Lecture 9: Smoothing and filtering data. Time series: smoothing, filtering, rejecting outliers, interpolation moving average, splines, penalized splines,

Lecture 9: Smoothing and filtering data

Time series:

smoothing, filtering, rejecting outliers, interpolationmoving average, splines, penalized splines, wavelets

autocorrelation in time seriesvariance increase, pattern generation;ar(), arima() …

Image data

-- OMS-- QCLS

sig=5x0=1:100; y0=1/(sig*sqrt(2*pi))*exp(-(x0-50)^2/(2*sig^2))plot(x0,y0,type="l",col="green",lwd=3,ylim=c(-.02,.1))#add noise to y0x=x0; y=y0+rnorm(100)/50points(x,y,pch=16,type="o”)

--- 5 pt moving average

--- 30 pt moving average

Some signal filtering concepts:

“What is” “feed-forward”?

More advanced filters.

Splines: Splines use a collection of basis functions (usually polynomials of order 3 or 4) to represent a functional form for the time series to be filtered. They are fitted piecewise, so that they are locally determined. We choose K points in the interior of the domain (“knots”) and subdivide into K+1 intervals.

spline of order m: piecewise m – 1 degree polynomial, continuous thru m – 2 derivatives.

Continuous derivatives gives a smooth function. More complex shapes emerge as we increase the degree of the spline and/or add knots.• Few knots/low degree: Functions may be too restrictive (biased) or smooth• Many knots/high degree: Risk of overfitting, false maxima, etc

Penalized Splines add a penalty for curvature, specifying the strength λ.(=0, regular spline/interpolation; = ∞, straight line, linear regression fit)

More advanced filters (continued).

Locally-weighted least-squares (“lowess”, “loess”): fit a polynomial (usually a straight line) to points in a sliding window, accepting as the smoothed value the central point on the line, with a taper to capture the ends. Points are usually weighted inversely as a function of distance, very often tri-cubic: (1 - |x|3)3 <in range -1,1 of the window>

Savitsky-Golay filter: Fits a polynomial of order n in a moving window, requiring that the fitted curve at each point have the same moments as the original data to order n-1. Partakes of lowess and penalized spline features. (Designed for integrating chromatographic peaks.) Nomencature: 4.11.11.0 ( n.nl.nr.o). Allows direct computation of the derivatives. Parameters are tabulated on the web or computed.

sig noisy_sig 10-point MA savgol.4.11.11.0 lowess pspline supsmu 0.010 0.012 NA NA 0.0117 0.012 0.0124 0.010 0.012 0.0132 0.0155 0.0117 0.012 0.0124

#Summary:#X Moving Average: crude, phase shift, peaks severely flattened, ends discarded <Don't use>

## Centered Moving Average: crude, peaks severely flattened, no phase shift*, feed forward >, ends discarded

## Block Averages: not too crude, not phase shifted*, no feed forward*, conserved properties*, information discarded (Maybe OK)

##Savitzky-Golay: not crude, not phase shifted*, small feed forward (localized), conserved properties, ends discarded; derivative

##locally weighted least squares (lowess/loess): not crude or phase shifted, nice taper at ends, no derivative

##supsmu: analytical properties murky, but a nice smoother for many signals; no derivative

##penalized splines: effective, differentiable; adjusting the parameters may be tricky

#Xregular splines: either false maxima, or oversmoothed--<Don't use>

Assessing different sources of variance: Extracting Trends, Cycles, etc

by Data Filtering and Conditional Averaging.

Measurement has low signal-to-noise ratio.

Measurement has high signal-to-noise ratio, but the system (e.g. the atmosphere) has a lot of variability.

EPS 236 Workshop: 2014

CO2

“Ancillary measurements”, conditional sampling and suitable filtering or averaging reveals the key features of the data when system variability is the key factor.

Zum=tapply(wlef[,"value"],list(wlef[,"yr"],wlef[,"mo"],wlef[,"hr"],wlef[,"ht(magl)"]),median,na.rm=T)

Noisy data: which filter is the “best” (for what purpose?)?Residuals? Events ?

Kalman filter

If spar is given:

Leave-one-out cross-validation

In the default mode, the sm.spline model is selected using “leave-one-out cross-validation”.See article by Rob Hyndman (http://robjhyndman.com/hyndsight/crossvalidation/) for a description.

Interpolation: linear (approx; predict.loess) penalized splines (akima’s aspline)

XX=HIPPO.1.1[lsel&l.uct,"UTC"]YY=HIPPO.1.1[lsel&l.uct,"CO2_OMS"]ZZ=HIPPO.1.1[lsel&l.uct,"CO2_QCLS"]YY[1379:1387] = NA

require(pspline)lna1=!is.na(YY)YY.i=approx(x=XX[lna1],y=YY[lna1],xout=XX)YY.spl=sm.spline(XX[lna1],YY[lna1])require(akima)YY.aspline= aspline(XX[lna1],YY[lna1],xout=XX)#YY.lowess=lowess(XX[lna1],YY[lna1],f=.1)ddd=data.frame(x=XX[lna1],y=YY[lna1])YY.loess=loess(y ~ x,data=ddd,span=.055)YY.loess.pred=predict(YY.loess,newdata=data.frame(x=XX,y=YY))

Minimize CV for “best” model

“Leave-one-out” CVSource: http://robjhyndman.com/hyndsight/crossvalidation/