Random intercept models | Centre for Multilevel Modelling | University of Bristol
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You previously purchased this article through ReadCube. Institutional Login. Log in to Wiley Online Library. Purchase Instant Access. View Preview. Learn more Check out. In Sect. We also calculate its moments and present explicit results for a special power spectrum. Then, we repeat the derivation for bivariate distributions in Sect. We go on to discuss possible numerical implementations of these results in Sect. Using numerical evaluation, we can discuss the properties of the distribution functions in more detail in Sect.
There, we comment on the general analytical properties of the uni- and bivariate functions. We also use the moments to construct an Edgeworth expansion of the univariate distribution, and we generalise our derivations and results to higher dimensions. We conclude the article in Sect. We describe a real Gaussian random field by its Fourier decomposition 2 The Fourier modes are independently distributed, each with a Gaussian probability distribution: 3 The mode dispersions are determined by the power spectrum, 4 The following derivations will be independent of the choice of a power spectrum, as long as it obeys the constraint of non-negativity.
A sufficiently large finite field will be representative of a field on the whole of R N dim if the random field has no power on scales larger than L. This is also equivalent to the assumption of statistical homogeneity on these scales. Together with isotropy, we know that the correlation function depends on the distance modulus, or lag parameter, only: 5 We will now concentrate on a one-dimensional field to keep expressions simple.
However, in Sect. For a real-valued random field, the Fourier components fulfil. Using this property, the mode expansion of the field can be split up as 8 Without loss of generality, we assume that the field has zero mean, since we can always achieve this by a simple transformation. Then, the zero mode g 0 cancels out. We can then insert his expansion into the estimator 6. For the spatial integrals, we can use the integral representation of the Kronecker delta symbol : 9 The correlation function is then given by 10 Executing the sum over m , only half of the terms survive, and the remaining exponentials give a cosine function: 11 Now that we have a convenient expression for the estimator of the correlation function, we need to take one more intermediate step before calculating its probability distribution.
This is the characteristic function, which, in general, is defined as the Fourier transform of a probability distribution function. For the given random field, we can calculate the characteristic function by means of an ensemble average: 12 Since the field is Gaussian, the modes are independently distributed, and the probability distribution factorises. On the other hand, some C n may be equal, and so there may be poles of higher order multiple poles.
We were unable to further simplify the limit of the infinite sum in the probability distribution function However, as long as the power spectrum decreases at least like k -2 for large k , our numerical implementation of the sum formulae, as described in Sect. In practice, it is therefore possible to truncate the series at some maximum mode number N without losing much precision. Also, it is obvious from Eq. We also note at this point that Eq.
When we present numerical results in the further course of this article, we therefore give these quantities only. In general, the steepness of the power spectrum determines which maximum wave number N is necessary: Inserting the wave numbers from Eq. In this section, we calculate the moments of the distribution. Apart from possible use in future applications, this is also useful as a check for the distribution function derived above, since we can derive the moments in two independent ways and compare results. Alternatively, we can also calculate the moments from the probability distribution function by the integrals 33 An important check for the sanity of the distribution function will be to re-obtain the normalisation as unity by this approach.
To calculate the higher moments, we make use of the integral 38 and of sum formulae of the type These follow from taking derivatives with respect to s in Eq. We have checked the results for mean, variance, skewness and kurtosis and have reproduced the results of the characteristic function approach, demonstrating the validity of the probability distribution function. Analogous expressions in the presence of higher-order poles could be obtained by integrating the corresponding probability density A. In general, the probability distribution function we found is a sum formula that needs to be evaluated numerically.
It is not clear to us yet whether this connection to elliptical functions is a coincidence for just this special case, or whether it points towards a possible reformulation of the probability distribution for general power spectra. From Eq. Then, inserting into Eq. For the same power spectrum, we can also explicitly calculate the moments. For power law power spectra with a different exponent, Eq.
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Regrettably, these do not yield any known functions for the full probability distribution, or for the moments, as far as we are aware. All preliminaries carry over from the univariate case, and the starting point of this calculation is 54 The characteristic function is now bivariate as well, 55 In the last step, we have defined a generalised shorthand for the factors to allow for the two different lag parameters x m.
But an important difference arises in this step, since the inversion now contains a double integration. From the resulting formula, the symmetry will not be immediately apparent. However, we have checked the equivalence of both approaches by also explicitly evaluating the other choice. Also, we will assume simple poles in both integrations, and will only briefly comment on the effects of multiple poles at the end of this section.
The poles for the inner integration are now located at 58 and, for simple poles, their residues are 59 Here we have simplified the expression by defining the determinant factor 60 The arguments as to the choice of contours apply exactly as before, since the imaginary part of the poles remains unchanged from Eqs. Thus, poles with positive C n 2 factors lie within the lower contour, whereas those with negative C n 2 lie within the upper contour. So we obtain the full integral as 61 The remaining task is to calculate the second integral. For simple poles, the residues are 64 Furthermore, the choice of contours is also very similar to the previous procedure.
But like we do for the univariate distribution in Appendix A , we can find corrections to get the most general result. In this case, they become rather unwieldy, and we will not present them in this article, since they can be easily avoided by choosing well-behaved power spectra and non-commensurable lag parameters. However, in Fig.
Using Excel to Calculate and Graph Correlation Data
The corrections were implemented in the numerical code, as described in Sect. The right panel of Fig.
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Still, the non-Gaussian nature is apparent in this case as well, since the isoprobability contours have kinks and straight segments following the prescription of the Heaviside terms. There are analogues of mean, variance and also of higher moments for multivariate distributions. As we did in Sect. Since two-dimensional integrals are more cumbersome, we restrict ourselves to the characteristic function approach for the bivariate moments. The mean is now a vector, but calculation and result are quite similar to the univariate case in Eq. In this work, we have derived both univariate and bivariate distribution functions, but no multivariate distributions of more than two correlation functions.
In principle, however, these could be obtained by exactly the same type of derivation. From the general multivariate characteristic function, Eq. This approach could be quite efficient for practical purposes, where the likelihood function is only needed at discrete points anyway. However, the efficiency and stability of this approach depends strongly on the input C nm factors, since the denominator in the integrand might prove to be numerically hard to handle.
Estimating the performance of such a calculation would require further studies. The numerical implementation of the probability distribution function, Eq. For a small number of modes and benign parameters number of dimensions, power spectrum, lag parameter , this can be done straightforwardly in any computer numerics system. In general, however, the summation of many terms with very different orders of magnitude due to the exponentials and the product factors leads to a high demand in numerical accuracy.
The higher the ratio of separation and field size or the number of significant modes, the higher the necessary precision. The figure shows that approx.
Scatterplots and Correlation Coefficients
The standard deviation is defined as the square root of the variance V. The variance is defined as the sum of the squared deviations from the mean, divided by n- 1. Operationally, there are several ways of calculation: 6. Warning: some programs use n rather than n- 1! Coefficient of variation Although the standard deviation of analytical data may not vary much over limited ranges of such data, it usually depends on the magnitude of such data: the larger the figures, the larger s.
Therefore, for comparison of variations e. The RSD is expressed as a fraction, but more usually as a percentage and is then called coefficient of variation CV. Often, however, these terms are confused.
Constrained probability distributions of correlation functions
When needed e. A single analysis of a test sample can be regarded as literally sampling the imaginary set of a multitude of results obtained for that test sample.
The uncertainty of such subsampling is expressed by 6. The critical values for t are tabulated in Appendix 1 they are, therefore, here referred to as t tab.
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