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https://github.com/btrotta/m4

Entry for the Makridakis 4 forecasting competition, using convolutional neural networks
https://github.com/btrotta/m4

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Entry for the Makridakis 4 forecasting competition, using convolutional neural networks

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README 

        
# Convolutional neural networks for time series forecasting



This is the code for my entry in the Makridakis 4 time series forecasting competition (https://www.m4.unic.ac.cy/).

It was the best pure machine learning entry, and placed 23rd overall. The competition results are

published in a paper of the International Journal of Forecasting

(https://www.sciencedirect.com/science/article/pii/S0169207018300785); see Table 1 for the rankings.



The approach is to use all the given sequences in each frequency type (yearly, weekly, etc)

to train our neural networks. This avoids the usual problem with using machine learning for

time series forecasting, i.e., that there isn't enough data in each time series to train a model.

Also, using a machine learning approach rather than traditional statistical models means that

we don't need to explicitly model the mathematical behaviour of each series (linear,

exponential, etc). Instead, the machine learning algorithm can implicitly learn the different

behaviours and how best to forecast them, by looking at thousands of different examples.



# Instructions for running the code



The code is written in Python 3.6.1, and uses the Keras package for building the neural

network models, with the Tensorflow backend. We use the pandas and numpy packages for

reading, writing, and preparing the data. A full specification of the environment is given in

environment.txt.

Copy all the python files into a directory. Make a subdirectory called data, and copy the csv

data files into it.



# Training



Run shuffle.py; this randomises the order of the data in each file and saves a copy with

"shuffle" appended to the name. This is necessary for doing proper validation, since in the

original files the series are ordered by type: finance, macro, etc.

To train all the models, run train_models.py. This script will run the training code twice. In the

first run, the first 20% of the rows in each file are excluded from training and used to

measure the model's performance. A log file is written containing the calculated metrics on

this validation set. The second run trains the models using all the data and saves the models

in a subfolder called trained_models. The total runtime of the script is around 5 hours on a

fairly powerful Windows laptop (i7 processor, 24Gb RAM).



# Prediction



Run predict.py. This will load the previously saved trained models (that you have either

downloaded or generated in the previous step) and make predictions for all series. The

result is saved is a csv file called Submission_fc_YYMMDD_HHMM.csv. The lower and

upper bounds are saved in Submission_lower_YYMMDD_HHMM.csv and

Submission_upper_YYMMDD_HHMM.csv. This takes around 10 minutes.



# Model structure



We use separate models (with different architectures) for each frequency (yearly, monthly,

etc). For all frequencies except yearly, we expect that there will be cyclic effects (for

example, with the hourly series we expect that there may be daily and weekly patterns). To

capture these effects we use convolutional neural networks. This means we pass a filter

having window size equal to the frequency over the inputs. This filter is essentially a set of

linear functions (in our case 3), whose coefficients are found during the training. Since the

same filter, with the same coefficients, is applied to each cycle (e.g. each 12-month period

for the monthly models), we can capture the seasonal patterns.



We train separate models for each period of the forecast horizon; for example, when

modelling the quarterly series, we train 8 versions of each neural network, using the same x

inputs, but different y values (leaving gaps of 0 to 7 periods between the end of the x data

and the y value to be forecast).



The loss function for the neural networks is mean squared error.



# Training data



Within each frequency type, there are many different series lengths. For example, the

monthly series range in length from 42 to 2794, with 95% lying between 68 and 450. We

expect that in general, the more historical data the model can use, the better the accuracy.

The neural networks architecture we use requires fixed-length inputs, so we train multiple

models for each frequency type. In the monthly case, we train three models using training

periods of 48, 120, and 240, in each case using the most recent data available, i.e. the

periods at the end of the series. (Since the forecast horizon in this case is 18, as explained

above, we actually train 18 versions of each of these models, so we require an additional 18

periods at the end of the training x data, i.e., these models require 66, 138 and 258 periods

of data.) For the longer series, this means that we actually have multiple models to choose

from. For example, for a series of length 200 we could use the model with 48 inputs or the

model with 120 inputs.



We choose the training lengths to be multiples of all the frequencies which we expect the

series to exhibit (e.g. for the hourly series, our training period is a multiple of 24 and 7). This

makes the calculation of the convolutional filters straightforward, since the filter length will

divide the input length, meaning that we repeat the filter an integer number of times over the

input.



Note that although the models with longer inputs have more features per training sample, on

the other hand there are fewer training samples, so these models do not necessarily

outperform the ones with shorter inputs. We obtained good results by blending (with equal

weights) all the available models for a series.

For the small number of series with less than the minimum amount, we add some synthetic

data at the beginning by filling the data backwards from the start of the series. To preserve

the frequency patterns, we copy from the first available period that is at the same point in the

cycle.



Since the training set consists of unrelated series with different magnitudes, we standardise

each series (i.e. each row of the training set) by subtracting the mean and dividing by the

standard deviation (where the mean and standard deviation are calculated over the x values

for each series). Also, we remove outliers from the y values by restricting each series' y

value to be at most +/-5 times the standard deviation of the series' x values.

For the series with fewer training samples (i.e. those with weekly or high frequency), we

augment the training data by shifting the training data by 1, 2, ..., n-1 periods where n is the

frequency as follows:



Yearly (no augmentation); Quarterly (no augmentation); Monthly (no augmentation); Weekly

(52); Daily (7; although there are also yearly patterns, adding them would result in an

unfeasibly large training set); Hourly (7 x 24 = 168).



# Details of model structure



All neural networks use a learning rate of 0.001, batch size of 1000, and 250 epochs. The

loss function is mean squared error and the model is trained using the Adam optmizer

implementation in Keras. Early stopping is enabled using a validation set consisting of 10%

of the training data. If the validation loss fails to improve for 10 epochs, training is stopped.

This decreases the training time and helps control overfitting.



The inner layers of each neural network use the relu (rectified linear) activation.



## Yearly



Training periods: [10, 20, 30]  

NN structure:  

* 2 dense hidden layers of 50 relu units feeding into linear output cell.



## Quarterly



Training periods: [20, 48, 100]  

NN structure:  

* Yearly averages feeding into a 2 successive densely connected hidden layers of 50

relu units then a single linear output cell representing the yearly average forecast.

* Quarterly differences from the yearly average are run through a trainable convolution

filter of length 4 with 3 channels.

* Convolution outputs are fed into 2 successive densely connected hidden layers of 50

relu units then a single linear output cell representing the forecast difference from the

yearly output.

* Yearly forecast and quarterly difference forecast are summed to produce the final

output.



## Monthly



Training periods: [48, 120 (= 12 * 10), 240 (= 12 * 20)]  

NN structure:  

* Yearly averages feeding into 2 successive densely connected hidden layers of 20

relu units then a single linear output cell representing the yearly average forecast.

* Monthly differences from the yearly average are run through a trainable convolution

filter of length 12 with 3 channels.

* Convolution outputs are fed into 2 successive densely connected hidden layers of 20

relu units then a single linear output cell representing the forecast difference from the

yearly output.

* Yearly forecast and monthly difference forecast are summed to produce the final

output.



## Weekly



Training periods: [52, 520 (= 52 * 10), 1040 (= 52 * 20)]  

NN structure for training length 52:  

* Weekly inputs fed into 2 successive densely connected hidden relu layers of 20 relu

units, then a single linear output cell.



NN structure for training lengths > 52:  

* Yearly averages feeding into 2 successive densely connected hidden relu layers of

20 relu units then a single linear output cell representing the yearly average forecast.

* Weekly differences from the yearly average are run through a trainable convolution

filter of length 52 with 3 channels. Convolution outputs are fed into 2 successive

densely connected hidden relu layers of 20 relu units then a single linear output cell

representing the forecast difference from the yearly output.

* Yearly forecast and monthly difference forecast are summed to produce the final

output.



## Daily



Training periods: [98 (= 14 * 7)]  

NN structure:  

* Weekly averages feeding into 2 successive densely connected hidden relu layers of

20 relu units then a single linear output cell representing the yearly average forecast.

* Daily differences from the weekly average are run through a trainable convolution

filter of length 7 with 3 channels. Convolution outputs are fed into 2 successive

densely connected hidden relu layers of 20 relu units then a single linear output cell

representing the forecast difference from the weekly output.

* Yearly forecast and weekly difference forecast are summed to produce the final

output.

I experimented with adding longer training lengths, and adding a yearly convolution for

these, but these models produced poor results during testing, which seemed to be mostly

caused by some problematic series having very large step changes.



## Hourly



Training periods: [672 (= 96 * 7 * 24)]  

NN structure:  

* Weekly averages feeding into 2 successive densely connected hidden relu layers of

20 relu units then a single linear output cell representing the weekly average

forecast.

* Daily differences from the weekly average are run through a trainable convolution

filter of length 7 with 3 channels. Convolution outputs are fed into 2 successive

densely connected hidden relu layers of 20 relu units then a single linear output cell

representing the forecast difference from the weekly output.

* Hourly differences from the daily average are run through a trainable convolution filter

of length 24 with 3 channels. Convolution outputs are fed into 2 successive densely

connected hidden relu layers of 20 relu units then a single linear output cell

representing the forecast difference from the daily output.

* Weekly forecast and daily difference and hourly forecast are summed to produce the

final output.



# Prediction intervals



To calculate the prediction interval, we consider all series of a given frequency type (yearly,

monthly, etc) together. We calculate the error over the the last h periods of the training data,

where h is the forecast horizon, and normalise it by dividing by the prediction over this

period. For each period of the forecast horizon, we calculate the 0.025th and 0.975th

quantiles, and use these to calculate the forecast intervals.