Development of the Laycock-Gott occupancy model

C.A. Pantjiaros P.J. Laycock G . F. G ott S.K. Chan

Indexing terms: Ionospheric propagation, Laycock-Gott spectrum occupancy model, Spectrum occupancy modelling

Abstract: The authors examine mathematical modelling of occupancy for the entire high- frequency (HF) spectrum. The theory of modelling the experimental occupancy data is presented, and an example of the modelling procedures is given.

1 introduction

This is a companion paper to ‘HF spectral occupancy at the solstices’ [l]. That paper describes the occu- pancy-measurement system, and gives models for H F spectral occupancy during stable-day and stable-night ionospheric conditions at the times of the winter and summer soistices, for a complete sunspot cycle. This paper describes the theory and procedures used to develop the models.

10.15 frequency, MHz

Fig. 1 Measurement of congestion

10.60

Occupancy has been measured as congestion values for each International Telecommunications Union (ITU) frequency allocation across the entire H F spec- trum [l]. Congestion is defined as the fraction of sig- nal-strength observations across each allocation for which the signal exceeds a defined threshold. This is illustrated in Fig. 1, which shows RMS-signal values 0 Crown Copyright, 1997 ZEE Proceedings online no. 19970982 Paper received 5th August 1996 C.A. Pantjiaros, G.F. Gott and S.K. Chan are with the Department of Electrical Engineering and Electronics, UMIST, PO Box 88, Manchester M60 lQD, UK P.J. Laycock is with the Department of Mathematics, UMIST

measured across an allocation, and a slicing-threshold level. For the results presented in this paper, and in the companion paper [ 11, the measurement bandwidth is 1 kHz, the frequency increment is 1 kHz, and the dwell time is 1s. The measurement antenna is an active cali- brated monopole [l].

2 Model specification

The available congestion data comprise day and night measurements made approximately at the times of all the winter and summer solstices from January 1982 until December 1995, a period which exceeds a sunspot cycle. For each set of measurements the HF spectrum is divided into 95 allocations, and congestion values are obtained for each allocation, for five threshold levels. Each set comprises about 5500 congestion values, and corresponds to about 400 000 channel observations for the measurement bandwidth of 1 kHz [l].

It is appropriate to attempt to model the occupancy measurements. This will provide a convenient summary of the data set, a knowledge of the dependence of occu- pancy on physical parameters such as threshold level, frequency, and sunspot number, and also the ability to predict occupancy. Systematic change in usage of the spectrum will also be revealed. It is hoped that such models will be a useful aid to communication-system design and operation, and to spectrum management.

The measured congestion values show features of regularity, known as the systematic component, cou- pled with features of irregularity, known as the random component. The random variation may be due to ran- dom fluctuations in the propagation characteristics of the ionosphere, or in the usage of the spectrum. A ‘full’ model is one in which all the variability of the data is attributed to systematic effects. Such a model fits the data precisely, and is typically achieved by having as many parameters as measured values. Conversely, a ‘null’ model for congestion assumes that all variations are random, and not dependent on changes in any other measured physical parameters. A ‘parsimonious’ model is one which includes as many parameters as are needed to describe the systematic variation, without treating random effects as systematic. This model gives a compromise between economy of parameters and accuracy of fit, which is a fundamental requirement of statistical modelling. Also, a parsimonious model offers better prediction than a model which includes unneces- sary extra parameters, and is typically more amenable to the introduction of additional data.

Ideally, the parsimonious model seeks to represent the systematic variation of the experimental data by an

33 IEE Proc.-Commun., Vol. 144, No. 1, February 1997

expression in which mean values of congestion for allo- cation k are given by

where xl, x2 ... represent parameters on which occu- pancy depends. Consider the choice of a linear function of the variables, i.e.

where xkp represent functions of the parameters, and bp their estimated coefficients.

The measured values of congestion are proportions, and consequently their values must lie in the range 0 - 1. Since a significant fraction of the experimental val- ues lies on, or close to, one of these boundary values, there is a risk that a linear model, as defined by eqn. 1 may give estimated values of congestion which lie out- side this range. To overcome this, the logit transforma- tion [2] has been used as a link function between the modelled congestion values and the linear function where

logit(Qk) = log, Yic

Hence

ey‘ Qic = ~ 1 + eyk

where 0 < Q k < 1 and --oo < y k < 00 (2)

Thus the linear function of the variables has been included in the index of an exponent. The logit model of eqn. 2, with the appropriate index function Y k , forms the Laycock-Gott model for spectral occupancy. It was originally proposed for this application by Dr. P.J. Laycock of the Department of Mathematics at UMIST, and subsequently developed by Gott, Ray, Morrell, and Dennington [3-61. The logit transforma- tion is shown in Fig. 2. It is reasonably linear for mid- range values of Qk, and enables the extreme values of 0 and 1 to be approximated relatively simply, as the index function Y k is then required only to be large and negative, or large and positive, respectively.

1.0,

-10 -5 0 5 10

logit transformed value, yk Fig. 2 Logit transformation for modelling congestion

There are other transformations which would restrict the range of Qk between 0 and I , such as the probit transformation [2] , which gives the following expression for Qk:

1 Q k = __ exp { $} dX

-cc

Very similar results have been obtained when using the

34

logit and probit transformations to model the same congestion data.

The differences between the measured and modelled congestion values define the ‘residuals’. These residuals provide an assessment of the random component of the data set, and also an estimate of the accuracy of the model fit, since, when appropriately scaled, they should form independent asymptotically standard normal vari- ates [2].

3 Model assumptions

Each value of congestion in the occupancy experiment represents the fraction of channels in a particular allo- cation whose signal level exceeds a given threshold, as shown in Fig. 1. The number of channels visited in that allocation is the number of trials N. The number of channels n is counted, which exceeds a given threshold level. Congestion is then given as nlN, where N is known and n is measured.

In fitting a model to experimental congestion data, it has been assumed that the individual measured conges- tion values emanate from binomial probability distribu- tions [2, 71 with mean value Nx, where 3t is the probability that a successful outcome will occur in a single trial.

This is equivalent to assuming that the probability of a given signal level exceeding a defined threshold is equal to n, and thereafter that the successive signal measurements within the chosen allocation are statisti- cally independent, but otherwise identically distributed. Although this particular assumption of independence is not completely satisfied, alternative models incorporat- ing serial dependence between successive signal obser- vations within an allocation have shown no improvement in terms of the overall objective of mod- elling mean congestion values across the entire HF spectrum, and under differing conditions. Since these particular models are also technically much more com- plex, they are not presented here. Note also that some of the other physical measurements used to model the behaviour of z, such as time of day and frequency, may only be approximately constant across a given alloca- tion, but the sheer scale of the data set means that some smoothing is required.

Another technical point concerns the form adopted for the joint distribution of successive congestion counts at different thresholds within a given allocation on any one occasion, which we have modelled as inde- pendent binomials. Strictly speaking, the successive increments in the congestion counts at the chosen levels of threshold will have a joint (multivariate) multino- mial distribution. Again, this considerably more com- plex distributional assumption has been tested on arbitrarily selected subsets of the data, and, as described above, for the serial dependence modifica- tion, found no improvement in terms of the overall objective of modelling mean congestion values. These arguments have led us to fit the independent binomial models presented in this paper to the experimental con- gestion values. With the assumptions described above, use can be made of standard theories for fitting gener- alised linear models, and these are conveniently imple- mented in a few of the more advanced, but nevertheless readily available, statistical computer packages. For this work the package Genstat 5 [8] was used, both on IBM compatible PCs and on a Cray CS6400 main- frame.

IEE PuocCommun., Vol. 144, No. 1, February 1997

4 Method of maximum likelihood

For each measured congestion value qi, there is an independent binomial variable ni. Since the values of ni are independent of each other, their joint likelihood L(xi .. zR: Ni .. NR, ni ..nR) is the product of their indi- vidual likelihoods [9, 101 . Hence, the log-likelihood function, less the constant term

For the purposes of statistical modelling, the measured congestion values qi are expressed in terms of the parameters (explanatory variables) on which congestion values are assumed to depend. This is achieved by modelling the probabilities ni (for i = 1 ... R). If nj were free to assume arbitrary values, the maximum likeli- hood criterion would dictate that xi be equal to qi. Under these circumstances, all variations in the data, including random, would be attributed to systematic variations. We would effectively be fitting a ‘full’ model in which fitted values are equal to the corresponding measured values. However, in general, not all the varia- tions in the data will be attributed to systematic varia- tions and a component of random variation will be assumed. The values of ~d are then constrained by the nature of the systematic specification of the model in question.

For linear logistic models with p parameters,

where g(q) is the logistic link function, xip represent functions of the parameters, and pp their coefficients. This equation becomes

The log-likelihood function of eqn. 3 then becomes log, L(T1..7rR : N;..NR,ni..nR)

R

i=l A property of the exponential family of distributions is that the maximum of the log-likelihood function is given uniquely by the solutions of simultaneous equa- tions obtained when the partial derivatives of the log likelihood with respect to pp are set to zero [2, 91. Set- ting the vector of partial derivatives to zero to maxim- ise the likelihood yields the estimating equation R r -

The p simultaneous equations for the partial derivatives of eqn. 4 are nonlinear, and have to be solved by numerical iterative methods. Maximum-likelihood esti- mates bp.of the model coefficients pp are obtained. In the algorithm used by Genstat 5, the iterative equations have the same form as the normal equations used in least-square regression, except that a weighted least- squares method is used, in which each response varia- ble is weighted by its variance [l 11. Thus the estimation procedure for the model parameters is referred to as iteratively reweighted least-squares regression [ 1 13.

IEE Proc.-Commun., Vol. 144, No. 1, February 1997

5 Significance of model parameters

A term should normally be included in a model only if its contribution towards overall model accuracy is sta- tistically significant at some chosen level of confidence.

Most often, statistical software packages which are used to provide estimates of model parameters provide also estimates of their standard error (SE) and a test statistic known as the t-ratio [12]. The t-ratio is defined as

where b is the estimate of the mean of the parameter, Po is the hypothesised null value of the parameter, typi- cally zero in this context, and SE(b) is the standard deviation of the estimate.

Since the measured congestion data contain a ran- dom component, the test statistic will also be a random variable. If the hypothesised value is correct, statistical theory shows that the calculated statistic follows a sta- tistical distribution known as the t-distribution [12]. This last statement is only valid if the data values can be regarded as independent random observations from some normal distribution, which will be asymptotically true for the maximum-likelihood estimates used here.

The hypothesis that the parameter is not significant is tested by setting Po = 0. This hypothesis is known as the null hypothesis. The test statistic of eqn. 5 then takes the simple form blSE(b).

If the null hypothesis is a valid one, it can be shown that the test statistic approximates the theoretical t- distribution with Y degrees of freedom [12], i.e.

where Y = number of measured congestion values - number of model parameters.

Numerically large negative or positive values of the test statistic would imply rejection of the null hypothe- sis, so the critical region comprises the values in the left-hand and right-hand tails of the distribution. The two critical values at the 1% level of significance, are it0.995(”). Hence the hypothesis Po = 0 would be accepted if the absolute value of b/SE(b) < t0.995(Y). Otherwise the alternative hypothesis is assumed which states that Po z 0. There is always a 1% risk of wrongly rejecting the null hypothesis.

6

The ‘residual scaled deviance’ or simply ‘residual devi- ance’ D (since the scale factor for a binomial distribu- tion is one) is defined to be twice the difference between the maximum achievable log likelihood and that attained under the fitted model [2]. The maximum achievable log likelihood is attained at the point 8 = n / N k Such a model is the perfect model or a ‘full’ model where all variation is attributed to systematic effects. In general, the deviance function for any given model, with fitted probabilities 2, will be given by

D ( n : %) = 21og, L(E : n) - 210g, L(? : n) The ‘residual mean scaled deviance’ [2] d is then given by

Goodness of a model fit

D d = - R - P

35

where R is the number of experimental congestion val- ues, and p is the number of independently estimated model parameters.

The addition of further significant parameters has the effect of reducing the residual deviance, and goodness of fit is indicated by closeness of the mean scaled devi- ance to unity. Nevertheless values as large as 10 have consistently been found to give excellent fits between the models and the experimental data.

The deviance function is most directly useful not as an absolute measure of goodness-of-fit but for compar- ing two nested models. For instance, one may wish to test whether the addition of further parameters signifi- cantly improves the fit. Let MO, denote the model under test containing p parameters with fitted values denoted as t?o and MA the extended model containing k - p additional parameters with fitted values denoted as

The reduction in deviance is D ( n : i i o ) - D(YL : ? A ) 2 log, L(?A : n) - 2 log, L(?o : YL)

This statistic is asymptotically distributed as x2 with k - p degrees of freedom, under the null hypothesis Ho:

There are other simpler measures of accuracy of fit. For example, a histogram of errors may be con- structed, and the quality of fit determined relative to the overall appearance of such a histogram. Alterna- tively, the absolute difference between each of the measured and fitted congestion values, denoted as q2 and Q, respectively, is a useful summary of the overall quality-of-fit afforded by a model. The absolute differ- ence or error e, defined in this context as

= ... = /3k = 0 [2].

can be usefully summarised by the 'frequency distribu- tion' of absolute errors and its visual counterpart, the error histogram.

Fig. 3, shows an example of an error histogram per- taining to the solstice model for summer stable-day occupancy, over the period 1982 to 1995 [l]. The model is derived from a total of about 5500 experimental con- gestion values, of which 54% are fitted by the model to give an absolute difference lying in the range 0 to 0.01, The cumulative-relative-frequency distribution axis shows the percentage of fitted values whose error lies between 0 and e.

0 0.1 0.2 03 O L 0 5 e r ro r e

Fig.3 modelfit

Example of an ewor histogram used as a measure of goodness of

7

7.1 Development o f the solstice modets For illustrative purposes, the maritime mobile allaca- tion 57 (16.860-17.410MHz), for winter stable day, will be examined. Initially the measured congestion values are plotted against various parameters, e.g. time of measurement, threshold and frequency.

An example of the typical variation of logit-trans- formed congestion with field-strength threshold (where there is occupancy at all thresholds) is shown in Fig. 4. The relationship between logit-transformed congestion and field-strength threshold in decibels is h e a r with a slope of approximately -0.1, implying a logistic distri- bution for field strength at each frequency in this range.

2 ,

I I I , I I I I 1 -130 -120 -110 -100 -90 -80

20 [oq0 E (V/rn)

Fig. 4 + sunspot 15, 1985 0 sunspot 154, 1989 0 sunspot 11, 1995

Logif- transformed congesdion against Jield-strength threshold

For the example presented in Fig. 4, the slope for 1985 when the sunspot is at a minimum (1 5) is -0, in 1989 when the sunspot is at a maximum (154) is -0 0954 and in 1995 when the sunspot number is t 1 the slope is -0.103. Such investigations suggest that the coefficient for threshold can be taken to be i d en1 of sunspot number. Note that, for change in sunspot number from 15 to 1 effect on congestion as a 15dB reducti level. Far comparable sunspot values in 1985 and Fig. 4 suggests that there is a diffix value of congestion which corresponds to a mately a 2dB change in threshold. This may be the random nature of the ionosphere at the times o€ the measurements, which is smoothed in the mod the use of the smoothed sunspot number, change in the usage of the spectrum, and possibIe measurement errors.

From such plots, it is deduced that the variation of lagit-transformed congestion with field str is approximately linear for allocatiozs which are ied at all thresholds, and that the gradient of the straight line does not vary significantly with sunspot number.

m. to

0.6pVJm, but at night such a low threshold typically intercepts the atmospheric noise for the bandwidth of 1 kHz. However, owing to the fact that

Also, results apply for the threshold rmge I- Daytime models have been shown to be vafi

IEE Pror -Commn, Vol 144, No I , Februmy I997 36

congestion values are bounded in the range 0-1, the relationship of congestion with threshold departs from linearity for allocations in which congestion assumes these bounds for a few thresholds at the extremes. Also, the logit transform is by definition nonlinear for values close to 0 or 1. The gradient of the transform close to the extremes of values of modelled congestion (approximately > 0.85 and < 0.15), is much lower than the gradient in the middle portion. One would expect therefore that, for allocations in which congestion is very low, the gradient of the linear function for logit- transformed congestion with field-strength threshold would also be low. For well occupied allocations the gradient is approximately -0.1, as indicated in Section 7.3.

160 -

kl 120- n 5 - 0

80- 3 5 -

LO-

0

7.2 Partial dependence of congestion on year of measurement Fig. 5 shows the sunspot number on the left y axis and the measured congestion values in allocation 57, at 2kV/m for stable day, on the right y axis. The positive correlation noted in this example is typical for alloca- tions above approximately 10MHz at stable day and is consistent with the changes in the MUF observed over the sunspot cycle. The negative correlation observed in the low end of the HF spectrum at stable day is in accordance with the changes in the absorption levels at different sunspots.

+

\+ e t

-0.5

- 0 L

-0.3 $ C

- 0 L

- 8 U

-0.2 f - E -0.1 1 U

0 1 I I I 80 0L 88 92 96

yeor Fig.5 year, for allocation 57 + congestion 0 sunspot

Variation of smoothed swzspot and measured congestion with

The strong systematic dependence of congestion on field-strength-threshold level Y (V/m), and on sunspot number, may be expressed in the model index function as

g57 = A + B20 log,, \k + Csunspot (6) and fitted to the logit-transformed mean. The resulting fit for this particular allocation has a residual mean deviance of 6.6. The following estimates for the model coefficients were obtained:

A = -12.2957 t-ratio = -21.2 B = -0.0935662 t-ratio = -17.8 C = 0.00949317 t-ratio = 9.9

For all the stable day measured congestion values for allocation 57, 27% were given by the model to an accu- racy of rO.01, 76% to an accuracy of r0.05 and 97% to an accuracy of 20.1. Measured and fitted values for field-strength threshold 2pV/m are shown in Fig. 6.

IEE Proc -Commun , Vol 144, No. 1, February 1997

I I I I 80 EL 88 92 96

year Variation of measured and fitted values of congestion with year, Fig.6

for allocation 57 fitted

0 measured

To determine whether this model fit is sufficient to account for the systematic variation in the data set without treating random effects as systematic, and therefore that no other terms currently excluded in the above equation are statistically significant, the set of residuals is examined. If all systematic variation is accounted for in the model, the scaled residuals should be distributed asymptotically normally, and a half-nor- mal plot of the residuals should lie on a straight line [2, 81. Such a plot for the above model is shown in Fig. 7, and supports the assumptions on which the model is based.

i: I , I I

0 1 2 3 expected half-norma4 order statistic

Fig. 7 Half-normal plot of ordered absolute standurdised residuals

When it is verified that the scaled residuals have an approximately normal distribution, it is evident that there are no significant systematic components which are absent from the model. To reinforce this, additional parameters were included in the model, for example (sunspot)2. The results showed that the reduction in residual mean deviance due to such additional parame- ters was insignificant and the resulting t-ratio for these parameters was lower than the critical value for statisti- cal significance.

11

7.3 Model for each individual allocation Analysis and modelling was carried out as described above for each of the 95 allocations. For the majority of allocations the model specification obtained for allo- cation 57 in eqn. 6 above gives a very good fit to the measured data. This general model specification is therefore fitted to each individual allocation and three sets of 95 coefficients are obtained. The coefficients are then plotted against allocation centre frequency and are shown in Figs. 8-10.

+ ++ + + + 9 ++ ++ +* 9 +%+&U++* +++#,' +

-lo{

++ ++ ++

+ + + +

-251 + +

+ +

t-

+ t t+

++'+ ++ '+ + ++ + ++

+ i+ ++ +

++ + +

+ + + +

+ + 0 10 20 30

allocation centre frequency, MHz Coeficient Ak against allocation centre fuequency

-30 I I I

Fig. 8

-O.OL,

-1 + + U + +

L + E -0.16

-o,201 +

+ + + + +

I I I I 0 10 20 30

allocation centre frequency, MHz Coeflcient Bk against allocation centre fvequency Fig. 9

The plot of the Ak coefficients (Fig. 8) reflects the mean value of congestion in each allocation for all thresholds over the period 1982-1995. The shape of the variation shows that, for stable day, occupancy at the low and high end of the H F spectrum is relatively low. The well occupied allocations occur in the frequency range 5-1 8 MHz.

The Bk terms (Fig. 9) show the effect of threshold on the mean value of congestion. A linear relationship of logit-transformed congestion and threshold is assumed for all allocations; however the gradient changes with frequency. The value of the threshold coefficient is approximately -0.1 for well congested allocations, i.e.

38

where Ak is high. The value of the threshold coefficient is lower where Ak is low due to the saturation effect in the congestion values and the nonlinear shape of the logit transformation for very low values.

The c k coefficients (Fig. lo) show how the mean value of congestion in each allocation changes with sunspot number. Above approximately lOMHZ the coefficients are positive, indicating positive correlation.

o'061 0.04

Y

0.02

U

0 00 -! I + -f

+ + -0.02 I I I

0 10 20 30 allocation centre frequency, MHz

Coeficient Ck against allocation centre frequency

I

Fig. 10

The model contains 95 Ak coefficients, one for each allocation. Attempts to model the Ak coefficients in terms of the parameter frequency results in a severe degradation of the model fit. This is due to the smooth- ing which occurs, particularly when allocations of high congestion reside next to allocations of low congestion. The Bk and C,< terms can be modelled in terms of fre- quency without degrading the model fit significantly, and polynomials of order 2 are able to account for the variation. Higher-order polynomials tend not to be sta- tistically significant, as indicated by the values of t- ratio, and by an insignificant reduction in residual mean deviance of the model fit.

8 Model for winter-day occupancy across the HF spectrum

From the analysis of the winter stable-day measured data in each allocation, model parameters are identi- fied. These parameters are then added to the null model for occupancy, using stepwise-generalised linear regression which selects significant model parameters on the basis of a significant reduction in the residual mean deviance. The resulting model-index function, which consists only of statistically significant parame- ters, is given by

Yk = A k + (Bo + Bific + &f2)2010g,, * + (CO + c1 f k + C2f;)sunspot

The estimated coefficients and t-ratio values are given in the companion paper [l].

The same procedure is repeated for the other three sets of solstice data. If the resulting models are correct, additional data may be added without affecting the model fit and model specification significantly. Predic- tions using such models should be almost as accurate in

IEE ProcCommun., Vol. 144, No. 1, February 1997

their fitted values when they are compared with meas- ured data. Such accuracy of prediction has been dem- onstrated [ 11.

9 Conclusions

Aspects of statistical analysis appropriate for the math- ematical modelling of congestion data have been pre- sented, and examples of modelling procedures have been described. The models derived express the signifi- cant dependence of occupancy on physical parameters such as field strength threshold, frequency,and sunspot number. The mathematical expressions fit the measured data accurately and enable a summary of the data set and predictions to be made [I]. The data and models may be used by HF operators, system designers and frequency managers.

The ability to model solstice data accurately in terms of physical parameters, and to predict future values, has motivated a more intense measurement programme which is at present being undertaken. In particular, present effort attempts to include in the model the parameters bandwidth, geographical location (within northern Europe), and week of year.

10 Acknowledgments

The long-term investigation of HF spectral occupancy was initiated by J.R. Guest of DRA, and we thank DRA for their support over the complete period of the work. Additional support was given by the EPSRC, and the facilities at Baldock Radio Station are pro- vided by the Radiocommunications Agency. The con- tributions of former UMIST researchers A.R. Ray, M.

Morrell, and J. Brown are also acknowledged. This work is published with the permission of the Controller of her Britannic Majesty’s Stationery Office.

11 References

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