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Uncertainty Hexadecimal (“Ux”) Data Format

Introduction​

Terminology

The Signaloid Uncertainty Hexadecimal data format is a digital representation for complete probability distributions. You can think of each Signaloid Ux number as representing a complete histogram of values.

Traditional computing platforms take integer and floating-point values as inputs and perform operations (e.g., ADD, SUBSTRACT, MULTIPLY, XOR, etc.) on these data. The results of these operations are other integers or floating-point values.

The Signaloid Cloud Compute Engine can take in as input the same kinds of integer and floating-point values and perform the same kinds of operations (e.g., ADD, SUBSTRACT, MULTIPLY, XOR, etc.). In addition to these capabilities of traditional computing platforms, Signaloid C0 uncertainty-tracking cores allow its users to optionally associate additional information about (potentially-different) probability distributions, with each individual input value. Then, when performing arithmetic operations such as those listed above, it simultaneously also performs distribution arithmetic on any associated probability distributions and generates the appropriate probability distribution to associate with the arithmetic operation’s result. The Signaloid Cloud Compute Engine also enables several valuable operations that act on distribution information, such as performing Bayesian inference or creating empirical probability distributions from data, e.g., using UxHwDoubleDistFromSamples.

Ux Data is a new way to represent uncertainty information (probability distributions) in a way that is data-efficient and makes computations on distributions also efficient. It exploits the mathematics of probability distributions to make possible uncertainty quantification that avoids the false choice between speed and correctness and sidesteps the challenges of convergence faced by traditional iterative uncertainty quantification such as Monte Carlo methods. With Ux Data representations, computing systems no longer need to make the false choice between speed (time to reach an answer) and convergence (e.g., how many standard deviations out from the expected value the uncertainty quantification has sampled, e.g., 3-sigma or 6-sigma).

Why Ux Data Exists and What It Contains​

If you've used fprintf() of a floating-point value into a file using the Signaloid Cloud Developer Platform, you might have come across a string like this:

0.785398Ux0400000000000000003FE921FB54442D18000000013FE921FB54442D188000000000000000

These strings begin with a number (0.785398 in this example), followed by the characters “Ux”, followed by a hexadecimal string. They are an instance of uncertainty hexadecimal data or "Ux Data”. We stylize this in initial capitals as we treat it as a proper noun. In the form of the example above, we refer such Ux Data as "Ux Strings". Ux Strings contain both traditional (point-valued or particle) numerical information (0.785398) as well as uncertainty distribution information associated with that value (0400...0000). The distribution information is a new representation, developed by Signaloid, which exploits knowledge of the mathematics behind Signaloid's efficient uncertainty-tracking computation methods to represent arbitrary empirical distributions. At the same time, it takes advantage of knowledge of the analytic form of the probability density function in the case of analytic distributions.

We use the term Ux Data to refer to the general form of this information and use the more specific term Ux String to refer to the ASCII-formatted version of the data (we use the term Ux Binary to refer to the binary representation of the information). The following sections of this overview describe these string and binary formats in more detail.

The Format of Ux Data: Ux Binary and Ux Hexadecimal Strings​

There are therefore two forms in which Ux Data might appear inline in a data stream (there are other ways in which distribution information can be stored and transmitted).

At the level of binary representation, the format of Ux Binary data is:

  1. "Ux" (2 ASCII characters).
  2. Representation type (8 bits).
  3. Number of samples (64 bits).
  4. Mean value of distribution (64 bits).
  5. Number of non-zero mass Dirac deltas (32 bits).
  6. A sequence of n pairs of (a) support position (64 bits); (b) probability mass (64 bits).

When we represent Ux Data as ASCII strings, the format, which we refer to as Ux Strings, is:

  1. "Ux" (2 ASCII characters).
  2. Representation type (hexadecimal value as 2 ASCII characters).
  3. Number of samples (hexadecimal value as 16 ASCII characters).
  4. Mean value of distribution (hexadecimal value as 16 ASCII characters).
  5. Number of non-zero mass Dirac deltas (hexadecimal value as 8 ASCII characters).
  6. A sequence of n pairs of (a) support position (hexadecimal value as 16 ASCII characters); (b) probability mass (hexadecimal value as 16 ASCII characters).

✏️   Example of storing a Ux string​

The following simple code example shows a distribution representing the uncertainty of a variable value whose samples are stored in the samples array. The simple example runs on the Signaloid C0 processor and uses the capabilities exposed above the ISA to inject distribution information into the processor, i.e., using UxHwDoubleDistFromSamples. The example assumes that you have mounted your Signaloid Cloud Storage at sd0. You can read the reference of the Code Editor for detailed steps on how to mount your Signaloid Cloud Storage. The example opens the uxString.txt file and stores the distributional information of value encoded as a Ux string. Figure 1 shows the visualization of the distribution of the variable value. The Ux String and binary digits that follow below show the representation of that distribution as hexadecimal and binary formatted uncertainty information in Signaloid’s uncertainty data format. Figure 2 shows the contents of the uxString.txt file as they appear in the Signaloid Cloud Data Storage in the Data and Sensors page.

#include <stdio.h>
#include <uxhw.h>

int
main(void)
{
    FILE *  fp;
    double  value, samples[] = {90, 120, 96, 20, 99, 102, 112, 68, 90, 109};
    
    value = UxHwDoubleDistFromSamples(samples, sizeof(samples)/sizeof(double));

    fp = fopen("sd0/uxString.txt", "w+");
    fprintf(fp, "value = %lf\n", value);
    printf("value = %lf\n", value);

    return 0;
}
Figure 1. The distribution representing the uncertainty of the variable value whose samples are stored in the samples array. The simple code example runs on the Signaloid C0 processor and uses the UxHwDoubleDistFromSamples to initialize the uncertainty of the variable value.

Ux String representing the distribution of the variable value (including particle value prefix):

90.600000Ux04000000000000000A4056A666666666670000000A40568000000000000CCCCCCCCCCCCCCD405E0000000000000CCCCCCCCCCCCCCD40580000000000000CCCCCCCCCCCCCCD40340000000000000CCCCCCCCCCCCCCD4058C000000000000CCCCCCCCCCCCCCD40598000000000000CCCCCCCCCCCCCCD405C0000000000000CCCCCCCCCCCCCCD40510000000000000CCCCCCCCCCCCCCD40568000000000000CCCCCCCCCCCCCCC405B4000000000000CCCCCCCCCCCCCCC
Ux Binary representing the distribution of the variable "value" (excluding particle value prefix and "Ux"):
00000100000000000000000000000000000000000000000000000000000000000000101001000000010101101010011001100110011001100110011001100110011001110000000000000000000000000000101001000000010101101000000000000000000000000000000000000000000000000000110011001100110011001100110011001100110011001100110011001101010000000101111000000000000000000000000000000000000000000000000000001100110011001100110011001100110011001100110011001100110011010100000001011000000000000000000000000000000000000000000000000000000011001100110011001100110011001100110011001100110011001100110101000000001101000000000000000000000000000000000000000000000000000000110011001100110011001100110011001100110011001100110011001101010000000101100011000000000000000000000000000000000000000000000000001100110011001100110011001100110011001100110011001100110011010100000001011001100000000000000000000000000000000000000000000000000011001100110011001100110011001100110011001100110011001100110101000000010111000000000000000000000000000000000000000000000000000000110011001100110011001100110011001100110011001100110011001101010000000101000100000000000000000000000000000000000000000000000000001100110011001100110011001100110011001100110011001100110011010100000001010110100000000000000000000000000000000000000000000000000011001100110011001100110011001100110011001100110011001100110001000000010110110100000000000000000000000000000000000000000000000000110011001100110011001100110011001100110011001100110011001100
Figure 2. The code example stored the distributional information of variable value encoded as a Ux String in the file uxString.txt in the Signaloid Cloud Data Storage. You can see the contents of the uxString.txt file as they appear in the Signaloid Cloud Data Storage in the Data and Sensors page.

✏️   Example of ingesting a Ux string​

The simple program shown below reads in the Ux String using the standard C library call sscanf, and by doing so, injects distribution information encoded as a Ux String into the variable value when executed on the Signaloid C0 processor.

#include <stdio.h>

int
main(void)
{
    double value;

    sscanf("90.600000Ux04000000000000000A4056A666666666670000000A40568000000000000CCCCCCCCCCCCCCD405E0000000000000CCCCCCCCCCCCCCD40580000000000000CCCCCCCCCCCCCCD40340000000000000CCCCCCCCCCCCCCD4058C000000000000CCCCCCCCCCCCCCD40598000000000000CCCCCCCCCCCCCCD405C0000000000000CCCCCCCCCCCCCCD40510000000000000CCCCCCCCCCCCCCD40568000000000000CCCCCCCCCCCCCCC405B4000000000000CCCCCCCCCCCCCCC", "%lf", &value);

    printf("value = %lf\n", value);

    return 0;
}