 # Forecasting time series and regression pdf

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## Forecasting, Time Series, and Regression (with CD-ROM). Richard O'Connell

### Selected Contributions from ITISE 2017

How to install R. This booklet itells you how to use the R statistical software to carry out some simple analyses that are common in analysing time series data. This booklet assumes that the reader has some basic knowledge of time series analysis, and the principal focus of the booklet is not to explain time series analysis, but rather to explain how to carry out these analyses using R.

The first thing that you will want to do to analyse your time series data will be to read it into R, and to plot the time series. You can read data into R using the scan function, which assumes that your data for successive time points is in a simple text file with one column.

Only the first few lines of the file have been shown. The first three lines contain some comment on the data, and we want to ignore this when we read the data into R. To read the file into R, ignoring the first three lines, we type:. To store the data in a time series object, we use the ts function in R. Sometimes the time series data set that you have may have been collected at regular intervals that were less than one year, for example, monthly or quarterly.

An example is a data set of the number of births per month in New York city, from January to December originally collected by Newton. We can read the data into R by typing:. Once you have read a time series into R, the next step is usually to make a plot of the time series data, which you can do with the plot. For example, to plot the time series of the age of death of 42 successive kings of England, we type:.

We can see from the time plot that this time series could probably be described using an additive model, since the random fluctuations in the data are roughly constant in size over time. We can see from this time series that there seems to be seasonal variation in the number of births per month: there is a peak every summer, and a trough every winter. Again, it seems that this time series could probably be described using an additive model, as the seasonal fluctuations are roughly constant in size over time and do not seem to depend on the level of the time series, and the random fluctuations also seem to be roughly constant in size over time.

Similarly, to plot the time series of the monthly sales for the souvenir shop at a beach resort town in Queensland, Australia, we type:. In this case, it appears that an additive model is not appropriate for describing this time series, since the size of the seasonal fluctuations and random fluctuations seem to increase with the level of the time series.

Thus, we may need to transform the time series in order to get a transformed time series that can be described using an additive model. For example, we can transform the time series by calculating the natural log of the original data:. Here we can see that the size of the seasonal fluctuations and random fluctuations in the log-transformed time series seem to be roughly constant over time, and do not depend on the level of the time series.

Thus, the log-transformed time series can probably be described using an additive model. Decomposing a time series means separating it into its constituent components, which are usually a trend component and an irregular component, and if it is a seasonal time series, a seasonal component. A non-seasonal time series consists of a trend component and an irregular component.

Decomposing the time series involves trying to separate the time series into these components, that is, estimating the the trend component and the irregular component. To estimate the trend component of a non-seasonal time series that can be described using an additive model, it is common to use a smoothing method, such as calculating the simple moving average of the time series.

For example, as discussed above, the time series of the age of death of 42 successive kings of England appears is non-seasonal, and can probably be described using an additive model, since the random fluctuations in the data are roughly constant in size over time:. Thus, we can try to estimate the trend component of this time series by smoothing using a simple moving average.

To smooth the time series using a simple moving average of order 3, and plot the smoothed time series data, we type:. There still appears to be quite a lot of random fluctuations in the time series smoothed using a simple moving average of order 3. Thus, to estimate the trend component more accurately, we might want to try smoothing the data with a simple moving average of a higher order.

This takes a little bit of trial-and-error, to find the right amount of smoothing. For example, we can try using a simple moving average of order The data smoothed with a simple moving average of order 8 gives a clearer picture of the trend component, and we can see that the age of death of the English kings seems to have decreased from about 55 years old to about 38 years old during the reign of the first 20 kings, and then increased after that to about 73 years old by the end of the reign of the 40th king in the time series.

A seasonal time series consists of a trend component, a seasonal component and an irregular component. Decomposing the time series means separating the time series into these three components: that is, estimating these three components. This function estimates the trend, seasonal, and irregular components of a time series that can be described using an additive model.

For example, as discussed above, the time series of the number of births per month in New York city is seasonal with a peak every summer and trough every winter, and can probably be described using an additive model since the seasonal and random fluctuations seem to be roughly constant in size over time:. For example, we can print out the estimated values of the seasonal component by typing:.

The estimated seasonal factors are given for the months January-December, and are the same for each year. The largest seasonal factor is for July about 1.

The plot above shows the original time series top , the estimated trend component second from top , the estimated seasonal component third from top , and the estimated irregular component bottom.

We see that the estimated trend component shows a small decrease from about 24 in to about 22 in , followed by a steady increase from then on to about 27 in If you have a seasonal time series that can be described using an additive model, you can seasonally adjust the time series by estimating the seasonal component, and subtracting the estimated seasonal component from the original time series.

You can see that the seasonal variation has been removed from the seasonally adjusted time series. The seasonally adjusted time series now just contains the trend component and an irregular component. If you have a time series that can be described using an additive model with constant level and no seasonality, you can use simple exponential smoothing to make short-term forecasts. The simple exponential smoothing method provides a way of estimating the level at the current time point.

Smoothing is controlled by the parameter alpha; for the estimate of the level at the current time point. The value of alpha; lies between 0 and 1. Values of alpha that are close to 0 mean that little weight is placed on the most recent observations when making forecasts of future values. We can read the data into R and plot it by typing:. You can see from the plot that there is roughly constant level the mean stays constant at about 25 inches. The random fluctuations in the time series seem to be roughly constant in size over time, so it is probably appropriate to describe the data using an additive model.

Thus, we can make forecasts using simple exponential smoothing. For example, to use simple exponential smoothing to make forecasts for the time series of annual rainfall in London, we type:. The output of HoltWinters tells us that the estimated value of the alpha parameter is about 0. This is very close to zero, telling us that the forecasts are based on both recent and less recent observations although somewhat more weight is placed on recent observations. By default, HoltWinters just makes forecasts for the same time period covered by our original time series.

In this case, our original time series included rainfall for London from , so the forecasts are also for The plot shows the original time series in black, and the forecasts as a red line. The time series of forecasts is much smoother than the time series of the original data here.

As a measure of the accuracy of the forecasts, we can calculate the sum of squared errors for the in-sample forecast errors, that is, the forecast errors for the time period covered by our original time series. It is common in simple exponential smoothing to use the first value in the time series as the initial value for the level.

For example, in the time series for rainfall in London, the first value is For example, to make forecasts with the initial value of the level set to As explained above, by default HoltWinters just makes forecasts for the time period covered by the original data, which is for the rainfall time series.

To use the forecast. When using the forecast. HoltWinters function, as its first argument input , you pass it the predictive model that you have already fitted using the HoltWinters function. For example, to make a forecast of rainfall for the years 8 more years using forecast. HoltWinters , we type:. The forecast. For example, the forecasted rainfall for is about To plot the predictions made by forecast.

We can only calculate the forecast errors for the time period covered by our original time series, which is for the rainfall data. As mentioned above, one measure of the accuracy of the predictive model is the sum-of-squared-errors SSE for the in-sample forecast errors. If the predictive model cannot be improved upon, there should be no correlations between forecast errors for successive predictions. In other words, if there are correlations between forecast errors for successive predictions, it is likely that the simple exponential smoothing forecasts could be improved upon by another forecasting technique.

To figure out whether this is the case, we can obtain a correlogram of the in-sample forecast errors for lags For example, to calculate a correlogram of the in-sample forecast errors for the London rainfall data for lags , we type:.

You can see from the sample correlogram that the autocorrelation at lag 3 is just touching the significance bounds. To test whether there is significant evidence for non-zero correlations at lags , we can carry out a Ljung-Box test. For example, to test whether there are non-zero autocorrelations at lags , for the in-sample forecast errors for London rainfall data, we type:. Here the Ljung-Box test statistic is To be sure that the predictive model cannot be improved upon, it is also a good idea to check whether the forecast errors are normally distributed with mean zero and constant variance.

To check whether the forecast errors have constant variance, we can make a time plot of the in-sample forecast errors:. The plot shows that the in-sample forecast errors seem to have roughly constant variance over time, although the size of the fluctuations in the start of the time series may be slightly less than that at later dates eg.

To check whether the forecast errors are normally distributed with mean zero, we can plot a histogram of the forecast errors, with an overlaid normal curve that has mean zero and the same standard deviation as the distribution of forecast errors. You will have to copy the function above into R in order to use it. You can then use plotForecastErrors to plot a histogram with overlaid normal curve of the forecast errors for the rainfall predictions:.

The plot shows that the distribution of forecast errors is roughly centred on zero, and is more or less normally distributed, although it seems to be slightly skewed to the right compared to a normal curve. However, the right skew is relatively small, and so it is plausible that the forecast errors are normally distributed with mean zero.

The Ljung-Box test showed that there is little evidence of non-zero autocorrelations in the in-sample forecast errors, and the distribution of forecast errors seems to be normally distributed with mean zero. This suggests that the simple exponential smoothing method provides an adequate predictive model for London rainfall, which probably cannot be improved upon. Smoothing is controlled by two parameters, alpha, for the estimate of the level at the current time point, and beta for the estimate of the slope b of the trend component at the current time point.

As with simple exponential smoothing, the paramters alpha and beta have values between 0 and 1, and values that are close to 0 mean that little weight is placed on the most recent observations when making forecasts of future values. We can see from the plot that there was an increase in hem diameter from about in to about in , and that afterwards the hem diameter decreased to about in ## Time Series Analysis and Forecasting

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Interesting It looks like this book is on our website. Bruce L. Bowerman, Richard T. O'Connell, Anne B. Pensumliste: Statistik HA Almen, 2. Bowerman] on. Request PDF | On Feb 1, , Carolyn Pillers Dobler and others published Forecasting, Time Series, and Regression: An Applied Approach (4th ed.), Bruce L.

A book like forecasting time series and regression solutions PDF would be quite hard to get but worry not cause Stuvera. With an emphasis on applications, this book provides both the conceptual development and practical motivation you need to effectively implement forecasts of your own. Bruce L. Bowerman is a professor of decision sciences at Miami University in Oxford, Ohio. He received his Ph.

### Forecasting, Time Series, and Regression (with CD-ROM). Richard O'Connell

Amongst the wealth of available machine learning algorithms for forecasting time series, linear regression has remained one of the most important and widely used methods, due to its simplicity and interpretability. A disadvantage, however, is that a linear regression model may often have higher error than models that are produced by more sophisticated techniques. In this paper, we investigate the use of a grouping based quadratic mean loss function for improving the performance of linear regression. In particular, we propose segmenting the input time series into groups and simultaneously optimizing both the average loss of each group and the variance of the loss between groups, over the entire series.

Pillers Dobler, Carolyn, Brief Reviews of Teaching Materials. Practical Genetic Algorithms 2nd ed. Haupt and Sue Ellen Haupt. Anderson-Cook, Christine M. Publications Events. Time series models pdf. #### Forecasting, Time Series, and Regression (with CD-ROM). Richard O'Connell

Amongst the wealth of available machine learning algorithms for forecasting time series, linear regression has remained one of the most important and widely used methods, due to its simplicity and interpretability. A disadvantage, however, is that a linear regression model may often have higher error than models that are produced by more sophisticated techniques. In this paper, we investigate the use of a grouping based quadratic mean loss function for improving the performance of linear regression. In particular, we propose segmenting the input time series into groups and simultaneously optimizing both the average loss of each group and the variance of the loss between groups, over the entire series. This aims to produce a linear model that has low overall error, is less sensitive to distribution changes in the time series and is more robust to outliers. We experimentally investigate the performance of our method and find that it can build models which are different from those produced by standard linear regression, whilst achieving significant reductions in prediction errors. The series of ITISE conferences provides a forum for scientists, engineers, educators and students to discuss the latest ideas and implementations in the foundations, theory, models and applications in the field of time series analysis and forecasting. It focuses on interdisciplinary and multidisciplinary research encompassing computer science, mathematics, statistics and econometrics. • #### Camelia B. 23.04.2021 at 01:31

Time lags. • Correlation over time (serial correlation, a.k.a. autocorrelation). • Forecasting models built on regression methods: oautoregressive (AR) models.

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How to install R.

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