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Mean Absolute Percentage Error
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. It usually expresses the accuracy as a ratio defined by the formula: : \mbox = \frac\sum_^n \left, \frac\ where is the actual value and is the forecast value. Their difference is divided by the actual value . The absolute value of this ratio is summed for every forecasted point in time and divided by the number of fitted points . MAPE in regression problems Mean absolute percentage error is commonly used as a loss function for regression problems and in model evaluation, because of its very intuitive interpretation in terms of relative error. Definition Consider a standard regression setting in which the data are fully described by a random pair Z=(X,Y) with values in \mathbb^d\times\mathbb, and i.i.d. copies (X_1, Y_1), ..., (X_n, Y_n) of (X,Y). Regression models aims at finding a good model ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Statistics
Statistics (from German language, German: ''wikt:Statistik#German, Statistik'', "description of a State (polity), state, a country") is the discipline that concerns the collection, organization, analysis, interpretation, and presentation of data. In applying statistics to a scientific, industrial, or social problem, it is conventional to begin with a statistical population or a statistical model to be studied. Populations can be diverse groups of people or objects such as "all people living in a country" or "every atom composing a crystal". Statistics deals with every aspect of data, including the planning of data collection in terms of the design of statistical survey, surveys and experimental design, experiments.Dodge, Y. (2006) ''The Oxford Dictionary of Statistical Terms'', Oxford University Press. When census data cannot be collected, statisticians collect data by developing specific experiment designs and survey sample (statistics), samples. Representative sampling as ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Regression Analysis
In statistical modeling, regression analysis is a set of statistical processes for estimating the relationships between a dependent variable (often called the 'outcome' or 'response' variable, or a 'label' in machine learning parlance) and one or more independent variables (often called 'predictors', 'covariates', 'explanatory variables' or 'features'). The most common form of regression analysis is linear regression, in which one finds the line (or a more complex linear combination) that most closely fits the data according to a specific mathematical criterion. For example, the method of ordinary least squares computes the unique line (or hyperplane) that minimizes the sum of squared differences between the true data and that line (or hyperplane). For specific mathematical reasons (see linear regression), this allows the researcher to estimate the conditional expectation (or population average value) of the dependent variable when the independent variables take on a given ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Measurable Function
In mathematics and in particular measure theory, a measurable function is a function between the underlying sets of two measurable spaces that preserves the structure of the spaces: the preimage of any measurable set is measurable. This is in direct analogy to the definition that a continuous function between topological spaces preserves the topological structure: the preimage of any open set is open. In real analysis, measurable functions are used in the definition of the Lebesgue integral. In probability theory, a measurable function on a probability space is known as a random variable. Formal definition Let (X,\Sigma) and (Y,\Tau) be measurable spaces, meaning that X and Y are sets equipped with respective \sigma-algebras \Sigma and \Tau. A function f:X\to Y is said to be measurable if for every E\in \Tau the pre-image of E under f is in \Sigma; that is, for all E \in \Tau f^(E) := \ \in \Sigma. That is, \sigma (f)\subseteq\Sigma, where \sigma (f) is the σ-algebra gen ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Mean Squared Error
In statistics, the mean squared error (MSE) or mean squared deviation (MSD) of an estimator (of a procedure for estimating an unobserved quantity) measures the average of the squares of the errors—that is, the average squared difference between the estimated values and the actual value. MSE is a risk function, corresponding to the expected value of the squared error loss. The fact that MSE is almost always strictly positive (and not zero) is because of randomness or because the estimator does not account for information that could produce a more accurate estimate. In machine learning, specifically empirical risk minimization, MSE may refer to the ''empirical'' risk (the average loss on an observed data set), as an estimate of the true MSE (the true risk: the average loss on the actual population distribution). The MSE is a measure of the quality of an estimator. As it is derived from the square of Euclidean distance, it is always a positive value that decreases as the error a ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Empirical Risk Minimization
Empirical risk minimization (ERM) is a principle in statistical learning theory which defines a family of learning algorithms and is used to give theoretical bounds on their performance. The core idea is that we cannot know exactly how well an algorithm will work in practice (the true "risk") because we don't know the true distribution of data that the algorithm will work on, but we can instead measure its performance on a known set of training data (the "empirical" risk). Background Consider the following situation, which is a general setting of many supervised learning problems. We have two spaces of objects X and Y and would like to learn a function \ h: X \to Y (often called ''hypothesis'') which outputs an object y \in Y, given x \in X. To do so, we have at our disposal a ''training set'' of n examples \ (x_1, y_1), \ldots, (x_n, y_n) where x_i \in X is an input and y_i \in Y is the corresponding response that we wish to get from h(x_i). To put it more formally, we assume ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Mean Absolute Error
In statistics, mean absolute error (MAE) is a measure of errors between paired observations expressing the same phenomenon. Examples of ''Y'' versus ''X'' include comparisons of predicted versus observed, subsequent time versus initial time, and one technique of measurement versus an alternative technique of measurement. MAE is calculated as the sum of absolute errors divided by the sample size: \mathrm = \frac =\frac. It is thus an arithmetic average of the absolute errors , e_i, = , y_i - x_i, , where y_i is the prediction and x_i the true value. Note that alternative formulations may include relative frequencies as weight factors. The mean absolute error uses the same scale as the data being measured. This is known as a scale-dependent accuracy measure and therefore cannot be used to make comparisons between series using different scales. The mean absolute error is a common measure of forecast error in time series analysis, sometimes used in confusion with the more standard de ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Quantile Regression
Quantile regression is a type of regression analysis used in statistics and econometrics. Whereas the method of least squares estimates the conditional ''mean'' of the response variable across values of the predictor variables, quantile regression estimates the conditional ''median'' (or other '' quantiles'') of the response variable. Quantile regression is an extension of linear regression used when the conditions of linear regression are not met. Advantages and applications One advantage of quantile regression relative to ordinary least squares regression is that the quantile regression estimates are more robust against outliers in the response measurements. However, the main attraction of quantile regression goes beyond this and is advantageous when conditional quantile functions are of interest. Different measures of central tendency and statistical dispersion can be useful to obtain a more comprehensive analysis of the relationship between variables. In ecology, quantile ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Consistency (statistics)
In statistics, consistency of procedures, such as computing confidence intervals or conducting hypothesis tests, is a desired property of their behaviour as the number of items in the data set to which they are applied increases indefinitely. In particular, consistency requires that the outcome of the procedure with unlimited data should identify the underlying truth.Dodge, Y. (2003) ''The Oxford Dictionary of Statistical Terms'', OUP. (entries for consistency, consistent estimator, consistent test) Use of the term in statistics derives from Sir Ronald Fisher in 1922. Use of the terms ''consistency'' and ''consistent'' in statistics is restricted to cases where essentially the same procedure can be applied to any number of data items. In complicated applications of statistics, there may be several ways in which the number of data items may grow. For example, records for rainfall within an area might increase in three ways: records for additional time periods; records for additional ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Mean Absolute Scaled Error
In statistics, the mean absolute scaled error (MASE) is a measure of the accuracy of forecasts. It is the mean absolute error of the forecast values, divided by the mean absolute error of the in-sample one-step naive forecast. It was proposed in 2005 by statistician Rob J. Hyndman and Professor of Decision Sciences Anne B. Koehler, who described it as a "generally applicable measurement of forecast accuracy without the problems seen in the other measurements." The mean absolute scaled error has favorable properties when compared to other methods for calculating forecast errors, such as root-mean-square-deviation, and is therefore recommended for determining comparative accuracy of forecasts. Rationale The mean absolute scaled error has the following desirable properties: # Scale invariance: The mean absolute scaled error is independent of the scale of the data, so can be used to compare forecasts across data sets with different scales. # Predictable behavior as y_ \rightarrow 0 ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Symmetric Mean Absolute Percentage Error
Symmetric mean absolute percentage error (SMAPE or sMAPE) is an accuracy measure based on percentage (or relative) errors. It is usually defined as follows: : \text = \frac \sum_^n \frac where ''A''''t'' is the actual value and ''F''''t'' is the forecast value. The absolute difference between ''A''''t'' and ''F''''t'' is divided by half the sum of absolute values of the actual value ''A''''t'' and the forecast value ''F''''t''. The value of this calculation is summed for every fitted point ''t'' and divided again by the number of fitted points ''n''. The earliest reference to similar formula appears to be Armstrong (1985, p. 348) where it is called "adjusted MAPE" and is defined without the absolute values in denominator. It has been later discussed, modified and re-proposed by Flores (1986). Armstrong's original definition is as follows: : \text = \frac 1 n \sum_^n \frac The problem is that it can be negative (if A_t + F_t < 0) or even undefined (if |
Mean Directional Accuracy (MDA)
Mean directional accuracy (MDA), also known as mean direction accuracy, is a measure of prediction accuracy of a forecasting method in statistics. It compares the forecast direction (upward or downward) to the actual realized direction. It is defined by the following formula: : \frac\sum_t \mathbf_ where ''A''''t'' is the actual value at time ''t'' and ''F''''t'' is the forecast value at time ''t''. Variable ''N'' represents number of forecasting points. The function \sgn(\cdot) is sign function and \mathbf is the indicator function. In simple words, MDA provides the probability that the under study forecasting method can detect the correct direction of the time series. MDA is a popular metric for forecasting performance in economics and finance. MDA is used in economics applications where the economist is often interested only in directional movement of variable of interest. As an example in macroeconomics, a monetary authority who wants to know the direction of the inflatio ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Least Absolute Deviations
Least absolute deviations (LAD), also known as least absolute errors (LAE), least absolute residuals (LAR), or least absolute values (LAV), is a statistical optimality criterion and a statistical optimization technique based minimizing the ''sum of absolute deviations'' (sum of absolute residuals or sum of absolute errors) or the ''L''1 norm of such values. It is analogous to the least squares technique, except that it is based on ''absolute values'' instead of squared values. It attempts to find a function which closely approximates a set of data by minimizing residuals between points generated by the function and corresponding data points. The LAD estimate also arises as the maximum likelihood estimate if the errors have a Laplace distribution. It was introduced in 1757 by Roger Joseph Boscovich. Formulation Suppose that the data set consists of the points (''x''''i'', ''y''''i'') with ''i'' = 1, 2, ..., ''n''. We want to find a function ''f'' such that f(x_i)\approx y_i. ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
