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近期要写一个arima模型的预测功能，假设现在已经求出三个参数，比如说是ARIMA(2,1,2)，具体的预测过程是怎样的？
我是要用java自己实现这个功能，不可以调用R包，也不可以用Eviews软件来计算。
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关于这里我首先得诚实地说一下, 目前我个人用的ARIMA的包都是R里面写好的. 我自己没有亲手写过计算的代码.
但是我进行了相关搜索. 如果只是AR部分的模型,用最小二乘就可以了.但是MA部分的误差项是具有相关性的, 所以用最小二乘不理想. 看到的几篇材料里说的是用极大似然.
我也去找到了一篇公式比较详尽的材料.希望能够帮助到你
http://math.usask.ca/~laverty/S349/S349Lectures/S349%2008.ppt
这个非常简单。公式我就不贴了。
请参照Duke的网页：http://people.duke.edu/~rnau/411arim.htm
具体思想就是将k+1期展开为第k期及之前期的展开式， 然后就搞掂啦！
jz.mraz 发表于
这个非常简单。公式我就不贴了。
请参照Duke的网页：http://people.duke.edu/~rnau/411arim.htm
具体思想 ...你所说的公式是下面这个吗？
ARIMA(1,1,2) without constant = damped-trend linear exponential smoothing:
Ŷt& &=& &Yt-1&&+&&ϕ1 (Yt-1 - Yt-2 ) - θ1et-1 - θ1et-1
可是里面的参数像ϕ1，θ1要怎么求解呢？
silent_strings 发表于
你所说的公式是下面这个吗？
ARIMA(1,1,2) without constant = damped-trend linear exponential smooth ...关于这里我首先得诚实地说一下, 目前我个人用的ARIMA的包都是R里面写好的. 我自己没有亲手写过计算的代码.
但是我进行了相关搜索. 如果只是AR部分的模型,用最小二乘就可以了.但是MA部分的误差项是具有相关性的, 所以用最小二乘不理想. 看到的几篇材料里说的是用极大似然.
我也去找到了一篇公式比较详尽的材料.希望能够帮助到你
http://math.usask.ca/~laverty/S349/S349Lectures/S349%2008.ppt
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论坛法律顾问：王进律师Moving average and exponential smoothing models
&(Here and elsewhere I will use the symbol
“Y-hat” to stand for a forecast of the time series Y made at the
earliest possible prior date by a given model.) This average is centered at period t-(m+1)/2, which
implies that the estimate of the local mean will tend to lag behind the true
value of the local mean by about (m+1)/2 periods. Thus, we say the average age of the data in the simple
moving average is (m+1)/2 relative to the period for which the forecast is
computed: this is the amount of time by which forecasts will tend to lag
behind turning points in the data. For example, if you are averaging the
last 5 values, the forecasts will be about 3 periods late in responding to
turning points. Note that if m=1, the simple moving average (SMA) model is
equivalent to the random walk model (without growth). If m is very large
(comparable to the length of the estimation period), the SMA model is
equivalent to the mean model. As with any parameter of a forecasting model, it
is customary to adjust the value of k in order to obtain the best &fit&
to the data, i.e., the smallest forecast errors on average.
The random
walk model responds very quickly to changes in the series, but in so doing it
picks much of the &noise& in the data (the random fluctuations) as
well as the &signal& (the local mean). If we instead try a simple
moving average of 5 terms, we get a smoother-looking set of forecasts:
The 5-term
simple moving average yields significantly smaller errors than the random walk
model in this case. The average age of the data in this forecast is 3
(=(5+1)/2), so that it tends to lag behind turning points by about three
periods. (For example, a downturn seems to have occurred at period 21, but the
forecasts do not turn around until several periods later.)
The confidence
limits computed by Statgraphics for the long-term forecasts of the simple
moving average do not get wider as the forecasting horizon increases.
This is obviously not correct! Unfortunately, there is no underlying
statistical theory that tells us how the confidence intervals ought to widen
for this model.& However, it is not
too hard to calculate& empirical
estimates of the confidence limits for the longer-horizon forecasts. For
example, you could set up a spreadsheet in which the SMA model would be used to
forecast 2 steps ahead, 3 steps ahead, etc., within the historical data sample.
You could then compute the sample standard deviations of the errors at each
forecast horizon, and then construct confidence intervals for longer-term
forecasts by adding and subtracting multiples of the appropriate standard
deviation.
a 9-term simple moving average, we get even smoother forecasts and more of a
lagging effect:
average age is now 5 periods (=(9+1)/2). If we take a 19-term moving average,
the average age increases to 10:
that, indeed, the forecasts are now lagging behind turning points by about 10
amount of smoothing is best for this series?& Here is a table that compares their
error statistics, also including a 3-term average:
Equivalently, we can express the next
forecast directly in terms of previous forecasts and previous observations, in
any of the following equivalent versions.&
In the first version, the forecast is an interpolation between previous forecast
and previous observation:
In the second version, the next forecast is
obtained by adjusting the previous forecast in the direction of the
previous error by a fractional amount
is the error made at time t.& In the third version,& the forecast is an exponentially
weighted (i.e. discounted) moving average with discount factor 1-α:
The interpolation version of the
forecasting formula is the simplest to use if you are implementing the model on
a spreadsheet: it fits in a single cell and contains cell references pointing
to the previous forecast, the previous observation, and the cell where the
value of α is stored.
if α=1, the SES model
is equivalent to a random walk model (without growth). If α=0, the SES model is equivalent to the mean
model, assuming that the first smoothed value is set equal to the mean.&
average age of the data in the simple-exponential-smoothing forecast is 1/α relative to the
period for which the forecast is computed. (This is not supposed to be obvious,
but it can easily be shown by evaluating an infinite series.) Hence, the simple
moving average forecast tends to lag behind turning points by about 1/α periods. For example, when α=0.5 the lag is 2 when α=0.2 the lag is 5 when α=0.1 the lag is 10 periods, and so on.
given average age (i.e., amount of lag), the simple exponential smoothing (SES)
forecast is somewhat superior to the simple moving average (SMA) forecast
because it places relatively more weight on the most recent observation--i.e.,
it is slightly more &responsive& to changes occuring in the recent
past.& For example, an SMA model with 9 terms and an SES model
with α=0.2 both have an
average age of 5 for the data in their forecasts, but the SES model
puts more weight on the last 3 values than does the SMA model and at the same
time it doesn’t entirely “forget” about values more than 9
periods old, as shown in this chart:CitationsSee all >15 References30.51 · University of Strathclyde25.22 · University of HoustonAbstractOver the past twenty years, damped trend exponential smoothing has performed well in numerous empirical studies, and it is now well established as an accurate forecasting method. The original motivation for this method was intuitively appealing, but said very little about why or when it provided an optimal approach. The aim of this paper is to provide a theoretical rationale for the damped trend method based on Brown’s original thinking about the form of underlying models for exponential smoothing. We develop a random coefficient state space model for which damped trend smoothing provides an optimal approach, and within which the damping parameter can be interpreted directly as a measure of the persistence of the linear trend.Discover the world's research12+ million members100+ million publications700k+ research projects
&Damped-Trend Exponential Smoothing assumes level and damped trend parameters. This model is appropriate for series with a linear trend that is dying out with no seasonality (Gardner and Mckenzie 1985). Its smoothing parameters are level, trend and damping trend. &ABSTRACT: ABSTRACT: During the process of evolution, changing lifestyle introduces new elements like accidents
effect mortality. The incidence of accidental deaths has shown a mixed trend with an increase of 51.8 percent
in the year 2012. From National Crime Report Bureau-2012 data this paper tries to investigate the statistical
model which method give best estimated value of forecasting accidental death. Time series model--Various
Exponential Smoothing and Auto Regressive Integrated Moving Average (ARIMA) tec it
is found that the trend of accidental cases showing a increasing pattern. The forecast value show that 438811
numbers of deaths may be in the year 2015 with 32.5 rate of increment as compared to 2006, if the rate will
be constant and there will be no change in patterns of mortality. So, on validation of Models, ARIMA
performed better than the DTES. This will be help for policy maker to control such type of incidence in future.
KEYWORDS: Damped trend Exponential Smoothing (DTES), Auto Regressive Integrated Moving Average
(ARIMA), Mean Absolute Percentage Error (MAPE), Bayesian Information Criterion (BIC), Root Mean
Square Error (RMSE) Full-text · Article · Apr 2015 &A damping parameter (φ) is added in Holt's formula to give more control over trend extrapolation (Taylor, 2003). The result is a method stationary in first differences, rather than second differences as in the Holt method (Gardner & McKenzie, 2010a).The damped trend exponential smoothing expressions are as follows
&ABSTRACT: The aim of this study is to model energy consumption and Manufacturing Value Added (MVA) in the industry level of five South Asian countries. Firstly, a cross-sectional model was developed by using R-statistical software to estimate the MVA with energy consumption being the independent variable. Secondly, a twenty years data series was analyzed to forecast volume of energy consumption in the manufacturing industry for five countries in a comparative manner. Thus, a prediction model was developed by using the time series forecasting system of the SAS statistical software and evaluated using Mean Square Error (MSE), Root Mean Square Error (RMSE), Mean Absolute Error (MAE) and Mean Absolute Percent Error (MAPE) with forecasts made up to year 2021. The forecasted energy consumption data might be used in the cross-sectional model to forecast MVA. Besides, based on the increasing trends in volume of energy, industry should prepare now for using efficient and clean energy in order to achieve an environment friendly and sustainable manufacturing industry. Full-text · Article · Jan 2013 &Although the state space models of Hyndman et al (2008) provide a theoretical rationale for exponential smoothing methods, we prefer the RCSS models of McKenzie and Gardner (2009) on the grounds that they are more realistic. Hyndman et al (2008) show that damped trend exponential smoothing is optimal for a single source of error state space model with constant coefficients: &ABSTRACT: The damped trend method of exponential smoothing is a benchmark that has been difficult to beat in empirical studies of forecast accuracy. One explanation for this success is the flexibility of the method, which contains a variety of special cases that are automatically selected during the fitting process. That is, when the method is fitted, the optimal parameters usually define a special case rather than the method itself. For example, in the M3-competition time series, the parameters defined the damped trend method only about 43% of the time using local initial values for the method components. In the remaining series, a special case was selected, ranging from a random walk to a deterministic trend. The most common special case was a new method, simple exponential smoothing with a damped drift term. Full-text · Article · Jun 2011 ArticleJune 2011 · Journal of the Operational Research Society · Impact Factor: 0.95ArticleApril 2015 · Journal of Business Research · Impact Factor: 1.48ArticleMarch 1989 · Management Science · Impact Factor: 2.48ArticleSeptember 1988 · Journal of the Operational Research Society · Impact Factor: 0.95Data provided are for informational purposes only. Although carefully collected, accuracy cannot be guaranteed. Publisher conditions are provided by RoMEO. Differing provisions from the publisher's actual policy or licence agreement may be applicable.This publication is from a journal that may support self archiving.Last Updated: 22 Dec 16
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