SMHS TimeSeries - Revision history

Dinov: /* Scientific Methods for Health Sciences - Time Series Analysis */

2026-04-09T18:45:29Z

‎Scientific Methods for Health Sciences - Time Series Analysis

Dinov: /* ARIMA Models */

2026-04-08T14:12:23Z

‎ARIMA Models

Dinov: /* ARIMA Models */

2026-04-08T14:12:03Z

‎ARIMA Models

Dinov: /* Modern ARIMA Modeling (Box-Jenkins Approach) */

2026-04-08T13:59:12Z

‎Modern ARIMA Modeling (Box-Jenkins Approach)

Dinov: /* Modern ARIMA Modeling (Box-Jenkins Approach) */

2026-04-07T16:08:40Z

‎Modern ARIMA Modeling (Box-Jenkins Approach)

Dinov: /* Modeling Seasonality and Trend= */

2026-04-07T15:13:10Z

‎Modeling Seasonality and Trend=

Dinov: /* Nonparametric Smoothing */

2026-04-07T15:12:49Z

‎Nonparametric Smoothing

Dinov: /* Problems */

2026-04-02T19:16:03Z

‎Problems

Dinov: /* Modern ARIMA Modeling (Box-Jenkins Approach) */

2026-04-02T19:14:38Z

‎Modern ARIMA Modeling (Box-Jenkins Approach)

Dinov: /* Modern ARIMA Modeling (Box-Jenkins Approach) */

2026-04-02T19:14:03Z

‎Modern ARIMA Modeling (Box-Jenkins Approach)

← Older revision		Revision as of 18:45, 9 April 2026
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	==[[SMHS\| Scientific Methods for Health Sciences]] - Time Series Analysis ==		==[[SMHS\| Scientific Methods for Health Sciences]] - Time Series Analysis ==
		+
		+	* See the [https://socr.umich.edu/DSPA2/DSPA2_notes/12_LongitudinalDataAnalysis.html '''DSPA2 Extensible Longitudinal Data Analysis Chapter'''].
		+	* [https://sda.statisticalcomputing.org/ The '''SOCR Statistical Data Analyzer (SDA)''' provides live app-based data and multiple tools for interactive time-varying data analytics].

	===Overview===		===Overview===

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 * Non-Seasonal Component: ''(p, d, q)'': These parameters handle the short-term relationships between consecutive observations.
 ** '''p''' (Autoregressive - AR): The number of lag observations included in the model. A value of 2 means the model uses the two previous time points to predict the next one.
 ** '''d''' (Degree of Differencing): The number of times the raw observations are differenced to make the data stationary (removing trends). A value of 1 means the model is looking at the change between points rather than the absolute values.

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 <math>\phi(B)(1-B)^d x_t = \theta(B)w_t</math>
 If seasonality is present, we extend this to the Seasonal ARIMA (SARIMA) model, which incorporates seasonal AR, MA, and differencing terms.
 ===Methods and Data Analysis===

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 library(forecast)
-dev.off()  # reset display, if necessary
+# dev.off()  # reset display, if necessary
 # auto.arima automatically identifies the best SARIMA model

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 fc <- forecast(fit, h=24)
 plot(fc, main="24-Month Forecast using SARIMA")
 </pre>

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 </pre>
-====Modeling Seasonality and Trend=====
+====Modeling Seasonality and Trend====
 =====Classical Decomposition=====

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 =====Nonparametric Smoothing=====
 * '''Loess (Local Polynomial Regression):''' Fits simple models to localized subsets of data. Controlled by the `span` parameter. A larger span yields a smoother curve.
-* '''Splines:** Piecewise polynomials joined smoothly at "knots." Controlled by degrees of freedom or a smoothing parameter (`spar`).
+* '''Splines: Piecewise polynomials joined smoothly at "knots." Controlled by degrees of freedom or a smoothing parameter (`spar`).
 <pre>

@@ Line 240: / Line 240: @@
 ) Consider a signal-plus-noise model of the general form <math>x_{t}=s_{t}+w_{t}</math>, where <math>w_{t}</math> is Gaussian white noise with <math>\sigma^{2}_{w} = 1</math>. Simulate and plot <math>n=200</math> observations from each of the following two models.
 (a) <math>x_{t} = s_{t}+w_{t}</math>, for <math>t=1,2,\dots,200</math>, where <math>s_{t}=\begin{cases} 0 & \text{ if } t=1,2,\dots,100 \\ 10\exp \left\{ -(t-100)/20 \right\} \cos\left( 2\pi t/4 \right) & \text{ if } t=101,102,\dots,200 \end{cases}</math>
 (b) <math>x_{t}=s_{t}+w_{t}</math>, for <math>t=1,2,\dots,200</math>, where <math>s_{t}=\begin{cases} 0 & \text{ if } t=1,2,\dots,100 \\ 10\exp \left\{ -(t-100)/200 \right\} \cos\left( 2\pi t/4 \right) & \text{ if } t=101,102,\dots,200 \end{cases}</math>
 (c) Compare the signal modulators <math>(a) \exp \left\{-t /20 \right\}</math> and <math>(b) \exp \left\{-t/200 \right\}</math>, for <math>t=1,2,\dots,100</math>. What is the primary difference in the persistence of the signal?
 ) (a) Generate <math>n=100</math> observations from the autoregression <math>x_{t}=-0.9 x_{t-2} + w_{t}</math>, with <math>\sigma_w=1</math>. Next, apply the moving average filter <math>v_{t}=\left(x_{t}+x_{t-1}+x_{t-2}+x_{t-3} \right)/4</math> to the data you generated. Now, plot <math>x_{t}</math> as a line and superimpose <math>v_{t}</math> as a dashed line. Comment on the behavior of <math>x_{t}</math> and how applying the moving average filter changes that behavior. [Hint: use <code>v <- filter(x, rep(1/4,4), sides=1)</code> for the filter].
 (b) Repeat (a) but with <math>x_{t}=\cos \left( 2 \pi t/4 \right)</math>.
 (c) Repeat (b) but with added <math>N(0,1)</math> noise, <math>x_{t}=\cos \left(2 \pi t/4 \right) + w_{t}</math>.
 (d) Compare and contrast the smoothing effects observed in <math>(a) - (c)</math>.
 ) For the two series, <math>x_{t}</math> in Problem 1 (a) and (b):
 (a) Compute and plot the mean functions <math>\mu_{x}(t)</math> for <math>t=1,2,\dots,200</math>.
 (b) Explain why these models are not stationary.
 ) Consider the time series <math>x_{t} = \beta_{1} + \beta_{2} t + w_{t}</math>, where <math>\beta_{1}</math> and <math>\beta_{2}</math> are known constants and <math>w_{t}</math> is a white noise process with variance <math>\sigma^{2}_{w}</math>.
 (a) Determine whether <math>x_{t}</math> is stationary. Justify your answer mathematically.
 (b) Show that the process <math>y_{t}=x_{t} - x_{t-1}</math> is stationary.
 (c) Show that the mean of the moving average <math>v_{t}= \frac{1}{2q+1} \sum_{j=-q}^{q} x_{t-j}</math> is <math>\beta_{1} + \beta_{2} t</math>, and provide a simplified expression for its autocovariance function.
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 ) Suppose we would like to predict a single stationary series <math>x_{t}</math> with zero mean and autocorrelation function <math>\gamma(h)</math> at some time in the future, say <math>t+l</math>, for <math>l>0</math>.
 (a) If we predict using only <math>x_{t}</math> and some scale multiple <math>A</math>, show that the mean-square prediction error <math>MSE(A)=E \left[ (x_{t+l}-A x_{t})^{2} \right]</math> is minimized by the value <math>A = \rho(l)</math>.
 (b) Show that the minimum mean-square prediction error is <math>MSE(A)= \gamma(0) \left[ 1- \rho^{2}(l) \right]</math>.
 (c) Show that if <math>x_{t+l} = Ax_{t}</math> perfectly, then <math>\rho(l) = 1</math> if <math>A > 0</math>, and <math>\rho(l) = -1</math> if <math>A < 0</math>.
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 ) Suppose <math>x_{1}, x_{2}, \dots, x_{n}</math> is a sample from the process <math>x_{t} = \mu + w_{t} - 0.8 w_{t-1}</math>, where <math>w_{t} \sim \text{WN}(0, \sigma^{2}_{w})</math>.
 (a) Show that the mean function is <math>E(x_{t})= \mu</math>.
 (b) Calculate the standard error of <math>\bar{x}</math> for estimating <math>\mu</math>.
 (c) Compare your result in (b) to the case where <math>x_{t}</math> is pure white noise. Is the standard error smaller or larger? Explain the result in the context of positive/negative autocorrelation.
 ) Although the model in Problem 1 (a) is not stationary, the sample ACF can still be computed and informative. For the data you generated in that problem, calculate and plot the sample ACF. Why does the ACF fail to decay to zero rapidly?
-) (a) Simulate a series of <math>n=500</math> Gaussian white noise observations and compute the sample ACF, <math>\hat{\rho}(h)</math>, to lag 20. Compare the sample ACF you obtain to the theoretical ACF <math>\rho(h)</math>. Do approximately 95% of the sample autocorrelations fall inside the bounds <math>\pm 2/\sqrt{500}</math>?
+)
 (b) Repeat part (a) using only <math>n=50</math>. How does changing <math>n</math> affect the variability of the sample ACF and the reliability of the confidence bounds?

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 . Diagnostic Checking: Ensure residuals are White Noise.
-The `forecast` package in `R` automates this efficiently:
+The forecast package in R automates this efficiently:
 <pre>