In this article, we are going to explore what is statistical modeling? and what are the types, applications, and benefits of it in detail?

Statistical modelingis a process to create a statistical learning process from the data using statistics, and mathematics.

## Important Goals of Statistical modeling

Statistical modeling has two main goals:

1. To enable the construction of mathematical or graphical models that represent complex phenomena and

2. To provide a basis for the formal statistical inference of hypotheses about the phenomena.

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## What is Statistical Model?

Statistical modeling is the process of creating a mathematical representation, called a model, of a real-life system.

The model may be a visual representation, such as a diagram or graph, or it may be in numerical form.

Statistical models are used to represent real systems, like the population of any country for Example India.

The purpose of statistical modeling is to make inferences about the nature and properties of the real system being modeled.

Statistical models are used to describe relationships between variables by using linear regression, logistic regression, Poisson regression, negative binomial regression, or multinomial logit, depending on the relationship between the dependent variable and explanatory variables.

There are many types of statistical models ranging from simple but unrealistic to complex but realistic.

The complexity of the model depends on the purpose for which it is built and the data available for building it. The most common types of statistical models include

**Static models:** In these models, the distribution (distribution function) of the random variable is assumed to be fixed and not changing with respect to time or other conditions.

**Time Series models:** These models assume that the random variable follows some kind of pattern over time.

**Panel Data models:** These are very general types of models in which there are two or more groups whose members are related in some categories. For example, income is affected by education, saving is affected by salary, etc.

**How does Statistical Modeling Work?**

Statistical models are usually specified mathematically and implemented using software tools, but there are other ways of specifying them, including via graphical models (in which the structure of the model is represented graphically rather than by equations).

It may be formulated for various purposes. Some statistical models are created for particular applications (such as regression analysis), others for theoretical investigation (such as Bayesian inference), and others for prediction (including predictive modeling).

Statistical models may be used to represent relationships between random variables at one or more points in time in a snapshot or may be used to represent relationships over time, for example, predicting future values based on past values.

**Important Theory about Statistical Models**

A statistical model is a mathematical model that is proposed to describe observed or existing data.

The use of statistical modeling has grown recently and is now used in many fields of application.

Tests based on statistical models are called “statistical hypothesis tests”, and their results are expressed as “p-values”.

#### 1. Different Usability:

Statistical models play an important role in various sciences (particularly econometrics, biostatistics, social sciences, and natural sciences) and these are also essential in marketing research.

#### 2. Problem Handling:

A statistical model provides a framework for understanding and characterizing the problem at hand.

It represents the problem in terms of random variables, which are quantities that vary according to specific probabilities.

These probabilities can be interpreted as the likelihood of values the variables could take on during an experiment or observation.

#### 3. Predictive Modeling:

There are three popular types of statistical models:

1. Regression

3. Clustering

Statistical models can be used to make predictions about future outcomes based on past observations.

For example, suppose we have data about the running times of marathon races in previous years.

We can use this data to make predictions about the expected running time for a future marathon runner, knowing how fast she has run in previous marathons.

However, we can also use the same data to make predictions about any other runner’s expected marathon time, given only her previous marathon times.

In fact, it is often possible to make predictions about a person’s performance without knowing anything else about her, simply by knowing her past performance.

#### 4. Backward Induction

It is a technique used in game theory and decision theory to choose between sequential actions when all actions have a value attached to them (the subjective value) and some actions are better than others (the objective value).

Backward induction works by selecting at each step in time an action that yields the best possible outcome from Tests and estimation procedures derived from statistical models form the basis of scientific inference.

These methods are used to construct inferences about unobserved populations and states using a limited amount of observed data.

#### 5. Descriptive Statistics

The observations are first summarized using descriptive statistics such as averages or percentages.

Such summary statistics typically suggest what form a model could take; for example, based on average prices and sales figures, one might guess that the distribution of prices is approximately normal.

This model may then be fitted to the data using statistical techniques such as regression analysis.

Once fitted, predictions may be made by computing estimates of unknown parameters; for example, one could use estimates of the mean and standard deviation of price to estimate prices for yet-to-be-observed houses.

Inference enables quantitative answers to questions like: How frequently does one group differ from another? What is the probability that an election outcome will occur? How much more risk do I need to buy insurance?

**Applications of Statistical Modeling**

- Statistical models are used in many fields and for many purposes. Examples include Mathematical models in the social sciences, for example in economics, demography, sociology, political science, and marketing;
- Models that express relationships between observational data and/or experimental data;
- Statistical models to fit curves to data (curve fitting); Statistical models that make predictions about data.
- Statistical models are used to make predictions about the future, and to help interpret the past.
- Statistical models supply a great deal of information in little space, but they must be interpreted with care.
- Statistical models are used in statistics, machine learning, computer simulation, pattern recognition, and related fields.
- The most common application of statistical modeling is in the construction and analysis of probability models for random phenomena. This is usually done using the tools of probability theory and statistics.
- Statistical modeling can also be applied to construct or analyze models for other types of observational data, such as spatial data (e.g., altitude measurements), Spatio-temporal data (e.g., wind speed measurements at different heights and times), and image data (e.g., medical images).

The term statistical modeling is sometimes loosely applied to any application of statistical inference to observational data, even if there is no clear connection with probability theory.

**Top Uses of Statistical Modeling**

#### 1. Science Area:

Statistical models are used in many areas of science, including in physical sciences such as physics and astronomy, life sciences such as biology and medicine, social sciences such as psychology and economics, and business disciplines such as operations research.

#### 2. Mathematical Modeling

The most common use of statistical modeling is in the construction of mathematical or graphical models that represent complex phenomena.

They are also used to make predictions and forecasts, Statistical modeling is closely related to causal modeling.

#### 3. Data Modeling

The term statistical model is also sometimes used to refer to a mathematical description of a set of data without any implication that the mathematical structure so described was derived from data.

The term statistical model is also used to refer to a process by which data are fitted to a pre-defined model.

These models are often used to gain insight into the relationships among variables in the real world.

#### 4. Econometrics

In fact, much of the early development of statistical modeling was carried out by members of the field of econometrics, which is concerned with techniques for causal modeling and forecasting.

#### 5. Statistics

In addition to constructing a model, statistical modeling also provides a basis for statistical inference.

Statistical inference typically involves testing hypotheses about the parameters (or unknown properties) in a predetermined statistical model.

For example, consider this simple model:

y = b0 + b1 x + e

Where y is an observed response variable, x is an observed predictor variable, b0 is an unknown constant whose value determines an intercept term for y, b1 is an unknown slope parameter for x, and e is an error term associated with y.

## Types of Statistical Models

### Regression Models

##### 1. Linear Regression

This model assumes that the relationship between the two variables is linear.

The model takes the form Y = a + by where a is the intercept and b is the slope.

The error term e is usually assumed to be normally distributed with mean 0 and variance σ2.

The simplest case involves just one independent variable (X) and one dependent variable (Y). Usually, several independent variables are considered simultaneously.

This type of model is also called simple linear regression or straight-line regression. More than one independent variable can be used in a single model; in fact, there may be no theoretical limit on the number of variables that can be considered together in a single multiple linear regression.

##### 2. Decision Tree

Decision tree learning is an algorithm for machine learning of predictive models. It builds a decision tree-based model trained on datasets consisting of cases (or examples) with features (attributes, also sometimes called variables).

From a given set of training examples, a decision tree learner forms a hypothesis (or hypothesis set) that divides the set into different subsets called decision trees.

### Classification Models

##### 1. Random Forest

Random forests are a type of ensemble learning used in machine-learning applications. They are used both for regression and classification problems, although most of the literature deals with their use for classification.

The random forest algorithm generates many decision trees at training time, ensembles them by calculating each prediction and choosing that of the majority.

Decision trees can be very prone to overfitting, especially when data is sparse. Random forests overcome this effect to some extent.

##### 2. Support Vector Machine

Support vector machines are becoming increasingly popular in the field of computer science and applications.

SVM is a useful tool for analyzing data and has been applied extensively in the area of text categorization, advertising, and credit scoring.

The prediction capabilities through machine learning have made SVM a very powerful and competitive tool for data analysis.

Support Vector Machine, otherwise known as SVM, is a type of supervised machine learning algorithm that helps users with the task of solving two-class classification problems.

The model gives users a high level of accuracy and speed when there are large sets of training examples.

### Clustering Models

##### Hierarchical Clustering

Clustering of similar items is the main aim of Data Mining and is used to increase the interpretability of data, exploratory analysis, summarization, and predictive modeling.

Grouping rows in a database table by similarity according to one or more attributes is called clustering.

Clustering has many applications such as image segmentation, medical diagnosis, and pattern recognition.

Some important applications are fraud detection in the banking sector, medical diagnosis, etc.

## Different Types of Statistical Modeling

Statistical modeling is a broad term that encompasses several different methods of conducting statistical analysis:

#### 1. Logical modeling:

This method of statistical modeling is also called deductive modeling. It uses logic to formulate assumptions about the population of interest, and then uses those assumptions to make predictions about the population.

This type of statistical model is also called “deductive inference” or “causal inference” because it focuses on making causal inferences from a set of assumptions.

One example of logical modeling is the theory of probability, which makes probabilistic inferences from axioms about random variables and random events.

#### 2. **Data-driven (also known as inductive) modeling**

This method involves determining relationships between variables based on the observed data.

Data-driven statistical models can be either deterministic or stochastic. Statistical models in which some variables are not observed directly but rather are inferred using other observed variables are called “hidden Markov models.”

#### 3. Data mining:

This is a non-statistical method that uses algorithms to find patterns in data, without necessarily having explicit hypotheses about the form (the relationship) of these patterns.

Data mining is often used by businesses and governments to uncover previously unknown relationships within large data sets.

## Example of Statistical modeling

So what’s statistical modeling? At its simplest, it’s a way to come up with an equation that represents the relationship between a set of predictor variables and a target variable.

The predictor variables are usually numerical quantities that you can measure or count, such as the age of a customer, the number of times she has purchased from you, how much she has spent in total, her education level, and so on.

The target variable is the one you want to predict. It could be sales next month, or profit next year, churn rate next month, or any other quantity you need to estimate.

It is sometimes called the dependent variable because it depends on the values of the predictor variables.

You can use your model to estimate the value of your target variable by plugging in values for your predictor variables.

You might not have any actual data for customers age 23 who have spent $4,000 with you – instead, you need to estimate what their sales would be if they existed.

Because it’s impossible (or at least impractical) to collect all this data on real people, this process is called estimation.

## Conclusion

Statistical modeling is of great importance in a wide variety of disciplines. It is used in the natural and social sciences as well as in business and engineering.

Statistical models are used for a wide range of applications, including the social, biological, and physical sciences.

The term statistical model may also refer to description terms of such a model.

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