Data transforms are intended to remove noise and improve the signal in time series forecasting.
It can be very difficult to select a good, or even best, transform for a given prediction problem.
There are many transforms to choose from and each has a different mathematical intuition. In this tutorial, you will discover how to explore different power-based transforms for time series forecasting with Python. After completing this tutorial, you will know:
- How to identify when to use and how to explore a square root transform.
- How to identify when to use and explore a log transform and the expectations on raw data.
- How to use the Box-Cox transform to perform square root, log, and automatically discover the best power transform for your dataset.
A. Airline Passengers Dataset
We will use the Airline Passengers dataset as an example. This dataset describes the total number of airline passengers over time.
# load and plot a time series
from pandas import read_csv
from matplotlib import pyplot
series = read_csv('airline-passengers.csv', header=0, index_col=0, parse_dates=True, squeeze=True)
pyplot.figure(1)
# line plot
pyplot.subplot(211)
pyplot.plot(series)
# histogram
pyplot.subplot(212)
pyplot.hist(series)
pyplot.show()
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| Line and density plots of the Airline Passengers dataset
|
The dataset is non-stationary, meaning that the mean and the variance of the observations change over time. This makes it difficult to model by both classical statistical methods, like ARIMA, and more sophisticated machine learning methods, like neural networks. This is caused by what appears to be both an increasing trend and a seasonality component.
In addition, the amount of change, or the variance, is increasing with time. This is clear when you look at the size of the seasonal component and notice that from one cycle to the next, the amplitude (from bottom to top of the cycle) is increasing.
B. Square Root Transform
A time series that has a quadratic growth trend can be made linear by taking the square root. Consider a series of the numbers 1 to 99 squared. The line plot of this series will show a quadratic growth trend and a histogram of the values will show an exponential distribution with a long tail.
# contrive a quadratic time series
from matplotlib import pyplot
series = [i**2 for i in range(1,100)]
pyplot.figure(1)
# line plot
pyplot.subplot(211)
pyplot.plot(series)
# histogram
pyplot.subplot(212)
pyplot.hist(series)
pyplot.show()
 |
| Line and density plots of the contrived quadratic series dataset
|
# square root transform a contrived quadratic time series
from matplotlib import pyplot
from numpy import sqrt
series = [i**2 for i in range(1,100)]
# sqrt transform
transform = series = sqrt(series)
pyplot.figure(1)
# line plot
pyplot.subplot(211)
pyplot.plot(transform)
# histogram
pyplot.subplot(212)
pyplot.hist(transform)
pyplot.show()
 |
| Line and density plots of the square root transformed quadratic series dataset
|
It is possible that the Airline Passengers dataset shows a quadratic growth. If this is the case, then we could expect a square root transform to reduce the growth trend to be linear and change the distribution of observations to be perhaps nearly Gaussian.
# square root transform a time series
from pandas import read_csv
from pandas import DataFrame
from numpy import sqrt
from matplotlib import pyplot
series = read_csv('airline-passengers.csv', header=0, index_col=0, parse_dates=True,
squeeze=True)
dataframe = DataFrame(series.values)
dataframe.columns = ['passengers']
dataframe['passengers'] = sqrt(dataframe['passengers'])
pyplot.figure(1)
# line plot
pyplot.subplot(211)
pyplot.plot(dataframe['passengers'])
# histogram
pyplot.subplot(212)
pyplot.hist(dataframe['passengers'])
pyplot.show()
 |
| Line and density plots of the square root transformed Airline Passengers dataset
|
We can see that the trend was reduced, but was not removed.
C. Log Transform
Time series with an exponential distribution can be made linear by taking the logarithm of the values. This is called a log transform. As with the square and square root case above, we can demonstrate this with a quick example. The code below creates an exponential distribution by raising the numbers from 1 to 99 to the value e, which is the base of the natural logarithms or Euler’s number (2.718...).
# create and plot an exponential time series
from matplotlib import pyplot
from math import exp
series = [exp(i) for i in range(1,100)]
pyplot.figure(1)
# line plot
pyplot.subplot(211)
pyplot.plot(series)
# histogram
pyplot.subplot(212)
pyplot.hist(series)
pyplot.show()
 |
| Line and density plots of the contrived exponential series dataset
|
Again, we can transform this series back to linear by taking the natural logarithm of the values.
# log transform a contrived exponential time series
from matplotlib import pyplot
from math import exp
from numpy import log
series = [exp(i) for i in range(1,100)]
transform = log(series)
pyplot.figure(1)
# line plot
pyplot.subplot(211)
pyplot.plot(transform)
# histogram
pyplot.subplot(212)
pyplot.hist(transform)
pyplot.show()
 |
| Line and density plots of the log transformed exponential dataset
|
The example below demonstrates a log transform of the Airline Passengers dataset
# log transform a time series
from pandas import read_csv
from pandas import DataFrame
from numpy import log
from matplotlib import pyplot
series = read_csv('airline-passengers.csv', header=0, index_col=0, parse_dates=True,
squeeze=True)
dataframe = DataFrame(series.values)
dataframe.columns = ['passengers']
dataframe['passengers'] = log(dataframe['passengers'])
pyplot.figure(1)
# line plot
pyplot.subplot(211)
pyplot.plot(dataframe['passengers'])
# histogram
pyplot.subplot(212)
pyplot.hist(dataframe['passengers'])
pyplot.show()
 |
| Line and density plots of the log transformed Airline Passengers dataset
|
Log transforms are popular with time series data as they are effective at removing exponential variance.
C. Box-Cox Transform
The square root transform and log transform belong to a class of transforms called power transforms. The Box-Cox transform is a configurable data transform method that supports both square root and log transform, as well as a suite of related transforms.
The scipy.stats library provides an implementation of the Box-Cox transform. The boxcox() function takes an argument, called lambda, that controls the type of transform to perform.
- lambda = -1.0 is a reciprocal transform.
- lambda = -0.5 is a reciprocal square root transform.
- lambda = 0.0 is a log transform.
- lambda = 0.5 is a square root transform.
- lambda = 1.0 is no transform
# manually box-cox transform a time series
from pandas import read_csv
from pandas import DataFrame
from scipy.stats import boxcox
from matplotlib import pyplot
series = read_csv('airline-passengers.csv', header=0, index_col=0, parse_dates=True, squeeze=True)
dataframe = DataFrame(series.values)
dataframe.columns = ['passengers']
dataframe['passengers'] = boxcox(dataframe['passengers'], lmbda=0.0)
pyplot.figure(1)
# line plot
pyplot.subplot(211)
pyplot.plot(dataframe['passengers'])
# histogram
pyplot.subplot(212)
pyplot.hist(dataframe['passengers'])
pyplot.show()
 |
| Line and density plots of the Box-Cox transformed Airline Passengers dataset
|
We can set the lambda parameter to None (the default) and let the function find a statistically tuned value.
# automatically box-cox transform a time series
from pandas import read_csv
from pandas import DataFrame
from scipy.stats import boxcox
from matplotlib import pyplot
series = read_csv('airline-passengers.csv', header=0, index_col=0, parse_dates=True,
squeeze=True)
dataframe = DataFrame(series.values)
dataframe.columns = ['passengers']
dataframe['passengers'], lam = boxcox(dataframe['passengers'])
print('Lambda: %f' % lam)
pyplot.figure(1)
# line plot
pyplot.subplot(211)
pyplot.plot(dataframe['passengers'])
# histogram
pyplot.subplot(212)
pyplot.hist(dataframe['passengers'])
pyplot.show()
-----Result-----
Lambda: 0.148023
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| Line and density plots of the Optimized Box-Cox transformed Airline Passengers dataset |
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