Previous in series:

In the previous two posts on machine learning, I presented a very basic introduction of an approach called "probabilistic graphical models". In this post I'd like to take a tour of some different techniques while creating code that will recognize handwritten digits.

The handwritten digits recognition is an interesting topic that has been explored for many years. It is now considered one of the best ways to start the journey into the world of machine learning.

##
Taking the Kaggle challenge

We'll take the "digits recognition" challenge as presented in Kaggle. It is an online platform with challenges for data scientists. Most of the challenges have their prizes expressed in real money to win. Some of them are there to help us out in our journey on learning data science techniques — so is the "digits recognition" contest.

##
The challenge

As explained on Kaggle:

MNIST ("Modified National Institute of Standards and Technology") is the de facto “hello world” dataset of computer vision.

The "digits recognition" challenge is one of the best ways to get acquainted with machine learning and computer vision. The so-called "MNIST" dataset consists of 70k images of handwritten digits - each one grayscaled and of a 28x28 size. The Kaggle challenge is about taking a subset of 42k of them along with labels (what actual number does the image show) and "training" the computer on that set. The next step is to take the rest 28k of images without the labels and "predict" which actual number they present.

Here's a short overview of how the digits in a set really look like (along with the numbers they represent):

I have to admit that for some of them I have a really hard time recognizing the actual numbers on my own :)

##
The general approach to supervised learning

Learning from labelled data is what is called "supervised learning". It's supervised because we're taking the computer by hand through the whole training data set and "teaching" it how the data that is linked with different labels "looks" like.

In all such scenarios we can express the data and labels as:

Y ~ X1, X2, X3, X4, ..., Xn

The

`Y`

is called a

**dependent variable** while each

`Xn`

are

**independent variables**. This formula holds both for classification problems as well as regressions.

Classification is when the dependent variable

`Y`

is so called

*categorical* — taking values from a concrete set without a meaningful order. Regression is when the

`Y`

is not categorical — most often continuous.

In the digits recognition challenge we're faced with the classification task. The dependent variable takes values from the set:

Y = { 0, 1, 2, 3, 4, 5, 6, 7, 8, 9 }

I'm sure the question you might be asking yourself now is: what are the independent variables

`Xn`

? It turns out to be the crux of the whole problem to solve :)

##
The plan of attack

A good introduction to computer vision techniques is a book by J. R Parker - "Algorithms for Image Processing and Computer Vision". I encourage the reader to buy that book. I took some ideas from it while having fun with my own solution to the challenge.

The book outlines the ideas revolving around computing image profiles — for each side. For each row of pixels, a number representing the distance of the first pixel from the edge is computed. This way we're getting our first independent variables. To capture even more information about digit shapes, we'll also capture the differences between consecutive row values as well as their global maxima and minima. We'll also compute the width of the shape for each row.

Because the handwritten digits vary greatly in their thickness, we will first preprocess the images to detect so-called skeletons of the digit. The skeleton is an image representation where the thickness of the shape has been reduced to just one.

Having the image thinned will also allow us to capture some more info about the shapes. We will write an algorithm that walks the skeleton and records the direction change frequencies.

Once we'll have our set of independent variables

`Xn`

, we'll use a classification algorithm to first learn in a supervised way (using the provided labels) and then to predict the values of the test data set. Lastly we'll submit our predictions to Kaggle and see how well did we do.

##
Having fun with languages

In the data science world, the lingua franca still remains to be the

`R`

programming language. In the last years

`Python`

has also came close in popularity and nowadays we can say it's the duo of

`R`

and

`Python`

that rule the data science world (not counting high performance code written e. g. in C++ in production systems).

Lately a new language designed with data scientists in mind has emerged -

`Julia`

. It's a language with characteristics of both dynamically typed scripting languages as well as strictly typed compiled ones. It compiles its code into efficient native binary via LLVM — but it's using it in a JIT fashion - inferring the types when needed on the go.

While having fun with the Kaggle challenge I'll use Julia and Python for the so called

**feature extraction** phase (the one in which we're computing information about our

`Xn`

variables). I'll then turn towards

`R`

for doing the classification itself. Note that I might use any of those languages at each step getting very similar results. The purpose of this series of articles is to be a bird eye fun overview so I decided that this way will be much more interesting.

##
Feature Extraction

The end result of this phase is the data frame saved as a CSV file so that we'll be able to load it in R and do the classification.

First let's define the general function in Julia that takes the name of the input CSV file and returns a data frame with features of given images extracted into columns:

using DataFrames
function get_data(name :: String, include_label = true)
println("Loading CSV file into a data frame...")
table = readtable(string(name, ".csv"))
extract(table, include_label)
end

Now the extract function looks like the following:

"""
Extracts the features from the dataframe. Puts them into
separate columns and removes all other columns except the
labels.
The features:
* Left and right profiles (after fitting into the same sized rect):
* Min
* Max
* Width[y]
* Diff[y]
* Paths:
* Frequencies of movement directions
* Simplified directions:
* Frequencies of 3 element simplified paths
"""
function extract(frame :: DataFrame, include_label = true)
println("Reshaping data...")
function to_image(flat :: Array{Float64}) :: Array{Float64}
dim = Base.isqrt(length(flat))
reshape(flat, (dim, dim))'
end
from = include_label ? 2 : 1
frame[:pixels] = map((i) -> convert(Array{Float64}, frame[i, from:end]) |> to_image, 1:size(frame, 1))
images = frame[:, :pixels] ./ 255
data = Array{Array{Float64}}(length(images))
@showprogress 1 "Computing features..." for i in 1:length(images)
features = pixels_to_features(images[i])
data[i] = features_to_row(features)
end
start_column = include_label ? [:label] : []
columns = vcat(start_column, features_columns(images[1]))
result = DataFrame()
for c in columns
result[c] = []
end
for i in 1:length(data)
if include_label
push!(result, vcat(frame[i, :label], data[i]))
else
push!(result, vcat([], data[i]))
end
end
result
end

A few nice things to notice here about Julia itself are:

- The function documentation is written in Markdown
- We can nest functions inside other functions
- The language is statically and strongly typed
- Types can be inferred from the context
- It is often desirable to provide the concrete types to improve performance (but that an advanced Julia related topic)
- Arrays are indexed from 1
- There's the nice
`|>`

operator found e. g. In Elixir (which I absolutely love)

The above code converts the images to be arrays of

`Float64`

and converts the values to be within 0 and 1 (instead of 0..255 originally).

A thing to notice is that in Julia we can vectorize operations easily and we're using this fact to tersely convert our number:

images = frame[:, :pixels] ./ 255

We are referencing the

`pixels_to_features`

function which we define as:

"""
Returns ImageFeatures struct for the image pixels
given as an argument
"""
function pixels_to_features(image :: Array{Float64})
dim = Base.isqrt(length(image))
skeleton = compute_skeleton(image)
bounds = compute_bounds(skeleton)
resized = compute_resized(skeleton, bounds, (dim, dim))
left = compute_profile(resized, :left)
right = compute_profile(resized, :right)
width_min, width_max, width_at = compute_widths(left, right, image)
frequencies, simples = compute_transitions(skeleton)
ImageStats(dim, left, right, width_min, width_max, width_at, frequencies, simples)
end

This in turn uses the

`ImageStats`

structure:

immutable ImageStats
image_dim :: Int64
left :: ProfileStats
right :: ProfileStats
width_min :: Int64
width_max :: Int64
width_at :: Array{Int64}
direction_frequencies :: Array{Float64}
# The following adds information about transitions
# in 2 element simplified paths:
simple_direction_frequencies :: Array{Float64}
end
immutable ProfileStats
min :: Int64
max :: Int64
at :: Array{Int64}
diff :: Array{Int64}
end

The

`pixels_to_features`

function first gets the skeleton of the digit shape as an image and then uses other functions passing that skeleton to them. The function returning the skeleton utilizes the fact that in Julia it's trivially easy to use Python libraries. Here's its definition:

using PyCall
@pyimport skimage.morphology as cv
"""
Thin the number in the image by computing the skeleton
"""
function compute_skeleton(number_image :: Array{Float64}) :: Array{Float64}
convert(Array{Float64}, cv.skeletonize_3d(number_image))
end

It uses the

`scikit-image`

library's function

`skeletonize3d`

by using the

`@pyimport`

macro and using the function as if it was just a regular Julia code.

Next the code crops the digit itself from the 28x28 image and resizes it back to 28x28 so that the edges of the shape always "touch" the edges of the image. For this we need the function that returns the bounds of the shape so that it's easy to do the cropping:

function compute_bounds(number_image :: Array{Float64}) :: Bounds
rows = size(number_image, 1)
cols = size(number_image, 2)
saw_top = false
saw_bottom = false
top = 1
bottom = rows
left = cols
right = 1
for y = 1:rows
saw_left = false
row_sum = 0
for x = 1:cols
row_sum += number_image[y, x]
if !saw_top && number_image[y, x] > 0
saw_top = true
top = y
end
if !saw_left && number_image[y, x] > 0 && x < left
saw_left = true
left = x
end
if saw_top && !saw_bottom && x == cols && row_sum == 0
saw_bottom = true
bottom = y - 1
end
if number_image[y, x] > 0 && x > right
right = x
end
end
end
Bounds(top, right, bottom, left)
end

Resizing the image is pretty straight-forward:

using Images
function compute_resized(image :: Array{Float64}, bounds :: Bounds, dims :: Tuple{Int64, Int64}) :: Array{Float64}
cropped = image[bounds.left:bounds.right, bounds.top:bounds.bottom]
imresize(cropped, dims)
end

Next, we need to compute the profile stats as described in our plan of attack:

function compute_profile(image :: Array{Float64}, side :: Symbol) :: ProfileStats
@assert side == :left || side == :right
rows = size(image, 1)
cols = size(image, 2)
columns = side == :left ? collect(1:cols) : (collect(1:cols) |> reverse)
at = zeros(Int64, rows)
diff = zeros(Int64, rows)
min = rows
max = 0
min_val = cols
max_val = 0
for y = 1:rows
for x = columns
if image[y, x] > 0
at[y] = side == :left ? x : cols - x + 1
if at[y] < min_val
min_val = at[y]
min = y
end
if at[y] > max_val
max_val = at[y]
max = y
end
break
end
end
if y == 1
diff[y] = at[y]
else
diff[y] = at[y] - at[y - 1]
end
end
ProfileStats(min, max, at, diff)
end

The widths of shapes can be computed with the following:

function compute_widths(left :: ProfileStats, right :: ProfileStats, image :: Array{Float64}) :: Tuple{Int64, Int64, Array{Int64}}
image_width = size(image, 2)
min_width = image_width
max_width = 0
width_ats = length(left.at) |> zeros
for row in 1:length(left.at)
width_ats[row] = image_width - (left.at[row] - 1) - (right.at[row] - 1)
if width_ats[row] < min_width
min_width = width_ats[row]
end
if width_ats[row] > max_width
max_width = width_ats[row]
end
end
(min_width, max_width, width_ats)
end

And lastly, the transitions:

function compute_transitions(image :: Image) :: Tuple{Array{Float64}, Array{Float64}}
history = zeros((size(image,1), size(image,2)))
function next_point() :: Nullable{Point}
point = Nullable()
for row in 1:size(image, 1) |> reverse
for col in 1:size(image, 2) |> reverse
if image[row, col] > 0.0 && history[row, col] == 0.0
point = Nullable((row, col))
history[row, col] = 1.0
return point
end
end
end
end
function next_point(point :: Nullable{Point}) :: Tuple{Nullable{Point}, Int64}
result = Nullable()
trans = 0
function direction_to_moves(direction :: Int64) :: Tuple{Int64, Int64}
# for frequencies:
# 8 1 2
# 7 - 3
# 6 5 4
[
( -1, 0 ),
( -1, 1 ),
( 0, 1 ),
( 1, 1 ),
( 1, 0 ),
( 1, -1 ),
( 0, -1 ),
( -1, -1 ),
][direction]
end
function peek_point(direction :: Int64) :: Nullable{Point}
actual_current = get(point)
row_move, col_move = direction_to_moves(direction)
new_row = actual_current[1] + row_move
new_col = actual_current[2] + col_move
if new_row <= size(image, 1) && new_col <= size(image, 2) &&
new_row >= 1 && new_col >= 1
return Nullable((new_row, new_col))
else
return Nullable()
end
end
for direction in 1:8
peeked = peek_point(direction)
if !isnull(peeked)
actual = get(peeked)
if image[actual[1], actual[2]] > 0.0 && history[actual[1], actual[2]] == 0.0
result = peeked
history[actual[1], actual[2]] = 1
trans = direction
break
end
end
end
( result, trans )
end
function trans_to_simples(transition :: Int64) :: Array{Int64}
# for frequencies:
# 8 1 2
# 7 - 3
# 6 5 4
# for simples:
# - 1 -
# 4 - 2
# - 3 -
[
[ 1 ],
[ 1, 2 ],
[ 2 ],
[ 2, 3 ],
[ 3 ],
[ 3, 4 ],
[ 4 ],
[ 1, 4 ]
][transition]
end
transitions = zeros(8)
simples = zeros(16)
last_simples = [ ]
point = next_point()
num_transitions = .0
ind(r, c) = (c - 1)*4 + r
while !isnull(point)
point, trans = next_point(point)
if isnull(point)
point = next_point()
else
current_simples = trans_to_simples(trans)
transitions[trans] += 1
for simple in current_simples
for last_simple in last_simples
simples[ind(last_simple, simple)] +=1
end
end
last_simples = current_simples
num_transitions += 1.0
end
end
(transitions ./ num_transitions, simples ./ num_transitions)
end

All those gathered features can be turned into rows with:

function features_to_row(features :: ImageStats)
lefts = [ features.left.min, features.left.max ]
rights = [ features.right.min, features.right.max ]
left_ats = [ features.left.at[i] for i in 1:features.image_dim ]
left_diffs = [ features.left.diff[i] for i in 1:features.image_dim ]
right_ats = [ features.right.at[i] for i in 1:features.image_dim ]
right_diffs = [ features.right.diff[i] for i in 1:features.image_dim ]
frequencies = features.direction_frequencies
simples = features.simple_direction_frequencies
vcat(lefts, left_ats, left_diffs, rights, right_ats, right_diffs, frequencies, simples)
end

Similarly we can construct the column names with:

function features_columns(image :: Array{Float64})
image_dim = Base.isqrt(length(image))
lefts = [ :left_min, :left_max ]
rights = [ :right_min, :right_max ]
left_ats = [ Symbol("left_at_", i) for i in 1:image_dim ]
left_diffs = [ Symbol("left_diff_", i) for i in 1:image_dim ]
right_ats = [ Symbol("right_at_", i) for i in 1:image_dim ]
right_diffs = [ Symbol("right_diff_", i) for i in 1:image_dim ]
frequencies = [ Symbol("direction_freq_", i) for i in 1:8 ]
simples = [ Symbol("simple_trans_", i) for i in 1:4^2 ]
vcat(lefts, left_ats, left_diffs, rights, right_ats, right_diffs, frequencies, simples)
end

The data frame constructed with the

`get_data`

function can be easily dumped into the CSV file with the

`writeable`

function from the

`DataFrames`

package.

You can notice that gathering / extracting features is a

**lot** of work. All this was needed to be done because in this article we're focusing on the somewhat "classical" way of doing machine learning. You might have heard about algorithms existing that mimic how the human brain learns. We're

**not** focusing on them here. This we will explore in some future article.

We use the mentioned

`writetable`

on data frames computed for both training and test datasets to store two files:

`processed_train.csv`

and

`processed_test.csv`

.

##
Choosing the model

For the task of classifying I decided to use the

`XGBoost`

library which is somewhat a hot new technology in the world of machine learning. It's an improvement over the so-called

`Random Forest`

algorithm. The reader can read more about XGBoost on its website:

http://xgboost.readthedocs.io/.

Both random forest and xgboost revolve around the idea called

*ensemble learning*. In this approach we're not getting just one learning model — the algorithm actually creates many variations of models and uses them to collectively come up with better results. This is as much as can be written as a short description as this article is already quite lengthy.

##
Training the model

The training and classification code in R is very simple. We first need to load the libraries that will allow us to load data as well as to build the classification model:

library(xgboost)
library(readr)

Loading the data into data frames is equally straight-forward:

processed_train <- read_csv("processed_train.csv")
processed_test <- read_csv("processed_test.csv")

We then move on to preparing the vector of labels for each row as well as the matrix of features:

labels = processed_train$label
features = processed_train[, 2:141]
features = scale(features)
features = as.matrix(features)

###
The train-test split

When working with models, one of the ways of evaluating their performance is to split the data into so-called train and test sets. We train the model on one set and then we predict the values from the test set. We then calculate the accuracy of predicted values as the ratio between the number of correct predictions and the number of all observations.

Because Kaggle provides the test set without labels, for the sake of evaluating the model's performance without the need to submit the results, we'll split our Kaggle-training set into local train and test ones. We'll use the amazing

`caret`

library which provides a wealth of tools for doing machine learning:

library(caret)
index <- createDataPartition(processed_train$label, p = .8,
list = FALSE,
times = 1)
train_labels <- labels[index]
train_features <- features[index,]
test_labels <- labels[-index]
test_features <- features[-index,]

The above code splits the set uniformly based on the labels so that the train set is approximately 80% in size of the whole data set.

###
Using XGBoost as the classification model

We can now make our data digestible by the XGBoost library:

train <- xgb.DMatrix(as.matrix(train_features), label = train_labels)
test <- xgb.DMatrix(as.matrix(test_features), label = test_labels)

The next step is to make the XGBoost learn from our data. The actual parameters and their explanations are beyond the scope of this overview article, but the reader can look them up on the XGBoost pages:

model <- xgboost(train,
max_depth = 16,
nrounds = 600,
eta = 0.2,
objective = "multi:softmax",
num_class = 10)

It's critically important to pass the

`objective`

as

`"multi:softmax"`

and

`num_class`

as

`10`

.

##
Simple performance evaluation with confusion matrix

After waiting a while (couple of minutes) for the last batch of code to finish computing, we now have the classification model ready to be used. Let's use it to predict the labels from our test set:

predicted = predict(model, test)

This returns the vector of predicted values. We'd now like to check how well our model predicts the values. One of the easiest ways is to use the so-called

**confusion matrix**.

As per Wikipedia, confusion matrix is simply:

(...) also known as an error matrix, is a specific table layout that allows visualization of the performance of an algorithm, typically a supervised learning one (in unsupervised learning it is usually called a matching matrix). Each column of the matrix represents the instances in a predicted class while each row represents the instances in an actual class (or vice versa). The name stems from the fact that it makes it easy to see if the system is confusing two classes (i.e. commonly mislabelling one as another).

The

`caret`

library provides a very easy to use function for examining the confusion matrix and statistics derived from it:

confusionMatrix(data=predicted, reference=labels)

The function returns an R list that gets pretty printed to the R console. In our case it looks like the following:

Confusion Matrix and Statistics
Reference
Prediction 0 1 2 3 4 5 6 7 8 9
0 819 0 3 3 1 1 2 1 10 5
1 0 923 0 4 5 1 5 3 4 5
2 4 2 766 26 2 6 8 12 5 0
3 2 0 15 799 0 22 2 8 0 8
4 5 2 1 0 761 1 0 15 4 19
5 1 3 0 13 2 719 3 0 9 6
6 5 3 4 1 6 5 790 0 16 2
7 1 7 12 9 2 3 1 813 4 16
8 6 2 4 7 8 11 8 5 767 10
9 5 2 1 13 22 6 1 14 14 746
Overall Statistics
Accuracy : 0.9411
95% CI : (0.9358, 0.946)
No Information Rate : 0.1124
P-Value [Acc > NIR] : < 2.2e-16
Kappa : 0.9345
Mcnemar's Test P-Value : NA
(...)

Each column in the matrix represents actual labels while rows represent what our algorithms predicted this value to be. There's also the accuracy rate printed for us and in this case it equals

`0.9411`

. This means that our code was able to predict correct values of handwritten digits for 94.11% of observations.

##
Submitting the results

We got 0.9411 of an accuracy rate for our local test set and it turned out to be very close to the one we got against the test set coming from Kaggle. After predicting the competition values and submitting them, the accuracy rate computed by Kaggle was 0.94357. That's quite okay given the fact that we're not using here any of the new and fancy techniques.

Also, we haven't done any

*parameter tuning* which could surely improve the overall accuracy. We could also revisit the code from the features extraction phase. One improvement I can think of would be to first crop and resize back - and only then compute the skeleton which might preserve more information about the shape. We could also use the confusion matrix and taking the number that was being confused the most, look at the real images that we failed to recognize. This could lead us to conclusions about improvements to our feature extraction code. There's always a way to extract more information.

Nowadays, Kagglers from around the world were successfully using advanced techniques like

*Convolutional Neural Networks* getting accuracy scores close to 0.999. Those live in somewhat different branch of the machine learning world though. Using this type of neural networks we don't need to do the feature extraction on our own. The algorithm includes the step that automatically gathers features that it later on feeds into the network itself. We will take a look at them in some of the future articles.

##
See also