Automatic algorithms for time series forecasting
I will describe some algorithms for automatically forecasting univariate time series that have been developed over the last 20 years. The role of forecasting competitions in comparing the forecast accuracy of these algorithms will also be discussed.Automatic algorithms for time series forecasting
D:\搜狗高速下载\时间序列预测的自动化算法 Automatic algorithms for time series forecasting_ by Rob Hyndman 幻灯片 重要2015.pdf
Automatic Time Series Forecasting
(D:\搜狗高速下载\时间序列 预测 代码工具 Automatic Time Series Forecasting The forecast Package for R Hyndman.pdf)Automatic time series forecasting
the Comprehensive R Archive Network at
http://CRAN.R-project.org/package=forecasting.hts: Hierarchical and grouped time series
Methods for analysing and forecasting hierarchical and grouped time series
(https://github.com/robjhyndman/hts/issues)
I will describe some algorithms for automatically forecasting univariate time series that have been developed over the last 20 years. The role of forecasting competitions in comparing the forecast accuracy of these algorithms will also be discussed.
2.1. Point forecasts for all methods
2.2. Innovations state space models
2.3. State space models for all exponential smoothing methods 6
2.4. Estimation
7
2.5. Model selection 7
2.6. Automatic forecasting 8
using a restricted set of exponential smoothing models, and demonstrated that the methodology is
particularly good at short term forecasts (up to about 6 periods ahead), and especially for
seasonal short-term series (beating all other methods in the competitions for these series).
3. ARIMA models
3.1. Choosing the model order using unit root tests and the AIC
3.2. A step-wise procedure for traversing the model space
Consequently, it is often not feasible to simply fit every potential model and choose the one with the lowest AIC. Instead, we need a way of traversing the space of models efficiently in order to arrive at the model with the
lowest AIC value.
We propose a step-wise algorithm as follows. 11
The algorithm is guaranteed to return a valid model because the model space is finite and at least one of the starting models will be accepted (the model with no AR or MA parameters).The selected model is used to produce forecasts.
3.3. Comparisons with exponential smoothing 11
The exponential smoothing state space models are all non-stationary。The philosophy of exponential smoothing is that the world is non-stationary.So if a stationary model is required, ARIMA models are better.
One advantage of the exponential smoothing models is that they can be non-linear. So
time series that exhibit non-linear characteristics including heteroscedasticity may be better
modelled using exponential smoothing state space models.
这样的时间序列显示出非线性特性，包括异使用指数平滑状态空间模型可以更好地来模拟。
It may be thought that the larger model class is advantageous.
The smaller exponential smoothing class is sufficiently rich to capture the dynamics of almost all
real business and economic time series.
较小的指数平滑类是足够丰富捕获几乎所有的动态真正的商业和经济时间序列。
4.4. The forecast() function
We also define a default predict() method which is used when no existing predict() function exists, and calls the relevant forecast() function. Thus, predict() methods parallel forecast() methods, but the latter provide consistent output that is more useable.
4.5. The forecast class
4.6. Other functions
croston() implements Croston’s (1972) method for intermittent demand forecasting。
theta() provides forecasts from the Theta method (Assimakopoulos and Nikolopoulos 2000).
meanf() returns forecasts based on the historical mean.
rwf() gives “na¨ıve” forecasts equal to the most recent observation assuming a random walk model. This function also allows forecasting using a random walk with drift.
tsdisplay() provides a time plot along with an ACF and PACF.
seasonplot() produces a seasonal plot as described in Makridakis et al. (1998).
4. The forecast package
4.1. Implementation of the automatic exponential smoothing algorithm
Although there is a lot of computation involved, it can be handled remarkably quickly on
modern computers. Each of the forecasts shown in Figure 1 took no more than a few seconds
on a standard PC. The US electricity generation series took the longest as there are no analytical prediction intervals available for the ETS(M,Md,N) model. Consequently, the prediction intervals for this series were computed using simulation of 5000 future sample paths.
Although it is nowhere documented, it appears that the prediction intervals produced by predict() for an object of class HoltWinters are based on an equivalent ARIMA model in the case of the (N,N), (A,N) and (A,A) methods, assuming additive errors.
虽然它是不通记载，看来由预测（）用于类HoltWinters的目的所产生的预测间隔是基于在第（N，N），（A，N）和的情况下的等效ARIMA模型（A，A）的方法，假设附加误差。
These prediction intervals are equivalent to the prediction intervals that arise from the (A,N,N), (A,A,N) and (A,A,A) state space models.
这些预测区间是相当于起源于（A，N，N），（A，A，N）和（A，A，A）状态空间模型的预测区间。
For the (A,M) method, the prediction interval provided by predict() appears to be based on Chatfield and Yar (1991) which is an approximation to the true prediction interval arising from the (A,A,M) model. Prediction intervals with multiplicative errors are not possible using the HoltWinters() function.
对于（A，M）的方法，通过预测提供的预测区间（）似乎是基于查特菲尔德和亚尔（1991年），这是一个近似的（A，A，M）模型所产生的真正的预测区间。预测区间与乘法错误是不可能使用HoltWinters（）函数。
Prediction intervals with multiplicative errors are not possible using the HoltWinters() function.
预测区间与乘法错误是不可能使用HoltWinters（）函数。
4.2. The HoltWinters() function
4.3. Implementation of the automatic ARIMA algorithm
Functional time series with applications in demography
Abstract:Functional time series are curves that are observed sequentially in time, one curve being observed in each time period. In demography, examples include curves formed by annual death rates as a function of age, or annual fertility rates as a function of age. In finance, functional time series can occur in the form of bond yield curves, for example, with each curve being the yield of a bond as a function of the maturity of a bond.
I will discuss methods for describing, modelling and forecasting such functional time series data. Challenges include:
developing useful graphical tools (I will illustrate a functional version of the boxplot);
dealing with outliers (e.g., death rates have outliers in years of wars or epidemics);
cohort effects (how can we identify and allow for these in the forecasts);
synergy between groups (e.g, we expect male and female mortality rates to evolve in a similar way in the future, and we expect different types of yield curves to behave similarly over time);
deriving prediction intervals for forecasts;
how to combine mortality and fertility forecasts to obtain forecasts of the total population;
how to use these ideas to simulate the age-structure of future populations and use the results to analyse proposed government policies.
Lectures:
Tools for functional time series analysis [Slides]
Automatic time series forecasting [Slides]
Forecasting functional time series [Slides]
Connections, extensions and applications [Slides]
Forecasting functional time series via PLS [Slides]
Coherent functional forecasting [Slides]
Common functional principal components [Slides]
Stochastic population forecasting [Slides]