Time series analysis and Spatial Statistics

1
Time series analysis and Spatial Statistics

• May 30 2005

2
Mauna Loa CO2 data (1958-2000)

• A Time Series is a list of observations of a
  variable (or variables) through time.
• Most analyses require equal spacing in time
• Usually violate assumptions of OLS and ANOVA.
• Adjacent observations are not independent
• In contrast to regression models, Time Series
  Analysis exploits temporal patterns in data.
• Major uses (1) analyze trends cycles (2)
  describe relationships within time series and (3)
  forecasting

3
Detrending

• Remove trends by differencing
• If trend in X is linear, then Y will be
  stationary no trend in time
• If trend in X is quadratic, then Y will have
  linear trend
• Difference Y to get a stationary variable
• Second order differencing of X

4
Seasonal Detrending

• Take differences between points one year apart
• For monthly data
• Takes into account effect of time of year by
  removing average seasonal cycle
• Could also do for daily, weekly, quarterly
  observations
• Other relevant cycles to treat as seasons
• Diurnal
• Tidal

5
Autocorrelation

• Autocorrelation function (ACF) measures
  correlation between points separated by a given
  amount of time (lag)
• Partial autocorrelation function measures
  residual correlations (after taking into account
  autocorrelation at lower lags)
• If X(t) is correlated with X(t-1), and X(t-1) is
  correlated with X(t-2), then X(t) will be
  correlated with X(t-2) just because of the lag 1
  correlations

6
ACFs of cyclic and nonstationary time series
7
ARIMA model

• Auto Regressive Integrated Moving Average
• Integrated means take differences
• Moving Average means replace X(t) with average
  of points nearby in time
• Need to specify orders for each of these
• of AR coefficients
• of times differenced
• Window for moving average
• Possible seasonality in each
• Choosing ARIMA model used to be a black art
• Now just try lots of possibilities and select one
  with lowest AIC

8
pH of Norwegian Lakes
9
Questions about Norwegian lake data

• Is there spatial autocorrelation in pH values?
• How can we interpolate and smooth those values?

10
Semivariogram

• Semivariance is half the average squared
  difference between pairs of lakes a certain
  distance apart
• Measures variance among sites as a function of
  distance
• Also called empirical variogram

11
Theoretical variogram
12
(No Transcript)
13
Kriging

• Minimum mean-squared-error method of spatial
  prediction
• Interpolates and smoothes
• Named after South African mining engineer D.G.
  Krige
• Uses a theoretical variogram
• Have to fit the theoretical variogram first
• Produces a smooth surface that fits the data

14
pH