Curve Fitting

Curve fitting refers to a process that involves making adjustments to inputs or parameters of a system or a model to ensure it better fits the data. In finance, it can help traders find the right settings for their strategy on the basis of the historical data available to them.

Although such a strategy offers decent backtest results, it fails when conducting real-time trading. Using this process, traders may often find patterns that are simply random noise. As the strategy finds new data, it often considers random noise as predictive patterns, leading to trading losses. There are multiple curve fitting techniques, for example, linear and polynomial regression.

• The curve fitting refers to a process in statistics that involves specifying the model or system providing the best fit to particular curves in a data set. It helps traders optimize the settings for their strategy to make prudent trading decisions.
• There are different curve fitting techniques that individuals can utilize to achieve their objectives. Some of them are linear regression, polynomial interpolation, and polynomial regression.
• A key difference between interpolation and curve fitting is that in the case of the former, all data points pass via the curve.

The curve fitting refers to a process that involves constructing a curve with the ideal fit to a sequence of data points, perhaps subject to constraints. It is a popular analysis tool examining the association between at least one independent variable or predictor and a dependent variable or response variable. The aim of the process is to define the relationship’s best-fit model. Ideally, this process can capture the trend in the available data and allow people to make predictions regarding the data series’ behavior in the future.

In the world of investments, this concept involves over-adjusting an investment strategy to the main or underlying data series. The modification of individual parameters takes place in a repetitive process until the selection of the one promising an optimal result occurs. The selection occurs after considering a significant number of combinations. That said, the choice is based on certain data history utilized for the computation.

Simply put, curve fitting is a phenomenon materializing when a strategy used by an individual fits the market noise, not the market behavior. This leads to failure with regard to live trading. Developers utilize different tricks to combat it. For example, they may test on out-of-sample data without optimizing criteria for the ideal back-tested results and ensuring only the least possible parameters exist in the system.

Let us look at the different curve fitting techniques.

#1 - Linear Regression

In this case, the easiest example involves fitting a straight line to a number of paired observations. For the straight line, the following is the mathematical equation:

y = ao + a1x+ e

Here,

• a1 and o are coefficients that represent the line’s slope and the y intercept, respectively.
• e is the residual or error between the observations and the model. One can represent it as e = y - ao - a1x

#2 - Polynomial Regression

Even though engineering data exhibit a marked pattern, a straight line poorly represents it. For such cases, a curve is better suitable to fit the data. One way to solve this type of issue involves fitting the data using a polynomial function. Note that this is known as polynomial regression.

#3 - Multiple Linear Regression

A useful extension in the case of a linear regression involves y as a linear function of over 1 variable. For example, let us say x₁ and x₂. The equation will be as follows:

#4 - Polynomial Interpolation

In different engineering applications, estimating the intermediate values between the precise data points is necessary. The most popular technique utilized is polynomial interpolation. In the case of n+1 data points, a one-of-a-kind polynomial of order n passes via every point. The equation looks like this:

Let us look at a curve fitting graph to understand the process better.

As one can observe, the graph compares two variables — stock price and time in days. The data points seem to fit the above curve close to perfectly. Hence, one can say that the association is very strong. Also, with time, the stock price increases, which demonstrates a positive relationship.

Let us look at a few curve fitting examples to understand the concept better.

Example #1

Suppose Sam, an experienced trader, accumulated historical data on ABC stock, which was on his watchlist for over a month. The data included the volume traded, closing price, opening price, etc. Then, he utilized a popular curve fitting technique known as polynomial regression to demonstrate the mathematical approximation concerning the association between ABC stock price and time.

After that, Sam utilized the curve to estimate future price movements on the basis of the historical data available to him. Since the curve showed a downtrend, he entered a short position in ABC stock.

Example #2

Suppose David, a trader in the stock market, decided to utilize polynomial regression, a widely used curve fitting technique, to analyze ABC stock. He applied the method to the historical stock price data, trying to fit a curve that would demonstrate the price movements over time in the best possible manner.

With this technique, he could not manage to factor in the complexity associated with the price trend. It turned out that the model had fit the noise or randomness in the price data instead of the underlying price trend. This resulted in David making inaccurate predictions regarding future price movements. As a result, he suffered significant losses.

Let us look at the applications of this process in finance.

• The use of this process in common in technical analysis, which involves individuals and organizations using historical data to make trading decisions.
• This process assists people in identifying complicated data patterns in financial markets.
• It helps in the identification of important factors that influence business data.
• People can utilize the process to understand trends in a market and make predictions.
• In business, this concept assists in data visualization.

Lastly, people can use it to summarize the association or relationship between at least two variables, for example, the number of customers and time.

Let us understand the benefits of this process by going through the following points.

• It can improve the reliability and accuracy of a system or model by minimizing the error existing between the data and output.
• The process can help in uncovering new relationships or patterns in the data through the exploration of various combinations comprising inputs or parameters.
• It can help fine tune a strategy by testing the sensitivity to multiple scenarios or factors.

The drawbacks associated with this process are as follows:

• A noteworthy drawback of this process is overfitting. It refers to a model becoming extremely complex or too specific to the available data, losing the capacity to generalize. It can result in high risk concerning real trading or poor performance because the strategy may not be able to capture the altering market conditions.
• This process can lead to data snooping, which, in turn, results in inflated metrics or statistics.
• It can consume significant resources and time because the individuals may have to run different iterations or tests to figure out the ideal fit and cope with the randomness or noise in the data.

The concepts of regression, interpolation, and curve fitting can be confusing. Let us look at how they differ to get a clear idea regarding their meaning.

This article has been a guide to what is Curve Fitting. We explain its methods, examples, application, graph, comparison with interpolation, and benefits. You may also find some useful articles here -

• Underfitting
• Time Series Analysis
• Local Regression