# Maximum Likelihood Analysis

Given an Np-pixel map of the sky temperature \Deltap and a measure of the pixel-pixel correlations in the noise Npq we want to find the most likely underlying CMB signal in the map, characterised by its power spectrum Cl . For a given power spectrum we calculate the associated pixel-pixel correlations in the CMB signal Spq. Assuming that the signal and the noise are uncorrelated, the pixel-pixel correlations in the map - Mpq - are simply the sum of those in the signal and the noise. Assuming that the CMB fluctuations are Gaussian the likelihood of the particular power spectrum is
Our goal is to find the power spectrum which maximizes this likelihood function. A short paper outlining the implementation of algorithms for locating the peak of the likelihood function for a general sky temperature map can be found here .

# Timing

For a map with Np pixels, and a target power spectrum sub-divided into Nb bins, the quadratic estimator algorithm requires
• (2 x Nb + 2) x Np2 x 4   bytes of disc storage.
• 2 x Np2 x 8  bytes of RAM.
• (2 x N2/3 ) x Np3  floating point operations.
assuming that
1. all the necessary matrices are simultaneously stored on disc in single (4-byte) precision.
2. matrices are loaded into memory no more than two at a time in double (8-byte) precision.
If the power spectrum is divided into 20 multipole bins, then the computational requirements for the current MAXIMA and BOOMERanG balloon experiments for a single iteration of the algorithm on (i) a 600 MHz workstation and (ii) the NERSC Cray T3E-900 (using the specified number of processors and running at 2/3 peak) are

 Dataset Map Size Disk RAM Flops Serial CPU Time T3E Time BOOMERanG N.America 26,000 110 Gb 11 Gb 7.1 x 1014 14 days 5 hours   (x 64) MAXIMA-1 32,000 170 Gb 17 Gb 1.3 x 1015 25 days 9 hours    (x 64) MAXIMA-2 80,000 * 1 Tb 100 Gb 2.1 x 1016 13 months 18 hours    (x 512) BOOMERanG Antarctica 450,000 * 30 Tb 3 Tb 3.7 x 1018 196 years 140 days    (x 512)
(* projected)

If we project further and consider MAP and Planck data sets, we find even larger numbers.
Assume that the power spectrum is divided into 1000 multipole bins,  then the computational requirements for the current MAP and PLANCK missions for a single iteration of the algorithm on (i) a 600 MHz workstation and (ii) the NERSC Cray T3E-900 (using the specified number of processors and running at 2/3 peak) are

 Dataset Map Size Disk RAM Flops Serial CPU Time T3E Time MAP  (single frequency) 106 8 x 103 Tb 16 Tb 2 x 1021 105 years 200 years   (x 512) MAP 105-106 80-105  Tb 0.1-16 Tb 2 x 1022 103-106 years 2-2000 years   (x 512) PLANCK  (LFI) 106-107 104-105  Tb 16-1600 Tb 2 x 1024 108 years 104 years    (x 1024) PLANCK  (HFI) 107 106  Tb 1600 Tb 2 x 1024 108 years 104 years    (x 1024)
(* projected)
Note that a single map containing Np the correlation matrix is Np by Np and requires storage size 4 Np2 in single precision
(not exploiting symmetry or same in double precision using the symmetry). Thus the correlation matrix for a single map with Np = 106-107, will be 4-400 Terabytes in size.

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