Normalized Percent Error
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may be challenged and removed. (December 2009) (Learn how and when to remove this template message) The mean absolute percentage error (MAPE), also known as mean absolute percentage deviation (MAPD), is a measure of prediction accuracy of a forecasting method in statistics, for example in trend percent difference formula estimation. It usually expresses accuracy as a percentage, and is defined by the formula: M =
Percent Difference Physics
100 n ∑ t = 1 n | A t − F t A t | , {\displaystyle {\mbox{M}}={\frac {100}{n}}\sum _{t=1}^{n}\left|{\frac {A_{t}-F_{t}}{A_{t}}}\right|,} where At
Percent Difference Vs Percent Change
is the actual value and Ft is the forecast value. The difference between At and Ft is divided by the Actual value At again. The absolute value in this calculation is summed for every forecasted point in time and
Percent Difference Chemistry
divided by the number of fitted pointsn. Multiplying by 100 makes it a percentage error. Although the concept of MAPE sounds very simple and convincing, it has major drawbacks in practical application [1] It cannot be used if there are zero values (which sometimes happens for example in demand data) because there would be a division by zero. For forecasts which are too low the percentage error cannot exceed 100%, but for forecasts which are too high there percent difference definition is no upper limit to the percentage error. When MAPE is used to compare the accuracy of prediction methods it is biased in that it will systematically select a method whose forecasts are too low. This little-known but serious issue can be overcome by using an accuracy measure based on the ratio of the predicted to actual value (called the Accuracy Ratio), this approach leads to superior statistical properties and leads to predictions which can be interpreted in terms of the geometric mean.[1] Contents 1 Alternative MAPE definitions 2 Issues 3 See also 4 External links 5 References Alternative MAPE definitions[edit] Problems can occur when calculating the MAPE value with a series of small denominators. A singularity problem of the form 'one divided by zero' and/or the creation of very large changes in the Absolute Percentage Error, caused by a small deviation in error, can occur. As an alternative, each actual value (At) of the series in the original formula can be replaced by the average of all actual values (Āt) of that series. This alternative is still being used for measuring the performance of models that forecast spot electricity prices.[2] Note that this is the same as dividing the sum of absolute differences by the sum of actual values, and is sometimes referred to as WAPE. Issues[edit] While MAPE is one of the most popular measures for forecasting error, there are many studi
here for a quick overview of the site Help Center Detailed answers to any questions you might have Meta Discuss the workings and policies of this site absolute change formula About Us Learn more about Stack Overflow the company Business Learn more relative change calculator about hiring developers or posting ads with us Mathematics Questions Tags Users Badges Unanswered Ask Question _ Mathematics Stack Exchange relative difference vs actual difference is a question and answer site for people studying math at any level and professionals in related fields. Join them; it only takes a minute: Sign up Here's how it works: https://en.wikipedia.org/wiki/Mean_absolute_percentage_error Anybody can ask a question Anybody can answer The best answers are voted up and rise to the top On normalized error measures up vote 1 down vote favorite I have function values $f_1,\ldots,f_n$ that are approximated by data $y_1,\ldots,y_n$. I am looking for a measure that describes the error in the data $y_1,\ldots,y_n$ and I want the measure to take values between $0$ http://math.stackexchange.com/questions/1565651/on-normalized-error-measures and $1$. I am familiar with the Root Mean Squared Error (RMSE) or RMSD (D for deviation): $$\mathrm{RMSE}=\sqrt{\frac{1}{n}\sum_{i=1}^n{\left(f_i-y_i\right)^2}} $$ Since this not normalized, I started searching for a normalized version, on which Wikipedia says: Normalizing the RMSE facilitates the comparison between datasets or models with different scales. Though there is no consistent means of normalization in the literature, the range of the measured data defined as the maximum value minus the minimum value is a common choice: $$\mathrm{NRMSE}=\frac{\mathrm{RMSE}}{y_{\mathrm{max}}-y_{\mathrm{min}}}$$ Not that I consider Wikipedia a trustworthy source, but seems to me this isn't normalizing, since this could even blow up the error measure (if $y_{\mathrm{max}}-y_{\mathrm{min}}$ is small). Another approach would be to not consider the absolute error, but the error as a percentage. Something of the form $$\frac{1}{n}\sum_{i=1}^n{\left|\frac{f_i-y_i}{f_i}\right|}\ \ \ \mbox{ or } \ \ \ \frac{1}{n}\sqrt{\sum_{i=1}^n{\left(\frac{f_i-y_i}{f_i}\right)^2}}.$$ But this again does not necessarily take values between $0$ and $1$. Now my question is: are there other (perhaps better) measures to describe a normalized error. In other words, given values $f_1,\ldots,f_n$ and approximations $y_1,\ldots,y_n$, is there a measure to describe the error in these approximations that always takes values between $0$
an eigenvalue of a matrix, Vector, such as the solution x of a linear system Ax=b, Matrix, such as a matrix inverse A-1, and Subspace, such as the space spanned by one or more eigenvectors of a matrix. This section provides measures for errors in these http://www.netlib.org/lapack/lug/node75.html quantities, which we need in order to express error bounds. First consider scalars. Let the scalar https://books.google.com/books?id=EEodkQqPGs4C&pg=PA237&lpg=PA237&dq=normalized+percent+error&source=bl&ots=cO9mSYb9i_&sig=3v2B0JUtbM4XmQkd8EPx7541qms&hl=en&sa=X&ved=0ahUKEwjigci35uPPAhWq7YMKHU2pDjsQ6AEIczAP be an approximation of the true answer . We can measure the difference between and either by the absolute error , or, if is nonzero, by the relative error . Alternatively, it is sometimes more convenient to use instead of the standard expression for relative error (see section4.2.1). If the relative error of is, say 10-5, then we say that is accurate to percent difference 5 decimal digits. In order to measure the error in vectors, we need to measure the size or norm of a vector x. A popular norm is the magnitude of the largest component, , which we denote . This is read the infinity norm of x. See Table4.2 for a summary of norms. Table 4.2: Vector and matrix norms Vector Matrix one-norm two-norm Frobenius norm |x|F = |x|2 infinity-norm If is an approximation to the exact vector x, we normalized percent error will refer to as the absolute error in (where p is one of the values in Table4.2), and refer to as the relative error in (assuming ). As with scalars, we will sometimes use for the relative error. As above, if the relative error of is, say 10-5, then we say that is accurate to 5 decimal digits. The following example illustrates these ideas: Thus, we would say that approximates x to 2 decimal digits. Errors in matrices may also be measured with norms. The most obvious generalization of to matrices would appear to be , but this does not have certain important mathematical properties that make deriving error bounds convenient (see section4.2.1). Instead, we will use , where A is an m-by-n matrix, or ; see Table4.2 for other matrix norms. As before is the absolute error in , is the relative error in , and a relative error in of 10-5 means is accurate to 5 decimal digits. The following example illustrates these ideas: so is accurate to 1 decimal digit. Here is some related notation we will use in our error bounds. The condition number of a matrix A is defined as , where A is square and invertible, and p is or one of the other possibilities in Table4.2. The condition number measures how sensitive A-1 is to changes in A; the larger the condition number, the more sensitive is A-1.
from GoogleSign inHidden fieldsBooksbooks.google.com - A typical optical system is composed of three basic components: a source, a detector, and a medium in which the optical energy propagates. Many textbooks cover sources and detectors, but very few cover propagation in a comprehensive way, incorporating the latest progress in theory and experiment concerning...https://books.google.com/books/about/Optical_Propagation_in_Linear_Media.html?id=EEodkQqPGs4C&utm_source=gb-gplus-shareOptical Propagation in Linear MediaMy libraryHelpAdvanced Book SearchGet print bookNo eBook availableOxford University PressAmazon.comBarnes&Noble.comBooks-A-MillionIndieBoundFind in a libraryAll sellers»Get Textbooks on Google PlayRent and save from the world's largest eBookstore. Read, highlight, and take notes, across web, tablet, and phone.Go to Google Play Now »Optical Propagation in Linear Media: Atmospheric Gases and Particles, Solid-State Components, and WaterMichael E. ThomasOxford University Press, Sep 7, 2006 - Science - 584 pages 0 Reviewshttps://books.google.com/books/about/Optical_Propagation_in_Linear_Media.html?id=EEodkQqPGs4CA typical optical system is composed of three basic components: a source, a detector, and a medium in which the optical energy propagates. Many textbooks cover sources and detectors, but very few cover propagation in a comprehensive way, incorporating the latest progress in theory and experiment concerning the propagating medium. This book fulfills that need. It is the first comprehensive and self-contained book on this topic. It is useful reference book for researchers, and a textbook for courses like Laser Light Propagation, Solid State Optics, and Optical Propagation in the Atmosphere. Preview this book » What people are saying-Write a reviewWe haven't found any reviews in the usual places.Selected pagesTitle PageTable of ContentsIndexReferencesContentsPRACTICAL MODELS FOR VARIOUS MEDIA261 Symbols and Units475 Special Functions479 Hilbert and Fourier Transforms483 Model Parameters for Gases Liquids and Solids491 Electromagnetic Field Quantization547 Index553 Copyright Other editions - View allOptical Propagation in Linear Media: Atmospheric Gases and Particles, Solid ...Michael E. ThomasLimited preview - 2006Common terms and phrasesA4.5 Solid-State Complex absorption bands absorption coefficient Allowed IR Modes angle atmosphere Bandgap based on Eq BR