Browse code

new fiels

Michi authored on03/06/2012 16:52:14
Showing15 changed files
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new file mode 100644
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Binary files /dev/null and b/3/.RData differ
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+read.table("M100_spec.txt",header=TRUE)
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+spec = read.table("M100_spec.txt",header=TRUE)
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+
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+spec.fit <- lm(spec[175:746,2] ~ [175:746,1])
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+spec.fit <- lm(spec[175:746,2] ~ spec[175:746,1])
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+summary(spec.fit)
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+abline(spec.fit)
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+plot(spec[:,1],spec[:,2])
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+plot(spec[,1],spec[,2])
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+abline(spec.fit)
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+plot(spec[,1],spec[,2],type=l)
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+plot(spec[,1],spec[,2],l)
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+plot(spec[,1],spec[,2],'l')
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+abline(spec.fit)
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+x <- array(1:20, dim=c(4,5))
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+x
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+spec[:5,1]
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+spec[1:5,1]
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+spec[1:5,]
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+spec[2:5,]
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+spec
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+spec = read.table("M100_spec.txt",header=FALSE)
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+spec[1:3,]
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+spec.boolmask = 21 > spec[,2] >17
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+spec.boolmask =  (spec[,2] >17) 
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+spec.boolmask =  (spec[,2] >17) & (spec[,2] <20)
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+spec.boolmask
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+spec.fit <- lm(spec[spec.boolmask,2] ~ [spec.boolmask,1])
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+spec.boolmask
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+spec.fit <- lm(spec[spec.boolmask,2] ~ spec[spec.boolmask,1])
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+abline(spec.fit)
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+summery(spec.fit)
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+summary(spec.fit)
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+coef(spec.fit)[1]
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+spec.zero = spec - coef(spec.fit)[1]
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+spec.zero = spec
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+spec.zero[,2] = spec[,2] - coef(spec.fit)[1]
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+plot(spec[,1],spec.zero[,2],'l')
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+plot(spec[,1],spec.zero[,2],'l')
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+spec.zero = spec - coef(spec.fit)[1]
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+spec.zero
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+spec.notNull = (spec != 0)
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+spec.notNull
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+spec.notNull = (spec[,2] != 0)
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+spec.notNull
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+ls
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+plot(spec[spec.notNull,1],spec.zero[notNull,2],'l')
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+plot(spec[spec.notNull,1],spec.zero[spec.notNull,2],'l')
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+save()
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+save(fiel="flux2zero.R")
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+q()
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@@ -0,0 +1,1202 @@
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+# lambda Flux ((10^{-17} erg s^{-1}"+aaangs+"^{-1} cm^{-2}))
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+       6248.6000      0.00000
3
+       6249.1540      0.00000
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+       6249.7080      0.00000
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+       6250.2620      0.00000
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+       6250.8160      0.00000
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+       6251.3700      0.00000
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+       6251.9240      0.00000
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+       6252.4780      0.00000
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+       6253.0320      0.00000
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+       6253.5860      0.00000
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+       6254.1400      0.00000
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+       6254.6940      0.00000
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+       6255.2480      0.00000
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+       6255.8020      0.00000
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+       6256.3560      0.00000
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+       6256.9100      0.00000
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+       6257.4640      0.00000
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+       6258.0180      0.00000
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+       6258.5720      0.00000
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+       6259.1260      0.00000
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+       6259.6800      0.00000
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+       6260.2340      0.00000
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+       6260.7880      0.00000
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+       6261.3420      0.00000
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+       6261.8960      0.00000
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+       6262.4500      0.00000
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+       6263.0040      0.00000
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+       6263.5580      0.00000
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+       6264.1120      0.00000
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+       6264.6660      0.00000
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+       6265.2200      0.00000
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+       6265.7740      0.00000
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+       6271.3140      0.00000
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+       6271.8680      0.00000
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+       6274.0840      0.00000
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+       6274.6380      0.00000
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+       6275.1920      0.00000
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+       6275.7460      0.00000
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+       6276.3000      0.00000
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+       6276.8540      0.00000
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+       6277.4080      0.00000
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+       6277.9620      0.00000
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+       6278.5160      0.00000
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+       6279.0700      0.00000
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+       6279.6240      0.00000
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+       6280.1780      0.00000
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+       6280.7320      0.00000
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+       6281.2860      0.00000
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+       6281.8400      0.00000
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+       6282.3940      0.00000
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+       6282.9480      0.00000
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+       6283.5020      0.00000
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+       6284.0560      0.00000
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+       6284.6100      0.00000
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+       6285.1640      0.00000
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+       6285.7180      0.00000
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+       6286.2720      0.00000
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+       6286.8260      0.00000
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+       6289.5960      0.00000
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+       6290.1500      0.00000
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+       6290.7040      0.00000
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+       6296.2440      0.00000
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+       6296.7980      28.1966
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+       6297.9060      18.9971
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210
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214
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215
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216
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911
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912
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913
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914
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915
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916
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917
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918
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919
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920
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921
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922
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923
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924
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925
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926
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927
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928
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929
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930
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931
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932
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933
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934
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935
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936
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937
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938
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939
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940
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941
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942
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943
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944
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945
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946
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947
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948
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949
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950
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951
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952
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953
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954
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955
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956
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957
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959
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960
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961
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962
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963
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964
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965
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966
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967
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968
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969
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970
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971
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972
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973
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974
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975
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976
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977
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978
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979
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980
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981
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982
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983
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984
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985
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986
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987
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988
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989
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990
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991
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992
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993
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994
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995
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996
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997
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998
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999
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1000
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1001
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1002
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1003
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1004
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1005
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1006
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1007
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1008
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1009
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1010
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1011
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1012
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1013
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1014
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1015
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1016
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1017
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1018
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1019
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1020
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1021
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1022
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1023
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1024
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1025
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1026
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1027
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1028
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1029
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1030
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1031
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1032
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1033
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1034
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1035
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1036
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1037
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1038
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1039
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1040
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1041
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1042
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1043
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1044
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1045
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1046
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1047
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1048
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1049
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1050
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1051
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1052
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1053
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1054
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1055
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1056
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1057
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1058
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1059
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1060
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1061
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1062
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1063
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1064
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1065
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1066
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1067
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1068
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1069
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1070
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1071
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1072
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1073
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1074
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1075
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1076
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1077
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1078
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1079
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1080
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1081
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1082
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1083
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1084
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1085
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1086
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1087
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1088
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1089
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1090
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1091
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1092
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1093
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1094
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1095
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1096
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1097
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1098
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1099
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1100
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1101
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1102
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1103
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1104
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1105
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1106
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1107
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1108
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1109
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1110
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1111
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1112
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1113
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1114
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1115
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0 1203
new file mode 100644
1 1204
Binary files /dev/null and b/3/coursework3.pdf differ
2 1205
new file mode 100755
... ...
@@ -0,0 +1,18 @@
1
+#1
2
+spec = read.table("M100_spec.txt",header=FALSE)
3
+spec.boolmask = (21 > spec[,2]) & (spec[,2] >17)
4
+spec.zero = spec
5
+spec.notnull = (spec[,2] != 0)
6
+spec.fit <- lm(spec[spec.boolmask,2] ~ spec[spec.boolmask,1])
7
+spec.zero[,2]=(spec[,2])-coef(spec.fit)[1]
8
+
9
+#2
10
+y=spec.zero[622:646,2]
11
+x=spec.zero[622:646,1]
12
+gbin <- cbind(spec.zero[622:646,])
13
+Sigma= var(gbin)
14
+mu = apply(gbin,2,mean)
15
+gcoeffs <-nls(y~(b/a)*exp(-(x-6599.2820)^2/(2*a**2)),start=list(a=1,b=200), trace=TRUE)
16
+yg=(231.24/1.73)*exp(-(x-6599.2820)^2/(2*1.73**2))
17
+plot(spec.zero[spec.notNull,1],spec.zero[spec.notNull,2],'l')
18
+lines(x,yg,col='red')
0 19
new file mode 100644
... ...
@@ -0,0 +1,115 @@
1
+#Exercise 1
2
+#a)
3
+#setwd("C:/Users/Studium/Desktop/Statistische Methoden/Sheet 4")
4
+input=scan(file = "sn_data_riess.dat", what = list(character(), double(), double(), double()), skip=1, multi.line=FALSE)
5
+data=cbind(input[[2]], input[[3]], input[[4]])
6
+data
7
+
8
+#b)
9
+#Constants
10
+H0=72.0 #km/s/Mpc
11
+c = 3*10^5 #km/s
12
+
13
+
14
+#Hubble's law
15
+Hubble=function(z,Omegam) H0*sqrt(Omegam*(1+z)^3+(1.0-Omegam))
16
+
17
+
18
+#H^(-1)
19
+Hinv=function(zint,Omegam) 1.0/Hubble(zint,Omegam)
20
+
21
+
22
+#Luminosity distance
23
+dL=function(z,Omegami){
24
+	dLsol=z
25
+	for (i in (1:length(z))){
26
+		zarg=z[i]
27
+		I=integrate(Hinv,0.0,zarg,Omegam=Omegami)
28
+		dLsol[i] = c*(1+zarg)*I$value
29
+		}
30
+	return(dLsol)
31
+	}
32
+
33
+m=function(z, Omegam, M) M + 5*log10(H0 * dL(z, Omegam))
34
+
35
+mag_th=NULL
36
+mag_th[186]=1
37
+for (i in (1:186)){
38
+	mag_th[i] = m(data[i,1], 1, data[i,2]) #Omegam = 1 for a flat universe 
39
+	}
40
+
41
+#c)
42
+chisquare=function(Omegam, M) {
43
+	chisqu=0
44
+	for (i in (1:186)){
45
+		chisqu=chisqu+(data[i,2] - m(data[i,1], Omegam, M))**2/data[i,3]**2
46
+	}
47
+	return(chisqu)
48
+}
49
+
50
+	
51
+#d)
52
+#create vectors Omegam und M
53
+Omegam_vec=seq(0.0,1.0, by=0.05)
54
+M_vec=seq(15.5, 16.6, 0.01)
55
+
56
+chisquare_min = chisquare(0,0) #set initial value
57
+Omegam_min=0
58
+M_min=0
59
+
60
+#zweifach verschachtelte Schleife f�r Omegam und M
61
+
62
+for (i in M_vec){
63
+	for (j in Omegam_vec){
64
+		if (chisquare(j,i) < chisquare_min){ #compare current value with known smallest value 
65
+			chisquare_min = chisquare(j,i)# set new smallest value
66
+			Omegam_min = j 
67
+			M_min = i
68
+		}
69
+	}
70
+}
71
+chisquare_min
72
+Omegam_min
73
+M_min
74
+
75
+#e)
76
+posterior = function(Omegam, M){
77
+	p = exp(-0.5*(chisquare(Omegam, M)- chisquare_min))
78
+	return(p)
79
+}
80
+
81
+#f)
82
+Atest=seq(0,1,by = 0.05)	
83
+Btest=seq(15.5,16.5,by = 0.01)
84
+
85
+jmax=length(Atest)
86
+kmax=length(Btest)
87
+j=1
88
+k=1
89
+
90
+postplot = matrix(nrow=jmax,ncol=kmax)
91
+while (j<=jmax){ k=1
92
+	while (k<=kmax) {
93
+	postplot[j,k]=posterior(Atest[j],Btest[k])
94
+	k=k+1}
95
+	j=j+1}
96
+contour(Atest,Btest,postplot,levels=c(1.0,0.8,0.6,0.4,0.2,0.0),drawlabels=FALSE,xlab='Omegam',ylab='M',xlim=c(0,3),ylim=c(0,3))
97
+
98
+#g)
99
+pMmarg=function(Mi) {I=integrate(posterior,13.0,18.0,A=Mi)
100
+	return(I$value)
101
+	}
102
+
103
+#h)
104
+pMplot=function(Mi) {dumplot=Mi
105
+	for (i in (1:length(Mi))) {
106
+	     Marg=Mi[i]*1.0
107
+	     dumplot[i]=pMmarg(Marg)
108
+	            }
109
+	return(dumplot)}
110
+	
111
+Mplot=seq(0.1,3.0,by = 0.02)
112
+
113
+normM=1.0/max(pMplot(Mplot))
114
+plot(Mplot,normM*pMplot(Mplot),type='l',lwd=2,xlab='M',ylab='prob(M|{N_k},I)')
115
+
0 116
new file mode 100644
... ...
@@ -0,0 +1,52 @@
1
+# EXAMPLE 3: Introduction to Statistics for Astrophysicists
2
+# SS 2012
3
+# JOCHEN WELLER
4
+
5
+
6
+
7
+
8
+# Sequence of probabilities (Hypothesis)
9
+H=seq(0,1,by = 0.01)
10
+
11
+# number of trials
12
+n=32
13
+
14
+# bias 
15
+bias=0.25
16
+
17
+# generate random sample with possible outcome 0,1 and bias: toss the (un-)fair coin
18
+coin=sample(c(0,1),n,replace=TRUE,c(1.0-bias,bias))
19
+
20
+# 
21
+
22
+# count heads of sample
23
+heads=sum(coin)
24
+
25
+
26
+# prior distribution 
27
+# uniform
28
+prior = 1.0
29
+
30
+
31
+#Gaussian approximation
32
+H0 = heads/n
33
+sigma=sqrt(H0*(1-H0)/n)
34
+likeapprox <- function(H) 1.0/sqrt(2.0*pi)/sigma*exp(-0.5*(H-H0)**2/sigma**2)
35
+
36
+#calculate normalization of binomial distribution
37
+sum = 0.0
38
+i = 0
39
+while (i < n-heads) {sum=sum+choose(n-heads,i)*((-1)^i)/(heads+i+1) 
40
+	i=i+1 }
41
+norm = abs(1.0/sum)
42
+
43
+# plot posterior (likelihood times prior)
44
+
45
+par(font.lab=2)
46
+par(font.axis=2)
47
+plot(H,norm*dbinom(heads,n,H)*prior,type='l', xlab='H',ylab='prob(H|{data},I)', lwd=2, main=paste("N=",n))
48
+lines(H,likeapprox(H)*prior,lty=2,lwd=2)
49
+abline(v=H0,lty=3,lwd=2,col='blue')
50
+abline(v=H0-sigma,lty=3,lwd=2,col='blue')
51
+abline(v=H0+sigma,lty=3,lwd=2,col='blue')
52
+legend("topright",legend=c("Binomial","Gaussian approximation","maximum, sigma"),lty=c(1,2,3),lwd=c(2,2,2),col=c("black","black","blue"))
0 53
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1
+# EXAMPLE 4: Introduction to Statistics for Astrophysicists
2
+# SS 2012
3
+# JOCHEN WELLER
4
+
5
+
6
+# The Lighthouse Problem
7
+alpha0 = 1 #km true position at coastline
8
+beta = 1 #km position out at see
9
+
10
+# Sequence of positions (use already prior range from -10 to 10 km)
11
+alpha=seq(-10,10,by = 0.05)
12
+
13
+# number of trials
14
+N=512
15
+
16
+# generate random sample distributed according to a Cauchy distribution (see lecture)
17
+# with N trial
18
+pos=rcauchy(N,alpha0,beta)
19
+
20
+# prior distribution 
21
+# uniform between -10 and +10 km
22
+prior = 1.0/20.0
23
+
24
+sum = 0.0
25
+i = 1
26
+
27
+logpost <- function(alpha) { while (i<=N){sum=sum+log(beta^2+(pos[i]-alpha)^2)
28
+		i=i+1 }; sum}
29
+
30
+#find minimum of log likelihood
31
+lmin=min(logpost(alpha))
32
+
33
+# normalized posterior
34
+post <- function(alpha) exp(-logpost(alpha)+lmin)
35
+
36
+# plot posterior 
37
+
38
+par(font.lab=2)
39
+par(font.axis=2)
40
+plot(alpha,post(alpha),type='l', xlab=expression(paste(alpha,"(km)")),ylab=expression(prob(~alpha~"|{"~x[k]~"}",~beta,I)), lwd=2, main=paste("N=",N),xlim=c(-12,12),ylim=c(0,1.2))
41
+
42
+# plot points of samples at vertical position 1.1
43
+
44
+normpoint = (pos-pos)+1.1
45
+points(pos,normpoint)
46
+
47
+# vertical line at mean of sample
48
+abline(v=mean(pos),lwd=2,lty=2)
49
+
50
+
51
+
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1
+# EXAMPLE 5: Introduction to Statistics for Astrophysicists
2
+# SS 2012
3
+# JOCHEN WELLER
4
+
5
+# BINOMIAL PROBABILITY
6
+
7
+N=20
8
+r=seq(0,N,by=1)
9
+
10
+p=0.1
11
+
12
+plot(r,dbinom(r,N,p),type='h',xlim=c(0,20),xlab='r',ylab='f(r;N,p)',main=paste("N=",N,"; p=",p))
13
+
14
+
15
+
16
+
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1
+# EXAMPLE 6: Introduction to Statistics for Astrophysicists
2
+# SS 2012
3
+# JOCHEN WELLER
4
+
5
+# Poisson PROBABILITY
6
+
7
+nu=2
8
+r=seq(0,20,by=1)
9
+
10
+plot(r,dpois(r,nu),type='h',xlim=c(0,20),xlab='r',ylab="f(r;nu)",main=paste("nu=",nu))
11
+
12
+
13
+
14
+
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1
+# EXAMPLE 7: Introduction to Statistics for Astrophysicists
2
+# SS 2012
3
+# JOCHEN WELLER
4
+
5
+# Exponential PROBABILITY
6
+
7
+
8
+x=seq(0,5,by=0.1)
9
+
10
+xi=1.0
11
+plot(x,exp(-x/xi)/xi,type='l',xlim=c(0,5),xlab='x',ylab="f(x;xi)",lwd=2)
12
+
13
+xi2=2.0
14
+lines(x,exp(-x/xi2)/xi2,lwd=2,lty=2)
15
+
16
+xi3=5.0
17
+lines(x,exp(-x/xi3)/xi3,lwd=2,lty=3)
18
+legend("topright",legend=c(paste("xi=",xi),paste("xi=",xi2),paste("xi=",xi3)),lty=c(1,2,3),lwd=c(2,2,2),col=c("black","black","black"))
19
+
20
+
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1
+# EXAMPLE 7: Introduction to Statistics for Astrophysicists
2
+# SS 2012
3
+# JOCHEN WELLER
4
+
5
+# LOG-NORMAL PROBABILITY
6
+
7
+
8
+x=seq(0,4,by=0.1)
9
+
10
+mu=0.0
11
+sigma=1.0
12
+norm=1.0/sqrt(2.0*pi)/sigma
13
+plot(x,norm*exp(-(log(x)-mu)^2/sigma^2)/x,type='l',xlim=c(0,4),ylim=c(0,1),xlab='x',ylab="f(x;mu,sigma)",lwd=2)
14
+
15
+
16
+
17
+mu=0
18
+sigma=1.5
19
+norm=1.0/sqrt(2.0*pi)/sigma
20
+lines(x,norm*exp(-(log(x)-mu)^2/sigma^2)/x,lwd=2,lty=2)
21
+
22
+mu=0
23
+sigma=0.5
24
+norm=1.0/sqrt(2.0*pi)/sigma
25
+lines(x,norm*exp(-(log(x)-mu)^2/sigma^2)/x,lwd=2,lty=3)
26
+
27
+mu=1
28
+sigma=1.0
29
+norm=1.0/sqrt(2.0*pi)/sigma
30
+lines(x,norm*exp(-(log(x)-mu)^2/sigma^2)/x,lwd=2,lty=4)
31
+
32
+
33
+legend("topright",legend=c("mu=0,sigma=1","mu=0,sigma=1.5","mu=0,sigma=0.5","mu=1,sigma=1"),lty=c(1,2,3,4),lwd=c(2,2,2,2),col=c("black","black","black","black"))
34
+
35
+
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1
+# EXAMPLE 9: Introduction to Statistics for Astrophysicists
2
+# SS 2012
3
+# JOCHEN WELLER
4
+
5
+# Poisson PROBABILITY
6
+
7
+nu=1.7
8
+r=seq(0,8,by=1)
9
+height=dpois(r,nu)
10
+
11
+barplot(height,xlab='Number of Counts N',ylab="prob(N|D=12.5)",axisnames=TRUE)
12
+
13
+
14
+
15
+
16
+
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1
+# EXAMPLE 10: Introduction to Statistics for Astrophysicists
2
+# SS 2012
3
+# JOCHEN WELLER
4
+
5
+x0=1
6
+#fwhm = 5 = 2.35*sigma
7
+w=2.13
8
+n0=33.333
9
+A=1
10
+B=2
11
+
12
+#First generate random data 
13
+#Datum is Gaussian, and measurement Poisson
14
+
15
+xmin=-7
16
+xmax=7
17
+x=seq(xmin,xmax,by=1)
18
+
19
+d=n0*(A*exp(-0.5*(x-x0)^2/w^2)+B)
20
+N=d
21
+imax=length(d)
22
+i=1
23
+while (i <= imax) {N[i]=rpois(1,d[i])
24
+	i=i+1}
25
+	
26
+# The minimum chi2 for later	
27
+logepmin=sum(N*log(d)-d)	
28
+	
29
+quartz()
30
+
31
+# Plot the data	
32
+par(font.lab=2)
33
+par(font.axis=2)	
34
+plot(x,N,type='s',xlim=c(xmin,xmax),lwd=2)
35
+
36
+
37
+# posterior function for arbitrary A and B
38
+posterior<- function(A,B) {
39
+	d=n0*(A*exp(-0.5*(x-x0)^2/w^2)+B)
40
+	dummy=exp(sum(N*log(d)-d)-logepmin)
41
+	return(dummy)}
42
+
43
+# scan the parameter space
44
+Atest=seq(0,3,by = 0.02)	
45
+Btest=seq(0,3,by = 0.02)
46
+
47
+jmax=length(Atest)
48
+kmax=length(Btest)
49
+j=1
50
+k=1
51
+# generate a posterior matrix for the contour plot
52
+postplot = matrix(nrow=jmax,ncol=kmax)
53
+while (j<=jmax){ k=1
54
+	while (k<=kmax) {
55
+	postplot[j,k]=posterior(Atest[j],Btest[k])
56
+	k=k+1}
57
+	j=j+1}
58
+quartz()
59
+contour(Atest,Btest,postplot,levels=c(0.9,0.7,0.5,0.3,0.1),drawlabels=FALSE,xlab='A',ylab='B',xlim=c(0,3),ylim=c(0,3))
60
+	
61
+	
62
+
63
+
64
+
65
+
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1
+# EXAMPLE 11: Introduction to Statistics for Astrophysicists
2
+# SS 2012
3
+# JOCHEN WELLER
4
+
5
+x0=1
6
+#fwhm = 5 = 2.35*sigma
7
+w=2.13
8
+n0=33.3333
9
+A=1
10
+B=2
11
+
12
+#First generate random data 
13
+#Datum is Gaussian, and measurement Poisson
14
+
15
+xmin=-7
16
+xmax=7
17
+x=seq(xmin,xmax,by=1.0)
18
+
19
+d=n0*(A*exp(-0.5*(x-x0)^2/w^2)+B)
20
+N=d
21
+imax=length(d)
22
+i=1
23
+while (i <= imax) {N[i]=rpois(1,d[i])
24
+	i=i+1}
25
+	
26
+# The minimum chi2 for later	
27
+logepmin=sum(N*log(d)-d)	
28
+	
29
+
30
+# posterior function for arbitrary A and B
31
+posterior=function (A,B) 
32
+{ summe=0.0*A
33
+	for (i in 1:length(x)) {
34
+	di=n0*(A*exp(-0.5*(x[i]-x0)^2/w^2)+B)
35
+	summe=summe+N[i]*log(di)-di
36
+	}
37
+	return(exp(summe-logepmin))
38
+}
39
+pAmarg=function(Ai) {I=integrate(posterior,0.0,Inf,A=Ai)
40
+	return(I$value)
41
+	}
42
+
43
+pAplot=function(Ai) {dumplot=Ai
44
+	for (i in (1:length(Ai))) {
45
+	     Aarg=Ai[i]*1.0
46
+	     dumplot[i]=pAmarg(Aarg)
47
+	            }
48
+	return(dumplot)}		
49
+	
50
+pBmarg=function(Bi) {I=integrate(posterior,0.0,Inf,B=Bi)
51
+	return(I$value)
52
+	}
53
+
54
+pBplot=function(Bi) {dumplot=Bi
55
+	for (i in (1:length(Bi))) {
56
+	     Barg=Bi[i]*1.0
57
+	     dumplot[i]=pBmarg(Barg)
58
+	            }
59
+	return(dumplot)}	
60
+	
61
+Aplot=seq(0.1,3.0,by = 0.02)
62
+Bplot=seq(0.1,3.0,by = 0.02)
63
+
64
+normA=1.0/max(pAplot(Aplot))
65
+plot(Aplot,normA*pAplot(Aplot),type='l',lwd=2,xlab='A',ylab='prob(A|{N_k},I)')
66
+normA2=1.0/max(posterior(Aplot,2.0))
67
+lines(Aplot,normA2*posterior(Aplot,2.0),lty=3,lwd=2)	
68
+
69
+quartz()
70
+normB=1.0/max(pBplot(Bplot))
71
+plot(Bplot,normB*pBplot(Bplot),type='l',lwd=2,xlab='B',ylab='prob(B|{N_k},I)')
72
+
73
+
74
+
75
+
76
+