Deep Learning Nonlinear Regression

In this article we put to work a perceptron to predict a high difficulty level nonlinear regression problem. The data has been generated using an exponential function with this shape:

Eckerle4_exponential_function_graph

The graph above corresponds to the values of the dataset that can be downloaded from the Statistical Reference Dataset of the Information Technology Laboratory of the United States on this link: http://www.itl.nist.gov/div898/strd/nls/data/eckerle4.shtml

Neural networks are especially appropriate to learn patterns and remember shapes. Perceptrons are very basic but yet very powerful neural networks types. Their structure is basically an array of weighted values that is recalculated and balanced iteratively. They can implement activation layers or functions to modify the output within a certain range or list of values.

In order to create the neural network we are going to use Keras, one of the most popular Python libraries. The code is as follows:

The first thing to do is to import the elements that we will use. We will not use aliases for the purpose of clarity:

# Numeric Python Library.
import numpy
# Python Data Analysis Library.
import pandas
# Scikit-learn Machine Learning Python Library modules.
#   Preprocessing utilities.
from sklearn import preprocessing
#   Cross-validation utilities.
from sklearn import cross_validation
# Python graphical library
from matplotlib import pyplot
 
# Keras perceptron neuron layer implementation.
from keras.layers import Dense
# Keras Dropout layer implementation.
from keras.layers import Dropout
# Keras Activation Function layer implementation.
from keras.layers import Activation
# Keras Model object.
from keras.models import Sequential

In the previous code we have imported the numpy and pandas libraries to manage the data structures and perform operations with matrices. The two scikit-learn modules will be used to scale the data and to prepare the test and train data sets.

The matplotlib package will be used to render the graphs.

From Keras, the Sequential model is loaded, it is the structure the Artificial Neural Network model will be built upon. Three types of layers will be used:

  1. Dense: Those are the basic layers made with weighted neurons that form the perceptron. An entire perceptron could be built with these type of layers.
  2. Activation: Activation functions transform the output data from other layers.
  3. Dropout: This is a special type of layer used to avoid over-fitting by leaving out of the learning process a number of neuron.

First we load the dataset already formatted as csv.

# Peraring dataset
# Imports csv into pandas DataFrame object.
Eckerle4_df = pandas.read_csv("Eckerle4.csv", header=0)
 
# Converts dataframes into numpy objects.
Eckerle4_dataset = Eckerle4_df.values.astype("float32")
# Slicing all rows, second column...
X = Eckerle4_dataset[:,1]
# Slicing all rows, first column...
y = Eckerle4_dataset[:,0]
 
# Data Scaling from 0 to 1, X and y originally have very different scales.
X_scaler = preprocessing.MinMaxScaler(feature_range=(0, 1))
y_scaler = preprocessing.MinMaxScaler(feature_range=(0, 1))
X_scaled = ( X_scaler.fit_transform(X.reshape(-1,1)))
y_scaled = (y_scaler.fit_transform(y.reshape(-1,1)))
 
# Preparing test and train data: 60% training, 40% testing.
X_train, X_test, y_train, y_test = cross_validation.train_test_split( \
    X_scaled, y_scaled, test_size=0.40, random_state=3)

The predictor variable is saved in variable X and the dependent variable in y. The two variables have values that differ several orders of magnitude; and the neural networks work better with values next to zero. For those two reasons the variables are scaled to remove their original magnitudes and put them within the same magnitude. Their values are proportionally transformed within 0 and 1.

The data is divided into two sets. One will be used to train the neural network, using 60% of all the samples; and the other will contain 40% of the data, that will be used to test if the model works well with out-of-the-sample data.

Now we are going to define the neural network. It will consist in an input layer to receive the data, several intermediate layers, to process the weights, and a final output layer to return the prediction (regression) results.

The objective is that the network learns from the train data and finally can reproduce the original function with only 60% of the data. It could be less, it could be more; I have chosen 60% randomly. In order to verify that the network has learnt the function, we will ask it to predict which response should return the test data that was not used to create the model.

Now let’s think about the neural network topology. If we study the chart, there are three areas that differ considerably. Those are the left tail, up to the 440 mark, a peak between the 440 and 465 marks approximately, and the second tail on the right, from the 465 mark on. For this reason we will use three neuron intermediate layers, so that the first one learns any of these areas, the second one other area, and the third one the final residuals that should correspond to the third area. We will have therefore 3 layers in our network plus one input and one output layer too. The basic layer structure of the neural network should be similar to this, a sequence of layers, from left to right with this topology:

INPUT LAYER(2) > [HIDDEN(i)] > [HIDDEN(j)] > [HIDDEN(k)] > OUTPUT(1)

An input layer that accepts two values X and y, a first intermediate layer that has i neurons, a second hidden layer that has j neurons, an intermediate layer that has k neurons, and finally, an output layer that returns the regression result for each sample X, y.

# New sequential network structure.
model = Sequential()
 
# Input layer with dimension 1 and hidden layer i with 128 neurons. 
model.add(Dense(128, input_dim=1, activation='relu'))
# Dropout of 20% of the neurons and activation layer.
model.add(Dropout(.2))
model.add(Activation("linear"))
# Hidden layer j with 64 neurons plus activation layer.
model.add(Dense(64, activation='relu'))
model.add(Activation("linear"))
# Hidden layer k with 64 neurons.
model.add(Dense(64, activation='relu'))
# Output Layer.
model.add(Dense(1))
 
# Model is derived and compiled using mean square error as loss
# function, accuracy as metric and gradient descent optimizer.
model.compile(loss='mse', optimizer='adam', metrics=["accuracy"])
 
# Training model with train data. Fixed random seed:
numpy.random.seed(3)
model.fit(X_train, y_train, nb_epoch=256, batch_size=2, verbose=2)

Now the model is trained by iterating 256 times through all the train data, taking each time two sampless.

In order to graphically see the accuracy of the model, now we apply the regression model to new data that has not been used to create the model. We will also plot the predicted values versus the actual values.

~ Predict the response variable with new data
predicted = model.predict(X_test)
 
# Plot in blue color the predicted adata and in green color the
# actual data to verify visually the accuracy of the model.
pyplot.plot(y_scaler.inverse_transform(predicted), color="blue")
pyplot.plot(y_scaler.inverse_transform(y_test), color="green")
pyplot.show()

And the produced graph shows that the network has adopted the same shape as the function:

Eckerle4_perceptron_predicted_vs_actual

This demonstrates the exceptional power of neural networks to solve complex statistical problems, especially those in which causality is not crucial such as image processing or speech recognition.

Exclusive and Independent Probability Events

There are two commonly used terms in Probability and generically in Statistics:

  • Exclusive or disjoint events.
  • Independent events.

A great number of theorems and applications within the Statistics field depend on whether the studied events are either mutually exclusive or not, and if they are either mutually independent or not as well.

Disjoint or mutually exclusive events

Two events are disjoint if they cannot occur at the same time. For instance, the age ranges probabilities for a customers are disjoint. It cannot occur simultaneously that a particular customer is more than twenty and less than twenty year old.

Other example is the status of an order. It may be in preparation, at the magazine, en route or delivered to the consignee; being those states mutually exclusive as well.

On the other hand non-disjoint events may coexist at the same point in time. A customer may live in a particular town and be at concurrently more than twenty year old. Those two conditions are not mutually exclusive. Those type of events are not disjoint or mutually exclusive. In the same way, an order may either be in preparation and being assembled or in preparation and ready for delivery all together.

Depending if two or more events are or not disjoint, the way to calculate their probabilities is different. And the outcome of the probabilistic calculus will vary therefore based on it.

Dependent events

Two events are independent when the outcome of one does not depend on the other. In terms of probability, two events are independent when the probability of one of them is not affected by the probability of the other event.

This is the case of the games of chance like lotteries and casinos. Every time the die is rolled the chances to obtain a particular outcome do not change; at each roll the probability of obtaining any of the six possible values for a six-sided die is equal to \(\)\(\frac{1}{6}\).

Conversely, dependent events are affected by their respective probabilities. In this case we talk about conditional probability and that probability is expressed using the nomenclature \(P(A|B)\). An example may be the probability of selling an on-line product when the user has already opened an account on the site and returns. It is different if the second event (account opened previously) occurs or not.

Python for Digital Signal Processing

Before starting with this post, you may want to learn the basics of Python. If you’re an experienced programmer and head Python for the first time, you will likely find it very easy to understand. One important thing about Python: Python requires perfect indentation (4 spaces) to validate the code. So if you get an error and you code seems perfect, review if you have indented correctly each line. Python has also a particular way to deal with arrays, more close the the R programming language than to C-like style.

Python’s core functionality is extended with thousands of available free libraries, many of them are incredibly handy. Even if Python is not a compiled language, many of its libraries are written in C, being Python a wrapper for them.

The libraries used on this article are:

  • scipy – Scientific library for Python.
  • numpy – Numeric library for Python.

To load a wav file in Python:

# Loads the scipy.io package for later usage of io.wavefile module.
from scipy import io
#
# Location of the wav file in the file system.
fileName = '/Downloads/Music_Genres/genres/country/country.00053.wav'
#
# Loads sample rate (bps) and signal data (wav). Notice that Python
# functions can return multiple variables at the same time.
sample_rate, data = io.wavfile.read(fileName)
#
# Print in sdout the sample rate, number of items. 
#and duration in seconds of the wav file. 
print "Sample rate:{0}, data size:{1}, duration:{2} seconds" \
    .format(sample_rate,data.shape,len(data)/sample_rate)

The output generated should seem like:

Sample rate:22050, data size:(661794L,), duration:30 seconds

The output shows that the wav file contains in all 661,794 samples (the data variable is an array with 661,794 elements). The sample rate is 22,050 samples per second. Dividing 661,794 elements by 22,050 samples per second, we obtain 30 seconds, the length in seconds of the wav file.

The Fourier Transform

The Fourier transform is the method that we will use to extract the prevalent frequencies from the wav file. Each frequency corresponds to a musical tone; knowing the frequencies from a particular time interval we are able to know which are the most frequent tones within that interval, being possible to infer the key and chords played during that time lapse.

This article is not going to enter into the details of the Fourier transform, only on how to use it to extract information regarding the frequency power from the wav signal analyzed. The video below is an intuitive introduction to the Fourier transform in case the reader is interested on it. It also includes examples of how to implement it algorithmically. It is quite advisable to watch it once now and then come back again to review it after the training in Fourier transform is completed.

Basically, given a signal, a wav file on this post, which is composed by a number n of samples \(x[n]\). We can get the frequency power within the signal with the FFT (Fast Fourier Transform) function. The FFT function is an improvement that optimizes the Fourier transform.

The FFT function receives two arguments, the signar \(x\) and the number of items to retrieve \(k, k\leq n\). The commonly choosen k value is \(\frac{n}{2}\) because the FFT result, \(fft[k]\) is usually symmetric around that length. This means that in order to calculate the FFT, only a half of the total signal length is required to retrieve the different frequencies occurrence. So, in plain words, if the original signal file has 100 samples, only 50 samples are needed to process the complete FFT transform.

In Python language there are two useful functions to calculate and get the Fourier transform from a sample array, like the one where the data variable from the wav file is stored:

  • fftfreq – Returns the frequency corresponding to each \(x_i\) sample from the signal data sample file \(x[n]\) corresponding to the power of the fourier transform. This is the frequency to which each fft element corresponds to.
  • fft – Returns the fourier transform data from the sample file. The position of the elements returned correspond to the position of the fftfreq, so that using both arrays the fft power elements correspond by position to the fftfreq frequencies.

For instance, if the fourier transform function returns fft = {0,0.5,1} and \(\)fftfreq = {100,200,300}\(\), it means that the signal has a power of 0 for frequency 100Hz, a power of 0.5 for 200Hz and a power of 1 within 300Hz; being 300Hz the frequency most frequent.

The following code would extract from a wav file the first 10 second, apply the fourier transform  and the frequencies associated to each item within  the spectral data.

import scipy.io
 
# Package that implements the fast
# fourier transform functions.
from scipy import fftpack
import numpy as np
 
# Loads wav file as array.
fileName = './country.00053.wav'
sample_rate, data = io.wavfile.read(fileName)
 
# Extracting 10 seconds. N is the numbers of samples to
# extract or elements from the array.
seconds_to_extract = 10
N = seconds_to_extract * sample_rate
 
# Knowing N and sample rate, fftfreq gets the frequency
# Associated to each FFT unit.
f = fftpack.fftfreq(N, 1.0/sample_rate)
 
# Extracts fourier transform data from the sample
# returning the spectrum analysis
subdata = data[:N]
F = fftpack.fft(subdata)

F contains the power and f the frequency each item within F is related to. The higher the power, the higher the frequency prevalence across the signal. Filtering the frequencies using the f matrix and extracting the power we could get a graph like the next one:fourier_transform_python

On the y-axis, \(|F|\) is the absolute value of each unit from F and the values of f are the Frequency (Hz) on the x-axis. The green and orange lines can be ignored. To get the subset of frequencies [200-900] displayed on the chart, the next code was used:

# Interval limits
Lower_freq = 200
Upper_freq = 900
 
# f (frequencies) between lower frequency AND
# f (frequencies) upper frequencies.
filter_subset = (f >= Lower_freq) * (f <= Upper_freq)
 
# Extracts filtered items from the frequency list.
f_subset = f[filter_subset]
 
# Extracts filtered items from the Fourier transform power list.
F_subset = F[filter_subset]

Spectral Analysis and Harmony

Chromatic scale tone frequencies

On the previous post, Spectral Analysis and Harmony, it is shown an elementary introduction to harmony and digital signal. We are now going to study the range of tones between A3 an A5. Our central axis is A tone (or A4) which frequency is equal to 440Hz.

The next table shows all the tones and frequencies within the chromatic scale belonging to the range between A3 and A5. The piano key number corresponding to each tone is also displayed.

 Scientific
name
Key
number
Helmholtz
name
Frequency
(Hz)
A3 37 a 220.000
A♯3/B♭3 38 a♯/b♭ 233.082
B3 39 b 246.942
C4 Middle C 40 c′ 1-line octave 261.626
C♯4/D♭4 41 c♯′/d♭′ 277.183
D4 42 d′ 293.665
D♯4/E♭4 43 d♯′/e♭′ 311.127
E4 44 e′ 329.628
F4 45 f′ 349.228
F♯4/G♭4 46 f♯′/g♭′ 369.994
G4 47 g′ 391.995
G♯4/A♭4 48 g♯′/a♭′ 415.305
A4 – A440 49 a′ 440.000
A♯4/B♭4 50 a♯′/b♭′ 466.164
B4 51 b′ 493.883
C5 Tenor C 52 c′′ 2-line octave 523.251
C♯5/D♭5 53 c♯′′/d♭′′ 554.365
D5 54 d′′ 587.330
D♯5/E♭5 55 d♯′′/e♭′′ 622.254
E5 56 e′′ 659.255
F5 57 f′′ 698.456
F♯5/G♭5 58 f♯′′/g♭′′ 739.989
G5 59 g′′ 783.991
G♯5/A♭5 60 g♯′′/a♭′′ 830.609
A5 61 a′′ 880.000

The difference or leap between two tones is called interval. One interesting feature of the chromatic scale is that it is composed by constant intervals. For instance, tone A3 is equal to 220Hz, tone A4 to 440Hz and tone A5 to 880Hz. Each tone frequency is double its analogue tone from the precedent respective octave.

The important idea is that we can analyze tones as numbers and operate with basic arithmetics with them with their frequencies. Who said emotions cannot be explained by Science? Do not be intimidated if you don’t know neither music theory nor Optical Physics; These texts will led you by the hand on a trip at which end you will know how to extract the waves, tones and emotions from digital music even without knowing none of those.

Frequency analysis in a nutshell

In order to analyze the frequencies that compose a piece of music, we take a part from it and extract a subset of frequencies. Like using an equalizer we filter the sound between two specific frequencies or tones. For instance, we could read the first ten seconds of a music mp3 file and generate a table displaying how many times tone A appears within that sequence. Going farther we could analyze how many tones appear and how many times each tone is played within those 10 first seconds.

As seen on the Emotions Within Digital Signals article, those tones can be used to define the chords and keys a piece of music is formed by.

In order to extract the signal frequency occurrences, we can use a frequency spectrum graph. This graph displays how many times a frequency appears on a signal and its power or prevalence other the rest. In this case, the signal is the first 10 seconds of music. Let’s see an example:

frequency_spectrum_graph
Signal and Frequency Spectrum Graphs

From the graph on the right we can see that the most used frequencies, those having higher \(|F|\), are one next to the 200Hz, another between the 300Hz and 400Hz and a third one between the 400Hz and the 500Hz. The x-axis shows the frequency spectrum (or range) we are analyzing, and the y-axis the power of the signal. The higher the line at a certain point on the x-axis, the more the power that signal has over that frequency.

To get an insight of the most used tones, the frequencies that have more power can be extracted, and in this case the dominant frequencies within the signal are in particular 220Hz, 246.942Hz, 329.628Hz and 440Hz. Rounding those frequencies to the nearest integer and comparing them to the ones in the table above, we can extract some of the main tones within the first ten seconds of the song.

Scientific
name
Frequency
(Hz)
A3 220.000
B3 247
E4 330
A4 A440 440

From the data above it can be determined that the dominant key within the first seconds is composed by tones A, B and  E. That key corresponds to chord A2Sus (A 2nd suspended). A chord is how it’s called the sound composed by multiple tones, multiple frequencies. The names of the different chords are not described in this article, since there are many of them.

In terms of music harmony A2Sus, or generically speaking 2nd suspended chords are tones that create a sensation of waiting for something to be resolved. The listener is holding on until the song resolves in something. We could say that the first ten seconds of this song are causing an emotion of expectation.

For more information on music and emotions, search in Google “emotions chords harmony”. For a good introduction to the matter I would recommend the paper Music and Emotions.

This article and the previous one, Emotions Within Digital Signals, set the basis to successfully tackle the problem of extracting emotions from music sequences. I will explain how to perform that task using Python language in the post Python for Digital Signal Processing.

Emotions Within Digital Signals

Music and artistic expression are conceived to provoke emotions to people. Music and visual arts travel in waves through the air across distances, from the transmitter to the receiver. Music is maybe the most influencing form of art, capable of producing deep emotional effects, evoking feelings and awakening memories when one is exposed to it.

The sound perceived is nothing else that the effect from the vibration of the eardrum hit by the sound waves traveling through the air. Like a pendulum, a fast one, the eardrum oscillates and that oscillation is felt as an emotion by our brain.

The fundamental unit in music is the tone. When one sings a song that one is reproducing a sequence of tones in a certain order to produce a melody. In music secondary tones usually follow the lead tone or principal melody. When multiple tones sound at the same time we may call it a chord. Chords define the temper of the music, and are in large part responsible for the emotions that individuals will appreciate when hearing the music.

The tone is the basic musical unit. Western music uses twelve typical tones (C, C#, D, D#, E, F, F#, G, G#, A, A#, B). That range of tones is called octave, and that tone structure is also commonly called chromatic scale.

Each chord is composed exclusively and always by two or more of those tones in western music. For instance, the C Major Chord I is formed by tones C, E and G played at the same time.

Same way we call chord to the sound of multiple tones at once, we call key to the group of tones the music evolves through. Key is similar to chord, and the basic difference is that keys are tones across time within the same space or plane, and chords are tones on the same instant but across different planes. To summarize we can assume that chords are multiple simultaneous tones and keys are multiple tones belonging always to the same space of tones.

For instance C Major chord would consist on tones C, E and G played at the same time for two seconds. C Major key could consist instead on C tone played on second 1, E tone played on second 2, and finally G tone played alone on second 3.

Remember that we said that music are just waves, in fact tones are waves too, and each tone has an unique corresponding wave. If we examine the most common waves within each part of a musical piece, we can find out which notes are defining that music within each time interval. We can therefore extract the tones, chords and the key of that music just by analyzing the frequency of the waves it is composed of.

Frequency is the time a wave completes a cycle. It is measured in hertzs. One hertz is equal to one cycle per second. Each tone has a fixed frequency that never changes. For instance, tone A corresponds to a frequency of 440Hz. Instruments are usually tempered using that tone A as a basis, meaning that all instruments that we can hear and produce notes will produce the same frequencies for the same tones.

In the next post, Spectral Analysis and Harmony, we will see how can we take advantage of wave analysis (Digital Signal Processing) and Music theory (Harmony) to programmatically identify feelings from music files.