Size intervals

A difference equation is a relation between the differences and the independent variable, A y, A " y,. . . , Ay, y, x) = 0, where ( ) is some given function. The general case in which the interval between the successive points is any real number h, instead of I, can be reduced to that with interval size I by the substitution x = hx. Hence all further difference-equation work will assume the interval size between successive points is I. 

Romberg s Method Rombergs method uses extrapolation techniques to improve the answer (Ref. 231). If we let Zi be the value of the integral obtained using intei val size h = Ax, and I9 be the value of I obtained when using interval size h/2, and Zq the true value of I, then the error in a methoa is approximately/ ", or... 

One can narrow the interval size h for greater precision, or increase it as the reaction proceeds for greater efficiency. A Runge-Kutta solution to Scheme I, Eq. (5-54), is of some interest, since this result could not have been obtained by integration. The val-... 

FIG. 2. Sister chromatid separation does not depend on the mitotic spindle. Light micrographs of mitosis in living flattened endosperm from Haemanthus katherinae BAK. treated with colchicine (c-mitosis). The micrographs were taken at 10 min intervals. Size bar, 10 /tm. Reprinted with permission from Mole-Bajer (1958). 

Indicator and sample selection are not the only choices a researcher has to make when using MAXCOV. A decision also has to be made about interval size, that is, how finely the input variable will be cut. Sometimes it is possible to use raw scores as intervals that is, each interval corresponds to one unit of raw score (e.g., the first interval includes cases that score one on anhedonia, the second interval includes cases that score two). This is what we used in the depression example. This approach usually works when indicators are fairly short and the sample size is very large, since it would allow for a sufficient number of cases with each raw score. In our opinion, this is the most defensible method of interval selection and should be used whenever possible. However, research data usually do not fit the requirements of this approach (e.g., the sample size is too small). Instead, the investigator can standardize indicators and make cuts at a fixed distance from each other (e.g.,. 25 SD), thereby producing intervals that encompass a few raw scores. 

We have to conclude that selecting an interval size a priori appears to be the best approach. However, things are more complicated in practice. When the data are far from ideal (e.g., the taxon has very low base rate, indicators are short, or sample size is small), an a priori approach may fail because it does not take these problems into account. For example, if indicators have only 12 levels (rather than the recommended 20), an interval size of. 25 may produce holes in the taxonic plots because the indicators are sliced too finely and some of them fall in between raw values (e.g., an interval ranging from 1.05 to 1.90 on anhedonia does not contain any cases). On the other hand, a low base rate taxon may need very fine cutting for the full peaks to emerge. Under these conditions, an interval size of. 25 may produce cusps because it allocates all of the taxon members to the final interval. 

Lenzenweger (1999) performed MAXCOV using the three scales as taxon indicators. He used an interval size of. 50 SD and MRIN of 15. Two of the plots showed clear peaks, and one produced a cusp. The one incomplete peak was probably a consequence of the interval size being set too high, which allowed for only seven intervals on the input variable. A lower interval size would have produced a finer gradation and probably allowed the cusp to turn into a full peak. However, this may not have been possible due to the modest size of the sample. The base rates estimates were. 11,. 22, and. 13. The author did not report a base rate consistency test, but one can easily calculate the SD of the three estimates to be. 06, which is somewhat high but probably acceptable. 

SSMAXCOV analyses were performed with the two sets of single indicators. Extreme values of the input indicators were combined in a single interval until there were at least 50 cases present in each interval, effectively setting MRIN at 50. MAXCOV was applied to the two sets of paired indicators with an interval size being set to one in raw score units that is, each interval corresponded to a specific raw score. MAXCOV was also applied to the theoretical indicators, and instead of using the same interval size (e.g.,. 25 SD) for all indicators, each marker was divided into ten equal intervals. This deviation from standard procedures does not seem to pose any obvious problems, but it has not been tested in simulation studies, so this particular set of findings should be interpreted with caution. MRINs were not reported for either of the MAXCOV analyses. 

The authors conducted MAMBAC and MAXCOV analyses. MRIN was set at 20 with an interval size of. 25 SD. Somewhat at odds with convention, the MAMBAC input variable was also cut in. 25 intervals, instead of making a cut after each case, but this appears to be an acceptable practice. MAMBAC mostly yielded peaking curves, but the positions of the peaks varied considerably. The mean base rate estimate was. 43 and the variance of individual estimates was quite high (SD =. 09), though this level was prob-... 

The above paragraph describes the forward option of the interval methods, where one starts with no variables selected, and sequentially adds intervals of variables until the stop criterion is reached. Alternatively, one could operate the interval methods in reverse mode, where one starts using all available x variables, and sequentially removes intervals of variables until the stop criterion is reached. Being stepwise selection methods, the interval methods have the potential to select local rather than global optima, and they require careful selection of the interval size (number of variables per interval) based on prior knowledge of the spectroscopy, to balance computation time and performance improvement. However, these methods are rather straightforward, relatively simple to implement, and efficient. 

A simulation results in a number (or a vector of numbers) at some time. Depending on the dimensionality of the problem, the simulation uses intervals in time ST and one or more space intervals. Often there is only one space interval, here given the symbol H. A result - a current, or a concentration, for example - will, due to truncation errors, have an error associated with it, that can be expressed in the following way. The discussion is, for the moment, restricted to an ode with interval size h. Then the simulated result at time t can be written as a polynomial... 

If we know an exact solution, Method 1 is used. First a result v, is calculated, using interval size h. From (14.49), we have... 

If we do not know an exact solution, we can still estimate the error order by Method 2, as described by Osterby [430], We must use one more interval size, a2/ . We then have a third result ... 

When screening experiments are used, it is generally anticipated that several effects may be nonzero. Hence, one ought to use statistical procedures that are known to provide strong control of error rates. It is not enough to control error rates only under the complete null distribution. This section discusses exact confidence intervals. Size-a tests are considered in Section 5. 

Here the scaling parameter ( is the dimensionless time interval size and N is the number of delta functions (relaxation acts) in that interval. However, a characteristic time constant of the CC process is the relaxation time r. Thus, the scaling parameter ( and the relaxation time should be proportional to each other ... 

Figure 2.1a shows a line chart of the midpoints of the data. Although the particle diameter distribution is plainly shown, it is possible to alter the shape of the distribution by changing the interval size. 

Diameter interval, im Percentage in interval Cumulative percentage less than upper interval size... 

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