---
title: "Introduction to Pattern Sequence based Forecasting (PSF) algorithm"
author: "Neeraj Bokde, Gualberto Asencio-Cortes and Francisco Martinez-Alvarez"
date: "`r Sys.Date()`"
output: rmarkdown::html_vignette
vignette: >
  %\VignetteIndexEntry{Vignette Title}
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---
## 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.

#### Install library

Download the Package and install with instruction:

```{r}
library(PSF)
```

#### Prediction results for ’nottem’ dataset with psf() function.

```{r}
a <- psf(data = nottem, n.ahead = 12)
a
```

#### Prediction results for ’sunspots’ dataset with psf() function.

```{r}
b <- psf(data = sunspots, n.ahead = 48)
b
```

To represent the prediction performance in plot format, the psf_plot() function is used as shown in
the following code.

#### Plot for 'nottem' dataset

```{r, fig.width = 7, fig.height = 4}
psf_plot(data = nottem, predictions = a$predictions)
```

#### Plot for 'sunspots' dataset

```{r, fig.width = 7, fig.height = 4}
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.


```{r}
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
  mean(PSF)
  
## Error values for ARIMA
  ARIMA
  mean(ARIMA)
  
## Error values for ETS
  ETS
  mean(ETS)
```

## 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.