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Time series is a sequence of observations of categorical or numeric variables indexed by a date, or timestamp. A clear example of time series data is the time series of a stock price. In the following table, we can see the basic structure of time series data. In this case the observations are recorded every hour.

Timestamp | Stock - Price |
---|---|

2015-10-11 09:00:00 | 100 |

2015-10-11 10:00:00 | 110 |

2015-10-11 11:00:00 | 105 |

2015-10-11 12:00:00 | 90 |

2015-10-11 13:00:00 | 120 |

Normally, the first step in time series analysis is to plot the series, this is normally done with a line chart.

The most common application of time series analysis is forecasting future values of a numeric value using the temporal structure of the data. This means, the available observations are used to predict values from the future.

The temporal ordering of the data, implies that traditional regression methods are not useful. In order to build robust forecast, we need models that take into account the temporal ordering of the data.

The most widely used model for Time Series Analysis is called **Autoregressive Moving Average** (ARMA). The model consists of two parts, an **autoregressive** (AR) part and a **moving average** (MA) part. The model is usually then referred to as the *ARMA(p, q)* model where *p* is the order of the autoregressive part and *q* is the order of the moving average part.

The *AR(p)* is read as an autoregressive model of order p. Mathematically it is written as −

$$X_t = c + \sum_{i = 1}^{P} \phi_i X_{t - i} + \varepsilon_{t}$$

where {φ_{1}, …, φ_{p}} are parameters to be estimated, c is a constant, and the random variable ε_{t} represents the white noise. Some constraints are necessary on the values of the parameters so that the model remains stationary.

The notation *MA(q)* refers to the moving average model of order *q* −

$$X_t = \mu + \varepsilon_t + \sum_{i = 1}^{q} \theta_i \varepsilon_{t - i}$$

where the θ_{1}, ..., θ_{q} are the parameters of the model, μ is the expectation of X_{t}, and the ε_{t}, ε_{t − 1}, ... are, white noise error terms.

The *ARMA(p, q)* model combines p autoregressive terms and q moving-average terms. Mathematically the model is expressed with the following formula −

$$X_t = c + \varepsilon_t + \sum_{i = 1}^{P} \phi_iX_{t - 1} + \sum_{i = 1}^{q} \theta_i \varepsilon_{t-i}$$

We can see that the *ARMA(p, q)* model is a combination of *AR(p)* and *MA(q)* models.

To give some intuition of the model consider that the AR part of the equation seeks to estimate parameters for X_{t − i} observations of in order to predict the value of the variable in X_{t}. It is in the end a weighted average of the past values. The MA section uses the same approach but with the error of previous observations, ε_{t − i}. So in the end, the result of the model is a weighted average.

The following code snippet demonstrates how to implement an *ARMA(p, q) in R*.

# install.packages("forecast") library("forecast") # Read the data data = scan('fancy.dat') ts_data <- ts(data, frequency = 12, start = c(1987,1)) ts_data plot.ts(ts_data)

Plotting the data is normally the first step to find out if there is a temporal structure in the data. We can see from the plot that there are strong spikes at the end of each year.

The following code fits an ARMA model to the data. It runs several combinations of models and selects the one that has less error.

# Fit the ARMA model fit = auto.arima(ts_data) summary(fit) # Series: ts_data # ARIMA(1,1,1)(0,1,1)[12] # Coefficients: # ar1 ma1 sma1 # 0.2401 -0.9013 0.7499 # s.e. 0.1427 0.0709 0.1790 # # sigma^2 estimated as 15464184: log likelihood = -693.69 # AIC = 1395.38 AICc = 1395.98 BIC = 1404.43 # Training set error measures: # ME RMSE MAE MPE MAPE MASE ACF1 # Training set 328.301 3615.374 2171.002 -2.481166 15.97302 0.4905797 -0.02521172

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