What is Nonlinear Regression

The nonlinear model is complex and, at the same time, creates accurate results. The analysis develops a curve depicting the relationship between variables based on the dataset provided. The model offering great flexibility can create a curve that best suits the scenario. This relationship can be anything from connecting time and population to investor sentiments and its nonlinear effect on stock market returns.

Key Takeaways

  • Nonlinear regression refers to a regression analysis where the regression model portrays a nonlinear relationship between dependent and independent variables.It is more accurate and flexible than a linear model. The model can accommodate diverse curves deriving complex relations between two or more variables.Examples of this statistical model include the portrayal of the relationship between gold price & US CPI inflation and investor sentiments & stock market returns.Its application is seen in diverse disciplines like machine learning, insurance, forestry research, etc.

How Does Nonlinear Regression Work?

In nonlinear regression, the experimental data are mapped to a model, and mathematical function representing variables (dependent and independent) in a nonlinear relationship that is curvilinear is formed and optimized. It is accepted as a flexible form of regression analysisRegression AnalysisRegression Analysis is a statistical approach for evaluating the relationship between 1 dependent variable & 1 or more independent variables. It is widely used in investing & financing sectors to improve the products & services further. read more. Examples of algorithms used to develop nonlinear models are Levenberg-Marquardt nonlinear least squares and Gauss-Newton algorithms.

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When the data shows a curvy relationship that is not a straight line applying a nonlinear model gives the accurate output. Unlike linear regressionLinear RegressionLinear regression represents the relationship between one dependent variable and one or more independent variable. Examples of linear regression are relationship between monthly sales and expenditure, IQ level and test score, monthly temperatures and AC sales, population and mobile sales.read more, there is no assumption of data linearity in the nonlinear model. In essence, when the data curve cannot be formed accurately using a linear method, the next option is the nonlinear method since it accommodates diverse types of curves. The fitness of the curve determines its correctness, and the curve illustrated reflects the curve’s accountability to the data.

The nonlinear model can be parametric or non-parametric. The parametric model portrays dependent or response variables as a function of nonlinear parameters and independent or predictor variables. The model can also accommodate multiple response variables. Furthermore, machine learning methods are more suitable when it comes to non-parametric linear regression.

Example

Let’s better understand the concept using the nonlinear relationship example between Gold and US CPI CPI The Consumer Price Index (CPI) is a measure of the average price of a basket of regularly used consumer commodities compared to a base year. The CPI for the base year is 100, and this is the benchmark  point.read moreinflation. Gold investment is an effective hedge Hedge Hedge refers to an investment strategy that protects traders against potential losses due to unforeseen price fluctuations in an assetread moreagainst inflation Inflation The rise in prices of goods and services is referred to as inflation. One of the measures of inflation is the consumer price index (CPI). Rate of inflation = (CPIx+1–CPIx )/CPIx. Where CPIx is the consumer price index of the initial year, CPIx+1 is the consumer price index of the following year.read moreand currency depreciationCurrency DepreciationCurrency depreciation is the fall in a country’s currency exchange value compared to other currencies in a floating rate system based on trade imports and exports. For example, an increase in demand for foreign products results in more imports, resulting in foreign currency investing, resulting in domestic currency depreciation.read more in many countries. Hence analyzing the gold price movement is of great importance.  

Even if the gold prices are stable to a great extent, they are affected by inflation, crude oil, etc. But the important one is the impact of inflation, and at the same time, gold prices can control the inflation instability. Therefore, a deep understanding of the relationship between inflation and gold price is a prerequisite. 

In this case, nonlinear regression analysis is employed for analyzing data. The dependent variable is gold price, and the independent variableIndependent VariableIndependent variable is an object or a time period or a input value, changes to which are used to assess the impact on an output value (i.e. the end objective) that is measured in mathematical or statistical or financial modeling.read more is inflation. The regression analysis results revealed that inflation impacts the gold price. The possible explanation why gold does not always move with CPI increases is that gold is considerably affected by large spikes in inflation. Still, small increases in inflation or dropping inflation have little impact on gold’s price upward or downward trajectory.

Application

The nonlinear regression models are predominantly used for prediction, financial modelingFinancial ModelingFinancial modeling refers to the use of excel-based models to reflect a company’s projected financial performance. Such models represent the financial situation by taking into account risks and future assumptions, which are critical for making significant decisions in the future, such as raising capital or valuing a business, and interpreting their impact.read more, and forecasting purposes. The nonlinear model is used in many fields and sectors like insurance, agriculture, finance, investing, machine learning AI, and understanding broader markets. Let’s look into some of the significant applications:

  • Since most biological processes are nonlinear in nature, we can find nonlinear model applications in forestry research. A simple power function to relate tree volume or weight in relation to its diameter or height is an example.The use of a nonlinear model in developing a wide-range colorless gas, HCFC-22 formulation is an example from the field of Chemistry.In research and developmentResearch And DevelopmentResearch and Development is an actual pre-planned investigation to gain new scientific or technical knowledge that can be converted into a scheme or formulation for manufacturing/supply/trading, resulting in a business advantage.read more, it is used in the process of formulation of the problem and deriving statistical solutions to the calibration problem.It is used in the insurance domain. For example, its usage can be seen in the computation of IBNR reserves.It is of great importance in agricultural research. Because many crops and soil processes are better captured by nonlinear than linear models.

This has been a Guide to What is Nonlinear Regression. We explain the nonlinear regression analysis, model, examples, applications like in machine learning. You may also have a look at the following articles to learn more –

One of the formulas used to represent the nonlinear model is listed below. Y = f(X,β) + ϵ. Where f is the regression function and ϵ is the error term while X β are vector parameters.Furthermore, performing nonlinear regression in excel is a famous method. It develops scatterplot and polynomial trendlines based on the given dataset.

The nonlinear model is more flexible and accurate. Although both models can accommodate curvature, the nonlinear model is significantly more versatile in terms of the forms of the curves it can accept. 

It’s a method for performing more flexible nonlinear analysis to obtain proper outputs such as choices, categorization, or inferences when similar future states or inputs are present. This method can provide impressive results and frequently beats people in performance, stability, and precision.

  • Least Squares RegressionMultiple Regression FormulaStochastic Modeling