FORECASTING: PRINCIPLES AND PRACTICE (3RD ED) Changes in the third edition. The most important change in edition 3 of the book is that we use the tsibble and fable packages rather than the forecast package. This allows us to integrate closely with the tidyverse collection of packages. As a consequence, we have replaced many examples to take advantage of the new facilities. FORECASTING: PRINCIPLES AND PRACTICE (2ND ED)FORECASTING: PRINCIPLES AND PRACTICEPREFACE1 GETTING STARTED2 TIME SERIES GRAPHICS Preface. This is the second edition of Forecasting: Principles & Practice, which uses the forecast package in R. The third edition, which uses the fable package, is also available.. Buy a print or downloadable version. Welcome to our online textbook on forecasting. This textbook is intended to provide a comprehensive introduction to forecasting methods and to present enough information

12.2 PROPHET MODEL

12.2 Prophet model. 12.2. Prophet model. A recent proposal is the Prophet model, available via the fable.prophet package. This model was introduced by Facebook ( S. J. Taylor & Letham, 2018), originally for forecasting daily data with weekly and yearly seasonality, plus holiday effects. It was later extended to cover more types of seasonal

data.

4.2 ACF FEATURES

4.2 ACF features. Autocorrelations were discussed in Section 2.8.All the autocorrelations of a series can be considered features of that series. We can also summarise the autocorrelations to produce new features; for example, the sum of the first ten squared autocorrelation coefficients is a useful summary of how much autocorrelation there is in a series, regardless of lag. 13.1 WEEKLY, DAILY AND SUB-DAILY DATA Figure 13.2: Forecasts for weekly US gasoline production using a dynamic harmonic regression model. ( 2 π j t 52.18)] + η t where ηt η t is an ARIMA (0,1,1) process. Because ηt η t is non-stationary, the model is actually estimated on the differences of the variables on both sides of this equation. There are 12 parameters to capture the 3.1 TRANSFORMATIONS AND ADJUSTMENTS 3.1. Transformations and adjustments. Adjusting the historical data can often lead to a simpler time series. Here, we deal with four kinds of adjustments: calendar adjustments, population adjustments, inflation adjustments and mathematical transformations. The purpose of these adjustments and transformations is to simplify the patterns in

the

7.7 FORECASTING WITH ETS MODELS In Table 7.8 we give the formulas for the additive ETS models, which are the simplest. Table 7.8: Forecast variance expressions for each additive state space model, where σ2 σ 2 is the residual variance, m m is the seasonal period, and k k is the integer part of (h−1)/m ( h − 1) / m (i.e., the number of complete years in the forecast 5.8 EVALUATING POINT FORECAST ACCURACY The accuracy of forecasts can only be determined by considering how well a model performs on new data that were not used when fitting the model. When choosing models, it is common practice to separate the available data into two portions, training and test data, where the training data is used to estimate any parameters of a forecasting

method

CHAPTER 11 FORECASTING HIERARCHICAL AND GROUPED TIME Chapter 11 Forecasting hierarchical and grouped time series. Time series can often be naturally disaggregated by various attributes of interest. For example, the total number of bicycles sold by a cycling manufacturer can be disaggregated by product type such as 8.9 季节性ARIMA模型 8.9 季节性ARIMA模型. 8.9. 季节性ARIMA模型. 之前我们一直将注意力集中在非季节性的数据和非季节性的 ARIMA 模型上。. 然而，ARIMA 模型同样可以用于很多季节性数据的建模。. 季节性的 ARIMA模型 在我们之前讨论的 ARIMA 模型多项式中引入了季节性的项。. 它可以表示 FORECASTING: PRINCIPLES AND PRACTICE (3RD ED) Changes in the third edition. The most important change in edition 3 of the book is that we use the tsibble and fable packages rather than the forecast package. This allows us to integrate closely with the tidyverse collection of packages. As a consequence, we have replaced many examples to take advantage of the new facilities. FORECASTING: PRINCIPLES AND PRACTICE (2ND ED)FORECASTING: PRINCIPLES AND PRACTICEPREFACE1 GETTING STARTED2 TIME SERIES GRAPHICS Preface. This is the second edition of Forecasting: Principles & Practice, which uses the forecast package in R. The third edition, which uses the fable package, is also available.. Buy a print or downloadable version. Welcome to our online textbook on forecasting. This textbook is intended to provide a comprehensive introduction to forecasting methods and to present enough information

12.2 PROPHET MODEL

12.2 Prophet model. 12.2. Prophet model. A recent proposal is the Prophet model, available via the fable.prophet package. This model was introduced by Facebook ( S. J. Taylor & Letham, 2018), originally for forecasting daily data with weekly and yearly seasonality, plus holiday effects. It was later extended to cover more types of seasonal

data.

4.2 ACF FEATURES

4.2 ACF features. Autocorrelations were discussed in Section 2.8.All the autocorrelations of a series can be considered features of that series. We can also summarise the autocorrelations to produce new features; for example, the sum of the first ten squared autocorrelation coefficients is a useful summary of how much autocorrelation there is in a series, regardless of lag. 13.1 WEEKLY, DAILY AND SUB-DAILY DATA Figure 13.2: Forecasts for weekly US gasoline production using a dynamic harmonic regression model. ( 2 π j t 52.18)] + η t where ηt η t is an ARIMA (0,1,1) process. Because ηt η t is non-stationary, the model is actually estimated on the differences of the variables on both sides of this equation. There are 12 parameters to capture the 3.1 TRANSFORMATIONS AND ADJUSTMENTS 3.1. Transformations and adjustments. Adjusting the historical data can often lead to a simpler time series. Here, we deal with four kinds of adjustments: calendar adjustments, population adjustments, inflation adjustments and mathematical transformations. The purpose of these adjustments and transformations is to simplify the patterns in

the

7.7 FORECASTING WITH ETS MODELS In Table 7.8 we give the formulas for the additive ETS models, which are the simplest. Table 7.8: Forecast variance expressions for each additive state space model, where σ2 σ 2 is the residual variance, m m is the seasonal period, and k k is the integer part of (h−1)/m ( h − 1) / m (i.e., the number of complete years in the forecast 5.8 EVALUATING POINT FORECAST ACCURACY The accuracy of forecasts can only be determined by considering how well a model performs on new data that were not used when fitting the model. When choosing models, it is common practice to separate the available data into two portions, training and test data, where the training data is used to estimate any parameters of a forecasting

method

CHAPTER 11 FORECASTING HIERARCHICAL AND GROUPED TIME Chapter 11 Forecasting hierarchical and grouped time series. Time series can often be naturally disaggregated by various attributes of interest. For example, the total number of bicycles sold by a cycling manufacturer can be disaggregated by product type such as 8.9 季节性ARIMA模型 8.9 季节性ARIMA模型. 8.9. 季节性ARIMA模型. 之前我们一直将注意力集中在非季节性的数据和非季节性的 ARIMA 模型上。. 然而，ARIMA 模型同样可以用于很多季节性数据的建模。. 季节性的 ARIMA模型 在我们之前讨论的 ARIMA 模型多项式中引入了季节性的项。. 它可以表示 13.1 WEEKLY, DAILY AND SUB-DAILY DATA Figure 13.2: Forecasts for weekly US gasoline production using a dynamic harmonic regression model. ( 2 π j t 52.18)] + η t where ηt η t is an ARIMA (0,1,1) process. Because ηt η t is non-stationary, the model is actually estimated on the differences of the variables on both sides of this equation. There are 12 parameters to capture the 8.4 MOVING AVERAGE MODELS 8.4. Moving average models. Rather than using past values of the forecast variable in a regression, a moving average model uses past forecast errors in a regression-like model. yt = c+εt +θ1εt−1 +θ2εt−2+⋯+θqεt−q, y t = c + ε t + θ 1 ε t − 1 + θ 2 ε t − 2 + ⋯ + θ q ε t − q, where εt ε 3.1 TRANSFORMATIONS AND ADJUSTMENTS 3.1. Transformations and adjustments. Adjusting the historical data can often lead to a simpler time series. Here, we deal with four kinds of adjustments: calendar adjustments, population adjustments, inflation adjustments and mathematical transformations. The purpose of these adjustments and transformations is to simplify the patterns in

the

13.8 FORECASTING ON TRAINING AND TEST SETS 13.8 Forecasting on training and test sets. Typically, we compute one-step forecasts on the training data (the “fitted values”) and multi-step forecasts on the test data. However, occasionally we may wish to compute multi-step forecasts on the training data, or one-step forecasts on the test data. 12.1 WEEKLY, DAILY AND SUB-DAILY DATA Figure 12.1: Forecasts for weekly US gasoline production using an STL decomposition with an ETS model for the seasonally adjusted data. An alternative approach is to use a dynamic harmonic regression model, as discussed in Section 9.5. In the following example, the number of Fourier terms was selected by minimising the AICc. 12.7 VERY LONG AND VERY SHORT TIME SERIES Forecasting very short time series. We often get asked how few data points can be used to fit a time series model.As with almost all sample size questions, there is no easy answer. It depends on the number of model parameters to be estimated and the amount of randomness in the data.The sample size required increases with the number of parameters to be estimated, and the amount of noise in the CHAPTER 9 DYNAMIC REGRESSION MODELS Chapter 9. Dynamic regression models. The time series models in the previous two chapters allow for the inclusion of information from past observations of a series, but not for the inclusion of other information that may also be relevant. For example, the effects of holidays, competitor activity, changes in the law, the wider economy,

or other

5.5 DISTRIBUTIONAL FORECASTS AND PREDICTION INTERVALS Prediction intervals. A prediction interval gives an interval within which we expect \(y_{t}\) to lie with a specified probability. For example, assuming that distribution of future observations is normal, a 95% prediction interval for the \(h\)-step forecast is \ where \(\hat\sigma_h\) is an estimate of the standard deviation of the \(h\)-step 5.8 EVALUATING POINT FORECAST ACCURACY The accuracy of forecasts can only be determined by considering how well a model performs on new data that were not used when fitting the model. When choosing models, it is common practice to separate the available data into two portions, training and test data, where the training data is used to estimate any parameters of a forecasting

method

10.1 HIERARCHICAL TIME SERIES 10.1 Hierarchical time series. Figure 10.1 shows a \(K=2\)-level hierarchical structure.At the top of the hierarchy (which we call level 0) is the “Total,” the most aggregate level of the data. The \(t\) th observation of the Total series is denoted by \(y_t\) for \(t=1,\dots,T\).The Total is disaggregated into two series at level 1, which in turn are divided into three and two series FORECASTING: PRINCIPLES AND PRACTICE (3RD ED) Changes in the third edition. The most important change in edition 3 of the book is that we use the tsibble and fable packages rather than the forecast package. This allows us to integrate closely with the tidyverse collection of packages. As a consequence, we have replaced many examples to take advantage of the new facilities. FORECASTING: PRINCIPLES AND PRACTICE (2ND ED)FORECASTING: PRINCIPLES AND PRACTICEPREFACE1 GETTING STARTED2 TIME SERIES GRAPHICS Preface. This is the second edition of Forecasting: Principles & Practice, which uses the forecast package in R. The third edition, which uses the fable package, is also available.. Buy a print or downloadable version. Welcome to our online textbook on forecasting. This textbook is intended to provide a comprehensive introduction to forecasting methods and to present enough information

12.2 PROPHET MODEL

12.2 Prophet model. 12.2. Prophet model. A recent proposal is the Prophet model, available via the fable.prophet package. This model was introduced by Facebook ( S. J. Taylor & Letham, 2018), originally for forecasting daily data with weekly and yearly seasonality, plus holiday effects. It was later extended to cover more types of seasonal

data.

4.2 ACF FEATURES

4.2 ACF features. Autocorrelations were discussed in Section 2.8.All the autocorrelations of a series can be considered features of that series. We can also summarise the autocorrelations to produce new features; for example, the sum of the first ten squared autocorrelation coefficients is a useful summary of how much autocorrelation there is in a series, regardless of lag. 13.1 WEEKLY, DAILY AND SUB-DAILY DATA Figure 13.2: Forecasts for weekly US gasoline production using a dynamic harmonic regression model. ( 2 π j t 52.18)] + η t where ηt η t is an ARIMA (0,1,1) process. Because ηt η t is non-stationary, the model is actually estimated on the differences of the variables on both sides of this equation. There are 12 parameters to capture the 3.1 TRANSFORMATIONS AND ADJUSTMENTS 3.1. Transformations and adjustments. Adjusting the historical data can often lead to a simpler time series. Here, we deal with four kinds of adjustments: calendar adjustments, population adjustments, inflation adjustments and mathematical transformations. The purpose of these adjustments and transformations is to simplify the patterns in

the

7.7 FORECASTING WITH ETS MODELS In Table 7.8 we give the formulas for the additive ETS models, which are the simplest. Table 7.8: Forecast variance expressions for each additive state space model, where σ2 σ 2 is the residual variance, m m is the seasonal period, and k k is the integer part of (h−1)/m ( h − 1) / m (i.e., the number of complete years in the forecast 5.8 EVALUATING POINT FORECAST ACCURACY The accuracy of forecasts can only be determined by considering how well a model performs on new data that were not used when fitting the model. When choosing models, it is common practice to separate the available data into two portions, training and test data, where the training data is used to estimate any parameters of a forecasting

method

CHAPTER 11 FORECASTING HIERARCHICAL AND GROUPED TIME Chapter 11 Forecasting hierarchical and grouped time series. Time series can often be naturally disaggregated by various attributes of interest. For example, the total number of bicycles sold by a cycling manufacturer can be disaggregated by product type such as 8.9 季节性ARIMA模型 8.9 季节性ARIMA模型. 8.9. 季节性ARIMA模型. 之前我们一直将注意力集中在非季节性的数据和非季节性的 ARIMA 模型上。. 然而，ARIMA 模型同样可以用于很多季节性数据的建模。. 季节性的 ARIMA模型 在我们之前讨论的 ARIMA 模型多项式中引入了季节性的项。. 它可以表示 FORECASTING: PRINCIPLES AND PRACTICE (3RD ED) Changes in the third edition. The most important change in edition 3 of the book is that we use the tsibble and fable packages rather than the forecast package. This allows us to integrate closely with the tidyverse collection of packages. As a consequence, we have replaced many examples to take advantage of the new facilities. FORECASTING: PRINCIPLES AND PRACTICE (2ND ED)FORECASTING: PRINCIPLES AND PRACTICEPREFACE1 GETTING STARTED2 TIME SERIES GRAPHICS Preface. This is the second edition of Forecasting: Principles & Practice, which uses the forecast package in R. The third edition, which uses the fable package, is also available.. Buy a print or downloadable version. Welcome to our online textbook on forecasting. This textbook is intended to provide a comprehensive introduction to forecasting methods and to present enough information

12.2 PROPHET MODEL

12.2 Prophet model. 12.2. Prophet model. A recent proposal is the Prophet model, available via the fable.prophet package. This model was introduced by Facebook ( S. J. Taylor & Letham, 2018), originally for forecasting daily data with weekly and yearly seasonality, plus holiday effects. It was later extended to cover more types of seasonal

data.

4.2 ACF FEATURES

4.2 ACF features. Autocorrelations were discussed in Section 2.8.All the autocorrelations of a series can be considered features of that series. We can also summarise the autocorrelations to produce new features; for example, the sum of the first ten squared autocorrelation coefficients is a useful summary of how much autocorrelation there is in a series, regardless of lag. 13.1 WEEKLY, DAILY AND SUB-DAILY DATA Figure 13.2: Forecasts for weekly US gasoline production using a dynamic harmonic regression model. ( 2 π j t 52.18)] + η t where ηt η t is an ARIMA (0,1,1) process. Because ηt η t is non-stationary, the model is actually estimated on the differences of the variables on both sides of this equation. There are 12 parameters to capture the 3.1 TRANSFORMATIONS AND ADJUSTMENTS 3.1. Transformations and adjustments. Adjusting the historical data can often lead to a simpler time series. Here, we deal with four kinds of adjustments: calendar adjustments, population adjustments, inflation adjustments and mathematical transformations. The purpose of these adjustments and transformations is to simplify the patterns in

the

7.7 FORECASTING WITH ETS MODELS In Table 7.8 we give the formulas for the additive ETS models, which are the simplest. Table 7.8: Forecast variance expressions for each additive state space model, where σ2 σ 2 is the residual variance, m m is the seasonal period, and k k is the integer part of (h−1)/m ( h − 1) / m (i.e., the number of complete years in the forecast 5.8 EVALUATING POINT FORECAST ACCURACY The accuracy of forecasts can only be determined by considering how well a model performs on new data that were not used when fitting the model. When choosing models, it is common practice to separate the available data into two portions, training and test data, where the training data is used to estimate any parameters of a forecasting

method

CHAPTER 11 FORECASTING HIERARCHICAL AND GROUPED TIME Chapter 11 Forecasting hierarchical and grouped time series. Time series can often be naturally disaggregated by various attributes of interest. For example, the total number of bicycles sold by a cycling manufacturer can be disaggregated by product type such as 8.9 季节性ARIMA模型 8.9 季节性ARIMA模型. 8.9. 季节性ARIMA模型. 之前我们一直将注意力集中在非季节性的数据和非季节性的 ARIMA 模型上。. 然而，ARIMA 模型同样可以用于很多季节性数据的建模。. 季节性的 ARIMA模型 在我们之前讨论的 ARIMA 模型多项式中引入了季节性的项。. 它可以表示 13.1 WEEKLY, DAILY AND SUB-DAILY DATA Figure 13.2: Forecasts for weekly US gasoline production using a dynamic harmonic regression model. ( 2 π j t 52.18)] + η t where ηt η t is an ARIMA (0,1,1) process. Because ηt η t is non-stationary, the model is actually estimated on the differences of the variables on both sides of this equation. There are 12 parameters to capture the 3.1 TRANSFORMATIONS AND ADJUSTMENTS 3.1. Transformations and adjustments. Adjusting the historical data can often lead to a simpler time series. Here, we deal with four kinds of adjustments: calendar adjustments, population adjustments, inflation adjustments and mathematical transformations. The purpose of these adjustments and transformations is to simplify the patterns in

the

13.8 FORECASTING ON TRAINING AND TEST SETS 13.8 Forecasting on training and test sets. Typically, we compute one-step forecasts on the training data (the “fitted values”) and multi-step forecasts on the test data. However, occasionally we may wish to compute multi-step forecasts on the training data, or one-step forecasts on the test data. 12.1 WEEKLY, DAILY AND SUB-DAILY DATA Figure 12.1: Forecasts for weekly US gasoline production using an STL decomposition with an ETS model for the seasonally adjusted data. An alternative approach is to use a dynamic harmonic regression model, as discussed in Section 9.5. In the following example, the number of Fourier terms was selected by minimising the AICc. 12.7 VERY LONG AND VERY SHORT TIME SERIES Forecasting very short time series. We often get asked how few data points can be used to fit a time series model.As with almost all sample size questions, there is no easy answer. It depends on the number of model parameters to be estimated and the amount of randomness in the data.The sample size required increases with the number of parameters to be estimated, and the amount of noise in the 8.4 MOVING AVERAGE MODELS 8.4. Moving average models. Rather than using past values of the forecast variable in a regression, a moving average model uses past forecast errors in a regression-like model. yt = c+εt +θ1εt−1 +θ2εt−2+⋯+θqεt−q, y t = c + ε t + θ 1 ε t − 1 + θ 2 ε t − 2 + ⋯ + θ q ε t − q, where εt ε 5.5 DISTRIBUTIONAL FORECASTS AND PREDICTION INTERVALS Prediction intervals. A prediction interval gives an interval within which we expect \(y_{t}\) to lie with a specified probability. For example, assuming that distribution of future observations is normal, a 95% prediction interval for the \(h\)-step forecast is \ where \(\hat\sigma_h\) is an estimate of the standard deviation of the \(h\)-step CHAPTER 9 DYNAMIC REGRESSION MODELS Chapter 9. Dynamic regression models. The time series models in the previous two chapters allow for the inclusion of information from past observations of a series, but not for the inclusion of other information that may also be relevant. For example, the effects of holidays, competitor activity, changes in the law, the wider economy,

or other

10.1 HIERARCHICAL TIME SERIES 10.1 Hierarchical time series. Figure 10.1 shows a \(K=2\)-level hierarchical structure.At the top of the hierarchy (which we call level 0) is the “Total,” the most aggregate level of the data. The \(t\) th observation of the Total series is denoted by \(y_t\) for \(t=1,\dots,T\).The Total is disaggregated into two series at level 1, which in turn are divided into three and two series 5.8 EVALUATING POINT FORECAST ACCURACY The accuracy of forecasts can only be determined by considering how well a model performs on new data that were not used when fitting the model. When choosing models, it is common practice to separate the available data into two portions, training and test data, where the training data is used to estimate any parameters of a forecasting

method

FORECASTING: PRINCIPLES AND PRACTICE (3RD ED) Changes in the third edition. The most important change in edition 3 of the book is that we use the tsibble and fable packages rather than the forecast package. This allows us to integrate closely with the tidyverse collection of packages. As a consequence, we have replaced many examples to take advantage of the new facilities. FORECASTING: PRINCIPLES AND PRACTICE (2ND ED)FORECASTING: PRINCIPLES AND PRACTICEPREFACE1 GETTING STARTED2 TIME SERIES GRAPHICS Preface. This is the second edition of Forecasting: Principles & Practice, which uses the forecast package in R. The third edition, which uses the fable package, is also available.. Buy a print or downloadable version. Welcome to our online textbook on forecasting. This textbook is intended to provide a comprehensive introduction to forecasting methods and to present enough information

12.2 PROPHET MODEL

12.2 Prophet model. 12.2. Prophet model. A recent proposal is the Prophet model, available via the fable.prophet package. This model was introduced by Facebook ( S. J. Taylor & Letham, 2018), originally for forecasting daily data with weekly and yearly seasonality, plus holiday effects. It was later extended to cover more types of seasonal

data.

5.9 CORRELATION, CAUSATION AND FORECASTING Multicollinearity and forecasting. A closely related issue is multicollinearity, which occurs when similar information is provided by two or more of the predictor variables in a multiple regression.. It can occur when two predictors are highly correlated with each other (that is, they have a correlation coefficient close to +1 or -1).

4.2 ACF FEATURES

4.2 ACF features. Autocorrelations were discussed in Section 2.8.All the autocorrelations of a series can be considered features of that series. We can also summarise the autocorrelations to produce new features; for example, the sum of the first ten squared autocorrelation coefficients is a useful summary of how much autocorrelation there is in a series, regardless of lag. 3.1 TRANSFORMATIONS AND ADJUSTMENTS 3.1. Transformations and adjustments. Adjusting the historical data can often lead to a simpler time series. Here, we deal with four kinds of adjustments: calendar adjustments, population adjustments, inflation adjustments and mathematical transformations. The purpose of these adjustments and transformations is to simplify the patterns in

the

9.11 EXERCISES

9.11 Exercises. Figure 9.32 shows the ACFs for 36 random numbers, 360 random numbers and 1,000 random numbers.. Explain the differences among these figures. Do 10.1 HIERARCHICAL TIME SERIES 10.1 Hierarchical time series. Figure 10.1 shows a \(K=2\)-level hierarchical structure.At the top of the hierarchy (which we call level 0) is the “Total,” the most aggregate level of the data. The \(t\) th observation of the Total series is denoted by \(y_t\) for \(t=1,\dots,T\).The Total is disaggregated into two series at level 1, which in turn are divided into three and two series 8.6 估计和阶数选择 信息准则. AIC信息准则（Akaike Information Criterion）在选择用于回归模型的变量时非常有用， 同样在确定ARIMA模型阶数时也可以发挥很大作用。. 它可以写作: AIC =−2log(L)+2(p+q+k +1), AIC = − 2 log. ⁡. ( L) + 2 ( p + q + k + 1), 这里的 L L 是数据的似然函数,当 c

≠ 0 c ≠ 0 时 k

8.9 季节性ARIMA模型 8.9 季节性ARIMA模型. 8.9. 季节性ARIMA模型. 之前我们一直将注意力集中在非季节性的数据和非季节性的 ARIMA 模型上。. 然而，ARIMA 模型同样可以用于很多季节性数据的建模。. 季节性的 ARIMA模型 在我们之前讨论的 ARIMA 模型多项式中引入了季节性的项。. 它可以表示 FORECASTING: PRINCIPLES AND PRACTICE (3RD ED) Changes in the third edition. The most important change in edition 3 of the book is that we use the tsibble and fable packages rather than the forecast package. This allows us to integrate closely with the tidyverse collection of packages. As a consequence, we have replaced many examples to take advantage of the new facilities. FORECASTING: PRINCIPLES AND PRACTICE (2ND ED)FORECASTING: PRINCIPLES AND PRACTICEPREFACE1 GETTING STARTED2 TIME SERIES GRAPHICS Preface. This is the second edition of Forecasting: Principles & Practice, which uses the forecast package in R. The third edition, which uses the fable package, is also available.. Buy a print or downloadable version. Welcome to our online textbook on forecasting. This textbook is intended to provide a comprehensive introduction to forecasting methods and to present enough information

12.2 PROPHET MODEL

12.2 Prophet model. 12.2. Prophet model. A recent proposal is the Prophet model, available via the fable.prophet package. This model was introduced by Facebook ( S. J. Taylor & Letham, 2018), originally for forecasting daily data with weekly and yearly seasonality, plus holiday effects. It was later extended to cover more types of seasonal

data.

5.9 CORRELATION, CAUSATION AND FORECASTING Multicollinearity and forecasting. A closely related issue is multicollinearity, which occurs when similar information is provided by two or more of the predictor variables in a multiple regression.. It can occur when two predictors are highly correlated with each other (that is, they have a correlation coefficient close to +1 or -1).

4.2 ACF FEATURES

4.2 ACF features. Autocorrelations were discussed in Section 2.8.All the autocorrelations of a series can be considered features of that series. We can also summarise the autocorrelations to produce new features; for example, the sum of the first ten squared autocorrelation coefficients is a useful summary of how much autocorrelation there is in a series, regardless of lag. 3.1 TRANSFORMATIONS AND ADJUSTMENTS 3.1. Transformations and adjustments. Adjusting the historical data can often lead to a simpler time series. Here, we deal with four kinds of adjustments: calendar adjustments, population adjustments, inflation adjustments and mathematical transformations. The purpose of these adjustments and transformations is to simplify the patterns in

the

9.11 EXERCISES

9.11 Exercises. Figure 9.32 shows the ACFs for 36 random numbers, 360 random numbers and 1,000 random numbers.. Explain the differences among these figures. Do 10.1 HIERARCHICAL TIME SERIES 10.1 Hierarchical time series. Figure 10.1 shows a \(K=2\)-level hierarchical structure.At the top of the hierarchy (which we call level 0) is the “Total,” the most aggregate level of the data. The \(t\) th observation of the Total series is denoted by \(y_t\) for \(t=1,\dots,T\).The Total is disaggregated into two series at level 1, which in turn are divided into three and two series 8.6 估计和阶数选择 信息准则. AIC信息准则（Akaike Information Criterion）在选择用于回归模型的变量时非常有用， 同样在确定ARIMA模型阶数时也可以发挥很大作用。. 它可以写作: AIC =−2log(L)+2(p+q+k +1), AIC = − 2 log. ⁡. ( L) + 2 ( p + q + k + 1), 这里的 L L 是数据的似然函数,当 c

≠ 0 c ≠ 0 时 k

8.9 季节性ARIMA模型 8.9 季节性ARIMA模型. 8.9. 季节性ARIMA模型. 之前我们一直将注意力集中在非季节性的数据和非季节性的 ARIMA 模型上。. 然而，ARIMA 模型同样可以用于很多季节性数据的建模。. 季节性的 ARIMA模型 在我们之前讨论的 ARIMA 模型多项式中引入了季节性的项。. 它可以表示 13.1 WEEKLY, DAILY AND SUB-DAILY DATA Figure 13.2: Forecasts for weekly US gasoline production using a dynamic harmonic regression model. ( 2 π j t 52.18)] + η t where ηt η t is an ARIMA (0,1,1) process. Because ηt η t is non-stationary, the model is actually estimated on the differences of the variables on both sides of this equation. There are 12 parameters to capture the 3.1 TRANSFORMATIONS AND ADJUSTMENTS 3.1. Transformations and adjustments. Adjusting the historical data can often lead to a simpler time series. Here, we deal with four kinds of adjustments: calendar adjustments, population adjustments, inflation adjustments and mathematical transformations. The purpose of these adjustments and transformations is to simplify the patterns in

the

8.4 MOVING AVERAGE MODELS 8.4. Moving average models. Rather than using past values of the forecast variable in a regression, a moving average model uses past forecast errors in a regression-like model. yt = c+εt +θ1εt−1 +θ2εt−2+⋯+θqεt−q, y t = c + ε t + θ 1 ε t − 1 + θ 2 ε t − 2 + ⋯ + θ q ε t − q, where εt ε 12.7 VERY LONG AND VERY SHORT TIME SERIES Forecasting very short time series. We often get asked how few data points can be used to fit a time series model.As with almost all sample size questions, there is no easy answer. It depends on the number of model parameters to be estimated and the amount of randomness in the data.The sample size required increases with the number of parameters to be estimated, and the amount of noise in the 10.1 HIERARCHICAL TIME SERIES 10.1 Hierarchical time series. Figure 10.1 shows a \(K=2\)-level hierarchical structure.At the top of the hierarchy (which we call level 0) is the “Total,” the most aggregate level of the data. The \(t\) th observation of the Total series is denoted by \(y_t\) for \(t=1,\dots,T\).The Total is disaggregated into two series at level 1, which in turn are divided into three and two series CHAPTER 9 DYNAMIC REGRESSION MODELS Chapter 9. Dynamic regression models. The time series models in the previous two chapters allow for the inclusion of information from past observations of a series, but not for the inclusion of other information that may also be relevant. For example, the effects of holidays, competitor activity, changes in the law, the wider economy,

or other

5.5 DISTRIBUTIONAL FORECASTS AND PREDICTION INTERVALS Prediction intervals. A prediction interval gives an interval within which we expect \(y_{t}\) to lie with a specified probability. For example, assuming that distribution of future observations is normal, a 95% prediction interval for the \(h\)-step forecast is \ where \(\hat\sigma_h\) is an estimate of the standard deviation of the \(h\)-step 3.3 RESIDUAL DIAGNOSTICS Residuals. The “residuals” in a time series model are what is left over after fitting a model. For many (but not all) time series models, the residuals are equal to the difference between the observations and the corresponding fitted values: \[ e_{t} = y_{t}-\hat{y}_{t}. Residuals are useful in checking whether a model has adequately captured the information in the data. CHAPTER 11 FORECASTING HIERARCHICAL AND GROUPED TIME Chapter 11 Forecasting hierarchical and grouped time series. Time series can often be naturally disaggregated by various attributes of interest. For example, the total number of bicycles sold by a cycling manufacturer can be disaggregated by product type such as 8.6 估计和阶数选择 极大似然估计. 一旦我们确定了模型的阶数(即 \(p\) 、 \(d\) 和 \(q\) 的取值)，我们就需要估计参数 \(c\) 、 \(\phi_1,\dots,\phi_p\) 、 \(\theta_1,\dots,\theta_q\) 了。 在R估计ARIMA模型时，它会采用极大似然估计(maximum

likelihood

estimation)。该方法通过最大化我们观测到的数据出现的概率来确定参数。 OTEXTSOTEXTSPRINCIPLES AND PRACTICEBUY OTexts is an online textbook publisher. Our books are freely available to students everywhere, whether for self-learning or as part of a university courses. For now, we are only hosting books written by Professor Rob J Hyndman and coauthors. FORECASTING: PRINCIPLES AND PRACTICE (3RD ED) Changes in the third edition. The most important change in edition 3 of the book is that we use the tsibble and fable packages rather than the forecast package. This allows us to integrate closely with the tidyverse collection of packages. As a consequence, we have replaced many examples to take advantage of the new facilities. FORECASTING: PRINCIPLES AND PRACTICE (2ND ED) Preface. This is the second edition of Forecasting: Principles & Practice, which uses the forecast package in R. The third edition, which uses the fable package, is also available.. Buy a print or downloadable version. Welcome to our online textbook on forecasting. This textbook is intended to provide a comprehensive introduction to forecasting methods and to present enough information

12.2 PROPHET MODEL

12.2 Prophet model. 12.2. Prophet model. A recent proposal is the Prophet model, available via the fable.prophet package. This model was introduced by Facebook ( S. J. Taylor & Letham, 2018), originally for forecasting daily data with weekly and yearly seasonality, plus holiday effects. It was later extended to cover more types of seasonal

data.

13.8 FORECASTING ON TRAINING AND TEST SETS 13.8 Forecasting on training and test sets. Typically, we compute one-step forecasts on the training data (the “fitted values”) and multi-step forecasts on the test data. However, occasionally we may wish to compute multi-step forecasts on the training data, or one-step forecasts on the test data. 7.4 SOME USEFUL PREDICTORS Note that trend() and season() are not standard functions; they are “special” functions that work within the TSLM() model formulae.. There is an average downward trend of -0.34 megalitres per quarter. On average, the second quarter has production of 34.7 megalitres lower than the first quarter, the third quarter has production of 17.8 megalitres lower than the first quarter, and the fourth 12.7 VERY LONG AND VERY SHORT TIME SERIES Forecasting very short time series. We often get asked how few data points can be used to fit a time series model.As with almost all sample size questions, there is no easy answer. It depends on the number of model parameters to be estimated and the amount of randomness in the data.The sample size required increases with the number of parameters to be estimated, and the amount of noise in the 11.1 HIERARCHICAL AND GROUPED TIME SERIES Hierarchical time series. Figure 11.1 shows a simple hierarchical structure. At the top of the hierarchy is the “Total,” the most aggregate level of the data. The \(t\) th observation of the Total series is denoted by \(y_t\) for \(t=1,\dots,T\).The Total is disaggregated into two series, which in turn are divided into three and two series respectively at the bottom level of the hierarchy. CHAPTER 11 FORECASTING HIERARCHICAL AND GROUPED TIME Chapter 11 Forecasting hierarchical and grouped time series. Time series can often be naturally disaggregated by various attributes of interest. For example, the total number of bicycles sold by a cycling manufacturer can be disaggregated by product type such as 4.5 EXPLORING AUSTRALIAN TOURISM DATA Each point on Figure 4.4 represents one series and its location on the plot is based on all 48 features. The first principal component (.fittedPC1) is the linear combination of the features which explains the most variation in the data.The second principal component (.fittedPC2) is the linear combination which explains the next most variation in the data, while being uncorrelated with the OTEXTSOTEXTSPRINCIPLES AND PRACTICEBUY OTexts is an online textbook publisher. Our books are freely available to students everywhere, whether for self-learning or as part of a university courses. For now, we are only hosting books written by Professor Rob J Hyndman and coauthors. FORECASTING: PRINCIPLES AND PRACTICE (3RD ED) Changes in the third edition. The most important change in edition 3 of the book is that we use the tsibble and fable packages rather than the forecast package. This allows us to integrate closely with the tidyverse collection of packages. As a consequence, we have replaced many examples to take advantage of the new facilities. FORECASTING: PRINCIPLES AND PRACTICE (2ND ED) Preface. This is the second edition of Forecasting: Principles & Practice, which uses the forecast package in R. The third edition, which uses the fable package, is also available.. Buy a print or downloadable version. Welcome to our online textbook on forecasting. This textbook is intended to provide a comprehensive introduction to forecasting methods and to present enough information

12.2 PROPHET MODEL

12.2 Prophet model. 12.2. Prophet model. A recent proposal is the Prophet model, available via the fable.prophet package. This model was introduced by Facebook ( S. J. Taylor & Letham, 2018), originally for forecasting daily data with weekly and yearly seasonality, plus holiday effects. It was later extended to cover more types of seasonal

data.

13.8 FORECASTING ON TRAINING AND TEST SETS 13.8 Forecasting on training and test sets. Typically, we compute one-step forecasts on the training data (the “fitted values”) and multi-step forecasts on the test data. However, occasionally we may wish to compute multi-step forecasts on the training data, or one-step forecasts on the test data. 7.4 SOME USEFUL PREDICTORS Note that trend() and season() are not standard functions; they are “special” functions that work within the TSLM() model formulae.. There is an average downward trend of -0.34 megalitres per quarter. On average, the second quarter has production of 34.7 megalitres lower than the first quarter, the third quarter has production of 17.8 megalitres lower than the first quarter, and the fourth 12.7 VERY LONG AND VERY SHORT TIME SERIES Forecasting very short time series. We often get asked how few data points can be used to fit a time series model.As with almost all sample size questions, there is no easy answer. It depends on the number of model parameters to be estimated and the amount of randomness in the data.The sample size required increases with the number of parameters to be estimated, and the amount of noise in the 11.1 HIERARCHICAL AND GROUPED TIME SERIES Hierarchical time series. Figure 11.1 shows a simple hierarchical structure. At the top of the hierarchy is the “Total,” the most aggregate level of the data. The \(t\) th observation of the Total series is denoted by \(y_t\) for \(t=1,\dots,T\).The Total is disaggregated into two series, which in turn are divided into three and two series respectively at the bottom level of the hierarchy. CHAPTER 11 FORECASTING HIERARCHICAL AND GROUPED TIME Chapter 11 Forecasting hierarchical and grouped time series. Time series can often be naturally disaggregated by various attributes of interest. For example, the total number of bicycles sold by a cycling manufacturer can be disaggregated by product type such as 4.5 EXPLORING AUSTRALIAN TOURISM DATA Each point on Figure 4.4 represents one series and its location on the plot is based on all 48 features. The first principal component (.fittedPC1) is the linear combination of the features which explains the most variation in the data.The second principal component (.fittedPC2) is the linear combination which explains the next most variation in the data, while being uncorrelated with the CHAPTER 5 TIME SERIES REGRESSION MODELS Chapter 5 Time series regression models. In this chapter we discuss regression models. The basic concept is that we forecast the time series of interest \(y\) assuming that it has a linear relationship with other time series \(x\).. For example, we might wish to forecast monthly sales \(y\) using total advertising spend \(x\) as a predictor. Or we might forecast daily electricity demand \(y 12.7 VERY LONG AND VERY SHORT TIME SERIES Forecasting very short time series. We often get asked how few data points can be used to fit a time series model.As with almost all sample size questions, there is no easy answer. It depends on the number of model parameters to be estimated and the amount of randomness in the data.The sample size required increases with the number of parameters to be estimated, and the amount of noise in the 5.9 CORRELATION, CAUSATION AND FORECASTING Multicollinearity and forecasting. A closely related issue is multicollinearity, which occurs when similar information is provided by two or more of the predictor variables in a multiple regression.. It can occur when two predictors are highly correlated with each other (that is, they have a correlation coefficient close to +1 or -1). 11.1 HIERARCHICAL AND GROUPED TIME SERIES Hierarchical time series. Figure 11.1 shows a simple hierarchical structure. At the top of the hierarchy is the “Total,” the most aggregate level of the data. The \(t\) th observation of the Total series is denoted by \(y_t\) for \(t=1,\dots,T\).The Total is disaggregated into two series, which in turn are divided into three and two series respectively at the bottom level of the hierarchy. CHAPTER 9 DYNAMIC REGRESSION MODELS Chapter 9. Dynamic regression models. The time series models in the previous two chapters allow for the inclusion of information from past observations of a series, but not for the inclusion of other information that may also be relevant. For example, the effects of holidays, competitor activity, changes in the law, the wider economy,

or other

3.3 RESIDUAL DIAGNOSTICS Residuals. The “residuals” in a time series model are what is left over after fitting a model. For many (but not all) time series models, the residuals are equal to the difference between the observations and the corresponding fitted values: \[ e_{t} = y_{t}-\hat{y}_{t}. Residuals are useful in checking whether a model has adequately captured the information in the data.

9.11 EXERCISES

9.11 Exercises. Figure 9.32 shows the ACFs for 36 random numbers, 360 random numbers and 1,000 random numbers.. Explain the differences among these figures. Do 9.4 STOCHASTIC AND DETERMINISTIC TRENDS 9.4 Stochastic and deterministic trends. 9.4. Stochastic and deterministic trends. There are two different ways of modelling a linear trend. A deterministic trend is obtained using the regression model yt =β0 +β1t +ηt, y t = β 0 + β 1 t + η t, where ηt η t is an ARMA process. A stochastic 5.8 EVALUATING POINT FORECAST ACCURACY The accuracy of forecasts can only be determined by considering how well a model performs on new data that were not used when fitting the model. When choosing models, it is common practice to separate the available data into two portions, training and test data, where the training data is used to estimate any parameters of a forecasting

method

OTEXTSWEB VIEW

Test Bank. Forecasting. Which simple forecasting method says the forecast is equal to the mean of the historical data? Average Method.

Naïve Method

FORECASTING: PRINCIPLES AND PRACTICE (3RD ED) Changes in the third edition. The most important change in edition 3 of the book is that we use the tsibble and fable packages rather than the forecast package. This allows us to integrate closely with the tidyverse collection of packages. As a consequence, we have replaced many examples to take advantage of the new facilities. 3.1 TRANSFORMATIONS AND ADJUSTMENTS 3.1. Transformations and adjustments. Adjusting the historical data can often lead to a simpler time series. Here, we deal with four kinds of adjustments: calendar adjustments, population adjustments, inflation adjustments and mathematical transformations. The purpose of these adjustments and transformations is to simplify the patterns in

the

CHAPTER 5 TIME SERIES REGRESSION MODELS Chapter 5 Time series regression models. In this chapter we discuss regression models. The basic concept is that we forecast the time series of interest \(y\) assuming that it has a linear relationship with other time series \(x\).. For example, we might wish to forecast monthly sales \(y\) using total advertising spend \(x\) as a predictor. Or we might forecast daily electricity demand \(y 3.3 RESIDUAL DIAGNOSTICS Residuals. The “residuals” in a time series model are what is left over after fitting a model. For many (but not all) time series models, the residuals are equal to the difference between the observations and the corresponding fitted values: \[ e_{t} = y_{t}-\hat{y}_{t}. Residuals are useful in checking whether a model has adequately captured the information in the data.

4.2 ACF FEATURES

4.2 ACF features. Autocorrelations were discussed in Section 2.8.All the autocorrelations of a series can be considered features of that series. We can also summarise the autocorrelations to produce new features; for example, the sum of the first ten squared autocorrelation coefficients is a useful summary of how much autocorrelation there is in a series, regardless of lag.

12.2 PROPHET MODEL

12.2 Prophet model. 12.2. Prophet model. A recent proposal is the Prophet model, available via the fable.prophet package. This model was introduced by Facebook ( S. J. Taylor & Letham, 2018), originally for forecasting daily data with weekly and yearly seasonality, plus holiday effects. It was later extended to cover more types of seasonal

data.

12.7 VERY LONG AND VERY SHORT TIME SERIES Forecasting very short time series. We often get asked how few data points can be used to fit a time series model.As with almost all sample size questions, there is no easy answer. It depends on the number of model parameters to be estimated and the amount of randomness in the data.The sample size required increases with the number of parameters to be estimated, and the amount of noise in the 7.3 HOLT-WINTERS’ SEASONAL METHOD The Holt-Winters seasonal method comprises the forecast equation and three smoothing equations — one for the level ℓt ℓ t, one for the trend bt b t, and one for the seasonal component st s t, with corresponding smoothing parameters α α, β∗ β ∗ and γ γ. We use m m to denote the frequency of the seasonality, i.e., the number

of

2.4 SEASONAL PLOTS

Multiple seasonal periods. Where the data has more than one seasonal pattern, the period argument can be used to select which seasonal plot is required. The vic_elec data contains half-hourly electricity demand for the state of Victoria, Australia. We can plot the daily pattern, weekly pattern or yearly pattern by specifying the period argument as shown in Figures 2.5–2.7.

OTEXTSWEB VIEW

Test Bank. Forecasting. Which simple forecasting method says the forecast is equal to the mean of the historical data? Average Method.

Naïve Method

FORECASTING: PRINCIPLES AND PRACTICE (3RD ED) Changes in the third edition. The most important change in edition 3 of the book is that we use the tsibble and fable packages rather than the forecast package. This allows us to integrate closely with the tidyverse collection of packages. As a consequence, we have replaced many examples to take advantage of the new facilities. 3.1 TRANSFORMATIONS AND ADJUSTMENTS 3.1. Transformations and adjustments. Adjusting the historical data can often lead to a simpler time series. Here, we deal with four kinds of adjustments: calendar adjustments, population adjustments, inflation adjustments and mathematical transformations. The purpose of these adjustments and transformations is to simplify the patterns in

the

CHAPTER 5 TIME SERIES REGRESSION MODELS Chapter 5 Time series regression models. In this chapter we discuss regression models. The basic concept is that we forecast the time series of interest \(y\) assuming that it has a linear relationship with other time series \(x\).. For example, we might wish to forecast monthly sales \(y\) using total advertising spend \(x\) as a predictor. Or we might forecast daily electricity demand \(y 3.3 RESIDUAL DIAGNOSTICS Residuals. The “residuals” in a time series model are what is left over after fitting a model. For many (but not all) time series models, the residuals are equal to the difference between the observations and the corresponding fitted values: \[ e_{t} = y_{t}-\hat{y}_{t}. Residuals are useful in checking whether a model has adequately captured the information in the data.

4.2 ACF FEATURES

4.2 ACF features. Autocorrelations were discussed in Section 2.8.All the autocorrelations of a series can be considered features of that series. We can also summarise the autocorrelations to produce new features; for example, the sum of the first ten squared autocorrelation coefficients is a useful summary of how much autocorrelation there is in a series, regardless of lag.

12.2 PROPHET MODEL

12.2 Prophet model. 12.2. Prophet model. A recent proposal is the Prophet model, available via the fable.prophet package. This model was introduced by Facebook ( S. J. Taylor & Letham, 2018), originally for forecasting daily data with weekly and yearly seasonality, plus holiday effects. It was later extended to cover more types of seasonal

data.

12.7 VERY LONG AND VERY SHORT TIME SERIES Forecasting very short time series. We often get asked how few data points can be used to fit a time series model.As with almost all sample size questions, there is no easy answer. It depends on the number of model parameters to be estimated and the amount of randomness in the data.The sample size required increases with the number of parameters to be estimated, and the amount of noise in the 7.3 HOLT-WINTERS’ SEASONAL METHOD The Holt-Winters seasonal method comprises the forecast equation and three smoothing equations — one for the level ℓt ℓ t, one for the trend bt b t, and one for the seasonal component st s t, with corresponding smoothing parameters α α, β∗ β ∗ and γ γ. We use m m to denote the frequency of the seasonality, i.e., the number

of

2.4 SEASONAL PLOTS

Multiple seasonal periods. Where the data has more than one seasonal pattern, the period argument can be used to select which seasonal plot is required. The vic_elec data contains half-hourly electricity demand for the state of Victoria, Australia. We can plot the daily pattern, weekly pattern or yearly pattern by specifying the period argument as shown in Figures 2.5–2.7.

OTEXTSWEB VIEW

Test Bank. Forecasting. Which simple forecasting method says the forecast is equal to the mean of the historical data? Average Method.

Naïve Method

OTEXTS

OTexts is an online textbook publisher. Our books are freely available to students everywhere, whether for self-learning or as part of a university courses. For now, we are only hosting books written by Professor Rob J Hyndman and coauthors. 5.9 CORRELATION, CAUSATION AND FORECASTING Multicollinearity and forecasting. A closely related issue is multicollinearity, which occurs when similar information is provided by two or more of the predictor variables in a multiple regression.. It can occur when two predictors are highly correlated with each other (that is, they have a correlation coefficient close to +1 or -1).

12.2 PROPHET MODEL

12.2 Prophet model. 12.2. Prophet model. A recent proposal is the Prophet model, available via the fable.prophet package. This model was introduced by Facebook ( S. J. Taylor & Letham, 2018), originally for forecasting daily data with weekly and yearly seasonality, plus holiday effects. It was later extended to cover more types of seasonal

data.

12.7 VERY LONG AND VERY SHORT TIME SERIES Forecasting very short time series. We often get asked how few data points can be used to fit a time series model.As with almost all sample size questions, there is no easy answer. It depends on the number of model parameters to be estimated and the amount of randomness in the data.The sample size required increases with the number of parameters to be estimated, and the amount of noise in the FORECASTING: PRINCIPLES AND PRACTICE (2ND ED) Preface. This is the second edition of Forecasting: Principles & Practice, which uses the forecast package in R. The third edition, which uses the fable package, is also available.. Buy a print or downloadable version. Welcome to our online textbook on forecasting. This textbook is intended to provide a comprehensive introduction to forecasting methods and to present enough information 13.8 FORECASTING ON TRAINING AND TEST SETS 13.8 Forecasting on training and test sets. Typically, we compute one-step forecasts on the training data (the “fitted values”) and multi-step forecasts on the test data. However, occasionally we may wish to compute multi-step forecasts on the training data, or one-step forecasts on the test data. 7.4 SOME USEFUL PREDICTORS Note that trend() and season() are not standard functions; they are “special” functions that work within the TSLM() model formulae.. There is an average downward trend of -0.34 megalitres per quarter. On average, the second quarter has production of 34.7 megalitres lower than the first quarter, the third quarter has production of 17.8 megalitres lower than the first quarter, and the fourth 8.7 ARIMA MODELLING IN R The auto.arima () function in R uses a variation of the Hyndman-Khandakar algorithm ( Hyndman & Khandakar, 2008), which combines unit root tests, minimisation of the AICc and MLE to obtain an ARIMA model. The arguments to auto.arima () provide for many variations on the algorithm. What is described here is the default

behaviour.

8.10 ARIMA VS ETS

8.10 ARIMA vs ETS. It is a commonly held myth that ARIMA models are more general than exponential smoothing. While linear exponential smoothing models are all special cases of ARIMA models, the non-linear exponential smoothing models have no equivalent ARIMA counterparts. 8.9 季节性ARIMA模型 8.9 季节性ARIMA模型. 8.9. 季节性ARIMA模型. 之前我们一直将注意力集中在非季节性的数据和非季节性的 ARIMA 模型上。. 然而，ARIMA 模型同样可以用于很多季节性数据的建模。. 季节性的 ARIMA模型 在我们之前讨论的 ARIMA 模型多项式中引入了季节性的项。. 它可以表示 FORECASTING: PRINCIPLES AND PRACTICE (3RD ED) Changes in the third edition. The most important change in edition 3 of the book is that we use the tsibble and fable packages rather than the forecast package. This allows us to integrate closely with the tidyverse collection of packages. As a consequence, we have replaced many examples to take advantage of the new facilities. 3.1 TRANSFORMATIONS AND ADJUSTMENTS 3.1. Transformations and adjustments. Adjusting the historical data can often lead to a simpler time series. Here, we deal with four kinds of adjustments: calendar adjustments, population adjustments, inflation adjustments and mathematical transformations. The purpose of these adjustments and transformations is to simplify the patterns in

the

CHAPTER 5 TIME SERIES REGRESSION MODELS Chapter 5 Time series regression models. In this chapter we discuss regression models. The basic concept is that we forecast the time series of interest \(y\) assuming that it has a linear relationship with other time series \(x\).. For example, we might wish to forecast monthly sales \(y\) using total advertising spend \(x\) as a predictor. Or we might forecast daily electricity demand \(y 3.3 RESIDUAL DIAGNOSTICS Residuals. The “residuals” in a time series model are what is left over after fitting a model. For many (but not all) time series models, the residuals are equal to the difference between the observations and the corresponding fitted values: \[ e_{t} = y_{t}-\hat{y}_{t}. Residuals are useful in checking whether a model has adequately captured the information in the data.

4.2 ACF FEATURES

4.2 ACF features. Autocorrelations were discussed in Section 2.8.All the autocorrelations of a series can be considered features of that series. We can also summarise the autocorrelations to produce new features; for example, the sum of the first ten squared autocorrelation coefficients is a useful summary of how much autocorrelation there is in a series, regardless of lag.

12.2 PROPHET MODEL

12.2 Prophet model. 12.2. Prophet model. A recent proposal is the Prophet model, available via the fable.prophet package. This model was introduced by Facebook ( S. J. Taylor & Letham, 2018), originally for forecasting daily data with weekly and yearly seasonality, plus holiday effects. It was later extended to cover more types of seasonal

data.

12.7 VERY LONG AND VERY SHORT TIME SERIES Forecasting very short time series. We often get asked how few data points can be used to fit a time series model.As with almost all sample size questions, there is no easy answer. It depends on the number of model parameters to be estimated and the amount of randomness in the data.The sample size required increases with the number of parameters to be estimated, and the amount of noise in the 7.3 HOLT-WINTERS’ SEASONAL METHOD The Holt-Winters seasonal method comprises the forecast equation and three smoothing equations — one for the level ℓt ℓ t, one for the trend bt b t, and one for the seasonal component st s t, with corresponding smoothing parameters α α, β∗ β ∗ and γ γ. We use m m to denote the frequency of the seasonality, i.e., the number

of

2.4 SEASONAL PLOTS

Multiple seasonal periods. Where the data has more than one seasonal pattern, the period argument can be used to select which seasonal plot is required. The vic_elec data contains half-hourly electricity demand for the state of Victoria, Australia. We can plot the daily pattern, weekly pattern or yearly pattern by specifying the period argument as shown in Figures 2.5–2.7.

OTEXTSWEB VIEW

Test Bank. Forecasting. Which simple forecasting method says the forecast is equal to the mean of the historical data? Average Method.

Naïve Method

FORECASTING: PRINCIPLES AND PRACTICE (3RD ED) Changes in the third edition. The most important change in edition 3 of the book is that we use the tsibble and fable packages rather than the forecast package. This allows us to integrate closely with the tidyverse collection of packages. As a consequence, we have replaced many examples to take advantage of the new facilities. 3.1 TRANSFORMATIONS AND ADJUSTMENTS 3.1. Transformations and adjustments. Adjusting the historical data can often lead to a simpler time series. Here, we deal with four kinds of adjustments: calendar adjustments, population adjustments, inflation adjustments and mathematical transformations. The purpose of these adjustments and transformations is to simplify the patterns in

the

CHAPTER 5 TIME SERIES REGRESSION MODELS Chapter 5 Time series regression models. In this chapter we discuss regression models. The basic concept is that we forecast the time series of interest \(y\) assuming that it has a linear relationship with other time series \(x\).. For example, we might wish to forecast monthly sales \(y\) using total advertising spend \(x\) as a predictor. Or we might forecast daily electricity demand \(y 3.3 RESIDUAL DIAGNOSTICS Residuals. The “residuals” in a time series model are what is left over after fitting a model. For many (but not all) time series models, the residuals are equal to the difference between the observations and the corresponding fitted values: \[ e_{t} = y_{t}-\hat{y}_{t}. Residuals are useful in checking whether a model has adequately captured the information in the data.

4.2 ACF FEATURES

4.2 ACF features. Autocorrelations were discussed in Section 2.8.All the autocorrelations of a series can be considered features of that series. We can also summarise the autocorrelations to produce new features; for example, the sum of the first ten squared autocorrelation coefficients is a useful summary of how much autocorrelation there is in a series, regardless of lag.

12.2 PROPHET MODEL

12.2 Prophet model. 12.2. Prophet model. A recent proposal is the Prophet model, available via the fable.prophet package. This model was introduced by Facebook ( S. J. Taylor & Letham, 2018), originally for forecasting daily data with weekly and yearly seasonality, plus holiday effects. It was later extended to cover more types of seasonal

data.

12.7 VERY LONG AND VERY SHORT TIME SERIES Forecasting very short time series. We often get asked how few data points can be used to fit a time series model.As with almost all sample size questions, there is no easy answer. It depends on the number of model parameters to be estimated and the amount of randomness in the data.The sample size required increases with the number of parameters to be estimated, and the amount of noise in the 7.3 HOLT-WINTERS’ SEASONAL METHOD The Holt-Winters seasonal method comprises the forecast equation and three smoothing equations — one for the level ℓt ℓ t, one for the trend bt b t, and one for the seasonal component st s t, with corresponding smoothing parameters α α, β∗ β ∗ and γ γ. We use m m to denote the frequency of the seasonality, i.e., the number

of

2.4 SEASONAL PLOTS

Multiple seasonal periods. Where the data has more than one seasonal pattern, the period argument can be used to select which seasonal plot is required. The vic_elec data contains half-hourly electricity demand for the state of Victoria, Australia. We can plot the daily pattern, weekly pattern or yearly pattern by specifying the period argument as shown in Figures 2.5–2.7.

OTEXTSWEB VIEW

Test Bank. Forecasting. Which simple forecasting method says the forecast is equal to the mean of the historical data? Average Method.

Naïve Method

OTEXTS

OTexts is an online textbook publisher. Our books are freely available to students everywhere, whether for self-learning or as part of a university courses. For now, we are only hosting books written by Professor Rob J Hyndman and coauthors. 5.9 CORRELATION, CAUSATION AND FORECASTING Multicollinearity and forecasting. A closely related issue is multicollinearity, which occurs when similar information is provided by two or more of the predictor variables in a multiple regression.. It can occur when two predictors are highly correlated with each other (that is, they have a correlation coefficient close to +1 or -1).

12.2 PROPHET MODEL

12.2 Prophet model. 12.2. Prophet model. A recent proposal is the Prophet model, available via the fable.prophet package. This model was introduced by Facebook ( S. J. Taylor & Letham, 2018), originally for forecasting daily data with weekly and yearly seasonality, plus holiday effects. It was later extended to cover more types of seasonal

data.

12.7 VERY LONG AND VERY SHORT TIME SERIES Forecasting very short time series. We often get asked how few data points can be used to fit a time series model.As with almost all sample size questions, there is no easy answer. It depends on the number of model parameters to be estimated and the amount of randomness in the data.The sample size required increases with the number of parameters to be estimated, and the amount of noise in the FORECASTING: PRINCIPLES AND PRACTICE (2ND ED) Preface. This is the second edition of Forecasting: Principles & Practice, which uses the forecast package in R. The third edition, which uses the fable package, is also available.. Buy a print or downloadable version. Welcome to our online textbook on forecasting. This textbook is intended to provide a comprehensive introduction to forecasting methods and to present enough information 13.8 FORECASTING ON TRAINING AND TEST SETS 13.8 Forecasting on training and test sets. Typically, we compute one-step forecasts on the training data (the “fitted values”) and multi-step forecasts on the test data. However, occasionally we may wish to compute multi-step forecasts on the training data, or one-step forecasts on the test data. 7.4 SOME USEFUL PREDICTORS Note that trend() and season() are not standard functions; they are “special” functions that work within the TSLM() model formulae.. There is an average downward trend of -0.34 megalitres per quarter. On average, the second quarter has production of 34.7 megalitres lower than the first quarter, the third quarter has production of 17.8 megalitres lower than the first quarter, and the fourth 8.7 ARIMA MODELLING IN R The auto.arima () function in R uses a variation of the Hyndman-Khandakar algorithm ( Hyndman & Khandakar, 2008), which combines unit root tests, minimisation of the AICc and MLE to obtain an ARIMA model. The arguments to auto.arima () provide for many variations on the algorithm. What is described here is the default

behaviour.

8.10 ARIMA VS ETS

8.10 ARIMA vs ETS. It is a commonly held myth that ARIMA models are more general than exponential smoothing. While linear exponential smoothing models are all special cases of ARIMA models, the non-linear exponential smoothing models have no equivalent ARIMA counterparts. 8.9 季节性ARIMA模型 8.9 季节性ARIMA模型. 8.9. 季节性ARIMA模型. 之前我们一直将注意力集中在非季节性的数据和非季节性的 ARIMA 模型上。. 然而，ARIMA 模型同样可以用于很多季节性数据的建模。. 季节性的 ARIMA模型 在我们之前讨论的 ARIMA 模型多项式中引入了季节性的项。. 它可以表示 Toggle navigation OTexts

*

* Books available

* About

* Contact

------------------------- Online - Open-Access - Textbooks

BOOKS AVAILABLE

------------------------- FORECASTING: PRINCIPLES AND PRACTICE BY ROB J HYNDMAN AND GEORGE ATHANASOPOULOS _2ND EDITION, MAY 2018_ A comprehensive introduction to the latest forecasting methods using the FORECAST package for R. Examples use R with many data sets taken from the authors' own consulting experience. In this second edition, all chapters have been updated to cover the latest research, and three new chapters have been added on dynamic regression forecasting, hierarchical forecasting and practical forecasting issues.

Read online Buy

FORECASTING: PRINCIPLES AND PRACTICE BY ROB J HYNDMAN AND GEORGE ATHANASOPOULOS _3RD EDITION, MAY 2021_ A comprehensive introduction to the latest forecasting methods using the FABLE package for R. Examples use R with many data sets taken from the authors' own consulting experience. In this third edition, all chapters have been updated to cover the latest research, and a new chapter has been added on time series features.

Read online

ABOUT

------------------------- OTexts is an online textbook publisher. Our books are freely available to students everywhere, whether for self-learning or as part of a

university courses.

For now, we are only hosting books written by Professor Rob J Hyndman

and coauthors.

CONTACT

-------------------------

Name

Email Address

Phone Number

Message

Send

(c) OTexts, Melbourne, Australia, 2018.

__