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Vishwadeep Khatri
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Non Linear Regression is a model building method used to understand the non linear relation between a response variable and one or more predictor variables. It is generally used when a suitable linear model cannot be obtained or if the researcher already has some prior insights into the underlying assumptions of data and their relationships.

 

An application-oriented question on the topic along with responses can be seen below. The best answer was provided by Moushmi Kandori and Gitarchana Roy.

 

Applause for all the respondents - Amit Kumar Shukla, Moushmi Kandori, Raghavendra Rao Althar, Gitarchana Roy, Ramjanam Singh.

Question

Q 565. Compare Non Linear Regression with Linear Regression highlighting benefits and challenges of each. What are the governing criteria to select non linear regression over linear regression?

 

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Linear Regression

Nonlinear Regression

Represents relationship between variables with a straight line

Represents relationship between variables with a curved line

Example: Defects vs. Rework

Example: Growth of Business i.e., Revenue with employee strength

Form of linear model is typically either the constant or a parameter multiplied by an independent variable. Simple Addition.

Rational function which is the ratio of 2 polynomial functions.

R-squared value is valid

R-squared value is invalid

Might not capture true relationships if they are complex.

Explains complex relationships

Data set must be homogeneous. Might be overlooked while creating models.

Better fit and prediction accuracy

Easy to understand.

Difficult to interpret and comprehend results.

 

Governing Criteria:

  1. If better model fit is essential, then nonlinear regression should be selected.
  2. If simple, easy to understand models need to be created then Linear models should be created.
  3. If prediction accuracy is important, then Nonlinear regression should be selected.
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Regression - A statistical method called regression links a dependent variable to one or more independent variables. It allows for the discovery of data trends. The method examines the relationship between a dependent variable and independent factors, and it is typically expressed as a graph. You can confidently establish which elements are most important, which ones can be ignored, and how these factors interact when you do a regression.  Regression analysis can assist you quantify your assumptions, for instance, that there is a relationship between the quantity of inventory received and the quantity of production.

Types of Regression – We have multiple types of regression models, and Linear Regression is one among them.

 

Linear Regression: A dependent variable (y) and an independent variable (x) are related in a linear regression model. Finding the correlation between dependent and independent variables is done statistically. Since the relationship between the two variables in a linear regression is one of a straight line, the term "linear regression" was coined. One of the most prevalent kinds of predictive analysis is this one. It uses a regression graph to examine the strength of the link between two variables. It chooses the data set's nearest points that correspond to a linear pattern. It is used in situations where a change in the value of a second variable has a considerable impact on the variance in the value of a first variable.

The linear equation for linear regression is always Y = a + bx, where x is the explanatory variable and Y denotes the dependent variable.

Subtypes - Simple and multiple regression are the two subtypes into which they are divided.

Simple Linear Regression: A straight line is used to evaluate the connection between one independent variable and one dependent variable in a simple linear regression model. Both variables ought to have numerical values.

Multiple linear regression is a type of regression analysis where the variation in two or more correlated independent variables affects the change in the dependent variable.

Refer to the example of linear regression below, which illustrates the link between production and various inventories (independent variable on x-axis, dependent variable on y-axis).

 

image.png.38011c25e634f931285ff32b99c4083e.png

 

Interpretation: From the above graph of regression, we can see there is a linear relationship between inventory and production numbers. As inventory increases, production numbers also increase. Thus, using this regression analysis, we can predict the value of the dependent variable (production i.e., ‘y’) based on the value of the independent variable (inventory i.e., ‘x’).

 

Advantages and Disadvantages of Linear Regression:

Benefits

Challenges

It is simpler to implement and interpret linear regression.

As boundaries in this method are linear, outliers might have a significant impact on the regression.

For data that can be separated linearly, linear regression works remarkably well.

It can be difficult to satisfy the assumption that dependent and independent variables are linear.

Easier to use, understand, and effectively train

It is quite often prone to noise

Using dimensionally reduced approaches, regularisation, and cross-validation, it manages overfitting rather well

The sensitivity of linear regression to outliers is high.

The ability to extrapolate beyond a particular data set is a further benefit

It is prone to multicollinearity

 

Non-Liner Regression: Regression analysis that depicts a nonlinear relationship between dependent and independent variables is referred to as non-linear regression. When using linear parameters, it is employed when the relationship cannot be accurately modelled. In contrast to linear analysis, nonlinear regression uses a curve to link the variables. In statistics, the word "nonlinearity" is used to describe a scenario in which an independent variable and a dependent variable do not have a straight-line or direct relationship. Changes in the output are not directly proportional to changes in any of the inputs in a nonlinear connection.

 

When predictors and responses have a certain function form, non-linear regression arises.

 

Y=f (X, β) + ϵ, where X is a vector of p predictors, is a vector of k parameters, f() is a recognised regression function, and is an error term with a potentially non-normal distribution.

View the example below to see how the dependent (y-axis) and independent variable (x-axis) are related nonlinearly.

 

image.png.6022d00a13add132ed36407a767289e1.png

 

Interpretation: From the above graph of regression, we can see there is a non-liner relationship exists between the variables in x and y axis. As per the scatter plot of changing production count over months of employee experience shows that there seems to be a relationship but that it is non-linear relationship which requires to use of a non-linear regression model to predict the data.

 

Advantages and Disadvantages of Non-Linear Regression:

 

Benefits

Challenges

The functionality for curve-fitting that is most flexible is offered by nonlinear regression.

Selecting the nonlinear function that produces the best fit for a specific curve shape can be very time-consuming.

Additionally, it can show oblique or nonlinear connections between variables.

Since nonlinear models can have several local minima and maxima, it can be challenging to determine the ideal parameters and function for the data.

In the data, nonlinear regression can detect more intricate patterns and behaviours.

When a model is too complicated for the data, overfitting can happen, leading to poor generalisation abilities.

It can also be applied to jobs involving forecasting, such as predicting future inventory based on current.

However, fitting a nonlinear model might require more computation and requires careful model parameter selection to prevent overfitting.

Most real-life phenomena typically involve nonlinear regression

Nonlinear models typically describe observable relationships more accurately, but at the cost of added complexity.

 

Governing criteria:  The following factors should be taken into consideration when selecting non-linear regression:

• Prioritise linear regression - Use linear regression first to see if it would fit the specific sort of curve in the data set. We must use nonlinear regression if satisfactory fit cannot be obtained using linear regression.

• R-Squared - Non-Linear regression cannot yield a reliable R-Squared value.

• P-Value - Non-Linear regression makes it impossible to calculate P-Value.

• Goal - To verify that the data fits a model and acquire the best-fit values for the parameters, or to compare the fits of other models, we must carefully choose a model (or two alternative models). We should also pay attention to all the outcomes.

 

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Non Linear Regression: Nonlinear regression is a statistical method used to model relationships between variables when the relationship is not linear. In contrast to linear regression, which assumes a linear relationship between the dependent and independent variables, nonlinear regression allows for more complex relationships by using nonlinear equations to fit the data.

Nonlinear regression models can take various forms, such as polynomial, exponential, logarithmic, power, or sigmoid functions, among others. The choice of the specific form depends on the nature of the data and the underlying theory.

To estimate the parameters of a nonlinear regression model, various techniques can be used, including iterative methods like the Gauss-Newton algorithm or the Levenberg-Marquardt algorithm. These methods iteratively adjust the model's parameters to minimize the difference between the predicted values and the observed data.

Nonlinear regression models are useful in many fields, including biology, economics, physics, engineering, and social sciences, where relationships between variables are often more complex than simple linear relationships. By employing nonlinear regression, researchers can gain insights into the underlying mechanisms and make predictions based on the observed data.

 

 

Linear regression is a statistical modeling technique used to explore and analyze the relationship between a dependent variable and one or more independent variables. It assumes a linear relationship between the variables, meaning that the dependent variable can be expressed as a linear combination of the independent variables.

In linear regression, the goal is to estimate the coefficients of the independent variables that minimize the difference between the observed data and the predicted values. This is typically done by minimizing the sum of squared differences, known as the least squares method.

The equation for a simple linear regression with one independent variable can be represented as:

y = β₀ + β₁x + ε

Where:

y is the dependent variable.

x is the independent variable.

β₀ is the intercept or constant term.

β₁ is the coefficient or slope of the independent variable.

ε is the error term representing the variability in the data not explained by the model.

Multiple linear regression extends this concept to include more than one independent variable. The equation becomes:

y = β₀ + β₁x₁ + β₂x₂ + ... + βx + ε

Where:

x₁, x₂, ..., x are the independent variables.

β₁, β₂, ..., β are the coefficients corresponding to each independent variable.

Linear regression is widely used in various fields, such as economics, finance, social sciences, and machine learning. It helps in understanding the relationship between variables, predicting values based on observed data, and identifying the most significant predictors.

 

 

Linear regression and nonlinear regression are two different statistical techniques used for modeling relationships between variables.

Linear regression assumes a linear relationship between the dependent variable and one or more independent variables. It models the data using a straight line or a hyperplane in higher dimensions. Linear regression is characterized by having a constant slope and intercept, and the relationship between the variables is described by a linear equation. Linear regression is often used when the relationship between variables is expected to be linear, and it provides a simple and interpretable model.

Nonlinear regression, on the other hand, allows for more complex relationships between variables by using nonlinear equations to model the data. It relaxes the assumption of linearity and can capture curved or nonlinear patterns in the data. Nonlinear regression models can take various forms, such as polynomial, exponential, logarithmic, power, or sigmoid functions. The choice of the specific form depends on the data and the underlying theory. Nonlinear regression requires estimating the parameters of the nonlinear equation, which is typically done using iterative methods.

In summary, the main difference between linear regression and nonlinear regression is the assumption about the relationship between the variables. Linear regression assumes a linear relationship, while nonlinear regression allows for more flexible and complex relationships. Linear regression provides a simpler model with interpretable coefficients, while nonlinear regression can capture more intricate patterns but may require more complex parameter estimation techniques. The choice between linear and nonlinear regression depends on the nature of the data and the underlying theory guiding the analysis.

example of Linear Regression

Let's say we have a dataset containing information about the number of hours studied and the corresponding scores achieved by a group of students. We want to understand the relationship between the number of hours studied and the scores obtained and create a linear regression model to predict scores based on the number of hours studied.

urs Studied (x) Scores (y) ------------------------------- 2 56 3 67 4 73 5 82 6 88 7 94 8 98

To perform linear regression, we fit a straight line to the data that represents the relationship between the hours studied (independent variable) and the scores (dependent variable). The equation for a simple linear regression model is:

y = β₀ + β₁x + ε

Where:

y is the predicted score.

x is the number of hours studied.

β₀ is the intercept or constant term.

β₁ is the coefficient or slope that represents the change in score per unit increase in hours studied.

ε is the error term representing the variability in the data not explained by the model.

By applying linear regression to the given data, we estimate the values of β₀ and β₁ that minimize the difference between the predicted scores and the actual scores. The resulting equation would be something like:

Score = 50.33 + 6.88 * Hours Studied

This equation represents the linear relationship between the number of hours studied and the predicted scores. We can use this equation to make predictions about scores for any given number of hours studied.

 

 

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Non linear regression will be useful when relationship between dependent and independent variable are not predictable. Linear regression provides the linear relation function that fits the dependent and independent variables. Non linear models helps to model the randomness in the relationship between the parameter. Non linear regression helps to capture complex relationship between the parameters. For real world phenomenon like economics, that have complex influencing factors, non linear regression will be useful. Non linear regression also helps to bring out the hidden relationship between parameters. Other benefits of non linear regression is in feature selection in prediction modeling, and also in building interpretability into prediction models. Challenges with non linear regression are the possibility of overfitting of the data. Since there are multiple local maxima and minima its harder to identify the optimal parameters for the prediction model. Also the models are more sensitive to the noise and the overall effect of noise would get amplified. Computational intensiveness and large data need for analysis are other limitations.

 

Governing factor for choosing right regression approach, is to start with linear regression to understand the fit of the data. In case if its not possible to have best fit. Then non linear regression is to be checked. Availability of large set of data is another need for being able to use non linear regression.

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Linear Regression: Benefits:

  • Linear regression model works well with all types of dataset sizes
  • Provides information with respect to the relevance of characteristics
  • Simple implementation and easy to interpret
  • Easily interpreted the results
  • Unfitting can be managed by regularisation

Linear Regression: Challenges:

  • Volatile to under fitting
  • Outliers have significant impact
  • Linear Borderlines
  • Assumptions for data independence

Non-Linear Regression: Benefits:

  • Very complex model but gives very accurate results compare to liner model
  • Analysis produces a curve for showcasing the connections between variables
  • This model provides great adaptability to cater larger scenarios
  • Very good applicability for AI, ML and wide range of industries
  • Provides the best flexi curve fitted model

Non-Linear Regression: Challenges:

  • Very complex and difficult to implement
  • Less flexible
  • Very time consuming while selecting the best fit model for the curve shape
  • Randomly data is saved in the history
  • Multiple data layers are involved, highly complex
  • Difficult to calculate P value

Selection Criteria:

The general principles is to use the linear regression initially to establish curve fitment with the available data set. Only when we are unable to achieve the fitting model with liner regression then we need to consider other non-linear regression models. Commonly linear regression is simple to use, interpret and achieve more statistics to help study the model. Best applicable when the relationship between dependent and independent variable.

 

Non-linear regression fitment is wide in nature and fit major type of curves. Best used for practical applications for providing solution to the real-world challenges.

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