Pairs of Graphs at 10% and 20% Hurst Exponent difference . 51

Table 4.2 shows, pairs of graphs with I.D.’s from 101 to 115, 10% and 20% Hurst exponent difference, between 0.1 and 0.9 Hurst exponent values. We observed that the estimated Hurst exponent values were close to the true Hurst exponent values and the rankings were also correct, meaning that we obtained correct judgement as to which graph was more persistent. The Thurstone-Mosteller model turned to over-estimate the values of H when the true Hurst exponent was between 0.5 (Included) and 0.1, and it also underestimate the values of H above 0.5 Hurst exponent.

The graph 4.3 is the visual representation of table 4.2 and here we have a quite accu-rate fit. We can see that both lines start at the same point when the Hurst exponent value is 0.1; this graph shows that between 0.1 and 0.5, the estimated graph is above the actual Hurst exponent graph and the estimated graph is below the actual Hurst exponent graph above 0.5 Hurst exponent. We have also observed that at certain

Class True H Value Weights from Weights after Linear T-M Algorithm Transformation

1 H = 0.1 16.9894 0.1

2 H = 0.2 14.0916 0.236455

3 H = 0.3 11.2103 0.372128

4 H = 0.4 9.80896 0.438115

5 H = 0.5 8.48998 0.500223

6 H = 0.6 7.15108 0.563269

7 H = 0.7 5.79261 0.627237

8 H = 0.8 2.87584 0.764582

9 H = 0.9 0. 0.9

Table 4.2: Results of Thurstone-Mosteller model at 10% and 20% H difference be-tween 0.1 and 0.9.

points when the estimated Hurst exponent graph is a bit far from the actual Hurst exponent graph, the respondents might have struggled to judge the pairs.

2 4 6 8 Class 0.2

0.4 0.6 0.8

Exponent Hurst Actual vs Estimated Hurst Exponent

Actual Estimated

Figure 4.3: Graph for Thurstone-Mosteller model at 10% and 20% H difference be-tween 0.1 and 0.9

The matrix A is a preference matrix for the responses from the comparisons of the pairs of graphs with 10% and 20% Hurst exponent difference from graph pair 101-115. In this matrix, aij is being compared with aji where i is the row number and j represents the column number. When aij > aji, this means that aij is the most preferred graph. The diagonals of this matrix are all zeros where aij = 0, because we assumed that a graph is not compared with itself. The Thurstone-Mosteller model suggests that aij = 1 if there was at least one comparison for a given object.

A=

In table 4.3 at 10% and 20% Hurst exponent difference, between 0.5 and 0.95 Hurst exponent values, looking at pairs of graphs from 116 to 132, we found similar results as in table 4.2. However, some of these estimated Hurst exponent values are not as close as the actual or true Hurst exponent values. The Thurstone-Mosteller model turned to overestimate the values of H when the true Hurst exponent is below 0.45 and it also under estimateed the values of H above 0.45 Hurst exponent. The rank-ings are also correct.

Class True H Value Weights from Weights after Linear T-M Algorithm Transformation

1 H = 0.05 18.3791 0.05

2 H = 0.15 15.4427 0.19379

3 H = 0.25 12.5675 0.334586

4 H = 0.35 11.1876 0.402156

5 H = 0.45 9.86952 0.466702

6 H = 0.55 8.48996 0.534257

7 H = 0.65 7.19984 0.597433

8 H = 0.75 5.82258 0.664875

9 H = 0.85 2.8876 0.808598

10 H = 0.95 0. 0.95

Table 4.3: Results of Thurstone-Mosteller model at 10% and 20% H difference be-tween 0.05 and 0.95.

The shape of graph 4.4 is exactly the same as that of graph 4.3 and the only difference are the limits and the points of intersection. This behaviour was expected, because the pairs of graphs that were being compared had the same difference Hurst exponent value. We have also got an accurate fit and we found instances where the estimated Hurst exponent graph was far from the actual Hurst exponent graph.

2 4 6 8 ^wx Class

Exponent Hurst Actual vs Estimated Hurst Exponent

Actual Estimated

Figure 4.4: Graph for Thurstone-Mosteller model at 10% and 20% H difference be-tween 0.05 and 0.95

The matrix B is another preference matrix for responses from the comparisons of the pairs of graphs with 10% and 20% Hurst exponent difference from graph pair 116-132.

Also bij + bji equals the total number of times that these two graphs were compared to each other. Judging from the number of preferences made by the respondents on the preference matrix A and B, I can safely say it was easier to judge these pairs of graphs at this Hurst exponent difference, because most respondents agreed with each other.

4.2.2 Pairs of Graphs at 5% and 15% Hurst Exponent dif-ference

Now we consider the pairs of graphs at 5% and 15% Hurst exponent difference in table 4.4, between 0.05 and 0.95 Hurst exponent values, looking at graph pairs from 201 to 232. We have found only two incorrect ranking in classes of {12,13} and {17,18}, where the H difference between the pairs was only 5%. The estimated Hurst exponent values are close to the true Hurst exponent values, the Thurstone-Mosteller model turned to overestimate the values of H between the intervals of the true Hurst exponent of (0.05,0.6] and [0.85,0.95), it also underestimate the values of H in this interval of [0.7,0.8] true Hurst exponent.

The graph 4.5 represents table 4.4 visually. The shape of this graph 4.5 is different to the two previous graphs at Hurst exponent difference of 10% and 20%, as expected.

However, we can confirm that we got an accurate fit. The pairs of graphs were judged better at 10% and 20% Hurst exponent difference than at 5% and 15% Hurst expo-nent difference. In this table 4.4, the shape of the estimated line is caused by the two incorrect rankings above H = 0.5. The respondents might have struggled to judge the pairs of graphs at these points.

This graph shows that between the intervals of (0.05,0.6] and [0.85,0.95), the esti-mated graph is above the actual Hurst exponent graph and in this Hurst exponent interval [0.7,0.8], the estimated graph is below the actual Hurst exponent graph. We have also observed this trend at certain points when the estimated Hurst exponent graph is a bit far from the actual Hurst exponent graph.

The matrix C is also a preference matrix for responses from the comparisons of the pairs of graphs with 5% and 15% Hurst exponent difference from graph pair 116-132.

It was also easy to judge these pairs of graphs according to the number of preferences the respondents agreed on in this matrix. However, comparing preference matrices

z {| {z Class

Exponent Hurst Actual vs Estimated Hurst Exponent

Actual Estimated

Figure 4.5: Graph for Thurstone-Mosteller model at 5% and 15% H difference be-tween 0.05 and 0.95

A and B with C it is still easier to judge the pairs of graphs at 10% and 20% Hurst exponent difference than at 5% and 15% Hurst exponent difference.

C=