THE MODWT ANALYSIS OF THE RELATIONSHIP BETWEEN MORTALITY AND AMBIENT TEMPERATURE FOR PRAGUE , CZECH REPUBLIC

Weather conditions infl uence the health of humans. Changing weather patterns may also cause considerable increase or decrease in the number of deaths. In this paper, we use daily data for Prague, Czech Republic, and the maximum overlap discrete wavelet transform to explore the time scale patterns in the relationship between the average ambient temperature and the number of deaths due to cardiovascular diseases. We summarize several well-known facts, give a short introduction to the maximum overlap discrete wavelet transform and study the relationship between several pairs of wavelet coeffi cients of various levels for the temperature time series and the number of deaths. The results are suggestive of a positive linear relationship between respective wavelet coeffi cients of the 4th level during summer. More specifi cally, we have estimated that a one-degree-Celsius increase in the weekly average temperature during summer is accompanied by an increase in the weekly average number of deaths by 0.37 on average. The odds in favor of an increase in the weekly average number of deaths during summer have been estimated to be 4.22 times higher if an increase in the weekly average temperature occurs when compared to the situation when a decrease in the weekly average temperature occurs. Further research might be desirable to verify and interpret the results.


Introduction and data
In this paper, we use the maximum overlap discrete wavelet transform to study the relationship between the average ambient temperature (T) and the number of deaths (D) due to cardiovascular diseases.The purpose of the use of the wavelet transform is to study the relationships in separate frequency bands.Such an approach is intriguing as it might enable us to reveal relationships that might otherwise remain undetected when explored by "classical" methods.
We use daily data for Prague, Czech Republic, during the period from January 1, 2000, to December 31, 2008.The data were provided by the Czech Statistical Offi ce (http://www.czso.cz/)and the Czech Hydrometeorological Institute (http://www.chmi.cz/).The respective time series are plotted in Figure 1.
As has been already demonstrated in a vast amount of literature, ambient temperature impacts the numbers of deaths considerably.There are several well-known aspects of this relationship.For example, in Figure 2 we plot D against T and smooth the scatterplot with the lowess smoother 1 .The plot slightly suggests the well-known V-shaped relationship between T and D with winter (i.e.low) temperatures accompanied on average by a higher number of deaths and spring and summer (i.e.high) temperatures accompanied on average by a lower number of deaths.The minimum number of deaths occurs close to 16°C.The seasonal relationship between T and D does not appear to be strong from Figure 2, since it is "buried" in the short-term dynamics.Later in the text, we show that when adjustment is made for these short-term dynamics the seasonal relationship is indeed strong.
The antiphase relationship in the seasonal dynamics of D and T -just revealed in Figure 2 for our data -has been identifi ed among others by Laschewski & Jendritzky (2002), who use data for SW Germany during the period 1968-1997 and show that mortality peaks in February and has its minimum in August.A similar V-shaped pattern for cardiovascular mortality against temperature was also identifi ed by Huynen et al. (2001), who use data for Netherlands during 1979-1997. Recently, Bašta et al. (2010) have confi rmed this seasonal type of relationship between T and D for Prague, Czech Republic, during the period 2000-2008.
Another well-known fact about the relationship between ambient temperature and mortality is that extremely hot events in summer cause an increase in the numbers of deaths.For example, Rooney et al. (1998) report that severely high temperatures in July and August 1995 in England and Wales increased mortality signifi cantly.The authors report that people with respiratory and cerebrovascular diseases were affected most noticeably.Laschewski & Jendritzky (2002) show that pronounced heat loads during the summer mortality minimum cause sharp increases in mortality.An increase in mortality during heat waves was also confi rmed by Huynen et al. (2001).They report that old people and those suffering from respiratory diseases were affected the most.Similarly, Grize et al. (2003) report that an all-case increase in mortality occurred in Switzerland during the heat waves in summer 2003.Again, elderly people were affected profoundly.
Not only heat waves but also cold spells affect the numbers of deaths.For example, Díaz et al. (2005) explore the effect of extremely cold weather on the mortality of elderly people in Madrid during the period 1986-1997 and report an impact (lagged by several days) of cold weather on the mortality from respiratory and circulatory diseases.Similarly, Keatinge & Donaldson (1995), studying British data, report that cold spells in winter cause an increase in mortality due to ischemic heart disease.Larsen (1990) studied temperature-related mortality for the United Stated from 1921 to 1985 using the data for six US states.She reports that unusually cold temperatures in winter, spring and autumn months cause higher mortality.
In our paper, we suggest the existence of a relationship between changes in weekly average temperatures and changes in weekly average number of deaths for summer months.For this purpose, we use the maximum overlap discrete wavelet transform, which will be briefl y introduced in the next section.The lowess smoothing of the scatterplot of D against T showing the V-shaped relationship between T and D.

Maximum overlap discrete wavelet transform
The maximum overlap discrete wavelet transform (MODWT) will be summarized in this section.Our summary of MODWT is based on Percival & Walden (2002) and Bašta (2010), which encompass details and proofs of the statements given below.Further books may be recommended too -e.g.Vidakovic (1999) gives a more technical and abstract introduction to wavelets.Gencay, Selcuk & Whitcher (2002) present a less technical introduction with applications in fi nance and economics.At fi rst, linear fi ltration will be introduced; afterwards, wavelet and scaling fi lters and MODWT wavelet and scaling coeffi cients will be defi ned and their properties discussed.
Let S X (.) and S Y (.) be the spectrum of the process {X t } and the process {Y t } defi ned as , ( ) exp( i2 ), for where i is the imaginary unit and f is a real-valued parameter called the frequency and the sequences {γ X,τ : τ = ...,-1,0,1, ...} and {γ Y,τ : τ = ...,-1,0,1, ...} are the autocovariance sequences of the processes {X t } and {Y t }.Since the spectrum is an even and periodic function of frequency, with the period equal to unity, further on it is suffi cient to study and explore spectra only in the range of frequencies [0, 1/2].We may informally say that the spectra S X (.) and S Y (.) inform us about the "frequency content" of the stochastic processes {X t } and {Y t }.The higher the spectrum in a given frequency interval, the larger the contribution of the "dynamics" of that frequency interval to the variance of the process.More specifi cally, it can be shown that The effect of the linear fi ltration with the linear fi lter {a t } of the stochastic process {X t } is given by the characteristics of the linear fi lter {a t }.An important characteristics is the frequency response F a (.) of this fi lter defi ned as the Fourier transform of {a t }, i.e. 1 0 ( ) exp( i2 ), for .
The function |F a (.)| 2 , i.e. the square of the modulus of F a (.), is called the squared gain function of the fi lter {a t }.The squared gain function is an even and periodic function of frequency, with the period equal to unity.Therefore, further on it is suffi cient to explore it only in the frequency range [0, 1/2].The squared gain function informs us about frequencies that are passed and about frequencies that are attenuated by the effect of the linear fi ltration.More specifi cally, it can be shown that Wavelet and scaling filters To introduce MODWT, we need to defi ne a special set of linear fi lters at fi rst.This special set may be created with the knowledge of two elementary linear fi lters: the fi lter {h 1,t : t = 0, ... , L 1 -1} and the fi lter {g 1,t : t = 0,..., L 1 -1}, both of lengths L 1 .
The linear fi lters {h 1,t } and {g 1,t } are interconnected via the so-called quadrature mirror relationship, i.e.
and fulfi ll certain specifi c conditions, e.g., In Figure 3, various pairs of fi lters {h 1,t } and {g 1,t } are presented, corresponding to various so-called families of wavelets.
An ideal fi lter with the nominal pass band [d, u], where d and u are real parameters satisfying 0 ≤ d ≤ u ≤ 1/2, is such a fi lter whose squared gain function is given as 2 1, for ( ) 0, for 0 and for 1 / 2 In Figure 4, the squared gain functions are presented for fi lters of Figure 3.It can be intuitively seen that {h 1,t } is approximately an ideal fi lter with the nominal pass band [1/4, 1/2].On the other hand, {g 1,t } is approximately an ideal low-pass3 fi lter with the nominal pass band [0, 1/4].
For a given pair of fi lters {h 1,t } and {g 1,t }, two sets of fi lters may be created via the so-called pyramid algorithm.This algorithm will not be specifi ed here, because it is technical and will be of no use for the purpose of this text (details on the algorithm can be found, e.g. in Percival & Walden, 2002;Gencay, Selcuk & Whitcher, 2002;Vidakovic, 1999;Bašta, 2010).Only the properties of the fi lters from both sets will be discussed in this text.

MODWT coefficients
Let {X t : t = 0, 1, ..., N -1} be a discrete-time stationary stochastic process.For j = 1, 2, ... let us defi ne a discrete-time stochastic process {W j,t : t = 0, 1, ..., N -1} and a discretetime stochastic process {V j,t : t = 0, 1, ..., N -1} as where mod denotes the operation modulo.The process {W j,t } is called the jth level MODWT wavelet coeffi cients and the process {V j,t } is called the jth level MODWT scaling coeffi cients.The modulo operation affects the construction of W j,t and V j,t for 0 ≤ t < L j -1.On the other hand, the construction of W j,t and V j,t for L j -1 ≤ t ≤ N -1 would remain unaffected if the modulo operation was left out in Equation 2 and Equation 3. Thus, we say that W j,t and V j,t for 0 ≤ t < L j -1 are affected by the circularity assumption, whereas W j,t and V j,t for L j -1 ≤ t ≤ N -1 are not.
Since the jth level wavelet coeffi cients {W j,t } have been obtained with the use of the linear fi lter {h j,t } -which is approximately an ideal fi lter with the nominal pass band [2 -(j + 1) , 2 -j ], it is intuitively clear that the jth level wavelet coeffi cients are associated with the same frequency range and capture the dynamics of the stochastic process {X t } in this frequency range.It can also be shown (see Percival & Walden, 2002) that the jth level wavelet coeffi cients {W j,t } are for many families of wavelets associated with changes between two adjacent weighted averages whereas these weighted averages have an effective length of 2 j -1 .
Similarly, jth level scaling coeffi cients {V j,t } have been obtained with the use of the linear fi lter {g j,t }, which is approximately an ideal low-pass fi lter for the frequency range [0, 2 -(j+1) ].Thus, it is intuitively clear that the jth level scaling coeffi cients are associated with the same frequency range and capture long-term dynamics of the stochastic process {X t } (since low frequencies correspond to high periods and thus trending motions).

The MODWT analysis of temperature and numbers of deaths
Now, let {T t : t = 0, ..., N -1} be an observed time series representing the daily average temperature in Prague, Czech Republic, during the period from January 1, 2000, to December 31, 2008 (see also Figure 1).The positive integer N is the length of the time series.A shorthand notation T may be used to denote this time series.Analogously, let {D t : t = 0, ..., N -1} be the observed time series of the daily number of deaths due to cardiovascular diseases in Prague, Czech Republic, during the period from January 1, 2000, to December 31, 2008 (see also Figure 1).A shorthand notation D may be used to denote this time series.
Since Haar wavelets have been used for the analysis, we can verify the notion in the MODWT coeffi cients section that "wavelet coeffi cients are associated with changes between two adjacent weighted averages whereas these averages have an effective length of 2 j -1 ".For example, the series of the 4th level wavelet coeffi cients for T has been calculated as follows: , for 15,..., 1. 16 16 Therefore, we can clearly see that this series is indeed associated with changes (the minus sign) between two adjacent averages (smoothers), whereas these averages have an effective length of 24 -1 = 8.For the ease of the further discussion we present the individual levels of wavelet coeffi cients and the corresponding frequency ranges and time scales associated with these levels in Table 1.
Table 1 Wavelet coeffi cients of different levels and the corresponding frequency and time scale ranges.

Level of wavelet coeffi cients
Corresponding frequency range Changes between averages which are calculated on the time scale of , contains the frequency year -1 128 days 9 2 -10 < f  2 -9 256 days 10 2 -11 < f  2 -10 512 days In our text below, our main interest will be focused on the series of wavelet coeffi cients for T and D. If the original time series T and D were stationary (i.e.realizations of stationary stochastic processes), then the series of corresponding wavelet coeffi cients would be stationary too (i.e.realizations of stationary stochastic processes) and their sample means would be close to zero (see also the section MODWT coeffi cients).Indeed, visual inspection of Figure 7 and Figure 8 suggests that the series of wavelet coeffi cients for T and D are indeed stationary.In Table 2, we give the summary statistics (specifi cally, sample means and sample variances 4 ) for the series of wavelet coeffi cients for T and D. Only those coeffi cients that are not affected by the circularity assumption have been used in the calculation.
The sample means of wavelet coeffi cients of different levels for T as well as for D are close to zero as might have been expected (if T and D themselves were stationary).Moreover, for D, the sample means of wavelet coeffi cients are slightly negative and tend to more negative values for higher levels.This suggests (due to the way the coeffi cients were calculated) a slight downward trend in D. This is in agreement with the visual inspection of Figure 1 and with the series of scaling coeffi cients of the 10th level for D which is plotted in Figure 8 -these scaling coeffi cients capture (due to their defi nition and properties) long-term movements in the original series D.
It can be shown (see e.g.Percival & Walden, 2002;or Bašta, 2010) that the sample variances of wavelet coeffi cients are related to the contribution of the respective frequency range (or time scale) -associated with these coeffi cients -to the sample variance of T (or D).Informally, we may say that the higher the sample variance of wavelet coeffi cients, the higher the contribution of the respective frequency range to the sample variance of T (or D).Therefore, it can be intuitively seen that whereas the dynamics of T has a strong seasonal component (i.e. the sample variance of the series of the 8th level wavelet coeffi cients is large), the dynamics of D has a weaker seasonal component and a strong high-frequency component (i.e. the sample variance of the series of the 1st level wavelet coeffi cients is large).The Haar MODWT wavelet coeffi cients of (from top to bottom) the 2nd, 4th, 6th and 8th levels, and the scaling coeffi cients of the 10th level for T. The solid vertical lines mark the position of the last coeffi cient affected by the circularity assumption.
It is also interesting to note that the series of wavelet coeffi cients of high levels for T and D (see e.g. level 6 of wavelet coeffi cients in Figure 7 and Figure 8) are not suffi ciently smooth as they would be if they had been obtained by the linear fi ltration with a true ideal linear fi lter for the respective frequency band.This is due to the fact that the Haar fi lter {h j,t } is only a rough approximation to the ideal fi lter of the nominal pass band [2 -(j + 1) , 2 -j ] (see also Figure 6).It can be shown that if other families (e.g., Daubechies-4, Daubechies-8, etc.) of wavelets had been used instead of Haar wavelets, a better approximation to ideal fi lters might have been achieved and the resultant wavelet coeffi cients of higher levels would be a bit smoother.However, the purpose of our text is not to achieve the best approximations to ideal fi lters, but to have results that can be interpreted easily at the end.Due to the simple structure of Haar fi lters, these fi lters are preferred in our case.The Haar MODWT wavelet coeffi cients of (from top to bottom) the 2nd, 4th, 6th and 8th levels, and the scaling coeffi cients of the 10th level for D. The solid vertical lines mark the position of the last coeffi cient affected by the circularity assumption.
Table 2 The summary statistics for the Haar MODWT wavelet coeffi cients for T and D. Only coeffi cients not affected by the circularity assumption have been used.In the calculation of the sample variance, the process mean was supposed to be equal to zero.

The relationship between wavelet coefficients
The seasonal dynamics The relationship between the wavelet coeffi cients for T and the wavelet coeffi cients for D may reveal interesting characteristics of the relationship between temperature and the number of deaths.For example, the seasonal pattern of the relationship between T and D might be easily captured within the relationship of the wavelet coeffi cients of the 8th level, which corresponds to seasonal variations (see Table 1).Thus, in Figure 9 we present a scatterplot of D.wav8 against T.wav8 smoothed by the lowess smoother.
One can clearly see that the seasonal changes in the temperature (i.e.changes caused by changing seasons of the year) are negatively correlated with the seasonal changes in the number of deaths (i.e.changes in the number of deaths exhibiting roughly a one-year period).
Such a result is in agreement with the V-shaped curve of Figure 2, which slightly suggested that winter (i.e.low) temperatures are accompanied on average by a higher number of deaths and spring and summer (i.e.high) temperatures are accompanied on average by a lower number of deaths.However, this seasonal relationship was overlaid by the dynamics of other frequency bands and was thus poorly seen in Figure 2.However, the plot of Figure 9 captures only the seasonal motions themselves with other frequency bands omitted.Therefore, the seasonal relationship which was poorly seen in Figure 2 is clearly seen in Figure 9.The scatterplot of D.wav8 against T.wav8 (smoothed by the lowess smoother).Only coeffi cients not affected by the circularity assumption have been used.

Relationship on intermediate time scales
The seasonal relationship captured between T and D is well-known (see also the section Introduction and data).Above, we have detected it with the use of the 8th level wavelet coeffi cients.Abrupt changes in temperature and the number of deaths might be explored with the use of 1st level wavelet coeffi cients.We will not pursue this goal in our text.On the other hand, we have decided to study the relationship between T and D on low and intermediate levels of wavelet coeffi cients, specifi cally on levels 2, 3, 4 and 5.
In Figure 10 we represent the matrix of scatterplots of D.wavX against T.wavX for X = 2, ...,5.Each of these plots might be informally interpreted as the relationship between T and D for the respective frequency band (e.g.frequency band [2 -5 , 2 -4 ] in case of X = 4).The 2nd, 3rd and 5th level wavelet coeffi cients do not suggest any interesting relationships.On the other hand, the situation seems to be slightly different for the 4th level wavelet coeffi cients.More specifi cally, as suggested by the lowess smoother, there seems to be a slight positive linear relationship between D.wav4 and T.wav4 for almost the whole sample range of T.wav4 -i.e. an increase in T.wav4 causes an increase in T.wav4.This relationship is broken (as suggested by the lowess smoother) only for large values of T.wav4, where an increase in T.wav4 causes a decrease in D.wav4 (however, the signifi cance of this break is likely to be low as there are only a few data points corresponding to large values of T.wav4).
On the other hand, this break in the relationship between T.wav4 and D.wav4 occurring for large values of T.wav4 may be suggestive of another variable being possibly important in explaining the variation of D.wav4.A possible choice of this variable is the season of the year.In our analysis, December, January and February represent winter; March, April, May represent spring; June, July and August represent summer; and September, October and November represent autumn.In Figure 11, the scatterplot of D.wav4 against T.wav4 is presented separately for winter, spring, summer and autumn.The estimates of the slopes of the linear relationship (obtained by ordinary least squares 5 ) are given in Table 3.These results suggest the relationship might be of practical importance for summer.The matrix of scatterplots of D.wavX against T.wavX for X = 2, 3, 4, 5 (smoothed by the lowess smoother).Only coeffi cients not affected by the circularity assumption have been used.
5 See also the text below for the adjustment for the correlation in errors.
Figure 11 also suggests that the relationship between D.wav4 against T.wav4 might be slightly nonlinear (at least for some seasons).Therefore, we will assess whether an increase in T.wav4 is more likely to cause an increase in D.wav4 (when compared with the decrease of T.wav4).We group the coeffi cients T.wav4 into two categories.
The dichotomization of T.wav4 at 0 is as follows: The fi rst category for T.wav4 corresponds to nonpositive values of T.wav4 (i.e.T.wav4 ≤ 0).The second category for T.wav4 corresponds to positive values of T.wav4 (i.e.T.wav4 > 0).Similarly, we group the coeffi cients of D.wav4 into two categories.The dichotomization of D.wav4 at 0 is as follows: The fi rst category for D.wav4 corresponds to nonpositive values of D.wav4 (i.e.D.wav4 ≤ 0).The second category for D.wav4 corresponds to positive values of D.wav4 (i.e.D.wav4 > 0).Such a dichotomization is very practical from the point of view of interpretation of the categories.The respective counts for the categories are given in Table 5 for each season.The ratio of odds (OR.4) in favor of D.wav4 being positive under the condition that T.wav4 is positive to the odds in favor of D.wav4 being positive under the condition that T.wav4 is nonpositive is defi ned as  4. The results support the idea of a relationship between T.wav4 and D.wav4 for summer, which was already found above.
The 95% confi dence intervals for the slopes of linear regression and for OR.4 in Table 3 and Table 4 are much wider than what the original data might imply 6 .The reason for this is the fact that the errors in the regressions are correlated.This stems from the fact that adjacent wavelet coeffi cients are highly correlated because they are in fact overlapping moving averages and thus use common data for their calculation.To correct for this fact, we have assumed in the calculation of confi dence intervals that the series of jth level wavelet coeffi cients decorrelates only after the length 2 j (see also Percival & Walden, 2002).This approach is conservative, in the sense that it is very likely that it produces wider 95% confi dence intervals than they truly are.
6 Also the signifi cance of the t test and Wald tests might be different from what the original data imply.

Figure 11
The scatterplot of D.wav4 against T.wav4 grouped by the seasons of the year (smoothed with the lowess smoother).Only coeffi cients not affected by the circularity assumption have been used.Table 3 The linear regression of D.wav4 against T.wav4 for different seasons of the year.The estimates of the slope of the regression and 95% confi dence intervals are provided.The calculation of the confi dence intervals adjusts for the correlation of wavelet coeffi cients (see the text for details).The signifi cance of the slope was tested with the t test (again adjustment for the correlation was applied), and the results are denoted as follows: -not signifi cant at 5 %, * signifi cant at 5 %.The diagnostics of the residuals was carried out involving the tests of homoscedasticity, autocorrelation (after having adjusted for it at fi rst) and normality and visual inspection of residual plots.An OK is to denote that no signifi cant violations of the assumptions (i.e.homoscedasticity of the error term, no autocorrelation in errors, and normality of errors) have been found.The temporal changes

Season
Wavelets are suitable for exploring different time scales as well as for studying temporal patterns in the data.We may easily assess whether the slopes of the linear regression of D.wav4 on T.wav4 for summer were stable during the years 2000 to 2008.In Figure 12, we present the scatterplots of D.wav4 against T.wav4 for the summers of years 2000 to 2008 together with the least squares lines.
To assess whether the slopes for individual years are signifi cantly different from each other, we may use the model and where β 0 , β 1 and γ i are parameters and ε t is an error term.The null hypothesis of γ 1 = γ 2 = ... = γ 8 = 0 cannot be rejected (at the 5% signifi cance level) -again the correlation of wavelet coeffi cients was taken into account.We may conclude that we are not able to reject the null hypothesis of stable slopes of the linear regression of D.wav4 on T.wav4.
The scatterplot of D.wav4 against T.wav4 for summers of different years (together with the least squares line).Only coeffi cients not affected by the circularity assumption have been used.

Discussion and conclusion
In this paper, we have used the Haar maximum overlap discrete wavelet transform to explore the relationship between the ambient temperature and the numbers of deaths due to cardiovascular diseases on different time scales.At fi rst, we verifi ed the seasonal antiphase relationship between temperature and the numbers of deaths.Afterwards, we studied the dynamics on other time scales.Our results suggest that a positive linear relationship is present between the 4th level of the wavelet coeffi cients of the time series of temperature and the 4th level of the wavelet coeffi cients of the time series of the numbers of deaths.
To interpret the results, we have to realize that Haar wavelet coeffi cients of level 4 are in fact differences of two adjacent non-overlapping moving averages, whereas both the moving averages are a one-half multiple of the simple moving average of 8 days.As 8 days are almost one week, we may informally interpret the results in the following manner: Changes in the weekly average temperature during summer are accompanied by the changes in the weekly average numbers of deaths due to cardiovascular diseases.More specifi cally, a one-degree-Celsius increase in the weekly average temperature in summer is accompanied on average by a 0.37 increase in the number of deaths due to cardiovascular diseases.We have also shown that no signifi cant temporal evolution in the slope of the linear relationship is present in the data set.
Moreover, the odds in favor of an increase in the weekly average number of deaths due to cardiovascular diseases given an increase in the weekly average temperature were found to be 4.22 times higher than the odds in favor of an increase in the weekly average number of deaths due to cardiovascular diseases given a decrease in the weekly average temperature.
It might be also interesting to pinpoint the exact time scale at which the relationship is the strongest, since in our study the chosen time scales were the powers of 2 (see Table 1).Therefore, we could have easily missed the true time scale which occurs somewhere in between the powers of 2.
Even though the nominal confi dence coeffi cients used in the data analysis have been 95 %, it is clear from the way the data have been analyzed (i.e.several levels of wavelet coeffi cients have been explored, the data were separated into subgroups, the "promising" group was selected for further analysis) that the true confi dence coeffi cient is "a bit" lower.Therefore, we may conclude that this paper suggests that an interesting relationship may exist between temperature and the number of deaths.Future verifications of the relationship on independent data sets (from other countries, temporal epochs) are necessary and so is the discussion on the plausibility of this relationship from the medical and climatological view.

Figure 3
Figure 3Pairs of linear fi lters {h 1,t } (in the top row) and {g 1,t } (in the bottom row) are presented in each column.The fi rst column (i.e. the fi rst pair) corresponds to the family of Haar fi lters, the second column to the family of Daubechies-4 fi lters, the third column to the family of Daubechies-6 fi lters, and the fourth column to the family of Daubechies-8 fi lters.

Figure 4
Figure 4 Squared gain functions are given for the fi lters of Figure 3, i.e. the top row corresponds to the squared gain function of the (from left to right) Haar, Daubechies-4, Daubechies-6 and Daubechies-8 {h 1,t } fi lter.Similarly, the bottom row corresponds to the squared gain function of the (from left to right) Haar, Daubechies-4, Daubechies-6 and Daubechies-8 {g 1,t } fi lter.The dotted rectangles depict the squared gain functions of ideal fi lters.

Figure 10
Figure 10 A|B) is the conditional probability of A, given B. The estimates of OR.4 for different seasons are given in Table

Table 5
Counts in individual categories for the four seasons of the year.