%clf;
clear all;
close all;
disp('Hubble Regression')
lw = 2;
set(0, 'DefaultAxesFontSize', 17);
fs = 17;
msize = 7;
distance = [.032 .034 .214 .263 .275 .275 ...
.45 .5 .5 .63 .8 .9 ...
.9 .9 .9 1.0 1.1 1.1 ...
1.4 1.7 2.0 2.0 2.0 2.0 ]';
velocity = [170 290 -130 -70 -185 -220 ...
200 290 270 200 300 -30 ...
650 150 500 920 450 500 ...
500 960 500 850 800 1090 ]';
%mod = x2fx(distance, [0; 1]);
s = regstats(velocity,distance,[0; 1],{'yhat','r','beta'});
figure(1)
plot(s.yhat,s.r,'o',...
'MarkerEdgeColor','k',...
'MarkerFaceColor','g',...
'MarkerSize',8)
hold on
plot([0 1000],0.*[0 1000], 'k--','linewidth',2)
xlabel('Fitted Values'); ylabel('Residuals');
axis tight
%
print -depsc 'C:\Springer\Reg\Regeps\hubbleresid.eps'
figure(2)
plot(distance, velocity, 'o', 'MarkerEdgeColor','k',...
'MarkerFaceColor','g',...
'MarkerSize',8)
hold on
b = regress(velocity,distance);
di=[0 2.1];
plot(di, b*di, 'r-','linewidth',2)
plot(di, s.beta(1) + s.beta(2)*di, 'b-','linewidth',2)
axis tight
ylabel('Velocity (km/s)')
xlabel('Distance (mps)')
print -depsc 'C:\Springer\Reg\Regeps\hubblescatter.eps'
% 'Q' Q from the QR Decomposition of the design matrix
% 'R' R from the QR Decomposition of the design matrix
% 'beta' Regression coefficients
% 'covb' Covariance of regression coefficients
% 'yhat' Fitted values of the response data
% 'r' Residuals
% 'mse' Mean squared error
% 'rsquare' R-square statistic
% 'adjrsquare' Adjusted R-square statistic
% 'leverage' Leverage
% 'hatmat' Hat (projection) matrix
% 's2_i' Delete-1 variance
% 'beta_i' Delete-1 coefficients
% 'standres' Standardized residuals
% 'studres' Studentized residuals
% 'dfbetas' Scaled change in regression coefficients
% 'dffit' Change in fitted values
% 'dffits' Scaled change in fitted values
% 'covratio' Change in covariance
% 'cookd' Cook's distance
% 'tstat' t statistics and p-values for coefficients
% 'fstat' F statistic and p-value
% 'dwstat' Durbin-Watson statistic and p-value
% 'all' Create all of the above statistics
% R has units of distance we discover that Ho has units of 1/time. Inverting the Hubble constant gives us an approximate age of the universe:
%
% Ho = 65 km.s-1.Mpc-1
%
% 1 Mpc = 3.26 million light years = 30841981340613408000km
%
% Ho = 65 km.s-1.Mpc-1 / 30841981340613408000km/Mpc
%
% 1/Ho = 30841981340613408000/65 seconds
%
% This gives an age for the universe of about 15 billion years. More accurate determinations of the constant would give more accurate universal ages - and vice versa.
1/424 * 3.086 * 10^19 /(60*60*24*365) %2.3079e+009
% The slope of the fitted line is 464 km/sec/Mpc,
% and is now known as the Hubble constant, Ho.
% Since both kilometers and Megaparsecs (1 Mpc = 3.086E19 km)
% are units of distance, the simplified units of Ho are 1/time,
% and the conversion is given by
% 1/Ho = (978 Gyr)/(Ho in km/sec/Mpc)
% Thus Hubble's value is equivalent to approximately 2 Gyr.
% Since this should be close to the age of the Universe,
% and we know (and it was known in 1929) that the age of
% the Earth is larger than 2 billion years, Hubble's value
% for Ho led to considerable skepticism about cosmological
% models, and motivated the Steady State model. However,
% later work found that Hubble had confused two different
% kinds of Cepheid variable stars used for calibrating distances,
% and also that what Hubble thought were bright stars in distant
% galaxies were actually H II regions. Correcting for these
% errors has led to a lowering of the value of the Hubble
% constant: there are now primarily two groups using Cepheids:
% the HST Distance Scale Key Project team
% (Freedman, Kennicutt, Mould etal) which gets 72+/-8 km/sec/Mpc,
% while the Sandage team, also using HST observations of
% Cepheids to calibrate Type Ia supernovae, gets 57+/-4 km/sec/Mpc.
% Other methods to determine the distance scale include the time
% delay in gravitational lenses and the Sunyaev-Zeldovich effect i
% n distant clusters: both are independent of the Cepheid calibration
% and give values consistent with the average of the two HST groups:
% 65+/-8 km/sec/Mpc. These results are consistent with a combination
% of results from CMB anisotropy and the accelerating expansion of
% the Universe which give 71+/-3.5 km/sec/Mpc. With this value for
% Ho, the "age" 1/Ho is 14 Gyr while the actual age from the
% consistent model is 13.7+/-0.2 Gyr.
%
% Intrinsic Redshifts and the Hubble Constant
% M.B. Bell1 and S.P. Comeau1
% ABSTRACT
% We show that the VCMB velocities of the Fundamental Plane (FP) clusters studied in the
% Hubble Key Project appear to contain the same discrete "velocities" found previously by us
% and by Ti to be present in normal galaxies. Although there is a particular Hubble constant
% associated with our findings we make no claim that its accuracy is better than that found by the
% Hubble Key Project. We do conclude, however, that if intrinsic redshifts are present and are not
% taken into account, the Hubble constant obtained will be too high.
%
%
% Cluster/Group D (Mpc) VCMB (km s?1)a Transit.(Disc.Vel.)(km s?1)b VH (km s?1)c
% Dorado 13.8 1131 ziG[1,7](145.2) 986
% Grm 15 47.4 4530 ziG[1,4](1157.9) 3372
% Hydra 49.1 4061 ziG[1,5](580.1) 3481
% Abell S753 49.7 4351 ziG[2,6](725.2) 3626
% Abell 3574 51.6 4749 ziG[1,4](1157.9) 3591
% Abell 194 55.9 5100 ziG[1,4](1157.9) 3942
% Abell S639 59.6 6533 ziG[1,3](2314.3) 4219
% Coma 85.8 7143 ziG[1,4](1157.9) 5985
% Abell 539 102.0 8792 ziG[2,7](1448.6) 7343
% DC 2345-28 102.1 8500 ziG[1,4](1157.9) 7342
% Abell 3381 129.8 11536 ziG[1,3](2314.3) 9222
D =[13.8 47.7 49.1 49.7 51.6 55.9 59.6 85.8 102.0 102.1 129.8]; %Mpc
VH=[986 3372 3481 3626 3591 3942 4219 5985 7343 7342 9222]; %km/sec
figure(3)
scatter(D, VH)
H = mean(D .* VH)/mean(VH) %82.6944
1/H * 3.086 * 10^19 /(60*60*24*365) % 1.1833e+010
% The universe is 11.8 billion years old