The derivative is positive when the curve heads uphill and is negative when the curve heads downhill. The derivative equals zero at peaks and troughs in the curve. After calculating the numerical derivative, Prism can smooth the results, if you choose.
The second derivative is the derivative of the derivative curve. The second derivative equals zero at the inflection points of the curve. If you give Prism a series of XY points that define a curve, it can compute the numerical derivative or integral from that series of points.
But if you give Prism an equation, it cannot compute a new equation that defines the derivative or integral. Don't confuse with a separate Prism analysis that computes a single value for the area under the curve.
If you import a curve from an instrument, you may wish to smooth the data to improve the appearance of a graph. Since you lose data when you smooth a curve, you should not smooth a curve prior to nonlinear regression or other analyses. Smoothing is not a method of data analysis, but is purely a way to create a more attractive graph. Prism gives you two ways to adjust the smoothness of the curve. You choose the number of neighboring points to average and the 'order' of the smoothing polynomial.
Order 3 generally has one or two hills or valleys. Order 4 generally has up to three. The following example shows an Order 2 polynomial trendline one hill to illustrate the relationship between speed and gasoline consumption. A power trendline is a curved line that is best used with data sets that compare measurements that increase at a specific rate — for example, the acceleration of a race car at one-second intervals.
You cannot create a power trendline if your data contains zero or negative values. In the following example, acceleration data is shown by plotting distance in meters by seconds. The power trendline clearly demonstrates the increasing acceleration. An exponential trendline is a curved line that is most useful when data values rise or fall at increasingly higher rates.
You cannot create an exponential trendline if your data contains zero or negative values. In the following example, an exponential trendline is used to illustrate the decreasing amount of carbon 14 in an object as it ages. Note that the R-squared value is 1, which means the line fits the data perfectly.
A moving average trendline smoothes out fluctuations in data to show a pattern or trend more clearly. A moving average trendline uses a specific number of data points set by the Period option , averages them, and uses the average value as a point in the trendline. Google Scholar 4. Google Scholar 6. Google Scholar 7.
Google Scholar 8. Google Scholar Google Scholar Download references. Glass Authors J. Glass View author publications. Rights and permissions Reprints and Permissions. About this article Cite this article Glass, J. Copy to clipboard. Pick any numbers in there, as long as the x values are distinct, a function exists that passes through them, but there are infinitely many choices, and here you cannot as easily know how to just pick the simplest solution, whatever that means to you.
As Walter points out, you can actually prove and this is trivial to do that it is possible to generate an infinite set of solutions. In fact, I used that standard trick to generate the various solutions I posed above. Just because you have true, real, exact function values, this means nothing. You cannot know which function created your data. There are infinitely many all equally valid choices of functions that could generate any set of points that represent some function.
Simply having the data points is NOT sufficient. Walter Roberson If you do not know the form of the equation, then this is not possible. There are provably an infinite number the infinity of continuous numbers, not just the infinity of integers of different equations that exactly fit any finite set of points that are stated to finite accuracy. In the past I have posted a constructive proof of this fact, showing how to construct an indefinite number of equations that exactly fit the finite list of points.
Stop adding answers every time you make a comment. Nothing you have said is an answer. Moved answer to a comment by Marwan:. Anyway I agree that many equation will pass throw these 3 point intersected point , but if we select another 3 points may another many equations could pass throw these new 3 points but at the end only one equation can pass throw the six points.
Please correct me if I am wrong. You are wrong. Flat out wrong. Did I mention that you are wrong? I picked 3 points before. What stopped me from picking sets of 6 arbitrary points? I could have picked sets of points. Still the theory is the same. There are infinitely many curves that pass through any set of points. You keep wanting to see some magical solution, where the magic wand of mathematics will give you the function you need. Sorry, but it does not exist. There are infinitely many ways to pass a curve exactly through ANY set of points.
All are equally as valid as any other. If you want more, then you need to invest the effort:. Choose a model, a family of functions that are consistent with your process. This choice should usually be driven by physical modeling considerations. So you need to understand your process. Or, if the model family is some well known family, i. Learn how to fit the model to your data.
Different function families are defined in different ways. Learn to use the mathematical tools to help you to fit the model. It may involve linear or nonlinear regression. It may involve spline modeling.
It may involve fourier transforms. Learn sufficient mathematics behind the fitting tools to be able to use those tools intelligently. But I'm sorry, there is no magic wand you can just wave and find the equation that generated any set of data points. I also show there that it is not sufficient to match derivatives, by showing how to construct a new equation whose derivative is the same as the original equation at each of the given points.
The argument I gave does not depend upon the number of points though I suppose it could be argued that it needs to be improved for the degenerate case of there being only one specified point. Alex Sha Sum of Squared Residual: Correlation Coef. R-Square: 0. Adjusted R-Square: 0. Determination Coef. Chi-Square: 0. F-Statistic: Parameter Best Estimate. Victor Zoratti Ferreira Alex Sha, why did you choose exactly this type of equation and how did you find the parameters? Start Hunting! Translated by.
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