# Conditional Dependencies and Position Fixing
## 5 Histogram Transformation  
A histogram displays a stepwise diagram of the distribution of the observed data. A histogram consists of adjacent rectangles that are primarily erected over nonoverlapping intervals. The histogram displays the relative frequencies, which are the empirical probabilities, and it shows the proportion of cases that fall into each bin. The problems that arise when using raw experimental data that are intended for the nautical scope of applications are as follows:  
1. How to evaluate the histograms;
2. How to convert the stepwise diagram to take the form of the locally injective function;
3. How to prepare a family of histograms for use when solving a particular problem (for example, position fixing). The exploration of a histogram layout results in its evaluation. The vertical and horizontal features are the bases for the judgement of the quality of the involved samples of the random data instances. Combining assignments that engage density data generates adequate and useful solutions once the injective concentration functions, or at least the locally injective concentration functions, are processed. However, we should not expect unique solutions when traditional histograms are involved. Converting the rectangular bins to fuzzy density sets changes the overall image of the histogram. A stepwise diagram might take the form of a continuous locally injective function, which consequently enables researchers to solve many practical problems with imprecise data. To obtain a new version of the histogram, we have to consider that, to some extent, a single discernment point belongs to each considered bin when treating them as fuzzy sets. Instead of a rectangular-shaped bin, an area limited by the membership function (MF) is considered. An inclusion diagram should feature some expectations to fulfil the equivalency requirements. The integral of a product of the belonging grades, and given the cell density, must be the same as the original bin's area (see Equation (5)), and its vertex should be in the middle of the cell. The value that it returns at the left and right limits of the reference uncertainty area should be 0.5. In the simplest case, the uncertainty range is equivalent to the bin's width. According to Equation (3), the cell's range (\(g_{Wi}\)) defines only one part of the histogram uncertainty. Another compound is related to the irregularity of the cell heights. Thus, the MF that reflects the uncertainty extends beyond the cell limits. I present examples of the three membership diagrams in Figure 5. We can observe the previously mentioned items for the sigmoidal function pairs that define the borders of the cells for the three histograms. We can obtain the resulting neighbourhood density as the output of the additive combination of the same point inclusions within all the bins, as follows:  
\[g(x_{l})=\sum_{i=1}^{n}f_{i}(x_{l})*h_{i}, \tag{4}\]  
Equation (4) presents the idea of the bin-to-bin additive concept that is crucial for the approach, where \(g(x_{l})\) is the modified histogram, which is a density function for the \(l\)-th fragment of evidence; \(f_{i}(x_{l})\) is the membership function, which is a grade of the \(x_{l}\) inclusion in the \(i\)-th bin meant as a fuzzy set; \(n\) is the number of cells; \(h_{i}\) is the height of the \(i\)-th cell. I present diagrams of the membership functions for the three bin histograms with the different uncertainty levels regarding each case in Figure 5. The total of the membership grades for each of the points uniformly distributed over a considered space multiplied by the bin densities results in the cumulated belonging. The value is a plausibility measure that results in the modified histograms in Figure 5. The MF reaches a value of 1 in the middle of a bin, and it might return values of 0.5 at the cell's limits (see the left presented  
Figure 5: Two examples of histograms, uncertainty-dependent membership functions, and transformed histograms.