This chapter covers the following topics:
Creating time series data
Plotting a time series object
Decomposing time series
Smoothing time series
Forecasting time series
Selecting an ARIMA model
Creating an ARIMA model
Forecasting with an ARIMA model
Predicting stock prices with an ARIMA model
The first example of time series analysis in human history occurred in ancient Egypt. The ancient Egyptians recorded the inundation (rise) and relinquishment (fall) of the Nile river every day, noting when fertile silt and moisture occurred. Based on these records, they found that the inundation period began when the sun rose at the same time as the Sirius star system became visible. By being able to predict the inundation period, the ancient Egyptians were able to make sophisticated agricultural decisions, greatly improving the yield of their farming activities.
As demonstrated by the ancient Egyptian inundation period example, time series analysis is a method that can extract patterns or meaningful statistics from data with temporal information. It allows us to forecast future values based on observed results. One can apply time series analysis to any data that has temporal information. For example, an economist can perform time series analysis to predict the GDP growth rate...
To begin time series analysis, we need to create a time series object from a numeric vector or matrix. In this recipe, we introduce how to create a time series object from the financial report of Taiwan Semiconductor (2330.TW) with the ts function.
Download the tw2330_finance.csv dataset from the following GitHub link:
https://github.com/ywchiu/rcookbook/raw/master/chapter10/tw2330_finance.csv
Please perform the following steps to create time series data:
First, read Taiwan Semiconductor's financial report into an R session:
> tw2330 = read.csv('tw2330_finance.csv', header=TRUE) > str(tw2330) 'data.frame': 32 obs. of 5 variables: $ Time : Factor w/ 32 levels "2008Q1","2008Q2",..: 1 2 3 4 5 6 7 8 9 10 ... $ Total.Income : int 875 881 930 646 395 742 899 921 922 1050 ... $ Gross.Sales : num 382 402 431 202 74.8 343 429 447 442 519 ... $ Operating.Income: num 291 304 329 120 12.1 251 320 336 341 405 ... $ EPS ...
Plotting a time series object will make trends and seasonal composition clearly visible. In this recipe, we introduce how to plot time series data with the plot.ts function.
Ensure you have completed the previous recipe by generating a time series object and storing it in two variables: m and m_ts.
Please perform the following steps to plot time series data:
First, use the plot.ts function to plot time series data, m:
> plot.ts(m)
Figure 1: A time series plot of single time series data
Also, if the dataset contains multiple time series objects, you can plot multiple time series data in a separate sub-figure:
> plot.ts(m_ts, plot.type = "multiple",)
Figure 2: A time series plot of multiple time series data
Alternatively, you can plot all four time series objects in a single figure:
> plot.ts(m_ts, plot.type = "single", col=c("red","green","blue", "orange"))
Figure 3: A multiple time series plot in different colors
Moreover, you can...
A seasonal time series is made up of seasonal components, deterministic trend components, and irregular components. In this recipe, we introduce how to use the decompose function to destruct a time series into these three parts.
Ensure you have completed the previous recipe by generating a time series object and storing it in two variables: m and m_ts.
Please perform the following steps to decompose a time series:
First, use the window function to construct a time series object, m.sub, from m:
> m.sub = window(m, start=c(2012, 1), end=c(2014, 4)) > m.sub Qtr1 Qtr2 Qtr3 Qtr4 2012 1055 1281 1414 1313 2013 1328 1559 1626 1458 2014 1482 1830 2090 2225 > plot(m.sub)
Figure 6: A time series plot in a quarter
Use the decompose function to destruct the time series object m.sub:
> components <- decompose(m.sub)
We can then use the names function to list the attributes of components:
> names(components) [1] "x" "seasonal...
Time series decomposition allows us to extract distinct components from time series data. The smoothing technique enables us to forecast the future values of time series data. In this recipe, we introduce how to use the HoltWinters function to smooth time series data.
Ensure you have completed the previous recipe by generating a time series object and storing it in two variables: m and m_ts.
Please perform the following steps to smooth time series data:
First, use HoltWinters to perform Winters exponential smoothing:
> m.pre <- HoltWinters(m) > m.pre Holt-Winters exponential smoothing with trend and additive seasonal component. Call: HoltWinters(x = m) Smoothing parameters: alpha: 0.8223689 beta : 0.06468208 gamma: 1 Coefficients: [,1] a 1964.30088 b 32.33727 s1 -51.47814 s2 17.84420 s3 146.26704 s4 70.69912
Plot the smoothing result:
> plot(m.pre)
Figure 9: A time series plot with Winters exponential smoothed...
After using HoltWinters to build a time series smoothing model, we can now forecast future values based on the smoothing model. In this recipe, we introduce how to use the forecast function to make a prediction on time series data.
In this recipe, you have to have completed the previous recipe by generating a smoothing model with HoltWinters and have it stored in a variable, m.pre.
Please perform the following steps to forecast Taiwan Semiconductor's future income:
Load the forecast package:
> library(forecast)
We can use the forecast function to predict the income of the next four quarters:
> income.pre <- forecast.HoltWinters(m.pre, h=4) > summary(income.pre) Forecast method: HoltWinters Model Information: Holt-Winters exponential smoothing with trend and additive seasonal component. Call: HoltWinters(x = m) Smoothing parameters: alpha: 0.8223689 beta : 0.06468208 gamma: 1 Coefficients: [,1] a 1964.30088 b ...
Using the exponential smoothing method requires that residuals are non-correlated. However, in real-life cases, it is quite unlikely that none of the continuous values correlate with each other. Instead, one can use ARIMA in R to build a time series model that takes autocorrelation into consideration. In this recipe, we introduce how to use ARIMA to build a smoothing model.
Please perform the following steps to select the ARIMA model's parameters:
First, simulate an ARIMA process and generate time series data with the arima.sim function:
> set.seed(123) > ts.sim <- arima.sim(list(order = c(1,1,0), ar = 0.7), n = 100) > plot(ts.sim)
Figure 14: Simulated time series data
We can then take the difference of the time series:
> ts.sim.diff <- diff(ts.sim)
Plot the differenced time series:
> plot(ts.sim.diff)
Figure 15: A differenced time series plot
Use the...
After determining the optimum p, d, and q parameters for an ARIMA model, we can now create an ARIMA model with the Arima function.
Ensure you have completed the previous recipe by generating a time series object and storing it in a variable, ts.sim.
Please perform the following steps to build an ARIMA model:
First, we can create an ARIMA model with time series ts.sim, with parameters p=1, d=1, q=0:
> library(forecast) > fit <- Arima(ts.sim, order=c(1,1,0)) > fit Series: ts.sim ARIMA(1,1,0) Coefficients: ar1 0.7128 s.e. 0.0685 sigma^2 estimated as 0.7603: log likelihood=-128.04 AIC=260.09 AICc=260.21 BIC=265.3
Next, use the accuracy function to print the training set errors of the model:
> accuracy(fit) ME RMSE MAE MPE Training set 0.004938457 0.863265 0.6849681 -41.98798 MAPE MASE ACF1 Training set 102.2542 0.7038325...
Based on our fitted ARIMA model, we can predict future values. In this recipe, we will introduce how to forecast future values with the forecast.Arima function in the forecast package.
Ensure you have completed the previous recipe by generating an ARIMA model and storing the model in a variable, fit.
Please perform the following steps to forecast future values with forecast.Arima:
First, use forecast.Arima to generate the prediction of future values:
> fit.predict <- forecast.Arima(fit)
We can then use the summary function to obtain the summary of our prediction:
> summary(fit.predict) Forecast method: ARIMA(1,1,0) Model Information: Series: ts.sim ARIMA(1,1,0) Coefficients: ar1 0.7128 s.e. 0.0685 sigma^2 estimated as 0.7603: log likelihood=-128.04 AIC=260.09 AICc=260.21 BIC=265.3 Error measures: ME RMSE MAE MPE Training set 0.004938457 0.863265...
As the historical prices of a stock are also a time series, we can thus build an ARIMA model to forecast future prices of a given stock. In this recipe, we introduce how to load historical prices with the quantmod package, and make predictions on stock prices with ARIMA.
In this recipe, we use the example of stock price prediction to review all the concepts we have covered in previous topics. You will require knowledge of how to create an ARIMA model and make predictions based on a built model to follow this recipe.
Please perform the following steps to predict Facebook's stock price with the ARIMA model:
First, install and load the quantmod package:
> install.packages("quantmod") > library(quantmod)
Download the historical prices of Facebook Inc from Yahoo with quantmod:
> getSymbols("FB",src="yahoo", from="2015-01-01")
Next, plot the historical stock prices as a line chart:
> plot(FB)
Figure 23: A historical...
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