traverse_trained_tree(self, tree, row):
if tree.left is None:
return tree.data
else:
index, threshold = tree.data
if row[index] > threshold:
return self._traverse_trained_tree(tree.left, row)
else:
return self._traverse_trained_tree(tree.right, row)

def predict(self, X):
return np.array(
list(map(lambda row: self._predict_one(row), X)), dtype=int)


Note that:

np.bincount(self._y[indices]).argmax()


simply returns the most common elements of y at the selected indices. The logic of navigating the tree is performed in _traverse_trained_tree(). For each node, if it is a splitter, follow the logic of the splitter (left or right according to the comparison the threshold). If the algorithm reaches a leaf (tree.left is None), return the indices stored in the leaf.

# Testing !

if __name__== "__main__":

from sklearn.datasets import make_classification
from sklearn.metrics import accuracy_score
import matplotlib.pyplot as plt

X, y = make_classification(n_samples=200, shuffle=False, n_redundant=3)
for max_depth in [1,2,5,10,15]:
cdt = CustomDecisionTree(accuracy_score, min_sample_size=1, max_depth=max_depth)
cdt.fit(X, y)
y_hat = cdt.predict(X)
score = accuracy_score(cdt.predict(X), y)
print("Max depth: ", max_depth, " score: ", score)


And tada! As expected, we reach a perfect accuracy if the depth is large enough!

Max depth:  1  score:  0.915
Max depth:  2  score:  0.92
Max depth:  5  score:  0.925
Max depth:  10  score:  0.975
Max depth:  15  score:  1.0


A more thorough testing would include benchmark on common datasets and a comparison to other implementations of decision trees. I will do it in a separate article.

# Learning more and stay tuned

I hope you liked this reading! Any comments regarding the code or the explanations is welcome! For those who want to stay tuned, I implemented a small form to leave me your email (which won’t be used for ads nor transmitted to any third party). It is in the “Subscribe” section of the navigation menu (small square on the top left).

The next article will go from the CustomDecisionTree to a CustomRandomForest and the following one will be about more detailed tests for these newly implemented classes.

# The code

from sklearn.utils.validation import check_X_y
import datetime
import numpy as np

class Tree:

def __init__(self):
self.left = None
self.right = None
self.data = None

def __str__(self, level=0):
ret = "\t"*level+repr(self.data)+"\n"
for child in [self.left, self.right]:
if child is not None:
ret += child.__str__(level+1)
return ret

def custom_print(self, f1, f2, level=0):
if self.left is None:
ret = "\t"*level+f2(self.data)+"\n"
else:
ret = "\t"*level+f1(self.data)+"\n"

if self.right is not None:
ret = self.right.custom_print(f1, f2, level+1) + ret
if self.left is not None:
ret += self.left.custom_print(f1, f2, level+1)

return ret

class bcolors:
OKBLUE = '\033[94m'
OKGREEN = '\033[92m'
WARNING = '\033[93m'
FAIL = '\033[91m'
ENDC = '\033[0m'
BOLD = '\033[1m'
UNDERLINE = '\033[4m'
CYAN = '\033[36m'

class CustomDecisionTree:

def __init__(self, penalty_function, max_depth=3, min_sample_size=3, max_thresholds=10,
verbose=False):
self._max_depth = max_depth
self._min_sample_size = min_sample_size
self._max_thresholds = max_thresholds
self._penalty_function = penalty_function
self._verbose = verbose
self._y = None

def fit(self, X, y, indices=None):
check_X_y(X, y)
self._y = y
self._tree = Tree()
splitters = self._build_splitters(X)

if indices is None:
indices = np.arange(X.shape[0])

if self._verbose:
self._print("{} splitters proposed".format(len(splitters)))

self._train(self._tree, indices, 0, splitters, 0, X, y)

return self

def predict(self, X):
return np.array(
list(map(lambda row: self._predict_one(row), X)), dtype=int)

def _penalty(self, indices, y):
predicted = [np.bincount(y[indices]).argmax()] * len(indices)
return self._penalty_function(y[indices], predicted)

def _print(self, input_str):
time = datetime.datetime.now().strftime("%Y-%m-%d %H:%M:%S")
print(bcolors.CYAN + "[CustomDecisionTree | " +
time + "] " + bcolors.ENDC + str(input_str))

def _find_indices_for_row(self, row):
return self._traverse_trained_tree(self._tree, row)

def _predict_one(self, row):
indices = self._find_indices_for_row(row)
return np.bincount(self._y[indices]).argmax()

def _traverse_trained_tree(self, tree, row):
if tree.left is None:
return tree.data
else:
index, threshold = tree.data
if row[index] > threshold:
return self._traverse_trained_tree(tree.left, row)
else:
return self._traverse_trained_tree(tree.right, row)

def _build_splitters(self, X):
splitters = []

for i, column in enumerate(X.T):
sorted_unique_values = np.sort(np.unique(column))
thresholds = (
sorted_unique_values[:-1] + sorted_unique_values[1:]) / 2
n_thresholds = len(thresholds)

if len(thresholds) > self._max_thresholds:
thresholds = thresholds[[round(
i*n_thresholds / self._max_thresholds) for i in range(self._max_thresholds)]]

for threshold in thresholds:
splitters.append((i, threshold))

return splitters

def _split(self, splitter, indices, X):
index, threshold = splitter
mask = X[indices, index] > threshold

def _splitter_score(self, splitter, indices, X, y):
indices_left, indices_right = self._split(splitter, indices, X)
n_left, n_right = len(indices_left), len(indices_right)

if n_left < self._min_sample_size:
return -100000

if n_right < self._min_sample_size:
return -100000

return (n_left * self._penalty(indices_left, y) +
n_right * self._penalty(indices_right, y)) / \
(n_left + n_right)

def _train(self, tree, indices, depth, splitters, current_score, X, y):
if depth >= self._max_depth:
tree.data = indices
else:
splitter_and_scores = list(
map(lambda ns: (ns, self._splitter_score(ns, indices, X, y)), splitters))
scores = list(map(lambda sp: sp[1], splitter_and_scores))
if len(scores) == 0:
tree.data = indices
return
max_score = max(scores)
max_index = scores.index(max_score)
non_trival_splitters_and_scores = list(
filter(lambda p: p[1] != -100000, splitter_and_scores))
non_trival_splitters = list(
map(lambda p: p[0], non_trival_splitters_and_scores))

best_splitter, best_score = splitter_and_scores[max_index]
indices_left, indices_right = self._split(
best_splitter, indices, X)

if len(indices_left) < self._min_sample_size or \
len(indices_right) < self._min_sample_size:
tree.data = indices

else:
tree.data = best_splitter

tree.left = Tree()
tree.right = Tree()

self._train(tree.left, indices_left, depth + 1,
non_trival_splitters, best_score, X, y)
self._train(tree.right, indices_right, depth + 1,
non_trival_splitters, best_score, X, y)

if __name__== "__main__":

from sklearn.datasets import make_classification
from sklearn.metrics import accuracy_score
import matplotlib.pyplot as plt

X, y = make_classification(n_samples=20, shuffle=False, n_redundant=3)
cdt = CustomDecisionTree(accuracy_score, verbose=True)

cdt.fit(X, y)

print(cdt._tree.custom_print(str,str))

X, y = make_classification(n_samples=200, shuffle=False, n_redundant=3)
for max_depth in [1,2,5,10,15]:
cdt = CustomDecisionTree(accuracy_score, min_sample_size=1, max_depth=max_depth)
cdt.fit(X, y)
y_hat = cdt.predict(X)
score = accuracy_score(cdt.predict(X), y)
print("Max depth: ", max_depth, " score: ", score)



# Python replace rarely occuring values in a pipeline

## Python pipeline series

Pipeline are probably one of the most convenient tools in scikit learn, and propose a simple way to write reusable models, for which all the hyperparameters, both of the learning and preprocessing part are in the exact same place. However, I do not see them that often on code snippets or in data science competitions.

### The problem

A good practice, when working with factors or categories in a dataframe is to replace values that appear a limited number of time.

A simple reason to do so is that a category appearing just once will be hard to generalize. Decision tree based methods will probably ignore it (as long as the min_sample_size is larger than the number of occurences of this value), so why bother keeping such variables.

### A simple solution

As stressed on this stackoverflow answer, a simple one-liner does the job:

df.loc[df[col].value_counts()[df[col]].values < 10, col] = "RARE_VALUE"


One liners are good. Easy to copy paste. Also easy to make mistakes with them.

Imagine, you are working with a messy dataset, figure out that it would be nice to have a function that takes care of the cleaning.

Copy pasting the above, you end up writing:

def clean_variables(data):
columns = ['Gender', 'Car_Category', 'Subject_Car_Colour',
'Subject_Car_Make', 'LGA_Name', 'State']

for column in columns:
data[column].fillna("empty", inplace=True)
data.loc[data[column].value_counts()[data[column]].values < 10, column] = "RARE_VALUE"

data["Age"] = data["Age"].apply(clip_age)

[...] # other stuff you may do

return data


### The issue

And then you forget about it, some day comes a test set and you blindly apply the clean_variables function on it. That’s what functions are for after all, reusing !

So you write:

train = clean_variables(train)
test = clean_variables(test)


And who knows what may happen from there on. If the test set is too small (less than 10 rows), all the factors will be turned into “RARE_VALUE”. Depending on the importance given to these features by the learning algorithms you applied later, the performance on the test set could be good, or very bad.

### A better solution

Instead, I would recommend putting all this in a pipeline. As far as I know,there is no simple class in scikit-learn that enable to do the removing, so I ended up writing the following class, which does the job:

class RemoveScarceValuesFeatureEngineer:

def __init__(self, min_occurences):
self._min_occurences = min_occurences
self._column_value_counts = {}

def fit(self, X, y):
for column in X.columns:
self._column_value_counts[column] = X[column].value_counts()
return self

def transform(self, X):
for column in X.columns:
X.loc[self._column_value_counts[column][X[column]].values
< self._min_occurences, column] = "RARE_VALUE"

return X

def fit_transform(self, X, y):
self.fit(X, y)
return self.transform(X)

if __name__ == "__main__":
import pandas as pd

sample_train = pd.DataFrame(
[{"a": 1, "s": "a"}, {"a": 1, "s": "a"}, {"a": 1, "s": "b"}])
print(sample_train)
print(20*"=")

sample_test = pd.DataFrame([{"a": 1, "s": "a"}, {"a": 1, "s": "b"}])
print(sample_test)


And executing the code:

   a  s
0  1  a
1  1  a
2  1  b
a           s
0  1           a
1  1           a
2  1  RARE_VALUE
====================
a  s
0  1  a
1  1  b
a           s
0  1           a
1  1  RARE_VALUE


you have the desired behavior: a is not replaced away with RARE_VALUE in the test set!

# Data science competition platforms

(Updated 22nd march 2021: added datasource.ai)

Here is, to my knowledge, the most complete list of data science competition platforms with sponsored (paid) competitions.

If you are not familiar with them, they are, to me, the best way to learn data science. Most of them have a dedicated community and many tutorials, starter kits for competitions. They are also a way to use your skills on topics your day job may not propose you.

## ML contests: an aggregator

ML contests

Clear. A list of competitions, by topics (NLP / supervised learning / vision …)

I am not sure all the platforms are showed here (I did not find references to numer.ai or Bitgrit, per example)

## Prediction

The principle here is simple, you have a train set and a test set to download (though the new trend is encouraging to push your code directly in a dedicated environment hosted by the platform).

The train set contains various columns, or images, or executables (or anything else probably) and the purpose is to predict another variable (which can be a label for classification problems, a value for regression, a set of labels for multiclass classification problem or other things, I am just focusing on the most common tasks)

Then, you upload your predictions (or your code, depending on the competition) and you get a value: the accuracy of your model on the test set. The ranking is immediate, making these platforms delightfully and dangerously addictive!

### Kaggle

Kaggle competitions

Kaggle is probably the largest platform hosting competitions, with the highest prizes and the largest community and resources. Beware, the higher the price, the harder the competition!

They also have the most complete set of learning resources and usable datasets.

### AIcrowd (or CrowdAI)

AIcrowd challenge page

Great platform, super active, many competitions and great topics! They are growing fast so expect even more competitions to happen here.

Besides, from the competitions I have seen here, they focus on less “classical” topics than the ones you would see on Kaggle. Some may like it, others may not, I personnally do.

AIcrowd enables data science experts and enthusiasts to collaboratively solve real-world problems, through challenges.

### Bitgrit

Bitgrit competitions

Launched in 2019, already showing 8 competitions with various topics, this platform looks promising! As said above, it does not seem referenced on mlcontests.

bitgrit is an AI competition and recruiting platform for data scientists, home to a community of over 25,000 engineers worldwide. We are developing bitgrit to be a comprehensive online ecosystem, centered around a blockchain-powered AI Marketplace.

### Drivendata

Driven data competition page

I never took part in their competitions, so I can’t say mcuh about it for now! But they have sponsored competitions.

DrivenData works on projects at the intersection of data science and social impact, in areas like international development, health, education, research and conservation, and public services. We want to give more organizations access to the capabilities of data science, and engage more data scientists with social challenges where their skills can make a difference.

### Crowdanalytix

Crowdanalytix

I never took part in their competitions, so I can’t say mcuh about it for now! They seemed less active recently, but they had sponsored competitions.

25,129 + Data Scientists

102,083 + Models Built

50 + Countries

### Numer.ai

https://numer.ai/

Focusing on predicting the stock market, with high quality data (which is usually a tedious task when you try to have quality data in finance). They claim to be the hardest platform in finance, and having worked there, I can confirm that finding the slightest valuable prediction is super hard!

Nice if you like finance, but be prepared to work with similar datasets!

Start with hedge fund quality data. It is clean and regularized, designed to be usable right away.

### Zindi

Zindi competition page

Data science platform with competitions which are related to Africa. The NLP part seems particularly exciting, as they are focus on languages which are not studied as often as English or Spanish! Looking forward to participate in one of their challenges!

We connect organisations with our thriving African data science community to solve the world’s most pressing challenges using machine learning and AI.

### Analytics Vidhya

Analytics Vidhya

India based.

Data science hackathons on DataHack enable you to compete with leading data scientists and machine learning experts in the world. This is your chance to work on real life data science problems, improve your skill set, learn from expert data science and machine learning professionals, and hack your way to the top of the hackathon leaderboard! You also stand a chance to win prizes and get a job at your dream data science company.

### Challengedata

https://challengedata.ens.fr/

Not sure the competitions are sponsored here. General topics, most of them seem to come from French companies and French institutions.

We organize challenges of data sciences from data provided by public services, companies and laboratories: general documentation and FAQ. The prize ceremony is in February at the College de France.

### Coda Lab

Codalab

French based.

CodaLab is an open-source platform that provides an ecosystem for conducting computational research in a more efficient, reproducible, and collaborative manner.

### Topcoder

Topcoder

Not focusing only on data science:

Access our community of world class developers, great designers, data science geniuses and QA experts for the best results

### InnoCentive

InnoCentive competitions

InnoCentive is the global pioneer in crowdsourced innovation. We help innovative organizations solve their important technology, science, business, A/I and data challenges by connecting them with a global network of expert problem solvers.

### Datasoure.ai

Datasoure

Young company, as the quote below shows (22nd march 2021). They seem to be focused on challenges for startups, but this may evolve!

At a glance

2 Team Members

1,692 Data Scientists

12 Companies

5.2% Weekly Growth

### Signate

Signate competitions

A Japanese competition platform. Most of the competitions are described in Japanese, but not all of them!

SIGNATE collaborates with companies, government agencies and research institutes in various industries to work on various projects to resolve social issues. We invite you to join SIGNATE’s project, which aims to make the world a better place through the power of open innovation.

### datasciencechallenge.org (probably down)

https://www.datasciencechallenge.org/

Unfortunately, I cannot reach the website any more…

Sponsored by the Defence Science and Technology Laboratory and other UK government departments.

### datascience.net (probably down for ever)

datascience.net

Used to be a French speaking data science competition for a while. However, the site has been down for a while now… Worth giving a look from time to time!

## Dataviz

Here, the idea is to provide the best vizualisation of datasets. The metric may therefore not be as absolute as the one for prediction problems and the skillset is really different!

Iron viz

### informationisbeautifulawards

informationisbeautifulawards

They are all the platforms I am aware of, if I missed any or if you have any relevant resources, please let me know!

I Hope you liked this article! If you plan to take part in any of these competitions, best of luck to you, and have fun competing and learning!

# Blog news

As I was renewing my domain name, I figured out I had been posting here for quite a while now :) at a completely irregular frequency, I must admit. Anyways, thank you to all of the readers for your support and interest in my articles ! It was really pleasant to read the various comments or mails you sent me.

As for the blog itself, well, I learnt a lot. About machine learning, obviously, but also about this thing called SEO. I am a little bit surprised by the success of some articles and the oblivion others fell into, but I guess this is how referencing works. One of my most read articles is python plot 3d scatter and density though it did not take much time to write, while a stacking tutorial in Python and theory behind model stacking seem to be invisible from a search engine point of view… But I am not here to rant.

The plan, if any, is to keep posting articles about all the aspects of machine learning which I consider interesting ! I have some material for the decision boundaries of common machine learning algorithms, some code for decision trees, random forest and parallel computations in OCaml and more data visualization snippets…

Another thing, if you like my content and want to support me, I joined the Brave creators program. So if you use this browser and want to help, I would gladly receive BAT tips! Thanks to the anonymous donors who already contributed.

And as usual, if you have some topics you are curious about, some tutorials you would like to read, just let me know in the comments or by mail, I will see what I can do!

Have a nice day!

# Why does staging works?

Model staging is a method that enables to produce (usually) the most competitive models, in terms of accuracy. As such, you will often find winning solutions to data science competitions to be 2-3 stages of models.

This article will assume some familiarity with cross validation. We will go through model averaging first, recall how staging works, observe the link between model averaging and model staging (also referred to as stacking or blending) and propose hypothesis (backed by some visualization) to explain why staging works so well.

Also my other post (more implementation oriented), a stacking tutorial in Python may help.

Going back to model staging, here are some verbatim about some winning solutions of data science competitions.

Per example, the Homesite quote conversion

Quick overview for now about the NMA approach:

10 variations of the dataset in total (factor combinations, factors mapped to response rates, replacing correlated pairs by differences etc) lots of models (xgboost, keras, ranger, logreg, even occasional svm - although that took forever) trained on various datasets and different params; stored as lvl1 metafeatures mix lvl1 metafeatures with: xgboost, nnet, hillclimbing, glmnet and ranger, stack - 5 lvl2 metafeatures mix the lvl2 metafeatures with hillclimbing bag at each stage as much as time permitted

Or, in the BNP Paribas Cardif Claims management:

We also produced many different base level models without much Feature engineering, just different input format types (like load all categorical variables as counts, or as onehot encoding etc).

Our ensemble was consisted of 223 models. Faron did a lot of work in removing noise and discarding many of these in order to get to our bets score with a lvl2 ensemble of geomean weights between an ET , 2NN and 2 Xgmodels.

Before understanding the mechanisms at work for model staging, let’s review the simpler “model averaging” approach.

# Model averaging

## What is model averaging?

Model averaging is a method that consists in averaging the predictions of different models.

It works particularly well in regression problems, or classification problem when the task consists in predicting probabilities of belonging to a specific class.

## Why does averaging works?

Imagine you are facing a regression problem, and have two models. One underestimates the true value, while the other one overestimates it. In this case, the average of the two predictions will be (much) closer to the truth than each individual prediction.

The two figures below illustrate it, in the case of a squared error penalty. The x-axis represents the difference between the true value and the estimated value. The red dots are the estimations of two different models (on the x axis) and the resulting error (on the y axis). The component of the blue dot on the x-axis is the average of the components of the red dots on the x-axis.

It is worth noting that blending will work better if the models have a similar performance (in terms of out-of-sample accuracy) and are as little correlated as possible.

## Jensen inequality

What is even better is that if both models overestimate (or underestimate) the true value, the penalty is still lower.

The graph below presents it:

The green point correpond to the average of the errors, while the blue point correspond of the error of the average.

Proofs can be found on Wikipedia for the purpose of the article, it is enough to convice oneself that this works with these simple graphs.

Jensen’s inequality applies to convex functions, but log loss, squared error (MSE), absolute value error (MAE) are convex functions.

The R code below can be used (just change the xs_points <- c(0.1, 0.8)) to reproduce the experiment with other values.

parabola = function(x) {
x * x
}
xs <- seq(-1, 1, 0.01)
xs_points <- c(0.1, 0.8)

plot(
xs,
parabola(xs),
xlab = expression(hat(x) - x),
ylab = "Square Error",
type = 'l',
main = "MSE as a function of the difference between the true value\n of x and its estimated value"
)

for (xs_point in xs_points) {
points(x = xs_point,
y = parabola(xs_point),
col = "red")
}

points(x = mean(xs_points),
y = parabola(mean(xs_points)),
col = "blue")
segments(
x0 = xs_points[1],
x1 = xs_points[2],
y0 = parabola(xs_points[1]),
y1 = parabola(xs_points[2]), lty = 11
)
points(x = mean(xs_points),
y = mean(parabola(xs_points)),
col = "green")



# Model stacking

## Historical note

As far as I know, the first presentation of stacking goes back to 1992: Stacked generalization, by David H. Wolpert (also famous for the result no free lunch in search and optimization)

[…] The conclusion is that for almost any real-world generalization problem one should use some version of stacked generalization to minimize the generalization error rate.

And almost 30 years later, it remains true! Quoting the wikipedia page:

This work was developed further by Breiman, Smyth, Clarke and many others, and in particular the top two winners of 2009 Netflix competition made extensive use of stacked generalization (rebranded as “blending”)

## How does it work?

### Generalizing averaging

So far, we have seen model averaging. You take two models (or $n$), and average them.

On the other hand, assuming you cross validate $n$ models and only use the best for future predictions, you have another strategy that forms one model from $n$ models.

So we have two schemes that put weights on different models. One puts an equal weight to every model, the other one puts all the weights on the best model.

Model staging consists in using another learning algorithm to choose the “weights” to give to each model. A linear regression “on top” of other models consists in giving an “ideal” (in the regression sense) weight to each model, but what is amazing is that you may use something else than a linear regression!

### Detailing staging procedure

The idea, to give a correct “weight” to each model is to perform a cross validation on the training set and return a dataset for which each element correspond to the unseen fold prediction. On this new dataset, you can fit the new model.

Per example, in the case of a regression problem, if you have $n$ rows in your data set, $p$ features and $k$ models, this step turns your training data from a $n,p$ matrix to a $n,k$ matrix.

Thus the element with index $i,j$ in this new matrix corresponds to the prediction of the $i$-th observation by the $j$-th model.

## Why does this work?

### General case of averaging

Averaging, because of Jensen’s inequality, usually improves the accuracy of the models. Staging, seen as a generalization of averaging, will also improve the accuracy of our learners.

### More general decision boundaries

#### Hypothesis

Another argument, which could be rephrased in more scientific terms is that it allows to obtain decision boundaries of a wider shape than the ones of each usual learning algorithm.

Put another way, the supervised learning problems can be seen as finding $f$ such that:

Where $f$ belongs to some function space. For the linear regressions (penalized or not), $f$ has to be a linear function, for a decision tree, $f$ belongs to a space of sum of indicator functions, for kernel functions, $f$ is a linear combination of kernels…

The hypothesis here is that, in the case of averaging, $f$ can be linear in some region, constant by pieces in another, etc, making the search space for $f$ larger than the search space of all the single families of models.

#### Visualization

As expected, the decision tree has a decision boundary consisting of segments parallel to the x and y axis.

The SVC on the other hand has a very smooth decision boundary.

And here comes the magic, the decision boundary is “a little bit of both”.

Same when we blend with a decision tree.

#### Code

I use the default stacking proposed by scikit-learn.

First a small class useful to plot decision boundaries.

import numpy as np
import matplotlib.pyplot as plt

class DecisionBoundaryPlotter:

def __init__(self, X, Y, xs=np.linspace(0, 1, 30),
ys=np.linspace(0, 1, 30)):
self._X = X
self._Y = Y
self._xs = xs
self._ys = ys

def _predictor(self, model):
model.fit(self._X, self._Y)
return (lambda x: model.predict_proba(x.reshape(1,-1))[0, 0])

def _evaluate_height(self, f):
fun_map = np.empty((self._xs.size, self._ys.size))
for i in range(self._xs.size):
for j in range(self._ys.size):
v = f(
np.array([self._xs[i], self._ys[j]]))
fun_map[i, j] = v
return fun_map

def plot_heatmap(self, model, name):
f = self._predictor(model)
fun_map = self._evaluate_height(f)

fig = plt.figure()
s = fig.add_subplot(1, 1, 1, xlabel='$x$', ylabel='$y$')
im = s.imshow(
fun_map,
extent=(self._ys[0], self._ys[-1], self._xs[0], self._xs[-1]),
origin='lower')
fig.colorbar(im)
fig.savefig(name + '_Heatmap.png')

def plot_contour(self, model, name):
f = self._predictor(model)
fun_map = self._evaluate_height(f)

fig = plt.figure()
s = fig.add_subplot(1, 1, 1, xlabel='$x$', ylabel='$y$')
s.contour(self._xs, self._ys, fun_map, levels = [0.5])
s.scatter(self._X[:,0], self._X[:,1], c = self._Y)
fig.suptitle(name)
fig.savefig(name + '_Contour.png')


The plots (yeah, the nested named models is not that elegant):

import numpy as np
from sklearn.ensemble import RandomForestClassifier, ExtraTreesClassifier, AdaBoostClassifier
from sklearn.tree import DecisionTreeClassifier
from sklearn.svm import SVR, SVC
from sklearn.neighbors import KNeighborsClassifier
from sklearn.linear_model import LogisticRegression
from sklearn.ensemble import StackingClassifier
from DecisionBoundaryPlotter import DecisionBoundaryPlotter

def random_data_classification(n, p, f):
predictors = np.random.rand(n, p)
return predictors, np.apply_along_axis(f, 1, predictors)

def parabolic(x, y):
return (x**2 + y**3 > 0.5) * 1

def parabolic_mat(x):
return parabolic(x[0], x[1])

X, Y = random_data_classification(300, 2, parabolic_mat)

dbp = DecisionBoundaryPlotter(X, Y)

named_classifiers = [ (DecisionTreeClassifier(max_depth=4), "DecisionTreeClassifier"),
(StackingClassifier(estimators=[
("svc", SVC(probability=True)),
("dt", DecisionTreeClassifier(max_depth=4))],
final_estimator=LogisticRegression()),
"Stacked (Linear)"),
(StackingClassifier(estimators=[
("svc", SVC(probability=True)),
("dt", DecisionTreeClassifier(max_depth=4))],
final_estimator=DecisionTreeClassifier(max_depth=4)),
"Stacked (Decision Tree)"),
(SVC(probability=True), "SVC")]

for named_classifier in named_classifiers:
print(named_classifier[1])
dbp.plot_contour(named_classifier[0], named_classifier[1])



## Learning more and references

The elements of statistical learning by Trevor Hastie, Robert Tibshirani, Jerome Friedman is a brilliant introduction to machine learning and will help you have a better understanding of cross validation and the learning algorithms presented here (SVC, Decision trees) but unfortunately does not treat model staging.

# Extract trees from a random forest in python

You may need to extract trees from a classifier for various reasons. In my case, I thought that the feature of xgboost ntree_limit was quite convenient when cross validating a gradient boosting method over the number of trees.

What it does is that it only uses the first ntree_limit trees to perform the prediction (instead of using all the fitted tree).

predict(data, output_margin=False, ntree_limit=0, pred_leaf=False,
pred_contribs=False, approx_contribs=False,
pred_interactions=False, validate_features=True, training=False)


And it is also available as an extra argument of .predict() if you use the scikit-learn interface :

ypred = bst.predict(dtest, ntree_limit=bst.best_ntree_limit)


Indeed, by doing so, if you want to find the optimal number of trees for your model, you do not have to fit the model for 50 trees, and then predict, then fit it for 100 trees and then predict. You may fit the model once and for all for 200 trees and then, playing with ntree_limit you can observe the performance of the model for various number of trees.

The RandomForest, as implemented in scikit-learn does not show this parameter in its .predict() method. However, this is something we can quickly fix. Indeed, the RandomForest exposes estimators_. You can modify it (beware, this is a bit hacky and may not work for other versions of scikit-learn).

rf_model = RandomForestRegressor()
rf_model.fit(x, y)

estimators = rf_model.estimators_

def predict(w, i):
rf_model.estimators_ = estimators[0:i+1]
return rf_model.predict(x)


And that’s it, the predict method now only looks at the first i trees ;)

# What is stacking and why using it

## Introduction

In the case of supervised learning, stacking is a process that enables to improve the performance of a predictor. It can be used for classification and regression problems. If you took part in statistics competitions, you may already be familiar with it, but the resources about this technique are quite scarce on the internet.

## From blending to stacking

As with blending, where a simple average between models with similar performance often proposes a model whose performance is higher than the one of each model in the blend, stacking combine models in a way that is dependant on the training set.

Stacking can be thought as “the sequel” of blending. Imagine you have two models with a similar performance on a dataset. The simplest blend consists in averaging the two models. However, you may become curious and propose different weight for each model. Per example, a blending with a weight of 0.7 for the first model predictions and 0.3 for the second may be better than the default 50/50 weights.

Now let’s suppose you have $n$ models and are looking for the best weights for these n models. The problem you are facing now becomes to performing a linear regression, doesn’t it ?

The only difference is that you cannot use the prediction on the training set directly (where some models like Random Forests usually have a perfect accuracy). This is where stacking comes in!

## When do you need stacking ?

Staging is usually the winning solution of many data science competitions.

Per example, the Homesite quote conversion

Quick overview for now about the NMA approach:

10 variations of the dataset in total (factor combinations, factors mapped to response rates, replacing correlated pairs by differences etc) lots of models (xgboost, keras, ranger, logreg, even occasional svm - although that took forever) trained on various datasets and different params; stored as lvl1 metafeatures mix lvl1 metafeatures with: xgboost, nnet, hillclimbing, glmnet and ranger, stack - 5 lvl2 metafeatures mix the lvl2 metafeatures with hillclimbing bag at each stage as much as time permitted

I will come back to the notions of lv1 / lvl2… stages

Or, in the BNP Paribas Cardif Claims management:

We also produced many different base level models without much Feature engineering, just different input format types (like load all categorical variables as counts, or as onehot encoding etc).

Our ensemble was consisted of 223 models. Faron did a lot of work in removing noise and discarding many of these in order to get to our bets score with a lvl2 ensemble of geomean weights between an ET , 2NN and 2 Xgmodels.

And there are plenty of other examples. So basically, stacking comes in when the accuracy of your classifier or regressor is the essence of your problem. It makes the interpretability of the model really low and is harder to implement and deploy than a simple machine learning pipe.

## Principles of stacking

The idea, to give a correct weight to each model is to perform a cross validation on the training set and return a dataset for which each element correspond to the unseen fold prediction. On this new dataset, you can fit the new model.

Per example, in the case of a regression problem, if you have $n$ rows in your data set, $p$ features and $k$ models, this step turns your training data from a $n,p$ matrix to a $n,k$ matrix.

In the case of a muli class problem, if you have $n$ rows in your data set, $p$ features, $m$ classes and $k$ models, this step turns your training data from a $n,p$ matrix to a $n,k \dot m$ matrix.

# An example

## The dataset

I will use the MNIST dataset, under a CSV format, which can be found here: on Kaggle and a logloss penalty. Of course, you will be able to play with other metrics / datasets using the code below!

## The stacking / CV class

The class below can be used for a multi-class learning problem. Some minor adaptations may be required, per example for regression problems.

from sklearn.model_selection import KFold
import datetime
import pandas as pd
import numpy as np
from time import time

class bcolors:
OKBLUE = '\033[94m'
OKGREEN = '\033[92m'
WARNING = '\033[93m'
FAIL = '\033[91m'
ENDC = '\033[0m'
BOLD = '\033[1m'
UNDERLINE = '\033[4m'

class ModelStager:

def __init__(self, penalty, n_folds,
verbose=1, shuffle=True, random_state=1):
self._penalty = penalty
self._n_folds = n_folds
self._verbose = verbose
self._random_state = random_state
self._shuffle = shuffle

def _print(self, input_str):
time = datetime.datetime.now().strftime("%Y-%m-%d %H:%M")
print(bcolors.HEADER + "[ModelStager | " + time + "] " + bcolors.ENDC + str(input_str))

def fit(self, X, y, model):
kfold = KFold(n_splits=self._n_folds, shuffle=self._shuffle,
random_state=self._random_state)

cv_scores = []
oof_predictions = pd.DataFrame(index=X.index, columns=range(y.nunique()))

fold_idx = 0

for tr_idx, val_idx in kfold.split(X):

X_tr = X.iloc[tr_idx]
X_val = X.iloc[val_idx]

y_tr = y.iloc[tr_idx]
y_val = y.iloc[val_idx]

if self._verbose:
self._print("Data_tr shape : " + str(X_tr.shape))

fold_idx = fold_idx + 1
t = time()

model.fit(X_tr, y_tr)

validation_prediction = model.predict_proba(X_val)

oof_predictions.iloc[val_idx] = validation_prediction

cv_score_model = self._penalty(y_val, validation_prediction)
cv_scores.append(cv_score_model)

if self._verbose:
self._print("Fold %.0f : TEST %.5f | TIME %.2fm (1-fold)" %
(fold_idx, cv_score_model, (time() - t) / 60))

self._print("TEST AVERAGE : %.5f" % (np.mean(cv_scores)))

return oof_predictions


As you can see, the ModelStager also performs cross validation. All the magic happens in oof_predictions, which is in charge of keeping track of the out-of-fold prediction and returning it. As mentioned earlier, it shares the index with X, and the columns correspond to the number of classes.

All the bcolors and custom printing function are just things I am used to work with, no need to bother about it.

## Example

### Random Forest and Extra trees

If you append this at the bottom of the previous class, you may re run the operations.

Two models are proposed, and their ensemble below (using a logistic regression).

if __name__ == "__main__":

from sklearn.metrics import log_loss
from sklearn.linear_model import LogisticRegression
from sklearn.ensemble import RandomForestClassifier, ExtraTreesClassifier
from xgboost import XGBClassifier

X = train_data.drop(["label"], axis=1)
y = train_data["label"]

stager = ModelStager(log_loss, 5)

print("RF model")
model_rf = RandomForestClassifier()
stage1_rf = stager.fit(X, y, model_rf)

print("ET model")
model_et = ExtraTreesClassifier()
stage1_et = stager.fit(X, y, model_et)

print("Stage 1 : (RF, ET) -> logistic model")
stage1_rf_et = pd.concat([stage1_rf, stage1_et], axis=1)
stager.fit(stage1_rf_et, y, LogisticRegression())


Results

RF model
[ModelStager | 2021-01-07 13:54] Data_tr shape : (4000, 784)
[ModelStager | 2021-01-07 13:54] Fold 1 : TEST 0.48133 | TIME 0.05m (1-fold)
[ModelStager | 2021-01-07 13:54] Data_tr shape : (4000, 784)
[ModelStager | 2021-01-07 13:55] Fold 2 : TEST 0.44262 | TIME 0.05m (1-fold)
[ModelStager | 2021-01-07 13:55] Data_tr shape : (4000, 784)
[ModelStager | 2021-01-07 13:55] Fold 3 : TEST 0.46714 | TIME 0.05m (1-fold)
[ModelStager | 2021-01-07 13:55] Data_tr shape : (4000, 784)
[ModelStager | 2021-01-07 13:55] Fold 4 : TEST 0.45846 | TIME 0.05m (1-fold)
[ModelStager | 2021-01-07 13:55] Data_tr shape : (4000, 784)
[ModelStager | 2021-01-07 13:55] Fold 5 : TEST 0.45377 | TIME 0.05m (1-fold)
[ModelStager | 2021-01-07 13:55] TEST AVERAGE : 0.46066
ET model
[ModelStager | 2021-01-07 13:55] Data_tr shape : (4000, 784)
[ModelStager | 2021-01-07 13:55] Fold 1 : TEST 0.44834 | TIME 0.04m (1-fold)
[ModelStager | 2021-01-07 13:55] Data_tr shape : (4000, 784)
[ModelStager | 2021-01-07 13:55] Fold 2 : TEST 0.44679 | TIME 0.04m (1-fold)
[ModelStager | 2021-01-07 13:55] Data_tr shape : (4000, 784)
[ModelStager | 2021-01-07 13:55] Fold 3 : TEST 0.43367 | TIME 0.04m (1-fold)
[ModelStager | 2021-01-07 13:55] Data_tr shape : (4000, 784)
[ModelStager | 2021-01-07 13:55] Fold 4 : TEST 0.43551 | TIME 0.04m (1-fold)
[ModelStager | 2021-01-07 13:55] Data_tr shape : (4000, 784)
[ModelStager | 2021-01-07 13:55] Fold 5 : TEST 0.42378 | TIME 0.04m (1-fold)
[ModelStager | 2021-01-07 13:55] TEST AVERAGE : 0.43762
Stage 1 : (RF, ET) -> logistic model
[ModelStager | 2021-01-07 13:55] Data_tr shape : (4000, 20)
[ModelStager | 2021-01-07 13:55] Fold 1 : TEST 0.20850 | TIME 0.01m (1-fold)
[ModelStager | 2021-01-07 13:55] Data_tr shape : (4000, 20)
[ModelStager | 2021-01-07 13:55] Fold 2 : TEST 0.16870 | TIME 0.00m (1-fold)
[ModelStager | 2021-01-07 13:55] Data_tr shape : (4000, 20)
[ModelStager | 2021-01-07 13:55] Fold 3 : TEST 0.21278 | TIME 0.00m (1-fold)
[ModelStager | 2021-01-07 13:55] Data_tr shape : (4000, 20)
[ModelStager | 2021-01-07 13:55] Fold 4 : TEST 0.20536 | TIME 0.00m (1-fold)
[ModelStager | 2021-01-07 13:55] Data_tr shape : (4000, 20)
[ModelStager | 2021-01-07 13:55] Fold 5 : TEST 0.18016 | TIME 0.00m (1-fold)
[ModelStager | 2021-01-07 13:55] TEST AVERAGE : 0.19510


This is quite a huge boost ;) to be honest, this part is a little bit of an artifact, as ensemble of decision trees are usually quite bad at predicting probabilities, the logloss is artificially high. And the logistic regression corrects this phenomenon.

### Random Forest, Extra Trees and Gradient Boosting

As stated above, the main performance gain comes from using an algorithm that is better at optimizing logloss.

Gradient boosting methods (most notably, xgboost) are good at predicting probability. This is illustrated when we perform the cross validation of a gradient boosting model over the original dataset.

print("XGB model")
model_xgb = XGBClassifier(use_label_encoder=False)
stage_1_xgb = stager.fit(X, y, model_xgb)


Yields the following results

XGB model
[ModelStager | 2021-01-07 13:55] Data_tr shape : (4000, 784)
[ModelStager | 2021-01-07 13:56] Fold 1 : TEST 0.21077 | TIME 1.05m (1-fold)
[ModelStager | 2021-01-07 13:56] Data_tr shape : (4000, 784)
[ModelStager | 2021-01-07 13:57] Fold 2 : TEST 0.16564 | TIME 1.15m (1-fold)
[ModelStager | 2021-01-07 13:57] Data_tr shape : (4000, 784)
[ModelStager | 2021-01-07 13:58] Fold 3 : TEST 0.25023 | TIME 1.09m (1-fold)
[ModelStager | 2021-01-07 13:58] Data_tr shape : (4000, 784)
[ModelStager | 2021-01-07 13:59] Fold 4 : TEST 0.24772 | TIME 1.03m (1-fold)
[ModelStager | 2021-01-07 13:59] Data_tr shape : (4000, 784)
[ModelStager | 2021-01-07 14:00] Fold 5 : TEST 0.18703 | TIME 1.12m (1-fold)
[ModelStager | 2021-01-07 14:00] TEST AVERAGE : 0.21228


Though much better than the single RandomForestClassifier or ExtraTreesClassifier alone, it does not beat the staged model.

Now let’s add the xgb features to the stage 1:

print("Stage 1 : (FR, ET, XGB) -> logistic model")
stage1_rf_et_xgb = pd.concat([stage1_rf, stage1_et, stage_1_xgb], axis=1)
stager.fit(stage1_rf_et_xgb, y, LogisticRegression())


And once again, the performance increases.

Stage 1 : (FR, ET, XGB) -> logistic model
[ModelStager | 2021-01-07 14:00] Data_tr shape : (4000, 30)
[ModelStager | 2021-01-07 14:00] Fold 1 : TEST 0.19343 | TIME 0.01m (1-fold)
[ModelStager | 2021-01-07 14:00] Data_tr shape : (4000, 30)
[ModelStager | 2021-01-07 14:00] Fold 2 : TEST 0.15602 | TIME 0.01m (1-fold)
[ModelStager | 2021-01-07 14:00] Data_tr shape : (4000, 30)
[ModelStager | 2021-01-07 14:00] Fold 3 : TEST 0.20996 | TIME 0.01m (1-fold)
[ModelStager | 2021-01-07 14:00] Data_tr shape : (4000, 30)
[ModelStager | 2021-01-07 14:00] Fold 4 : TEST 0.20830 | TIME 0.01m (1-fold)
[ModelStager | 2021-01-07 14:00] Data_tr shape : (4000, 30)
[ModelStager | 2021-01-07 14:00] Fold 5 : TEST 0.17053 | TIME 0.01m (1-fold)
[ModelStager | 2021-01-07 14:00] TEST AVERAGE : 0.18765


And if we include the gradient boosting predictions in the stage 1 features, the logloss drops from 0.19510 to 0.18765

## More stage 1 features

I only presented 3 models in the stage 1. I could have added plenty of others, such as nearest neighbors, linear models… However, I strongly recommend to play with the code below and try to add these models, I am pretty sure than much better scores can be obtained ;)

Or I could also have performed some feature engineering for some models and not for others. As you can see, the number of combination becomes really huge. The “secret” to have a good performance after stacking is to have models that are as different (surprisingly, the performance of each model is not that important) as possible.

## Beyond the linear model

I mostly referred to the stage 1 as a weighting operation, but it does not have to be a linear model. You can also use other model on top of your stage one features! Per example, another gradient boosting model, or a neural network. You can even repeat the above to produce stage 2 features, and train another model on this stage 2.

# The code

from sklearn.model_selection import KFold
import datetime
import pandas as pd
import numpy as np
from time import time

class bcolors:
OKBLUE = '\033[94m'
OKGREEN = '\033[92m'
WARNING = '\033[93m'
FAIL = '\033[91m'
ENDC = '\033[0m'
BOLD = '\033[1m'
UNDERLINE = '\033[4m'

class ModelStager:

def __init__(self, penalty, n_folds,
verbose=1, shuffle=True, random_state=1):
self._penalty = penalty
self._n_folds = n_folds
self._verbose = verbose
self._random_state = random_state
self._shuffle = shuffle

def _print(self, input_str):
time = datetime.datetime.now().strftime("%Y-%m-%d %H:%M")
print(bcolors.HEADER + "[ModelStager | " + time + "] " + bcolors.ENDC + str(input_str))

def fit(self, X, y, model):
kfold = KFold(n_splits=self._n_folds, shuffle=self._shuffle,
random_state=self._random_state)

cv_scores = []
oof_predictions = pd.DataFrame(index=X.index, columns=range(y.nunique()))

fold_idx = 0

for tr_idx, val_idx in kfold.split(X):

X_tr = X.iloc[tr_idx]
X_val = X.iloc[val_idx]

y_tr = y.iloc[tr_idx]
y_val = y.iloc[val_idx]

if self._verbose:
self._print("Data_tr shape : " + str(X_tr.shape))

fold_idx = fold_idx + 1
t = time()

model.fit(X_tr, y_tr)

validation_prediction = model.predict_proba(X_val)

oof_predictions.iloc[val_idx] = validation_prediction

cv_score_model = self._penalty(y_val, validation_prediction)
cv_scores.append(cv_score_model)

if self._verbose:
self._print("Fold %.0f : TEST %.5f | TIME %.2fm (1-fold)" %
(fold_idx, cv_score_model, (time() - t) / 60))

self._print("TEST AVERAGE : %.5f" % (np.mean(cv_scores)))

return oof_predictions

if __name__ == "__main__":

from sklearn.metrics import log_loss
from sklearn.linear_model import LogisticRegression
from sklearn.ensemble import RandomForestClassifier, ExtraTreesClassifier
from xgboost import XGBClassifier

X = train_data.drop(["label"], axis=1)
y = train_data["label"]

stager = ModelStager(log_loss, 5)

print("RF model")
model_rf = RandomForestClassifier()
stage1_rf = stager.fit(X, y, model_rf)

print("ET model")
model_et = ExtraTreesClassifier()
stage1_et = stager.fit(X, y, model_et)

print("Stage 1 : (RF, ET) -> logistic model")
stage1_rf_et = pd.concat([stage1_rf, stage1_et], axis=1)
stager.fit(stage1_rf_et, y, LogisticRegression())

print("XGB model")
model_xgb = XGBClassifier(use_label_encoder=False)
stage_1_xgb = stager.fit(X, y, model_xgb)

print("Stage 1 : (FR, ET, XGB) -> logistic model")
stage1_rf_et_xgb = pd.concat([stage1_rf, stage1_et, stage_1_xgb], axis=1)
stager.fit(stage1_rf_et_xgb, y, LogisticRegression())


# What are p-adic numbers ?

P-adic numbers are an original way to look at the (limit of sequence of) elements in $\mathbb{Z}$.

More precisely, just like $\mathbb{R}$ represents the limits of Cauchy sequences in $\mathbb{Q}$ endowed with the distance : $d(x,y)=x-y$, $\mathbb{Z}_{p}$ represents the limits of Cauchy sequences in $\mathbb{Z}$ with another distance : $d_p(x, y)$, where $d_p$ is detailed below.

For $p$ a prime number, define $\mathrm{ord}_p(a)$ as the exponent of $p$ in the decomposition of $a$ in a product of prime factors. Also define $\mathrm{ord}_p(0)=\infty$

Then $d_p(a,b)=p^{-\mathrm{ord}_p(a-b)}$ is a distance on integers.

In $\mathbb{Z}$ with the distance $d_3$, note that the sequence $(3^n)_n$ converges towards $0$.

## Why they matter

Various results can be proved using p-adic numbers. I discovered them in “Introduction to number theory”, where they are used to determine whether an ellipse has rationnal points. They also enable to give a meaning to $\sum 5^i = -\frac{1}{4}$

# Visualization

## The idea

A p-adic number can be written $\sum_{i} p^i a_i$ where the sum might be infinite. Though it seems weird because the terms are growing, note that the sequence $(p^i)_i$ actually tends to $0$ really quickly in $\mathbb{Z}_{p}$

A traditionnal way to picture p-adic numbers is with co-centric circles, like below:

All the credit goes to: Heiko Knopse for this illustration, more are available on his site

My idea is to take this idea to the limit. Formally, for $n=\sum_{i} p^i a_i$, the complex number $z=\sum_{i} l^i \exp \left( a_i \frac{2i\pi}{p} \right)$ is associated to $n$.

$l$ is a parameter between $0$ and $1$ used to ensure convergence.

## Results

### Convergence

An interesting property is that $\mathrm{ord}_p(a+b) \geq \min(\mathrm{ord}_p(a), \mathrm{ord}_p(b))$. It is illustrated below. As you can see, addition in the p-addic representation shifts numbers to the right.

# Learning more

For those interested in number theory, I strongly recommend the following books, they are the reason I discovered p-adic integers and they motivated me to explore them (and write this article!)

Number Theory 1: Fermat’s Dream by Kazuya Kato, Nobushige Kurokawa and Takeshi Saito

Number Theory 2: Introduction to Class Field Theory by the same authors which requires more knowledge in algebra and group theory.

“One square and an odd number of triangles”, a problem from Proofs from the book also makes an amazing use of p-adic valuations. The problem itself is simple to state:

is it possible to dissect a square into an odd number $n$ of triangles of equal area?

And this concept appears here, quite surprisingly.

## Code

from cmath import *

def __init__(self, p, l, output_length=30):
self._p = p
self._l = l
self._output_length = output_length

def to_plane(self, n):
l = self._l
p = self._p
decomposed_int = self._completed_int_to_base(n)
complex_coordinates = sum(
[l ** n * exp(1j * c * 2 * pi / p) for n, c in enumerate(decomposed_int)])
return complex_coordinates.real, complex_coordinates.imag

def transform_sample(self, ns):
xs, ys = [], []

for n in ns:
x, y = self.to_plane(n)
xs.append(x)
ys.append(y)

return xs, ys

def _int_to_base(self, n):
p = self._p
i = 0
decomposition = []
while n > 0:
residual = n % p
n = (n - residual) / p
decomposition.append(residual)
return decomposition

def _completed_int_to_base(self, n):
decomposed_int = self._int_to_base(n)
return decomposed_int + [0] * (self._output_length - len(decomposed_int))


The first visualization being obtaining using the following:

import matplotlib.pyplot as plt
plt.rcParams["figure.figsize"] = (8,8)

n_points = 3**10
p = 3
small_sample_size = 55
l = 0.45

xs, ys = par.transform_sample(range(n_points))

fig, ax = plt.subplots()

ax.hist2d(xs, ys, bins = 500, cmap = 'Greys')

ax.scatter(xs[0:small_sample_size], ys[0:small_sample_size], c='black')
for i in range(small_sample_size):
ax.annotate(str(i), (xs[i] - 0.03 , ys[i] + 0.05))

plt.axis('off')
plt.show()


# OCaml Bigarray vs array

A simple benchmark

I was recently wondering if I could speed up the access to the element of a matrix in OCaml, using Bigarrays. As the benchmark below shows, it turns out that no, Bigarray was actually slower than matrix operations.

However, a couple of interesting observations can also be made: inlining manually provided a huge increase of the performance for nested arrays, and Owl is actually the fastest solution available (but not by a lot).

open Owl

let n = 10000

let array_at m i j =
Array.unsafe_get (Array.unsafe_get m i ) j

let sum_array_no_inline n m =
let a = ref 0. in
for i = 0 to (n-1) do
for j = 0 to (n-1) do
a := !a +. array_at m i j
done;
done;
!a

let sum_array n m =
let a = ref 0. in
for i = 0 to (n-1) do
for j = 0 to (n-1) do
a := !a +. (Array.unsafe_get (Array.unsafe_get m i ) j )
done;
done;
!a

let sum_big_array n m =
let a = ref 0. in
for i = 0 to (n-1) do
for j = 0 to (n-1) do
a := !a +. (Bigarray.Array2.unsafe_get m i j);
done;
done;
!a

let sum_owl_array n m =
let a = ref 0. in
for i = 0 to (n-1) do
for j = 0 to (n-1) do
a := !a +. Mat.get m i j;
done;
done;
!a

let sum_owl_array_lib n m =
Mat.sum' m

let time f x =
let start = Unix.gettimeofday () in
let res = f x in
let stop = Unix.gettimeofday () in
Printf.printf "Execution time: %fs\n%!" (stop -. start);
res

let () =
let arr = Array.init n (fun i -> Array.init n (fun j -> (Random.float 1.0))) in

print_string "[ArrayNoInline] ";
let a1 = time (sum_array_no_inline n) arr in

print_string "[Array] ";
let a2 = time (sum_array n) arr in

let big_arr = Bigarray.Array2.of_array Bigarray.float32 Bigarray.c_layout arr in
print_string "[BigArray] ";
let b = time (sum_big_array n) big_arr in

let owl_arr = Mat.of_arrays arr in
print_string "[OwlArray] ";
let c = time (sum_owl_array n) owl_arr in
print_string "[OwlArrayLib] ";
let d = time (sum_owl_array_lib n) owl_arr in

print_string "\n";
print_float a1;

print_string "\n";
print_float a2;

print_string "\n";
print_float b;

print_string "\n";
print_float c;

print_string "\n";
print_float d;
()


And the results

[ArrayNoInline] Execution time: 0.432230s
[Array] Execution time: 0.105445s
[BigArray] Execution time: 3.037937s
[OwlArray] Execution time: 2.177349s
[OwlArrayLib] Execution time: 0.080217s