SMHS TimeSeriesAnalysis - Revision history

Tdlee: Text replacement - "{{translate|pageName=http://wiki.stat.ucla.edu/socr/" to ""{{translate|pageName=http://wiki.socr.umich.edu/"

2020-03-03T19:36:06Z

Text replacement - "{{translate|pageName=http://wiki.stat.ucla.edu/socr/" to ""{{translate|pageName=http://wiki.socr.umich.edu/"

Pineaumi: /* Footnotes */

2016-05-24T14:29:03Z

‎Footnotes

Imoubara: /* Exponential Models */

2016-05-23T13:49:44Z

‎Exponential Models

Imoubara: /* Seasonal Decomposition */

2016-05-23T13:49:32Z

‎Seasonal Decomposition

Imoubara: /* Seasonal ARIMA models */

2016-05-23T13:47:59Z

‎Seasonal ARIMA models

Dinov: /* Non-seasonal ARIMA models */

2016-05-21T17:55:36Z

‎Non-seasonal ARIMA models

Dinov: /* Non-seasonal ARIMA models */

2016-05-21T17:55:16Z

‎Non-seasonal ARIMA models

Dinov at 17:54, 21 May 2016

2016-05-21T17:54:24Z

Dinov: /* Non-seasonal ARIMA models */

2016-05-21T17:54:03Z

‎Non-seasonal ARIMA models

Dinov at 17:53, 21 May 2016

2016-05-21T17:53:32Z

@@ Line 524: / Line 524: @@
 <hr>
 * SOCR Home page: http://www.socr.ucla.edu
-{{translate|pageName=http://wiki.stat.ucla.edu/socr/index.php?title=SMHS_TimeSeriesAnalysis}}
+"{{translate|pageName=http://wiki.socr.umich.edu/index.php?title=SMHS_TimeSeriesAnalysis}}

@@ Line 517: / Line 517: @@
 * <sup>2</sup> https://umich.instructure.com/courses/38100/files
 * <sup>3</sup> https://cran.r-project.org/web/packages/mgcv/mgcv.pdf
 ==See also==

@@ Line 405: / Line 405: @@
 ====Exponential Models====
-<li>The '''HoltWinters()''' function ('''stats''' package), and the '''ets()''' function ('''forecast''' package) can fit exponential models.</li>
+*The '''HoltWinters()''' function ('''stats''' package), and the '''ets()''' function ('''forecast''' package) can fit exponential models.
   # simple exponential - models level
   fit_HW <- HoltWinters(sim_ts, beta=FALSE, gamma=FALSE)

@@ Line 386: / Line 386: @@
 ====Seasonal Decomposition====
-<li>The additive and seasonal trends, and irregular components, of time-series may be decomposed using the stl() function. Series with multiplicative effects can by transformed into series with additive effects through a log transformation (i.e., '''ln_sim_ts <- log(sim_ts)).</li>
+*The additive and seasonal trends, and irregular components, of time-series may be decomposed using the stl() function. Series with multiplicative effects can by transformed into series with additive effects through a log transformation (i.e., '''ln_sim_ts <- log(sim_ts)).
   # Seasonal decomposition
   fit_stl <- stl(sim_ts, s.window="period")   '''# Seasonal Decomposition of Time Series by Loess'''

@@ Line 445: / Line 445: @@
 If 2 of the 3 terms are trivial, the model is abbreviated using the non-zero parameter, skipping the "AR", "I" or "MA" from the acronym. For example,
-<li>ARIMA(1,0,0) = AR(1), a stationary and auto-correlated series can be predicted as a multiple of its own previous value, plus a constant. $X_t=μ + φ_1 × X_{t-1}+ \epsilon_t.$ Note that $ε_t=X_t-\hat{X}_t.$</li>
+*ARIMA(1,0,0) = AR(1), a stationary and auto-correlated series can be predicted as a multiple of its own previous value, plus a constant. $X_t=μ + φ_1 × X_{t-1}+ \epsilon_t.$ Note that $ε_t=X_t-\hat{X}_t.$
-<li>An ARIMA(0,1,0) = I(1) model, not stationary series, a limiting case of an AR(1) model, the auto-regressive coefficient is equal to 1, i.e., a series with infinitely slow mean reversion,  $X_t=μ+X_{t-1}+ε_t,$ a 1-step random walk.</li>
+*An ARIMA(0,1,0) = I(1) model, not stationary series, a limiting case of an AR(1) model, the auto-regressive coefficient is equal to 1, i.e., a series with infinitely slow mean reversion,  $X_t=μ+X_{t-1}+ε_t,$ a 1-step random walk.
 <B>For more complex models:</B>
-<li>An ARIMA(1,1,0), differenced first-order auto-regressive model.  $X_t=μ+X_{t-1}+α×(X_{t-1}-X_{t-2})+ε_t.$  </li>
+*An ARIMA(1,1,0), differenced first-order auto-regressive model.  $X_t=μ+X_{t-1}+α×(X_{t-1}-X_{t-2})+ε_t.$
-<li>An ARIMA(0,2,2) model is given by $X_t=2X_{t-1}-X_{t-2}+α×ε_{t-1}+β×ε_{t-2}+ ε_t,$ where $α$ and $β$ are the MA(1) and MA(2) coefficients (sometimes these are defined with negative signs). This is a general linear exponential smoothing model that uses exponentially weighted moving averages to estimate both a local level and a local trend in the series.  The long-term forecasts from this model converge to a straight line whose slope depends on the average trend observed toward the end of the series.
+*An ARIMA(0,2,2) model is given by $X_t=2X_{t-1}-X_{t-2}+α×ε_{t-1}+β×ε_{t-2}+ ε_t,$ where $α$ and $β$ are the MA(1) and MA(2) coefficients (sometimes these are defined with negative signs). This is a general linear exponential smoothing model that uses exponentially weighted moving averages to estimate both a local level and a local trend in the series.  The long-term forecasts from this model converge to a straight line whose slope depends on the average trend observed toward the end of the series.
-<li>ARIMA(1,1,2), $X_t=μ+X_{t-1}+(X_{t-1}+X_{t-2})+α×ε{t}+β×ε_{t-1}$</li>
+*ARIMA(1,1,2), $X_t=μ+X_{t-1}+(X_{t-1}+X_{t-2})+α×ε{t}+β×ε_{t-1}$
 The '''arima'''() function ('''stats''' package) can be used to fit an <b><u>auto-regressive integrated moving averages</u></b> model. Other useful functions include:

@@ Line 435: / Line 435: @@
 The Non-seasonal ARIMA models are denoted by ARIMA(p, d, q), where parameters p, d, and q are positive integers,
 * p = order of the auto-regressive model,
-* d = degree of differencing, when ''d''=2, the '''''d<sup>th</sup>'' difference''' is
+* d = degree of differencing, when ''d''=2, the '''''d<sup>th</sup>'' difference''' is $(X_t-X_{t-1})-(X_{t-1}-X_{t-2})= X_t-2X_{t-1}+X_{t-2}$. That is, the second difference of ''X'' (d=2) is not the difference between the current period and the value 2 periods ago.  It is the first-difference-of-the-first difference, the discrete analog of a second derivative, representing the local acceleration of the series rather than its local trend (first derivative).
 * q = order of the moving-average model.

@@ Line 428: / Line 428: @@
 There are 2 types of ARIMA time-series models: <BR>
-$$ X_t= \mu+ \underbrace{\sum_{i=1}^{p}{φ_iX_{t-i}}}_\text{auto-regressive (p) part} +
+$ X_t= \mu+ \underbrace{\sum_{i=1}^{p}{φ_iX_{t-i}}}_\text{auto-regressive (p) part} +
 \underbrace{\sum_{j=1}^{q}{θ_jε_{t-j}}}_\text{moving-average (q) part} +
-\underbrace{ ε_t }_\text{error term}.$$
+\underbrace{ ε_t }_\text{error term}.$
 ====Non-seasonal ARIMA models====

@@ Line 423: / Line 423: @@
   # clear plot:
   # dev.off()
 <center>[[Image:SMHS_TimeSeries18.png|500px]]</center>
 ===Auto-regressive Integrated Moving Average (ARIMA) Models<sup>4</sup> ===
 There are 2 types of ARIMA time-series models: <BR>
-'''1) Non-seasonal ARIMA models''', ARIMA(p, d, q), where parameters p, d, and q are positive integers,
+====Non-seasonal ARIMA models====
 <blockquote>
-<li>p = order of the auto-regressive model, </li>
+* p = order of the auto-regressive model,
-<li>d = degree of differencing, when ''d''=2, the '''''d<sup>th</sup>'' difference''' is</li>
+* d = degree of differencing, when ''d''=2, the '''''d<sup>th</sup>'' difference''' is
 $(X_t-X_{t-1})-(X_{t-1}-X_{t-2})= X_t-2X_{t-1}+X_{t-2}$
@@ Line 442: / Line 438: @@
 <li>q = order of the moving-average model.</li></blockquote>
-'''2) Seasonal ARIMA models''', ''ARIMA(p, d, q)(P, D, Q)<sub>m</sub>,''
+====Seasonal ARIMA models====
-<blockquote>
+The Seasonal AMIMA models are denoted by ''ARIMA(p, d, q)(P, D, Q)<sub>m</sub>,''
-<li>m = number of periods in each season, </li>
+* m = number of periods in each season,
-<li>uppercase P, D, Q represent the auto-regressive, differencing, and moving average terms for the seasonal part of the ARIMA model, and the lower case (p,d,q) are as with non-seasonal ARIMA. </li></blockquote>
+* uppercase P, D, Q represent the auto-regressive, differencing, and moving average terms for the seasonal part of the ARIMA model, and the lower case (p,d,q) are as with non-seasonal ARIMA.
 If 2 of the 3 terms are trivial, the model is abbreviated using the non-zero parameter, skipping the "AR", "I" or "MA" from the acronym. For example,
@@ Line 460: / Line 455: @@
 <li>ARIMA(1,1,2), $X_t=μ+X_{t-1}+(X_{t-1}+X_{t-2})+α×ε{t}+β×ε_{t-1}$</li>
 The '''arima'''() function ('''stats''' package) can be used to fit an <b><u>auto-regressive integrated moving averages</u></b> model. Other useful functions include:
+* lag(sim_ts, k) &nbsp;&nbsp;&nbsp;&nbsp; lagged version of time series, shifted back k observations
-<li>lag(sim_ts, k) &nbsp;&nbsp;&nbsp;&nbsp; lagged version of time series, shifted back k observations</li>
+* diff(sim_ts, differences=d) &nbsp;&nbsp;&nbsp;&nbsp; difference the time series d times
+* ndiffs(sim_ts) &nbsp;&nbsp;&nbsp;&nbsp; Number of differences required to achieve stationarity (from the forecast package)
-<li>diff(sim_ts, differences=d) &nbsp;&nbsp;&nbsp;&nbsp; difference the time series d times</li>
+* acf(sim_ts) &nbsp;&nbsp;&nbsp;&nbsp; auto-correlation function
+* pacf(sim_ts) &nbsp;&nbsp;&nbsp;&nbsp; partial auto-correlation function
-<li>ndiffs(sim_ts) &nbsp;&nbsp;&nbsp;&nbsp; Number of differences required to achieve stationarity (from the forecast package)</li>
+* adf.test(sim_ts) &nbsp;&nbsp;&nbsp;&nbsp; Augmented Dickey-Fuller test. Rejecting the null hypothesis suggests that a time series is stationary (from the tseries package)</li>
+* Box.test(x, type="Ljung-Box") &nbsp;&nbsp;&nbsp;&nbsp; Portmanteau test that observations in vector or time series x are independent.
 The '''forecast''' package has alternative versions of '''acf()''' and '''pacf()''' called '''Acf()''' and '''Pacf()''' respectively.
 &#35; fit an '''ARIMA(P, D, Q) model''' of order:
-<li>P, represents the AR order</li>
+ fit_arima1 <- arima(sim_ts, order=c(3, 1, 2))
   # predictive accuracy
   library(forecast)
@@ Line 517: / Line 498: @@
   fit_ets$\$$aic; fit_arima2$\$$aic
   accuracy(fit_ets); accuracy(fit_arima2);
 '''Akaike’s Information Criterion (AIC)''' = ''-2Log(Likelihood)+2p,'' where ''p'' is he number of estimated parameters.
   summary(fit_ets); summary(fit_arima2)
 ACF plot of the residuals from the ARIMA(3,1,2) model shows all correlations within the threshold limits indicating that the residuals are behaving like white noise. A portmanteau test returns a large p-value, also suggesting the  residuals are white noise.
   &#35; acf computes (and by default plots) estimates of the autocovariance or autocorrelation function
   acf(residuals(fit_ets))
@@ Line 536: / Line 514: @@
   &#35; more on ARIMA https://www.otexts.org/fpp/8/7
+===Footnotes===
+* <sup>1</sup> https://umich.instructure.com/files/689861/download?download_frd=1
-<sup>1</sup> https://umich.instructure.com/files/689861/download?download_frd=1
+* <sup>2</sup> https://umich.instructure.com/courses/38100/files
+* <sup>3</sup> https://cran.r-project.org/web/packages/mgcv/mgcv.pdf
 ==See also==