Introduction to Pattern Sequence based Forecasting (PSF) algorithm

Introduction

The Algorithm Pattern Sequence based Forecasting (PSF) was first proposed by Martinez Alvarez, et al., 2008 and then modified and suggested improvement by Martinez Alvarez, et al., 2011. The technical detailes are mentioned in referenced articles. PSF algorithm consists of various statistical operations like:

• Data Normalization/ Denormalization
• Calculation of optimum Window size (W)
• Calculation of optimum cluster size (k)
• Pattern Sequence based Forecasting
• RMSE/MAE Calculation, etc..

Example

This section discusses about the examples to introduce the use of the PSF package and to compare it with auto.arima() and ets() functions, which are well accepted functions in the R community working over time series forecasting techniques. The data used in this example are ’nottem’ and ’sunspots’ which are the standard time series dataset available in R. The ’nottem’ dataset is the average air temperatures at Nottingham Castle in degrees Fahrenheit, collected for 20 years, on monthly basis.

Similarly, ’sunspots’ dataset is mean relative sunspot numbers from 1749 to 1983, measured on monthly basis. First of all, the psf() function from PSF package is used to forecast the future values. For both datasets, all the recorded values except for the final year are considered as training data, and the last year is used for testing purposes. The predicted values for final year with psf() function for both datasets are now discussed.

library(PSF)

a <- psf(data = nottem, n.ahead = 12)
a

## $predictions
##  [1] 38.97692 38.71538 42.49231 46.32308 52.91538 57.97692 61.87692 60.19231
##  [9] 57.03846 49.42308 43.23846 40.21538
## 
## $k
## [1] 2
## 
## $w
## [1] 1

b <- psf(data = sunspots, n.ahead = 48)
b

## $predictions
##  [1] 71.60714 71.48214 64.80357 66.20357 64.06786 64.75357 60.67500 60.93214
##  [9] 59.48571 51.31429 50.25000 52.05714 71.60714 71.48214 64.80357 66.20357
## [17] 64.06786 64.75357 60.67500 60.93214 59.48571 51.31429 50.25000 52.05714
## [25] 71.60714 71.48214 64.80357 66.20357 64.06786 64.75357 60.67500 60.93214
## [33] 59.48571 51.31429 50.25000 52.05714 70.42792 70.53474 64.57002 65.02760
## [41] 63.40011 63.61093 59.50379 59.58474 57.27446 50.83766 49.68030 51.53550
## 
## $k
## [1] 2
## 
## $w
## [1] 4

psf_plot(data = nottem, predictions = a$predictions)

psf_plot(data = sunspots, predictions = b$predictions)

Comparison of psf() with auto.arima() and ets() functions:

Example below shows the comparisons for psf(), auto.arima() and ets() functions when using the Root Mean Square Error (RMSE) parameter as metric, for ’sunspots’ dataset. In order to avail more accurate and robust comparison results, error values are calculated for 5 times and the mean value of error values for methods under comparison are also shown. These values clearly state that ‘psf()’ function is able to outperform the comparative time series prediction methods. Additionally, the reader might want to refer to the results published in the original work Martinez Alvarez et al. (2011), in which it was shown that PSF outperformed many different methods when applied to electricity prices and demand forecasting.

library(PSF)
library(forecast)
options(warn=-1)
  
## Consider data `sunspots` with removal of last years's readings
# Training Data
x <- sunspots[1:2772]

# Test Data
y <- sunspots[2773:2820]

PSF <- NULL
ARIMA <- NULL
ETS <- NULL

for(i in 1:5)
{
  set.seed(i)
  
  # for PSF
  a <- psf(data = x, n.ahead = 48)$predictions
  
  # for ARIMA
  b <- forecast(auto.arima(x), 48)$mean
  
  # for ets
  c <- as.numeric(forecast(ets(x), 48)$mean)
  
  ## For Error Calculations
  # Error for PSF
  PSF[i] <- sqrt(mean((y - a)^2))
  # Error for ARIMA
  ARIMA[i] <- sqrt(mean((y - b)^2))
  # Error for ETS
  ETS[i] <- sqrt(mean((y - c)^2))

}

## Error values for PSF
  PSF

## [1] 61.08512 61.08512 61.08512 61.08512 61.08512

  mean(PSF)

## [1] 61.08512

## Error values for ARIMA
  ARIMA

## [1] 103.0719 103.0719 103.0719 103.0719 103.0719

  mean(ARIMA)

## [1] 103.0719

## Error values for ETS
  ETS

## [1] 70.66647 70.66647 70.66647 70.66647 70.66647

  mean(ETS)

## [1] 70.66647

References

Martínez-Álvarez, F., Troncoso, A., Riquelme, J.C. and Ruiz, J.S.A., 2008, December. LBF: A labeled-based forecasting algorithm and its application to electricity price time series. In Data Mining, 2008. ICDM’08. Eighth IEEE International Conference on (pp. 453-461). IEEE.

Martinez Alvarez, F., Troncoso, A., Riquelme, J.C. and Aguilar Ruiz, J.S., 2011. Energy time series forecasting based on pattern sequence similarity. Knowledge and Data Engineering, IEEE Transactions on, 23(8), pp.1230-1243.