**UQ** is the method in which input distributions for the uncertainty quantification are created. Type of input distribution can be selected form the list of featured distributions or there is a possibility for the user to create an arbitrary distribution shape according to his actual needs in a separate graphic user interface. Then, a set of numbers for each variable is generated where values are randomized according to prescribed probability density function. Also, adjustment of the so-called nominal sample can be made here as well as re-setting of computation domain’s boundaries. Set of generated distributions, location of the nominal solution, and domain’s boundaries are being stored in separate files after the preprocess is done.

List of distribution types featured in the** Input preprocessor** is shown in the section below. Besides the basic information and examples acceptable ranges of input values are given also.

## How to use it: #

*Fig. 1: Input Preprocessor – Initial state of the UQ method*

Appearance of the** Preprocess window** with the** UQ** activated is on Fig. 1 showing its initial state.

There are two possible ways how input distribution can be defined:

First, an input **.txt** file can be loaded via main window menu item *File -> Load*. Its formatting can be seen on Fig. 2 with default input setting for in-built distribution types. Parameters of each distribution are in curly bracket, semicolon separated, introduced with parameter label and a colon symbol. It also contains the information about the number of MC samples representing the size of resulting distributions and about the distribution’s nature – continuous or discrete. When wrong type of file is selected, or some instruction is missing or incomplete, a pop-up error message is raised providing loading process details.

Other way to define input distribution parameters is to fill-up and confirm values in entry fields of the *Main Frame*.

*Fig. 2: Example of input file*

In its initial state two of them are available:

**Number of inputs:**Number of input distributions of the case. Must be integer number between 1 and 1,000. Value can be also set using the slider on the right.**MC Samples:**Number of samples for Monte Carlo simulation, size input distributions. Must be integer number between 1,000 and 1,000,000.

Above mentioned limits prevents excessive size of *Input Preprocessor* output files.

*Fig. 3: Input Preprocessor – Number of inputs set*

Once the *Number of inputs* is set and confirmed with the* return *key, matching number of new entry rows appear. For each input distribution, there is a possibility to change its naming according to your variable and to pick the distribution via *Pick a distribution* combobox widget. Choosing the type of distribution will access entry fields with preset default values of its parameters. Only the* User defined *distribution has a *Define* button instead, starting the separate GUI tool. State of the *Main Frame *before and after this action can be seen in Fig. 3 and Fig. 4. Names of parameters of each distribution type are listed together with their meaning and limits in the **Featured distribution types **section of this document. Setting of all necessary values activates the *Prepare* button which appears at the bottom of the* Main Frame*.

*Fig. 4: Input Preprocessor – Distribution types chosen*

Clicking on the *Prepare button* pre-generates all input distributions as defined and the *Main Frame* is split into two halves – names, types and parameters of input distributions on the left, their boundaries and nominal samples on the right. The *Prepare* button will appear at the bottom part of the right window section. This button creates all output files of the **Input preprocessor **and the program stores them in the project directory. These files are:

**InputSet.txt:**Contains general information about each input distribution and its parameters.**XDis.txt:**Matrix of input distribution values – each column for one input distribution.**InputDomain.txt:**Values of input distribution boundaries together with coordinates of the nominal solution.

Proposed values of input domains’ boundaries are minimal and maximal values of prepared input distributions. The user can redefine these simply by overwriting selected entries, but new values need to be also within the range of already prepared input distribution values. Values of the nominal sample can be changed too, but with some additional restrictions applied. It must be within the range of each input variable and not to be equal to its boundaries. It is recommended for the nominal sample to have inputs close to the mean value of the variable‘s input distribution. Nominal sample too far away from variable means may lead to increased interpolation errors on edges of the computational domain. Therefore, if there is a user-defined value of the nominal sample more than 10% of variable’s range far from the mean value, an alert is raised to inform the user that a special caution will be needed when processing results.

For input variables having a symmetrical input distribution, extra caution is required. When their output function is expected to be of symmetrical shape also, it is highly recommended not to set their nominal value exactly to the location of the mean value of the corresponding input distribution! A typical example can be an angular position of the crankshaft, wave phase, etc.

*Fig. 5: Input Preprocessor – Distributions prepared*

## Featured distribution types: #

### Normal Distribution #

Normal distribution, also known as Gaussian distribution, is a continuous probability distribution which is symmetric around its mean value. According to so-called empirical rule, 99.7% of all its values lie within three standard deviations of the mean value. It is expected for physical quantities that are a sum of many independent processes affected by random errors, e.g. material properties.

*Fig. 6: Normal distribution, 100000 random samples, Mean = 0.0, Std = 1.0*

### Uniform Distribution #

It is considered as continuous symmetric distribution where the probability of occurrence in the distribution is equal for all values from given range. It is a type of distribution well suitable for all input parameters dependent on designer’s decision.

*Fig. 7: Uniform distribution, 100000 random samples, a = 0.0, b = 1.0*

### Lognormal Distribution #

It is a continuous probability distribution where logarithm of its values is distributed normally. This distribution consists of real positive values only. It can be observed among various phenomena in the field of e.g. biology, sociology, chemistry, technology, or finance.

*Fig. 8: Lognormal distribution, 100000 random samples, Mean = 0.0, Std = 1.0*

### Gumbel Distribution #

It is a continuous distribution describing the distribution of the maximum (or the minimum) values of a series of distributions, e.g. maximum rainfall rate in a series of distributions measured over the time. Since it is related to the duration of time up to an event, its use can be in e.g. reliability analysis in engineering.

*Fig. 9: Gumbel distribution, 100000 random samples, Location = 0.0, Scale = 1.0*

### Laplace Distribution #

It is a symmetric continuous probability distribution also known as the double exponential distribution, since it can be seen as an exponential distribution mirrored around the location parameter. It is the distribution of differences between two independent varieties with identical exponential distributions. It can be used for e.g. describing the distribution of errors in observed values.

*Fig. 10: Laplace distribution, 100000 random samples, Location = 0.0, Scale = 1.0*

### Beta Distribution #

It is a continuous probability distribution defined on the interval (0,1), thus, it can be used to model random variables of prescribed length. Settings of its two parameters can result in wide variety of shapes of the distribution (e.g. it turns to be the uniform distribution with default values of parameters, as seen on Fig. 11), therefore, there is a wide variety of its use.

*Fig. 11: Beta distribution, 100000 random samples, shape a = 1.0, shape B= 1.0*

### Logistic Distribution #

It is a symmetric continuous probability distribution very similar to the normal distribution in its appearance, but having heavier tails, which can increase the robustness of analyses based on it when compared to those using the normal distribution. It is used for logistic regression model predicting the likelihood of an event based on an input variable.

*Fig. 12: Logistic distribution, 100000 random samples, Location = 0.0, Scale = 1.0*

### Poisson Distribution #

It is a discrete probability distribution used to represent integer-valued counts, such as the number of events observed within an interval of time or other specified dimensions such as distance, area or volume. Therefore, it is used for applications in many fields related to counting individual objects, e.g. astronomy, biology, or telecommunication.

*Fig. 13: Poisson distribution, 100000 random samples, Rate = 1.0*

### Logarithmic Distribution #

Also known as the logarithmic series distribution, is a discrete probability distribution based on the standard power series expansion of the natural logarithm function. It can be used in cases involving number of event occurrences within a time period or a space.

*Fig. 14: Logarithmic distribution, 100000 random samples, Shape = 0.5*

### Power Distribution #

It is a continuous probability distribution defined on the interval (0,1), also known as the power function distribution. It also may be seen as the inverse of the Pareto distribution, thus, its application can cover similar range of problems.

*Fig. 15: Power distribution, 100000 random samples, Shape = 2.0*

### Discrete Distribution #

As the name suggests, it is a discrete type of distribution, in this case completely arbitrary according to user’s needs. As inputs, two vectors of coma separated values are required. One is the *Vector of positions* setting distribution values, the other is the *Vector of weights* setting ratios of values in the input distributions. At least three distinct position values must be given, weights must be greater than zero.

### User Defined Distribution #

Principle is similar to the previous distribution, but this custom set distribution is continuous. Its parameters are set through standalone GUI whose features are described in a separate sheet available with the tool.