Filter to Reveal Trend

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Class Web Page

1. Moving Average Filter

For your choice of \(m\), \[ M_t = \frac{1}{2q+1} \sum_{i=-q}^q X_{t+i} \]

This will filter the noise out, exposing overall trend.

It could also be \[ M_t = \frac{1}{q} \sum_{i=0}^q X_{t-i} \]

  set.seed(235)                         # set seed for RS generation (you don't need to do this)
  X <- abs(round(rnorm(100)*10, 0))

  library(forecast)                     # install.packages("forecast")
  M <- ma(X, 3)                         # This order=3 goes from t-1 to t+1
  head(cbind(X,M))

## Time Series:
## Start = 1 
## End = 6 
## Frequency = 1 
##    X         M
## 1 21        NA
## 2  3 10.333333
## 3  7  9.333333
## 4 18  8.666667
## 5  1 11.000000
## 6 14 10.000000

  plot(X, type="o")
  lines(M, col="red")

  plot(X, type="o")
  lines(ma(X, 5), col="red")

2. Exponential Smoothing

For your choice of \(\alpha\), \((0<\alpha<1)\), let \(S_0=X_0\) and \[ S_t = \alpha X_t + (1-\alpha) S_{t-1} \hspace{10mm} \mbox{ for } t>0. \]

This can be a one step forecast: \[ \hat X_{t+1} = S_t \]

  set.seed(235)                         # set seed for RS generation (you don't need to do this)
  X <- abs(round(rnorm(100)*10, 0))

  library(forecast)                         # install.packages("forecast")

  S <- ses(X, h=10, alpha=0.2, initial='simple')
  head(cbind(X, S$fitted))

## Time Series:
## Start = 1 
## End = 6 
## Frequency = 1 
##    X S$fitted
## 1 21  21.0000
## 2  3  21.0000
## 3  7  17.4000
## 4 18  15.3200
## 5  1  15.8560
## 6 14  12.8848

  plot(X, type="o")
  lines(S$fitted, col="red")