Nonlinear fitting

The nonlinear fit was carried out using the built-inMathematica function

“NonLinearRegress”. It uses the Levenberg-Marquardt method [135,136] to find the parameters (aij,bi,ci,y,ν,Wc), which provide the best fit of the data (tM, W,ΛMu)

to the model ˜f given in Eqs. (2.19)-(2.22) with fixed values of nr, ni,mr, mi. The

data points are weighted according to_Λ2 1

MpδγNmin{γminqN 2

, which is the inverse variance of ΛM. Importantly, it also provides errors forWc andν and statistical measurements

of the goodness of fit, which indicates the validity of the model. This information enables us to choose the most appropriate values ofnr, ni, mr, mi. The goodness

of fit should be maximised and the errors minimised whilst simultaneously keeping the number of parameters low. The number of parameters (not including Wc and

ν) ispni 1qpnr 1q mi mr 2 for the caseni¡0 andnr mr 1 when ni0.

One of the outputs of “NonLinearRegress” is χ2 [137]. It is related to the issue of how to quantitatively judge how well a set of parameters fit the data for a particular model. This is done using the concept ofmaximum likelihood estima- tors. One calculates the probability that the data set could have occured, given a particular set of parameters. Suppose the data is ttx1, y1u,tx2, y2u, . . . ,txn, ynuu

and the goodness of fit of the model ypxq is to be tested. For data which may take continuous values, such as our localisation lengths, the aforementioned prob- ability is zero. Instead we calculate the probability that a data set contained

in ttx1, y1∆yu,tx2, y2∆yu, . . . ,txn, yn∆yuu could have occured, given the

modelypxq, where ∆y!1 is fixed. This is

P9 n ¹ i1 exp 1 2 yiypxiq σi 2 ∆y. (2.23)

The expression for P assumes that each data point, yi, is normally distributed

about ypxiq with standard deviation σi and that the errors are independent. The

assumption of a normal distribution should hold due to the central limit theorem [138]. However, one should bare in mind that this is not always the case in practice. Maximising Eq. (2.23) is equivalent to minimising

χ2 n ¸ i1 yiypxiq σi 2 . (2.24)

The method of minimisingχ2 to find the best-fit parameters, is refered to as aleast- squares fit. The Levenberg-Marquardt method [135, 136] minimises χ2 iteratively. An initial guess for the parameters is used to calculate an initial value forχ2. The parameters are then altered to further lowerχ2_{. This is done using either the inverse}

Hessian method or method of steepest descent, depending on how well the function χ2paq (a is a vector containing the parameters) approximates the behaviour of χ2 close to its minimum value.

To examine the probability distribution of different values of χ2, we must define the incomplete gamma function:

Γppa, xq 1 Γpaq »x 0 dt ta1et, (2.25) where Γpaq »₈ 0 dt ta1et (2.26)

is the ordinary Gamma function. Letµbe the number of degrees of freedom:

µ#data points#parameters. (2.27)

Then Γppµ₂,χ

2

2 qgives the probability that the observed chi-squared value for a correct

model is less thanχ2. It is usually the complement of Γp, i.e. Γq 1Γp, which

is considered. The closer Γqpµ₂,χ

2

2 q is to 1, the better the model. However, values

Γqpµ₂,χ

2

2 q 1 are suspicious and may mean that the errors have been overestimated.

-12 -8 -4 0 4 8 12 E 0 5 10 15 20 25 W

high precision FSS results E=-E _{symmetry points} W_{=0 (analytical)}

extended localised

Figure 2.2: Phase diagram for the BCC lattice. The cyan region represents the approximate location of the phase boundary. Its edges (the solid black lines) were determined by comparing localisation lengths with errors ¤ 10% for system sizes M 7 andM 9 in the pE, W) plane. The solid red squares () are points calcu- lated by performing high-precision FSS on localisation data with an error¤0.1%. The hollow squares () are reflections of the solid squares in the E 0 axis. The diamonds () denote the band edges at W 0. They have been joined to the phase boundary edges calculated for higher disorders as a guide to the eye. The dashed lines are the theoretical band edgespZ W{2q, whereZ is the coordination num- ber. The horizontal dotted line is the BCC estimate 21.13 forWcof Ref. [139]. The

light grey, shaded area in the centre contains extended states; states outside the phase boundary are localised. Error bars are within symbol size for,.

2.2

Calculations and results