In addition, for sigma restricted models (e.g., in *General Stepwise Regression*; some software offers the user a choice between the sigma restricted and overparameterized models), we propose a *Type VI sums of squares* option; this approach is identical to what is described as the *effective hypothesis *method in Hocking (1996). For details regarding these methods, refer to the *Six types of sums of squares* topic of the General Linear Model chapter.

Efficient score statistic.
See *Score statistic*.

Ellipse, Prediction Interval (Area) and Range.

**Prediction Interval (Area) Ellipse:** This type of ellipse is useful for establishing confidence intervals for the prediction of single new observations (prediction intervals). Such bivariate confidence or control limits are, for example, often used in the context of multivariate control charts for industrial quality control (see, for example, Montgomery, 1996; see also Hotelling T-square chart).

The ellipse is determined based on the assumption that the two variables follow the bivariate normal distribution. The orientation of this ellipse is determined by the sign of the linear correlation between two variables (the longer axis of the ellipse is superimposed on the regression line). The probability that the values will fall within the area marked by the ellipse is determined by the value of the coefficient that defines the ellipse (e.g., 95%). For additional information see, for example, Tracy, Young, and Mason (1992), or Montgomery 1996); see also the description of the prediction interval ellipse.

**Range Ellipse:** This type of ellipse is a fixed size ellipse determined such that the length of its horizontal and vertical projection onto the X- and Y-axis (respectively) is equal to the mean (Range * I) where the mean and range refer to the X or Y variable, and I is the current value of the coefficient field.

Endogenous Variable.
An endogenous variable is a variable that appears as a dependent variable in at least one equation in a structural model. In a path diagram, endogenous variables can be recognized by the fact that they *have at least one arrow pointing to them*.

Enterprise Resource Planning (ERP).
*Enterprise Resource Planning (ERP)* software offers an integrated set of applications that help manage various aspects of the business activity of a company (e.g., orders, production schedules, supply chain and inventories, accounting and bookkeeping, human resource management). Usually, an *ERP* application consists of a number of specialized modules that still offer a consistent user interface and integration of all relevant parts of the company data base system. Examples of widely used *ERP* applications are products of SAP and Oracle.

See also, Enterprise-wide Software Systems.

Enterprise SPC.
*Enterprise SPC* is a groupware based process control system (see SPC), designed to work in enterprise-wide environment and allowing engineers and supervisors to share data, chart specifications (and other QC criteria), reports, and database queries. *Enterprise SPC* systems always include central QC data bases and if properly integrated, they allow the managers to maintain quality standards for all products/processes in a given corporation.

See also, *Statistical Process Control*, *Quality Control*, *Process Analysis* and *STATISTICA Enterprise-wide SPC System (SEWSS)*.

For more information on process control systems, see the ASQC/AIAG's *Fundamental statistical process control reference manual* (1991).

Enterprise-Wide Software Systems.
Software applications designed to work in enterprise computer envirnoments (e.g., such as large corporate computer systems). Such applications typically feature extensive groupware functionality, and they are usually well integrated with large repositories of data stored in corporate data warehouses.
See also Data Warehousing. For more information on enterprise-wide systems available from StatSoft, see *STATISTICA* Enterprise-Wide Systems.

Epoch in Neural Networks.
During iterative training of a neural network, an *Epoch* is a single pass through the entire training set, followed by testing of the verification set.

For more information, see the Neural Networks chapter.

Error Bars (2D Box Plots). In this style of 2D Box plots, the ranges or error bars are calculated from the data. The central tendency (e.g., median or mean), and range or variation statistics (e.g., min-max values, quartiles, standard errors, or standard deviations) are computed for each variable and the selected values are presented as error bars.

The diagram above illustrates the ranges of outliers and extremes in the "classic" box and whisker plot (for more information about box plots, see Tukey, 1977).

Error Bars (2D Range Plots). In this style of 2D Range Plot, the ranges or error bars are defined by the raw values in the selected variables. The midpoints are represented by point markers. One range or error bar is plotted for each case. In the simplest instance, three variables need to be selected, one representing the mid-points, one representing the upper limits and one representing the lower limits.

Error Bars (3D Box Plots). In this style of 3D Sequential Box Plot, the ranges of values of selected variables are plotted separately for groups of cases defined by values of a categorical (grouping) variable. The central tendency (e.g., median or mean), and range or variation statistics (e.g., min-max values, quartiles, standard errors, or standard deviations) are computed for each variable and for each group of cases and the selected values are presented as error bars.

*3D Range plots* differ from *3D Box plots* in that for Range plots, the ranges are the values of the selected variables (e.g., one variable contains the minimum range values and another variable contains the maximum range values) while for Box plots the ranges are calculated from variable values (e.g., standard deviations, standard errors, or min-max value).

Error Bars (3D Range Plots). In this style of 3D Sequential Range Plot, the error bars are not calculated from data but defined by the raw values in the selected variables. The midpoints are represented by point markers. One error bar is plotted for each case. The range variables can be interpreted either as absolute values or values representing deviations from the midpoint depending on the current setting of the Mode option in the graph definition dialog. Single or multiple variables can be represented in the graph.

3D Range plots differ from 3D Box plots in that for *Range plots*, the ranges are the values of the selected variables (e.g., one variable contains the minimum range values and another variable contains the maximum range values) while for *Box plots*, the ranges are calculated from variable values (e.g., standard deviations, standard errors, or min-max values).

Error Function in Neural Networks. Used to determine how well a neural network is performing during iterative training and execution. The derivative of the error function is used by the iterative training algorithms (Bishop, 1995).

For more information, see the Neural Networks chapter. See also, loss function.

Estimable Functions.
In *general linear models *and *generalized linear models*, if the *X**'*** X **matrix (where

f=**Lb**

where ** L** is a vector of coefficients, will also in general not be unique. However,

**L**=**L(X'X)`X'X**

is invariant for all possible *generalized inverses*, and is therefore called an *estimable function*.

See also general linear model, generalized linear model, design matrix, matrix rank, generalized inverse; for additional details, see also *General Linear Models*.

Exogenous Variable.
An exogenous variable is a variable that never appears as a dependent variable in any equation in a structural model. In a path diagram, exogenous variables can be recognized by the fact that they *have no arrows pointing to them*.

Experimental Design (DOE, Industrial Experimental Design).
In industrial settings, *Experimental design (DOE)* techniques apply analysis of variance principles to product development. The primary goal is usually to extract the maximum amount of unbiased information regarding the factors affecting a production process from as few (costly) observations as possible. In industrial settings, complex interactions among many factors that influence a product are often regarded as a "nuisance" (they are often of no interest; they only complicate the process of identifying important factors, and in experiments with many factors it would not be possible or practical to identify them anyway). Hence, if you review standard texts on experimentation in industry (Box, Hunter, and Hunter, 1978; Box and Draper, 1987; Mason, Gunst, and Hess, 1989; Taguchi, 1987) you will find that they will primarily discuss designs with many factors (e.g., 16 or 32) in which interaction effects cannot be evaluated, and the primary focus of the discussion is how to derive unbiased main effect (and, perhaps, two-way interaction) estimates with a minimum number of observations.

For more information, see the Experimental Design chapter.

Explained variance. The proportion of the variability in the data which is accounted for by the model (e.g., in Multiple Regression, ANOVA, Nonlinear Estimation, Neural Networks) .

Exploratory Data Analysis (EDA).
As opposed to traditional *hypothesis testing* designed to verify *a priori* hypotheses about relations between variables (e.g., *"There is a positive correlation between the AGE of a person and his/her RISK TAKING disposition"*), *exploratory data analysis (EDA)* is used to identify systematic relations between variables when there are no (or not complete) *a priori* expectations as to the nature of those relations. In a typical exploratory data analysis process, many variables are taken into account and compared, using a variety of techniques in the search for systematic patterns.

For more information, see Exploratory Data Analysis (EDA) and Data Mining Techniques.

Exponential Function. This fits to the data, an exponential function of the form:

y = b*exp(q*x)

Exponential Distribution. The exponential distribution function is defined as:

f(x) = * e^{-}^{x}

0 x < ,
> 0

where

(*lambda*) is an exponential function parameter (an alternative parameterization is scale parameter *b=1/*)

e is the
base of the natural logarithm, sometimes called Euler's e (2.71...)

The graphic above shows the shape of the *Exponential* distribution when lambda equals 1.

Exponential Family of Distributions.

A family of probability distributions with exponential terms, which includes many of the most important distributions encountered in real (neural network) problems (including the normal, or Gaussian distribution, and the alpha and beta distributions). See also, Neural Networks.

Exponentially Weighted Moving Average Line. This type of moving average can be considered to be a generalization of the simple moving average. Specifically, we could compute each data point for the plot as:

z_{t} = *x-bar_{t} + (1-)*z _{t-1}

In this formula, each point *z _{t}* is computed as (lambda) times the respective mean

Extrapolation. Predicting the value of unknown data points by projecting a function beyond the range of known data points.

Extreme Values (in Box Plots).
Values which are "far" from the middle of the distribution are referred to as outliers and *extreme values* if they meet certain conditions.

A data point is deemed to be an *extreme* value if the following conditions hold:

data point value > UBV + 2*o.c.*(UBV - LBV)

or

data point value < LBV - 2*o.c.*(UBV - LBV)

where

UBV is the upper value of the box in the box plot (e.g., the mean + standard error or the 75th percentile).

LBV is the lower value of the box in the box plot (e.g., the mean - standard error or the 25th percentile).

o.c. is the outlier coefficient (when this coefficient equals 1.5, the extreme values are those which are outside the 3 box length range from the upper and lower value of the box).

For example, the following diagram illustrates the ranges of outliers and extremes in the "classic" box and whisker plot (for more information about box plots, see Tukey, 1977).

Extreme Value Distribution. The extreme value (Type I) distribution (the term first used by Lieblein, 1953) has the probability density function:

f(x) = 1/b * e^{-(x-a)/b} * e^{-e-(x-a) / b}

- < x <

b > 0

where

a is the location parameter

b is the scale parameter

e is the
base of the natural logarithm, sometimes called Euler's e (2.71...)

This distribution is also sometimes referred to as the distribution of the largest extreme.

See also, Process Analysis.

The graphic above shows the shape of the *extreme value* distribution when the location parameter equals 0 and the scale parameter equals 1.

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