# Minimal Variance Weights

### Weighted Distribution

#### The COMPOSITE peculiar velocity catalogue distribution on the sky in Galactic coordinates. Points are colour coded by peculiar velocity with red outgoing and blue infalling. While it is customary in such plots to encode the amplitude of the peculiar velocity by the size of the symbol, here the symbol area is proportional to the weight. This is presented as a function of RI=[10,60] h-1Mpc.

 Click on the image to animate.
 Top panel (the Aitoff projection): The green square is the direction to of the bulk flow, its size is proportional to the magnitude of the velocity. Bottom right panel The red and blue histograms show the redshift and distance distributions, respectively. The green histogram shows the weighted histogram (renormalized to the same area) but using the usual MLE weights. Notice that most of the signal is driven by very nearby objects. The magenta histogram is also weighted, but using MV weights. The smooth black curve shows the expected weighted radial distribution for an ideal survey i.e. ∝ r2 exp[-r2/(2 R_I2)]. This shows that our MV weighting scheme produces the desired radial distribution. Bottom left panel The red and blue histograms show the velocity and velocity errors distributions, respectively.

### Moment weights vs. directions

The weights are indexed by the direction and the galaxy index wp,n where
• p = 1..19 denotes the direction (x, y, z, xx, yy, zz, xy, yz, zx, xxx, yyy, zzz, xxy, yyz, zzx, xyy, yzz, zxx, xyz)
• n = 1..N is the galaxy index in a catalogue with N galaxies.
• The color code is given at the top right-hand side of each figure
• The depth of the survey is dynamic and is shown at the left-hand side of the plot.
• Each plot shows the weights for each galaxy in the catalogue as a function of direction.
• The colors represent the weight moment displayed at the top right side of each plot.
• The depth is shown at the top left side of each plot. We display the depth RI=[10,60] h-1Mpc.
• The weight values (y axis) are normalized.

To see large format (1920 x 1080) animation click here
To see small format (768 x 576) animation click here

Hume Feldman