Consulting and
Education Services in
Mathematical Modeling
Statistical Methods

## Mathematical Modeling

Mathematical modeling can be divided into two main categories: mechanistic and empirical models.

### Mechanistic Models

In ideal cases, experimental work may be substituted by using mechanistic models, e.g. following situations:

• Design of process unites, scale-up, and optimization
• Development of reaction-kinetic Models
• Description of biological systems
• E.g., modeling environmental systems
• Model-based designing, optmizing and scaling of control
• Hazop studies
• It is cheaper to explode a virtual factory than a real one
• Studying scenarios

Profmath has plenty of experience in mechanistic modeling, and assists in all common problems related to mechanistic modeling.

### Empirical models

Empirical models are applied in situations where building mechanistic models is not possible. This includes cases such as:

• The phenomenon under study is so complex that the required theory is unknown, or the theoretical description of the system is impossible.
• The phenomenon under study is so complex that empirical modeling becomes less expensive.

Empirical models can be used for the same purposes as mechanistic models, but the spectrum of applications is wider. Some examples:

• Analyzing results of questionnaries
• Analyzing process data
• Fault diagnosis and preventive control
• Image analysis
• Pattern recognition, used, e.g., in quality control
• Sound analysis
• E.g., for the needs of anticipative maintenance
• Spectral analysis and other similar measurement signals
• The objective may be determination of chemical or physical properties, for example, determining particle-size-distributions or concentrations of chemical substances

There are many cases in which solving a problem requires both empirical and mechanistic models. ProfMath has plenty of experience in such applications, and our researchers have developed new methods in this field.

### Statistical Design of Experiments (DOE)

Empirical models can be built either based on data from designed experiments or from gathered process data. In the former case, we speak about statistical design of experiments (DOE), and in the latter, about data analysis (or machine learning). Generally, models based on DOE are more reliable than models based on arbitrarily gathered data (often called passive data). In empirical modeling, it is always important know the nature of data used, and the limitations of the models used.

Using passive experimental data, or data based intuitively designed experiments, may easily lead to lead to a situation where a lot money as been spent without results that could be used for reliable conclusions.

ProfMath has plenty of experience on DOE and data analysis in various fields of applications. Our special strengths include optimization, also multiresponse optimization, of processes of process units using so-called response surface methodology (RSM).