Month: March 2021

Percentage games

Let us consider a hypothetical situation where the authors of a paper promise the following:

“In the next figure, we compare our Quantum Ultra-Enhanced Superior-Throughput Intelligent Objectively-Novel Adaptive Beamformed Low-Latency Emission (QUESTIONABLE) scheme with the Arbitrarily-Massive Antenna Zero-Innovation Neural Grandiloquence (AMAZING) scheme, which is the best previously known scheme. As the figure shows, the QUESTIONABLE scheme outperforms the AMAZING scheme by 6122% at a threshold of 2 dB.”

They refer to this figure:

Fig. 1: Fair comparison of our QUESTIONABLE scheme with the AMAZING scheme reported elsewhere. As is clearly apparent, the reliability gain at 2 dB is very high, unquestionably.

A gain of more than 6,000% sounds, well, amazing. But it also questionable: Is it good practice to state this figure of 6122%? Is it good marketing? Is it acceptable? Should it be discouraged? Is it even unethical?
If claiming 6122% at 2 dB is acceptable, how about a 25119% gain at 10 dB? Here the absolute reliability of AMAZING is 6.3e-9, and that of QUESTIONABLE is 1.60e-6. Not a practical regime, but a more than 25000% gain nonetheless.

Conversely, at -1.5 dB, the improvement is from 0.996 (which is AMAZING) to 0.9999999 (which is QUESTIONABLE). This is a modest 0.4% improvement both in absolute and relative terms. However, focusing on outage instead of reliability, the factor between the two outage probabilities is 62763, or 6,276,267%. So the 0.4% gap becomes a 6M% gap (that’s mega-percent, admittedly an usual combination of a metric prefix with a unit).

Independently of such amazing and questionable transmission schemes, is an improvement from 10% to 13% a 3% improvement or a 30% improvement? Frequently we see such a gain reported as a 30% improvement. Is it fair to use the ratio to measure improvement, or should it be the gap (difference)? Equivalently, should the percentage improvement be measure in percent of the underlying quantity (in which case it is 3%) or in percent of the percentages (in which case it is 30%)? The problem is that terms like “gain”, “increase”, “improvement”, “boost”, “raise” are imprecise as they may refer to differences or ratios.

How about an improvement from 10 dB to 13 dB? Here, most people would agree that the gain is 3 dB. The reason why there is no ambiguity here is that 3 dB is both the difference 13 dB-10 dB and also the ratio of the underlying quantities expressed in dB, 20/10. Thanks to the logarithm in the dB scale, both lead to the same result. With percentages, this is not the case.

The ambiguity also manifests itself when we compare percentages with physical units such as meters or seconds. The difference between 10 m and 13 m is 3 m, and 13 m is 30% longer than 10 m. Percentage gains always measure ratios in this case. But if “meter” is replaced by “percent”, the situation becomes murky. By analogy then the difference between 10% and 13% is 3%, so 13% is 3% more than 10%, in the same way that 13 m is 3 m more than 10 m. But at the same time, 13% is 30% more than 10%, in the same way that 13 m is 30% more than 10 m.

So how do we resolve this? Here are some suggestions:

  • Always state whether we refer to ratios or difference. For instance, instead of saying “The increase is 30%” say the “increase is a factor of 1.3 or 30%”.
  • Avoid ambiguity by using fractions instead of percentages. For instance, the difference between 0.1 and 0.13 is 0.03, and the ratio is 1.3.
  • Avoid aggressive marketing when quantities are in very impractical regimes. For instance, in the example above, advertising huge gains at reliabilities below 1e-5 is not good practice. Conversely, in regimes of interest, such as the high-reliablity regime, it is perfectly acceptable to report a gain of 100 in the outage performance if the outage probability is reduced from 0.01 to 0.0001. (However, talking about 10,000% is not recommended.)
  • Consider using horizontal distances (gaps) instead of vertical ones. Especially with steep (almost step-like) curves like the one in Fig. 1, the vertical gap strongly depends on where it is measured along the x axis. Conversely, the horizontal gap may be fairly constant. In the case of Fig. 1, the horizontal gap is, in fact, constant. It is exactly 3 dB at all reliabilities. A quasi-constant gap is often observed in SIR distributions (cdfs) of cellular networks when different transmission schemes are compared. The theory behind is explained in this letter (here on IEEE Xplore) and in this paper (here on IEEE Xplore). In this case, the gains are either factors (in linear scale) or gaps (in the dB) scale. This method of reporting gains is the one adopted by the coding theory community – coding gains are reported in dB (horizontally), not as gains in the bit error performance (vertically).

On cell slicing

Network slicing is a warm topic these days. Here we discuss cell slicing, where a polygon is cut in three pieces (sub-polygons) by two lines through its nucleus and a random point, respectively. First, as a sequel to this post, we focus on the 0-cell in the Poisson-Voronoi tessellation, which is the Voronoi cell of a PPP that contains the origin.
As in that earlier post, we rotate and shift the 0-cell so that its nucleus is at the origin and the point located uniformly randomly in the cell is to the right (on the positive x axis). In a cellular network application, the nucleus (now at the origin) is the location of the base station, while the random point is the typical user. Then we draw vertical lines through the origin and the random point and calculate the areas of the polygons to the left of the origin and to the left of the random point. This movie shows a number of realizations of this setup:

Movie 1. Realizations of 0-cells in Poisson-Voronoi tessellation sliced into three sub-polygons.

For intensity λ=1, the 0-cell in the Poisson-Voronoi tessellation has a mean area of 1.280176, obtained by the numerical evaluation of an integral. The polygon to the left of the nucleus o, shown in green in the movie, has a mean area of 0.517649, also obtained by numerical integration. The red polygon, to the right of o and to the left of x, has a mean area of 0.529111, obtained by simulation. Relative to the total mean area, we thus have:

  • Mean area to the left of o: 40.4%
  • Mean area to the left of x: 81.8%
  • Mean area to the right of x: 18.2%

Hence, on average, almost 60% of the (other) users in the cell are on the same side of the base station as the typical user, and the mean area to the left of the typical user is larger than the entire mean area of the typical cell (which is 1 for λ=1). Also, more than 18% of the users are “behind” the typical user.

Not surprisingly, the area fractions are quite similar in the typical Poisson-Voronoi cell. (Note that the uniformly random point in the typical cell does not correspond to the typical user of a user point process that is independent of the base station process – see this post). These percentages (area fractions) for the typical cell are all simulated:

  • Mean area to the left of o: 39.8%
  • Mean area to the left of x: 81.3%
  • Mean area to the right of x: 18.7%

How unusual are these area fractions? Let us compare them with those in the disk, which is, in some sense, the ideal cell shape. In the disk, with the nucleus o at the center, the mean area fraction to the left of o is trivially 1/2, while the area to the left of a uniformly random point is easily determined to be 7/8. This shows that the (roughly) 2/5 – 2/5 – 1/5 split in the typical Poisson-Voronoi cells is relatively far from the 1/2 – 3/8 – 1/8 split of the disk. How about regular polygons with a finite number of sides? The second movie shows some realizations for 3,4, … ,10,12,15,20,32 sides.

Movie 2. Realizations of regular polygons sliced into three sub-polygons.

As expected, for a larger number of sides the area fractions approach those of the disk. Here is a plot of the relative deviations to the disk of regular polygons, where o is the centroid:

Figure 1. Percentage deviation of mean areas left of o and left of x in regular polygons relative to the disk.

For instance, for the triangle, the area fraction to the left of o is 0.516, which is 3.2% larger than for the disk. Hence the first blue point in Fig. 1 (top left), corresponding to 3 sides, is at 3.2. To better see how the deviations behave for 5 or more sides, here is another version of the figure that shows only pentagons and higher.

Figure 2. Same as Fig. 1 but starting at pentagons.

From the blue curve it is apparent that the area to the left of the center is always exactly one half if the number of sides is even. The reason is that with an even number of sides, the polygons to the left of o and to the right of o are always congruent irrespective of the location of the random point. We also see that the hexagon is quite close to the disk already, with a mere 0.15% deviation from the 7/8 area fraction (to the left of x) of the disk.