Aufgaben:Exercise 4.11Z: C Program "acf2"

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C program  $2$  for ACF–calculation

You can see on the right the C program  "acf2"  for calculating the discrete ACF values  $\varphi_x(k)$  with index  $k = 0$, ... , $l$.

In contrast to the program  "acf1"  from  Exercise 4.11,  here the algorithm described in the theory part is applied directly.  It should be noted:

• Let the long value passed to the program be  $l=10$.
• The calculated ACF values  $\varphi_x(0)$, ... , $\varphi_x(10)$  are returned to the main program with the float field  $\rm ACF[ \ ]$.  In lines 7 and 8,  this field is prefilled with zeros.
• The random variable  $x( \ )$  is defined as a float function in line 4,  as is an auxiliary field  ${\rm H}[10000 ]$,  into which the  $N = 10000$  samples  $x_\nu$  are entered  (lines 9 and 10).
• The names of the control variables in line 6 are adapted to the given algorithm.
• The actual ACF calculation is done from line 11.  This program part is marked red in the program code.

Hints:

• This exercise belongs to the chapter  Auto-Correlation Function.
• In particular, reference is made to the pages 
  • Numerical ACF determination,
  • Accuracy of the numerical ACF calculation.

Questions

1 On how many summands  ($S$)  is the ACF calculation based for index  $k=0$   resp.  for  $k=10$ ?

$S_{k=0} \ = \ $

$S_{k=10} \ = \ $

2 Which of the following statements are correct?

	The calculation time increases linearly with  $l + 1$,  i.e. with the number of ACF values to be calculated.
	The calculation time increases quadratically with the number  $N$  of samples considered.
	The calculation becomes more accurate as  $N$  increases.
	If a float variable is represented with  $\rm 4 \ bytes$,  "acf2"  needs at least  $4 \cdot N$  bytes of memory.

3 Which of the following statements are true?

	The stronger the internal statistical bindings of the process are,  the more  inexact  is the ACF result for a given  $N$.
	The stronger the internal statistical bindings of the process are,  the more  accurate  is the ACF result for a given  $N$.
	If the process has statistical bindings,  the errors of the numerical ACF calculation are also correlated.     Example:     If the value  $\varphi_x(k=5)$  is too large,  it is very likely that  $\varphi_x(k=4)$  and  $\varphi_x(k=6)$  will also be too large.

Solution

Solution

(1)  To calculate the ACF value  $\varphi_x(0)$  you have to average over  $\underline{N =10000}$  summands, for  $\varphi_x(10)$  only over  $\underline{N = 9990}$.

(2)  Correct are  the proposed solutions 1, 3 and 4:

• The computation time increases approximately linearly with  $N$  and  $l + 1$  as can be seen from the ACF calculation highlighted in red.
• In contrast,  the computation time for the other parts of the program can be neglected.
• Naturally,  the calculation also becomes more accurate with increasing  $N$.
• This goes here – in contrast to the program  "acf1"  of Exercise 4.11 – however at the expense of the necessary memory requirement.
• Since each float variable takes up exactly four bytes, the auxiliary field  ${\rm H}[10000 ]$  alone requires a memory of 40 kByte.

(3)  Correct are the  suggested solutions 1 and 3:

• The stronger the statistical bindings within the random process,  the less accurate is the ACF calculation for a given  $N$.
• This fact can be illustrated for example by the power calculation  $($ACF value at  $k=0)$ :
  • If all  $N$  samples are statistically independent,  then all contributions provide the maximum information about the ACF value  $\varphi_x(k=0)$.
  • However,  if there are statistical bindings between  $x_\nu$  and  $x_{\nu+1}$,  but not between  $x_\nu$  and  $x_{\nu+2}$,  only half of all samples provide the full information  about  $\varphi_x(k=0)$  and the others provide only limited information.
• This loss of information based on correlations can only be compensated in this example by doubling  $N$ .
• The last statement is also true,  as explained in detail on the page  "Accuracy of the numerical ACF calculation"  in the theory section.