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Image of random data plus trend, with best-fit line and different smoothings


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The data is 1000 points, with a trend of 1-in-100, with random normal noise of SD 10 superimposed. The red-line is the same data but averaged every 10 points. The blue line is every 100 points.

The r2 fit for the raw data is 0.08; for the 10-pt-filtered, 0.57; for 100-pt-filtered, 0.97.

For all series, the least squares fit line is virtually the same, with a slope of 0.01, as expected.

Ignoring autocorrelation, a confidence limit for the fit line is [0.0082, 0.0127] for the raw data (which include 0.01, as it should). For the 10-pt-filtered the limits are slightly narrower at [0.0084, 0.0125] and for the 100pt-filtering the limits are again slightly narrower.

So what does that all mean?

  • for the raw data, the simple trend line explains almost none of the variance of the time series (only 8%).
  • for the 100-pt filtering, the trend line explains almost all of the data (97%).
  • nonetheless, the trend lines are almost identical as are the confidence levels.

The time series are, of course, very closely related: the same except for the filtering. This shows that a low r2 value should not be interpreted as evidence of lack of trend.

[edit] Source code

Source id in IDL. pp_regress and reg_explain not given.

n=1000

data=10*randomn(seed,n)+indgen(n)/100.
y=indgen(n)
y1=y(indgen(n/10)*10+5)
y2=y(indgen(n/100)*100+5*10)

ret=pp_regress(y,data)
print,reg_explain(ret)

data1=reform(data,10,n/10)
data1=avg(data1,0)

ret1=pp_regress(y1,data1)
print,reg_explain(ret1)

data2=reform(data,100,n/100)
data2=avg(data2,0)

ret2=pp_regress(y2,data2)
print,reg_explain(ret2)

plot,y,data,yr=[-20,30]
pp_regress_plot,ret,th=3

oplot,y1,data1,col=2,th=3
oplot,y2,data2,col=3,th=3
date/time username edit summary
21:25, 20 December 2004 en:User:Quadell (tagged)
22:13, 14 August 2004 en:User:Danakil (fmt)
21:17, 14 August 2004 en:User:William M. Connolley (Add code.)
14:05, 12 August 2004 en:User:William M. Connolley (I bumped up the SD to make the point obvious.)
14:00, 12 August 2004 en:User:William M. Connolley (Comments)
13:50, 12 August 2004 en:User:William M. Connolley (...partial before reload)
13:32, 12 August 2004 en:User:William M. Connolley (Image of random data plus trend, with best-fit line and different smoothings)

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