In mathematical modeling, overfitting is "the production of an analysis that corresponds too closely or exactly to a particular set of data, and may therefore fail to fit to additional data or predict future observations reliably". An overfitted model is a mathematical model that contains more parameters than can be justified by the data. In a mathematical sense, these parameters represent the degree of a polynomial. The essence of overfitting is to have unknowingly extracted some of the residual variation as if that variation represented underlying model structure.
Validity is the main extent to which a concept, conclusion, or measurement is well-founded and likely corresponds accurately to the real world. The word "valid" is derived from the Latin validus, meaning strong. The validity of a measurement tool is the degree to which the tool measures what it claims to measure. Validity is based on the strength of a collection of different types of evidence described in greater detail below.
In mathematics, a time series is a series of data points indexed in time order. Most commonly, a time series is a sequence taken at successive equally spaced points in time. Thus it is a sequence of discrete-time data. Examples of time series are heights of ocean tides, counts of sunspots, and the daily closing value of the Dow Jones Industrial Average.
Cross-validation, sometimes called rotation estimation or out-of-sample testing, is any of various similar model validation techniques for assessing how the results of a statistical analysis will generalize to an independent data set. Cross-validation is a resampling method that uses different portions of the data to test and train a model on different iterations. It is mainly used in settings where the goal is prediction, and one wants to estimate how accurately a predictive model will perform in practice. In a prediction problem, a model is usually given a dataset of known data on which training is run, and a dataset of unknown data against which the model is tested. The goal of cross-validation is to test the model's ability to predict new data that was not used in estimating it, in order to flag problems like overfitting or selection bias and to give an insight on how the model will generalize to an independent dataset.
Linear trend estimation is a statistical technique that aids in the interpretation of data. When a series of measurements of a process are treated as a sequence or time series, trend estimation can be used to make and justify statements about tendencies in the data by relating the measurements to the times at which they occurred. This model can then be used to describe the behavior of the observed data.
In statistical modeling, regression analysis is a set of statistical processes for estimating the relationships between a dependent variable and one or more independent variables. The most common form of regression analysis is linear regression, in which one finds the line that most closely fits the data according to a specific mathematical criterion. For example, the method of ordinary least squares computes the unique line that minimizes the sum of squared differences between the true data and that line. For specific mathematical reasons, this allows the researcher to estimate the conditional expectation of the dependent variable when the independent variables take on a given set of values. Less common forms of regression use slightly different procedures to estimate alternative location parameters or estimate the conditional expectation across a broader collection of non-linear models.
In statistics, the coefficient of determination, denoted R2 or r2 and pronounced "R squared", is the proportion of the variation in the dependent variable that is predictable from the independent variable(s).
In machine learning, a common task is the study and construction of algorithms that can learn from and make predictions on data. Such algorithms function by making data-driven predictions or decisions, through building a mathematical model from input data. These input data used to build the model are usually divided into multiple data sets. In particular, three data sets are commonly used in different stages of the creation of the model: training, validation, and test sets.
In statistics, ordinary least squares (OLS) is a type of linear least squares method for choosing the unknown parameters in a linear regression model by the principle of least squares: minimizing the sum of the squares of the differences between the observed dependent variable in the input dataset and the output of the (linear) function of the independent variable.
Structural equation modeling (SEM) is a diverse set of methods used by scientists doing both observational and experimental research. SEM is used mostly in the social and behavioral sciences but it is also used in epidemiology, business, and other fields. A definition of SEM is difficult without reference to technical language, but a good starting place is the name itself.
In robust statistics, robust regression seeks to overcome some limitations of traditional regression analysis. A regression analysis models the relationship between one or more independent variables and a dependent variable. Standard types of regression, such as ordinary least squares, have favourable properties if their underlying assumptions are true, but can give misleading results otherwise. Robust regression methods are designed to limit the effect that violations of assumptions by the underlying data-generating process have on regression estimates.
Model selection is the task of selecting a model from among various candidates on the basis of performance criterion to choose the best one. In the context of machine learning and more generally statistical analysis, this may be the selection of a statistical model from a set of candidate models, given data. In the simplest cases, a pre-existing set of data is considered. However, the task can also involve the design of experiments such that the data collected is well-suited to the problem of model selection. Given candidate models of similar predictive or explanatory power, the simplest model is most likely to be the best choice.
In statistics, resampling is the creation of new samples based on one observed sample. Resampling methods are:
- Permutation tests
- Bootstrapping
- Cross validation
In statistics, stepwise regression is a method of fitting regression models in which the choice of predictive variables is carried out by an automatic procedure. In each step, a variable is considered for addition to or subtraction from the set of explanatory variables based on some prespecified criterion. Usually, this takes the form of a forward, backward, or combined sequence of F-tests or t-tests.
In statistics, model specification is part of the process of building a statistical model: specification consists of selecting an appropriate functional form for the model and choosing which variables to include. For example, given personal income together with years of schooling and on-the-job experience , we might specify a functional relationship as follows:
In statistics, regression validation is the process of deciding whether the numerical results quantifying hypothesized relationships between variables, obtained from regression analysis, are acceptable as descriptions of the data. The validation process can involve analyzing the goodness of fit of the regression, analyzing whether the regression residuals are random, and checking whether the model's predictive performance deteriorates substantially when applied to data that were not used in model estimation.
The following outline is provided as an overview of and topical guide to regression analysis:
Measurement invariance or measurement equivalence is a statistical property of measurement that indicates that the same construct is being measured across some specified groups. For example, measurement invariance can be used to study whether a given measure is interpreted in a conceptually similar manner by respondents representing different genders or cultural backgrounds. Violations of measurement invariance may preclude meaningful interpretation of measurement data. Tests of measurement invariance are increasingly used in fields such as psychology to supplement evaluation of measurement quality rooted in classical test theory.
In mathematics, statistics, and computational modelling, a grey box model combines a partial theoretical structure with data to complete the model. The theoretical structure may vary from information on the smoothness of results, to models that need only parameter values from data or existing literature. Thus, almost all models are grey box models as opposed to black box where no model form is assumed or white box models that are purely theoretical. Some models assume a special form such as a linear regression or neural network. These have special analysis methods. In particular linear regression techniques are much more efficient than most non-linear techniques. The model can be deterministic or stochastic depending on its planned use.
In statistics, linear regression is a linear approach for modelling a predictive relationship between a scalar response and one or more explanatory variables, which are measured without error. The case of one explanatory variable is called simple linear regression; for more than one, the process is called multiple linear regression. This term is distinct from multivariate linear regression, where multiple correlated dependent variables are predicted, rather than a single scalar variable. If the explanatory variables are measured with error then errors-in-variables models are required, also known as measurement error models.
