fitting non-parametric models

The non-paramtetric luminosity functions were fit using the Powell function minimising algorithm and the maximum likelihood statistic applied to the non-parametric sum-of- Gaussians form: Φ(M_{|z) ∝ 10}0.4(z−z0)P X k Φ_kexp  − 1₂(M− Mk + (z− z0)Q) 2 σ2 M  , (3.9)

where P and Q are redshift evolution parameters following the convention of Lin et al (Lin et al.,1999).

Figure3.3shows an example of the results when fitting equation3.9:

Note that in Fig.3.3there is an issue with edge effects in the fitting of the Gaussians. Near the edges of the fit the density of Gaussians as a function of magnitude falls off, this leads to the “sum of Gaussians” line representing the initial guess at the LF decreasing rapidly at the edges, especially visible on the faint edge. It is possible that this then affects the fit as the Gaussians at the (faint) edge increase disproportionately to counter this effect and/or as the tails of the other Gaussians attempt to compensate. In an effort to counter this effect the fit has been extended to fainter magnitudes than are to be used in the final analysis, that is pushing the regions of the LF most affected by the edge effects

CHAPTER 3. LUMINOSITY FUNCTIONS, THEORY AND CALCULATIONS 52

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Figure 3.3: Example SOG fit, large 1/Vmaxgalaxies causing upturn in faint tail.

past the point where the final cuts in absolute magnitude are made . The points are a 1/Vmax data estimate of the luminosity function and act as an initial starting point for the fitting algorithm by defining an initial sum-of-Gaussians function, the dashed line. Later recursive fits then use previous maximum likelihood results as a starting point. The displacement between the final and initial functions occurs after renormalisation and is entirely due to the large amplitude of the final Gaussian at the faint end, the introduction of faint magnitude cuts in the fitting range resolves this issue. The curves shown under the function represent the individual Gaussians comprising the final function.

Note that in their fitting Blanton et al.(2003a) convolved his SOG luminosity with a Gaussian of fixed width ∆m . This ∆m represents the estimated difference between the SDSS magnitudes and the true AB magnitudes, the value is fixed for a given band (u,g,r,i,z), since the convolution broadens all the Gaussians by a fixed amount and our σM is fixed, not fitted, this does not seem applicable in this case.

CHAPTER 3. LUMINOSITY FUNCTIONS, THEORY AND CALCULATIONS 53

Figure 3.4: A proof of good fit for the SOG fits. The initial starting points for the fits, shown by the lines, were randomised. The points represent 1/VMAX estimates of the luminosity function. The divergence of the two measurement techniques at the data-poor faint end of the slope is responsible for the apparent discrepancy in results.

tests of non-parametric function fitting

To verify the robustness of our fitting routine the algorithm was given a large number of random initial values for the parameters to be fitted. Convergence on the best fit solution was found to be good, an example illustration of this is given in Fig. 3.4 where each colour line corresponds to a different set of initial random starting points, the scatter at the faint end is due to the lack of data in this region. The reason for the apparent discrepancy between the data points and the models is one of normalisation and the limits of the ”non-parametric” model: if the amplitude of the faint en Gaussians were boosted to match the shape of the data this would would also introduce an incorrect boost to nearby fainter magnitudes, as the fainter magnitudes contain more galaxies this boost is rejected in favour of failing to fit the data-poor extreme faint tail - the normalisation of the models and data match, so the large faint end discrepancy (on the logarithmic vertical axis) causes the models and data to appear discordant around M⋆, however the shape of the models is actually a reasonable fit to the data.

It is clear that this model is not truly “non-paramtetric” but rather has a large number of parameters capable of fitting a large range of shapes in the data distribution. However,

CHAPTER 3. LUMINOSITY FUNCTIONS, THEORY AND CALCULATIONS 54 there are limits to this model. The steepness of the bright end slope which can be repre- sented by the model is dictated by the width of the individual Gaussians. There is also a need to have significant overlap between Gaussians in order to obtain a smooth resultant function. These two facts together mean the a large number of Gaussians can be neces- sary to provide an accurate, smooth fit to the data, of the order _{∼ 50 − 200. This large} number of highly degenerate parameters is computationally intensive to fit. We find that over the range of luminosities in which we are interested the Schechter function provides an equally good fit for an order of magnitude fewer parameters. Because of this we now exclusively consider fits to the Schechter function.