new fiels
Michi commited on 2012-06-03 16:52:14
Zeige 15 geänderte Dateien mit 1731 Einfügungen und 0 Löschungen.
- 3/.RData 0000000..b226da8
- 3/.Rhistory 0000000..6dadcd4
- 3/M100_spec.txt 0000000..4f0c8a8
- 3/coursework3.pdf 0000000..73b1a09
- 3/speczeroing.r 0000000..8b88130
- 4$visoluton.r 0000000..c38fa26
- examples/Lecture2a.R 0000000..fd77495
- examples/Lecture2b.R 0000000..bb1fd14
- examples/Lecture2c.R 0000000..68a847b
- examples/Lecture2d.R 0000000..5bbddf6
- examples/Lecture2e.R 0000000..94abff8
- examples/Lecture2f.R 0000000..a6ec755
- examples/Lecture3a.R 0000000..e62cf9a
- examples/Lecture3b.R 0000000..6b769d4
- examples/Lecture3c.R 0000000..ed63ad7
| ... | ... |
@@ -0,0 +1,51 @@ |
| 1 |
+read.table("M100_spec.txt",header=TRUE)
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| 2 |
+spec = read.table("M100_spec.txt",header=TRUE)
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| 3 |
+ |
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| 4 |
+spec.fit <- lm(spec[175:746,2] ~ [175:746,1]) |
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| 5 |
+spec.fit <- lm(spec[175:746,2] ~ spec[175:746,1]) |
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| 6 |
+summary(spec.fit) |
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| 7 |
+abline(spec.fit) |
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| 8 |
+plot(spec[:,1],spec[:,2]) |
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| 9 |
+plot(spec[,1],spec[,2]) |
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| 10 |
+abline(spec.fit) |
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| 11 |
+plot(spec[,1],spec[,2],type=l) |
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+plot(spec[,1],spec[,2],l) |
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| 13 |
+plot(spec[,1],spec[,2],'l') |
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+abline(spec.fit) |
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| 15 |
+x <- array(1:20, dim=c(4,5)) |
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| 16 |
+x |
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| 17 |
+spec[:5,1] |
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| 18 |
+spec[1:5,1] |
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| 19 |
+spec[1:5,] |
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| 20 |
+spec[2:5,] |
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| 21 |
+spec |
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| 22 |
+spec = read.table("M100_spec.txt",header=FALSE)
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| 23 |
+spec[1:3,] |
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| 24 |
+spec.boolmask = 21 > spec[,2] >17 |
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| 25 |
+spec.boolmask = (spec[,2] >17) |
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| 26 |
+spec.boolmask = (spec[,2] >17) & (spec[,2] <20) |
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| 27 |
+spec.boolmask |
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| 28 |
+spec.fit <- lm(spec[spec.boolmask,2] ~ [spec.boolmask,1]) |
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| 29 |
+spec.boolmask |
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| 30 |
+spec.fit <- lm(spec[spec.boolmask,2] ~ spec[spec.boolmask,1]) |
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| 31 |
+abline(spec.fit) |
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| 32 |
+summery(spec.fit) |
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| 33 |
+summary(spec.fit) |
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| 34 |
+coef(spec.fit)[1] |
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| 35 |
+spec.zero = spec - coef(spec.fit)[1] |
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| 36 |
+spec.zero = spec |
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| 37 |
+spec.zero[,2] = spec[,2] - coef(spec.fit)[1] |
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| 38 |
+plot(spec[,1],spec.zero[,2],'l') |
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| 39 |
+plot(spec[,1],spec.zero[,2],'l') |
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| 40 |
+spec.zero = spec - coef(spec.fit)[1] |
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| 41 |
+spec.zero |
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| 42 |
+spec.notNull = (spec != 0) |
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| 43 |
+spec.notNull |
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| 44 |
+spec.notNull = (spec[,2] != 0) |
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| 45 |
+spec.notNull |
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| 46 |
+ls |
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| 47 |
+plot(spec[spec.notNull,1],spec.zero[notNull,2],'l') |
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| 48 |
+plot(spec[spec.notNull,1],spec.zero[spec.notNull,2],'l') |
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| 49 |
+save() |
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| 50 |
+save(fiel="flux2zero.R") |
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| 51 |
+q() |
| ... | ... |
@@ -0,0 +1,1202 @@ |
| 1 |
+# lambda Flux ((10^{-17} erg s^{-1}"+aaangs+"^{-1} cm^{-2}))
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| 2 |
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|
| 1083 |
+ 6847.4740 19.4858 |
|
| 1084 |
+ 6848.0280 16.9752 |
|
| 1085 |
+ 6848.5820 20.6931 |
|
| 1086 |
+ 6849.1360 18.8379 |
|
| 1087 |
+ 6849.6900 18.3726 |
|
| 1088 |
+ 6850.2440 17.2251 |
|
| 1089 |
+ 6850.7980 18.3125 |
|
| 1090 |
+ 6851.3520 18.8181 |
|
| 1091 |
+ 6851.9060 19.6145 |
|
| 1092 |
+ 6852.4600 18.0518 |
|
| 1093 |
+ 6853.0140 16.4574 |
|
| 1094 |
+ 6853.5680 16.1572 |
|
| 1095 |
+ 6854.1220 18.3224 |
|
| 1096 |
+ 6854.6760 19.9573 |
|
| 1097 |
+ 6855.2300 19.1848 |
|
| 1098 |
+ 6855.7840 19.2895 |
|
| 1099 |
+ 6856.3380 20.0645 |
|
| 1100 |
+ 6856.8920 20.0335 |
|
| 1101 |
+ 6857.4460 18.8627 |
|
| 1102 |
+ 6858.0000 17.7411 |
|
| 1103 |
+ 6858.5540 19.4334 |
|
| 1104 |
+ 6859.1080 19.3225 |
|
| 1105 |
+ 6859.6620 18.8582 |
|
| 1106 |
+ 6860.2160 18.1860 |
|
| 1107 |
+ 6860.7700 20.4508 |
|
| 1108 |
+ 6861.3240 20.7856 |
|
| 1109 |
+ 6861.8780 20.1865 |
|
| 1110 |
+ 6862.4320 18.7809 |
|
| 1111 |
+ 6862.9860 18.9147 |
|
| 1112 |
+ 6863.5400 20.6800 |
|
| 1113 |
+ 6864.0940 18.4504 |
|
| 1114 |
+ 6864.6480 18.6347 |
|
| 1115 |
+ 6865.2020 24.5428 |
|
| 1116 |
+ 6865.7560 0.00000 |
|
| 1117 |
+ 6866.3100 0.00000 |
|
| 1118 |
+ 6866.8640 0.00000 |
|
| 1119 |
+ 6867.4180 0.00000 |
|
| 1120 |
+ 6867.9720 0.00000 |
|
| 1121 |
+ 6868.5260 0.00000 |
|
| 1122 |
+ 6869.0800 0.00000 |
|
| 1123 |
+ 6869.6340 0.00000 |
|
| 1124 |
+ 6870.1880 0.00000 |
|
| 1125 |
+ 6870.7420 0.00000 |
|
| 1126 |
+ 6871.2960 0.00000 |
|
| 1127 |
+ 6871.8500 0.00000 |
|
| 1128 |
+ 6872.4040 0.00000 |
|
| 1129 |
+ 6872.9580 0.00000 |
|
| 1130 |
+ 6873.5120 0.00000 |
|
| 1131 |
+ 6874.0660 0.00000 |
|
| 1132 |
+ 6874.6200 0.00000 |
|
| 1133 |
+ 6875.1740 0.00000 |
|
| 1134 |
+ 6875.7280 0.00000 |
|
| 1135 |
+ 6876.2820 0.00000 |
|
| 1136 |
+ 6876.8360 0.00000 |
|
| 1137 |
+ 6877.3900 0.00000 |
|
| 1138 |
+ 6877.9440 0.00000 |
|
| 1139 |
+ 6878.4980 0.00000 |
|
| 1140 |
+ 6879.0520 0.00000 |
|
| 1141 |
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|
| 1142 |
+ 6880.1600 0.00000 |
|
| 1143 |
+ 6880.7140 0.00000 |
|
| 1144 |
+ 6881.2680 0.00000 |
|
| 1145 |
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|
| 1146 |
+ 6882.3760 0.00000 |
|
| 1147 |
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|
| 1148 |
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|
| 1149 |
+ 6884.0380 0.00000 |
|
| 1150 |
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|
| 1151 |
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|
| 1152 |
+ 6885.7000 0.00000 |
|
| 1153 |
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|
| 1154 |
+ 6886.8080 0.00000 |
|
| 1155 |
+ 6887.3620 0.00000 |
|
| 1156 |
+ 6887.9160 0.00000 |
|
| 1157 |
+ 6888.4700 0.00000 |
|
| 1158 |
+ 6889.0240 0.00000 |
|
| 1159 |
+ 6889.5780 0.00000 |
|
| 1160 |
+ 6890.1320 0.00000 |
|
| 1161 |
+ 6890.6860 0.00000 |
|
| 1162 |
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|
| 1163 |
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|
| 1164 |
+ 6892.3480 0.00000 |
|
| 1165 |
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|
| 1166 |
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|
| 1167 |
+ 6894.0100 0.00000 |
|
| 1168 |
+ 6894.5640 0.00000 |
|
| 1169 |
+ 6895.1180 0.00000 |
|
| 1170 |
+ 6895.6720 0.00000 |
|
| 1171 |
+ 6896.2260 0.00000 |
|
| 1172 |
+ 6896.7800 0.00000 |
|
| 1173 |
+ 6897.3340 0.00000 |
|
| 1174 |
+ 6897.8880 0.00000 |
|
| 1175 |
+ 6898.4420 0.00000 |
|
| 1176 |
+ 6898.9960 0.00000 |
|
| 1177 |
+ 6899.5500 0.00000 |
|
| 1178 |
+ 6900.1040 0.00000 |
|
| 1179 |
+ 6900.6580 0.00000 |
|
| 1180 |
+ 6901.2120 0.00000 |
|
| 1181 |
+ 6901.7660 0.00000 |
|
| 1182 |
+ 6902.3200 0.00000 |
|
| 1183 |
+ 6902.8740 0.00000 |
|
| 1184 |
+ 6903.4280 0.00000 |
|
| 1185 |
+ 6903.9820 0.00000 |
|
| 1186 |
+ 6904.5360 0.00000 |
|
| 1187 |
+ 6905.0900 0.00000 |
|
| 1188 |
+ 6905.6440 0.00000 |
|
| 1189 |
+ 6906.1980 0.00000 |
|
| 1190 |
+ 6906.7520 0.00000 |
|
| 1191 |
+ 6907.3060 0.00000 |
|
| 1192 |
+ 6907.8600 0.00000 |
|
| 1193 |
+ 6908.4140 0.00000 |
|
| 1194 |
+ 6908.9680 0.00000 |
|
| 1195 |
+ 6909.5220 0.00000 |
|
| 1196 |
+ 6910.0760 0.00000 |
|
| 1197 |
+ 6910.6300 0.00000 |
|
| 1198 |
+ 6911.1840 0.00000 |
|
| 1199 |
+ 6911.7380 0.00000 |
|
| 1200 |
+ 6912.2920 0.00000 |
|
| 1201 |
+ 6912.8460 0.00000 |
|
| 1202 |
+ 6913.4000 0.00000 |
| ... | ... |
@@ -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') |
| ... | ... |
@@ -0,0 +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 |
+ |
| ... | ... |
@@ -0,0 +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 |
\ No newline at end of file |
| ... | ... |
@@ -0,0 +1,51 @@ |
| 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 |
+ |
| ... | ... |
@@ -0,0 +1,16 @@ |
| 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 |
+ |
| ... | ... |
@@ -0,0 +1,20 @@ |
| 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 |
+ |
| ... | ... |
@@ -0,0 +1,35 @@ |
| 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 |
+ |
| ... | ... |
@@ -0,0 +1,16 @@ |
| 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 |
+ |
| ... | ... |
@@ -0,0 +1,65 @@ |
| 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 |
+ |
| ... | ... |
@@ -0,0 +1,76 @@ |
| 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 |
+ |
|
| 0 | 77 |