See also, standard residual value, deleted residual and Cook’s distance.

Mallow's CP. If p regressors are selected from a set of k, Cp is defined as:

S (y-yp)² / s² - n+2p

where
y_p    is the predicted value of y from the p regressors
s²    is the residual mean square after regression on the complete set of k
n      is the sample size

The model is then chosen to give a minimum value of the criterion, or a value that is acceptably small. It is essential a special case of Akaike Information Criterion. Mallow's CP is used in General Regression Models (GRM) as the criterion for choosing the best subset of predictor effects when a best subset regression analysis is being performed. This measure of the quality of fit for a model tends to be less dependent (than the R-square) on the number of effects in the model, and hence, it tends to find the best subset that includes only the important predictors of the respective dependent variable. See Best Subset Regression Options in GRM for further details.

Mann-Scheuer-Fertig Test. This test, proposed by Mann, Scheuer, and Fertig (1973), is described in detail in, for example, Dodson (1994) or Lawless (1982). The null hypothesis for this test is that the population follows the Weibull distribution with the estimated parameters. Nelson (1982) reports this test to have reasonably good power, and this test can be applied to Type II censored data. For computational details refer to Dodson (1994) or Lawless (1982); the critical values for the test statistic have been computed based on Monte Carlo studies, and have been tabulated for n (sample sizes) between 3 and 25; for n greater than 25, this test is not computed.

The Mann-Scheuer-Fertig test is used in Weibull and Reliability/Failure Time Analysis; see also, Hollander-Proschan Test and Anderson-Darling Test.

Marginal Frequencies. In a Multi-way table, the values in the margins of the table are simply one-way (frequency) tables for all values in the table. They are important in that they help us to evaluate the arrangement of frequencies in individual columns or rows. The differences between the distributions of frequencies in individual rows (or columns) and in the respective margins inform us about the relationship between the crosstabulated variables.

For more information on Marginal frequencies, see the Crosstabulations section of the Basic Statistics chapter.

Markov Chain Monte Carlo (MCMC). The term "Monte Carlo method" (suggested by John von Neumann and S. M. Ulam, in the 1940s) refers to simulation of processes, using random numbers. The term Monte Carlo (a city long known for its gambling casinos) derived from the fact that "numbers of chance" (i.e., Monte Carlo simulation methods) were used in order to solve some of the integrals of the complex equations involved in the design of the first nuclear bombs (integrals of quantum dynamics). By generating large samples of random numbers from, for example, mixtures of distributions, the integrals of these (complex) distributions can be approximated from the (generated) data.

Complex equations with difficult to solve integrals are often involved in Bayesian Statistics Analyses. For a simple example of the MCMC method for generating bivariate normal random variables, see the description of the Gibbs Sampler.

For a detailed discussion of MCMC methods, see Gilks, Richardson, and Spiegelhalter (1996). See also the description of the Gibbs Sampler, and Bayesian Statistics (Analysis).

Mass. The term mass in correspondence analysis is used to denote the entries in the two-way table of relative frequencies (i.e., each entry is divided by the sum of all entries in the table). Note that the results from correspondence analysis are still valid if the entries in the table are not frequencies, but some other measure of correspondence, association, similarity, confusion, etc. Since the sum of all entries in the table of relative frequencies is equal to 1.0, one could say that the table of relative frequencies shows how one unit of mass is distributed across the cells of the table. In the terminology of correspondence analysis, the row and column totals of the table of relative frequencies are called the row mass and column mass, respectively.

Manifest Variable. A manifest variable is a variable that is directly observable or measurable. In path analysis diagrams used in structural modeling (see Path Diagram), manifest variables are usually represented by enclosing the variable name within a square or a rectangle.

Matching Moments Method. This method can be employed to determine parameter estimates for a distribution (see Quantile- Quantile Plots, Probability-Probability Plots, and Process Analysis). The method of matching moments sets the distribution moments equal to the data moments and solves to obtain estimates for the distribution parameters. For example, for a distribution with two parameters, the first two moments of the distribution (the mean and variance of the distribution, respectively, e.g., and , respectively) would be set equal to the first two moments of the data (the sample mean and variance, respectively, e.g., the unbiased estimators x-bar and s**2, respectively) and solved for the parameter estimates. Alternatively, you could use the Maximum Likelihood Method to estimate the parameters. For more information, see Hahn and Shapiro, 1994.

Matrix Ill-conditioning. Matrix ill-conditioning is a general term used to describe a rectangular matrix of values which is unsuitable for use in a particular analysis.

This occurs perhaps most frequently in applications of linear multiple regression when the matrix of correlations for the predictors is singular and thus the regular matrix inverse cannot be computed. In some modules (i.e., in Factor Analysis) this problem is dealt with by issuing a respective warning and then artificially lowering all correlations in the correlation matrix by adding a small constant to the diagonal elements of the matrix, and then restandardizing it. This procedure will usually yield a matrix for which the regular matrix inverse can be computed.

Note that in many applications of the general linear model and the generalized linear model, matrix singularity is not abnormal (i.e., when the overparameterized model is used to represent effects for categorical predictor variables) and is dealt with by computing a generalized inverse rather than the regular matrix inverse.

Another example of matrix ill-conditioning is intransitivity of the correlations in a correlation matrix. If in a correlation matrix variable A is highly positively correlated with B, B is highly positively correlated with C, and A is highly negatively correlated with C, this "impossible" pattern of correlations signals an error in the elements of the matrix.

See also matrix singularity, matrix inverse, generalized inverse.

Matrix Inverse. The regular inverse of a rectangular matrix of values is an extension of the concept of a numeric reciprocal. For a nonsingular matrix A, its inverse (denoted by a superscript of -1) is the unique matrix that satisfies

A^-1AA=A

No such regular inverse exists for singular matrices, but generalized inverses (an infinite number of them) can be computed for any singular matrix.

See also matrix singularity, generalized inverse.

Matrix Plots. Matrix graphs summarize the relationships between several variables in a matrix of true X-Y plots. The most common type of matrix plot is the scatterplot, which can be considered to be the graphical equivalent of the correlation matrix.

Matrix Plots - Columns. In this type of Matrix Plot, columns represent projections of individual data points onto the X- axis (showing the distribution of the maximum values), arranged in a matrix format. Histograms representing the distribution of each variable are displayed along the diagonal of the matrix (in square matrices, see example below) or along the edges (in rectangular matrices).

Matrix Plots - Lines. In this type of Matrix Plot, a matrix of X-Y (i.e., nonsequential) line plots (similar to a scatterplot matrix) is produced in which individual points are connected by a line in the order of their appearance in the data file. Histograms representing the distribution of each variable are displayed along the diagonal of the matrix (in square matrices) or along the edges (in rectangular matrices, see example below).

Matrix Plots - Scatterplot. In this type of Matrix Plot, 2D Scatterplots are arranged in a matrix format (values of the column variable are used as X coordinates, values of the row variable represent the Y coordinates). Histograms representing the distribution of each variable are displayed along the diagonal of the matrix (in square matrices, see example below) or along the edges (in rectangular matrices).

See also, Data Reduction.

Matrix Singularity. A rectangular matrix of values (e.g., a sums of squares and cross-products matrix) is singular if the elements in a column (or row) of the matrix are linearly dependent on the elements in one or more other columns (or rows) of the matrix. For example, if the elements in one column of a matrix are 1, -1, 0, and the elements in another column of the matrix are 2, -2, 0, then the matrix is singular because 2 times each of the elements in the first column is equal to each of the respective elements in the second column. Such matrices are also said to suffer from multicollinearity problems, since one or more columns are linearly related to each other.

A unique, regular matrix inverse cannot be computed for singular matrices, but generalized inverses (an infinite number of them) can be computed for any singular matrix.

See also matrix inverse.

Matrix Rank. The column (or row) rank of a rectangular matrix of values (e.g., a sums of squares and cross-products matrix) is equal to the number of linearly independent columns (or rows) of elements in the matrix. If there are no columns that are linearly dependent on other columns, then the rank of the matrix is equal to the number of its columns and the matrix is said to have full (column) rank. If the rank is less than the number of columns, the matrix is said to have reduced (column) rank and is singular.

See also matrix singularity.

Maximum Likelihood Loss Function. An common alternative to the least squares loss function is to maximize the likelihood or log-likelihood function (or to minimize the negative log-likelihood function; the term maximum likelihood was first used by Fisher, 1922a). These functions are typically used when fitting non-linear models. In most general terms, the likelihood function is defined as:

L=F(Y,Model)=ⁿ_i=1 { p[y_i , Model Parameters(x_i)]}

Maximum Likelihood Method. The method of maximum likelihood (the term first used by Fisher, 1922a) is a general method of estimating parameters of a population by values that maximize the likelihood (L) of a sample. The likelihood L of a sample of n observations x₁, x₂, ..., x_n, is the joint probability function p(x₁, x₂, ..., x_n) when x₁, x₂, ..., x_n are discrete random variables. If x₁, x₂, ..., x_n are continuous random variables, then the likelihood L of a sample of n observations, x₁, x₂, ..., x_n, is the joint density function f(x₁, x₂, ..., x_n).

Let L be the likelihood of a sample, where L is a function of the parameters ₁, ₂, ... _k. Then the maximum likelihood estimators of ₁, ₂, ... _k are the values of ₁, ₂, ... _k that maximize L.

Let be an element of . If is an open interval, and if L() is differentiable and assumes a maximum on W, then the MLE will be a solution of the following equation: (dL())/d = 0. For more information, see Bain and Engelhardt (1989) and Neter, Wasserman, and Kutner (1989).

See also, Nonlinear Estimation or Variance Components and Mixed Model ANOVA/ANCOVA.

Maximum Unconfounding. Maximum unconfounding is an experimental design criterion that is subsidiary to the criterion of design resolution. The maximum unconfounding criterion specifies that design generators should be chosen such that the maximum number of interactions of less than or equal to the crucial order, given the resolution, are unconfounded with all other interactions of the crucial order. It is an alternative to the minimum aberration criterion for finding the "best" design of maximum resolution. . For discussions of the role of design criteria in experimental design see 2**(k-p) fractional factorial designs and 2**(k-p) Maximally Unconfounded and Minimum Aberration Designs.

MD (Missing data). Same as Missing values.

Mean. The mean is a particularly informative measure of the "central tendency" of the variable if it is reported along with its confidence intervals. Usually we are interested in statistics (such as the mean) from our sample only to the extent to which they are informative about the population. The larger the sample size, the more reliable its mean. The larger the variation of data values, the less reliable the mean (see also Elementary Concepts).

Mean = (x_i)/n

where
n      is the sample size.

See also, Descriptive Statistics

Mean/S.D. An algorithm (used in neural networks) to assign linear scaling coefficients for a set of numbers. The mean and standard deviation of the set are found, and scaling factors selected so that these are mapped to desired mean and standard deviation values.

See also Neural Networks.

Mean Substitution of Missing Data. When you select Mean Substitution, the missing data will be replaced by the means for the respective variables during an analysis.

See also, Casewise vs. pairwise deletion of missing data

Median. A measure of central tendency, the median (the term first used by Galton, 1882) of a sample is the value for which one-half (50%) of the observations (when ranked) will lie above that value and one-half will lie below that value. When the number of values in the sample is even, the median is computed as the average of the two middle values.

See also, Descriptive Statistics.

Meta-Learning. The concept of meta-learning applies to the area of predictive data mining, to combine the predictions from multiple models. It is particularly useful when the types of models included in the project are very different. In this context, this procedure is also referred to as Stacking (Stacked Generalization).

Suppose your data mining project includes tree classifiers, such as C&RT and  CHAID, linear discriminant analysis (e.g., GDA), and Neural Networks. Each computes predicted classifications for a crossvalidation sample, from which overall goodness-of-fit statistics (e.g., misclassification rates) can be computed. Experience has shown that combining the predictions from multiple methods often yields more accurate predictions than can be derived from any one method (e.g., see Witten and Frank, 2000). The predictions from different classifiers can be used as input into a meta-learner, which will attempt to combine the predictions to create a final best predicted classification. So, for example, the predicted classifications from the tree classifiers, linear model, and the neural network classifier(s) can be used as input variables into a neural network meta-classifier, which will attempt to "learn" from the data how to combine the predictions from the different models to yield maximum classification accuracy.

One can apply meta-learners to the results from different meta-learners to create "meta-meta"-learners, and so on; however, in practice such exponential increase in the amount of data processing, in order to derive an accurate prediction, will yield less and less marginal utility.

See also, Neural Networks.

Minimum Aberration. Minimum aberration is an experimental design criterion that is subsidiary to the criterion of design resolution. The minimum aberration design is defined as the design of maximum resolution "which minimizes the number of words in the defining relation that are of minimum length" (Fries & Hunter, 1980). Less technically, the criterion apparently operates by choosing design generators that produce the smallest number of pairs of confounded interactions of the crucial order. For example, the minimum aberration resolution IV design would have the minimum number of pairs of confounded 2-factor interactions. For discussions of the role of design criteria in experimental design see 2**(k-p) fractional factorial designs and 2**(k-p) Maximally Unconfounded and Minimum Aberration Designs.

Missing values. Values of variables within data sets which are not known. Although such cases that contain missing data are incomplete, they can still be used in data analysis. Various methods exist to substitute missing data (e.g., by mean substitution, various types of interpolations and extrapolations). Also, pariwise deletion of missing data can be used. See also, Pairwise deletion of missing data, Casewise (Listwise) deletion of missing data, Pairwise deletion of missing data vs. mean substitution, and Casewise vs. pairwise deletion of missing data.

Mode. A measure of central tendency, the mode (the term first used by Pearson, 1895) of a sample is the value which occurrs most frequently in the sample.

See also, Descriptive Statistics.

Model Profiles (in Neural Networks). Model profiles are concise text strings indicating the architecture of networks and ensembles. A profile consists of a type code followed by a code giving the number of input and output variables and number of layers and units (networks) or members (ensembles). For time series networks, the number of steps and the lookahead factor are also given. The individual parts of the profile are:

Model Type. The codes are:

Network architecture. This is of the form I:N-N-N:O, where I is the number of input variable, O the number of output variables, and N the number of units in each layer.

Example. 2:4-6-3:1 indicates a network with 2 input variables, 1 output variable, 4 input neurons, 6 hidden neurons, and 3 output neurons.

For a time series network, the steps factor is prepended to the profile, and signified by an "s."

Example. s10 1:10-2-1:1 indicates a time series network with steps factor (lagged input) 10.

Ensemble architecture. This is of the form I:[N]:O, where I is the number of input variable, O the number of output variables, and N the number of members of the ensemble.

One such model, CRISP (Cross-Industry Standard Process for data mining) was proposed in the mid-1990s by a European consortium of companies to serve as a non-proprietary standard process model for data mining. This general approach postulates the following (perhaps not particularly controversial) general sequence of steps for data mining projects:

Another approach - the Six Sigma methodology - is a well-structured, data-driven methodology for eliminating defects, waste, or quality control problems of all kinds in manufacturing, service delivery, management, and other business activities. This model has recently become very popular (due to its successful implementations) in various American industries, and it appears to gain favor worldwide. It postulated a sequence of, so-called, DMAIC steps -

- that grew up from the manufacturing, quality improvement, and process control traditions and is particularly well suited to production environments (including "production of services," i.e., service industries).

Another framework of this kind (actually somewhat similar to Six Sigma) is the approach proposed by SAS Institute called SEMMA -

- which is focusing more on the technical activities typically involved in a data mining project.

All of these models are concerned with the process of how to integrate data mining methodology into an organization, how to "convert data into information," how to involve important stake-holders, and how to disseminate the information in a form that can easily be converted by stake-holders into resources for strategic decision making.

Some software tools for data mining are specifically designed and documented to fit into one of these specific frameworks.

The general underlying philosophy of StatSoft's STATISTICA Data Miner is to provide a flexible data mining workbench that can be integrated into any organization, industry, or organizational culture, regardless of the general data mining process-model that the organization chooses to adopt. For example, STATISTICA Data Miner can include the complete set of (specific) necessary tools for ongoing company wide Six Sigma quality control efforts, and users can take advantage of its (still optional) DMAIC-centric user interface for industrial data mining tools. It can equally well be integrated into ongoing marketing research, CRM (Customer Relationship Management) projects, etc. that follow either the CRISP or SEMMA approach - it fits both of them perfectly well without favoring either one. Also, STATISTICA Data Miner offers all the advantages of a general data mining oriented "development kit" that includes easy to use tools for incorporating into your projects not only such components as custom database gateway solutions, prompted interactive queries, or proprietary algorithms, but also systems of access privileges, workgroup management, and other collaborative work tools that allow you to design large scale, enterprise-wide systems (e.g., following the CRIPS, SEMMA, or a combination of both models) that involve your entire organization.

See also Data Mining Techniques

For each Monte Carlo replication, the computer:

1. Simulates a random sample from the population,
2. Analyzes the sample,
3. Stores the results.

MPatt Bar. Multi-pattern bar plots may be used to represent individual data values of the X variable (the same type of data as in pie charts), however, consecutive data values of the X variable are represented by the heights of sequential vertical bars, each of a different color and pattern (rather than as pie wedges of different widths).

Multidimensional Scaling. Multidimensional scaling (MDS) can be considered to be an alternative to factor analysis (see Factor Analysis), and it is typically used as an exploratory method. In general, the goal of the analysis is to detect meaningful underlying dimensions that allow the researcher to explain observed similarities or dissimilarities (distances) between the investigated objects. In factor analysis, the similarities between objects (e.g., variables) are expressed in the correlation matrix. With MDS one may analyze not only correlation matrices but also any kind of similarity or dissimilarity matrix (including sets of measures that are not internally consistent, e.g., do not follow the rule of transitivity).

For more information, see the Multidimensional Scaling chapter.

Multilayer Perceptrons. Feedforward neural networks having linear PSP functions and (usually) non-linear activation functions.

Multimodal Distribution. A distribution that has multiple modes (thus two or more "peaks").

Multimodality of the distribution in a sample is often a strong indication that the distribution of the variable in population is not normal. Multimodality of the distribution may provide important information about the nature of the investigated variable (i.e., the measured quality). For example, if the variable represents a reported preference or attitude, then multimodality may indicate that there are several pronounced views or patterns of response in the questionnaire. Often however, the multimodality may indicate that the sample is not homogenous and the observations come in fact from two or more "overlapping" distributions. Sometimes, multimodality of the distribution may indicate problems with the measurement instrument (e.g, "gage calibration problems" in natural sciences, or "response biases" in social sciences).

See also unimodal distribution, bimodal distribution.

Multinomial Distribution. The multinomial distribution arises when a response variable is categorical in nature, i.e., consists of data describing the membership of the respective cases to a particular category. For example, if a researcher recorded the outcome for the driver in accidents as "uninjured, "injury not requiring hospitalization", "injury requiring hospitalization", or "fatality", then the distribution of the counts in these categories would be multinomial (see Agresti, 1996). The multinomial distribution is a generalization of the binomial distribution to more than two categories.

If the categories for the response variable can be ordered, then the distribution of that variable is referred to as ordinal multinomial. For example, if in a survey the responses to a question are recorded such that respondents have to choose from the pre-arranged categories "Strongly agree", "Agree", "Neither agree nor disagree", "Disagree", and "Strongly disagree", then the counts (number of respondents) that endorsed the different categories would follow an ordinal multinomial distribution (since the response categories are ordered with respect to increasing degrees of disagreement).

Specialized methods for analyzing multinomial and ordinal multinomial response variables can be found in the Generalized Linear Models chapter.

Multinomial Logit and Probit Regression. The multinomial logit and probit regression models are extensions of the standard logit and probit regression models to the case where the dependent variable has more than two categories (e.g., not just Pass - Fail, but Pass, Fail, Withdrawn), i.e., when the dependent or response variable of interest follows a multinomial distribution rather than binomial distribution. When multinomial responses contain rank-order information, they are also called ordinal multinomial responses (see ordinal multinomial distribution).

For additional details, see also the discussion of Link Functions, Probit Transformation and Regression, Logit Transformation and Regression, or the Generalized Linear Models chapter.

Multiple Axes in Graphs. An arrangement of axes (coordinate scales) in graphs, where two or more axes are placed parallel to each other, in order to either:

1. represent different units in which the variable(s) depicted in the graph can be measured (e.g., a Celsius and Fahrenheit scales of temperature), or
   
2. allow for a comparison of trends or shapes between several plots placed in one graph (e.g., one axis for each plot) which otherwise would be obscured by incompatible measurement units or ranges of values for each variable (that is an extension of the common "double-Y" type of graph).

The latter instance, which requires the appropriate plot legends to be attached to each axis, is illustrated in the graph above.

Multiple Dichotomies. One possible coding scheme that can be used when more than one response is possible from a given question is to code responses using Multiple dichotomies . For example, as part of a larger market survey, suppose you asked a sample of consumers to name their three favorite soft drinks. The specific item on the questionnaire may look like this:

Write down your three favorite soft drinks:
1:__________    2:__________    3:__________

	COKE	PEPSI	SPRITE	. . . .
case 1 case 2 case 3 . . .	1   . . .	1 1   . . .	1 . . .

In other words, one variable was created for each soft drink, then a value of 1 was entered into the respective variable whenever the respective drink was mentioned by the respective respondent. Note that each variable represents a dichotomy; that is, only "1"s and "not 1"s are allowed (we could have entered 1's and 0's, but to save typing we can also simply leave the 0's as blanks or as missing values). When tabulating these variables, we would like to compute the number and percent of respondents (and responses) for each soft drink. In a sense, we "compact" the three variables Coke, Pepsi, and Sprite into a single variable (Soft Drink) consisting of multiple dichotomies.

For more information on Multiple dichotomies, see the Multiple Response Tables section of the Basic Statistics chapter.

Multiple Histogram. Multiple histograms present frequency distributions of more than one variable in one 2D graph. Unlike the Double-Y Histograms, the frequencies for all variables are plotted against the same left-Y axis.

Also, the values of all examined variables are plotted against a single X-axis, which facilitates comparisons between analyzed variables.

Multiple R. The coefficient of multiple correlation (Multiple R) is the positive square root of R-square (the coefficient of multiple determination, see Residual Variance and R-Square). This statistic is useful in multivariate regression (i.e., multiple independent variables) when you want to describe the relationship between the variables.

Multiple Regression. The general purpose of multiple regression (the term was first used by Pearson, 1908) is to analyze the relationship between several independent or predictor variables and a dependent or criterion variable.

The computational problem that needs to be solved in multiple regression analysis is to fit a straight line (or plane in an n-dimensional space, where n is the number of independent variables) to a number of points. In the simplest case -- one dependent and one independent variable -- one can visualize this in a scatterplot (scatterplots are two-dimensional plots of the scores on a pair of variables). It is used as either a hypothesis testing or exploratory method.

For more information, see the Multiple Regression chapter.

Multiple Response Variables. Coding the responses to Multiple response variables is necessary when more than one response is possible from a given question. For example, as part of a larger market survey, suppose you asked a sample of consumers to name their three favorite soft drinks. The specific item on the questionnaire may look like this:

Write down your three favorite soft drinks:
1:__________    2:__________    3:__________

	Resp. 1	Resp. 2	Resp. 3
case 1 case 2 case 3  . . .	COKE SPRITE PERRIER  . . .	PEPSI SNAPPLE GATORADE  . . .	JOLT DR. PEPPER MOUNTAIN DEW  . . .

For more information, see the Multiple Response Tables section of the Basic Statistics chapter.

Multiple-response Tables. Multiple-response tables are Crosstabulation tables used when the categories of interest are not mutually exclusive. Such tables can accommodate Multiple response variables as well as Multiple dichotomies.

For more information, see the Multiple Response Tables section of the Basic Statistics chapter.

Multiple Stream Group Charts. Variable and attribute control charts (see also Quality Control) can be computed for multiple-stream processes (e.g., operators, machines, assembly lines); the resulting multiple stream group chart summarizes the measurements for all streams simultaneously. These charts can also be produced for short production runs, and the measurements summarized in short run group charts. In addition to the standard parameters for determining the control limits and other characteristics of the control charts, the number of consecutive points r from the same process stream (i.e., "runs" of length r) to be highlighted in the chart can be specified.

Multiplicative Season, Damped Trend. In this Time Series model, the simple exponential smoothing forecasts are "enhanced" both by a damped trend component (independently smoothed with the single parameter ; this model is an extension of Brown's one-parameter linear model, see Gardner, 1985, p. 12-13) and a multiplicative seasonal component (smoothed with parameter ). For example, suppose we wanted to forecast from month to month the number of households that purchase a particular consumer electronics device (e.g., VCR). Every year, the number of households that purchase a VCR will increase, however, this trend will be damped (i.e., the upward trend will slowly disappear) over time as the market becomes saturated. In addition, there will be a seasonal component, reflecting the seasonal changes in consumer demand for VCR's from month to month (demand will likely be smaller in the summer and greater during the December holidays). This seasonal component may be multiplicative, for example, sales during the December holidays may increase by factor of 1.4 (or 40%) over the average annual sales. To compute the smoothed values for the first season, initial values for the seasonal components are necessary. Also, to compute the smoothed value (forecast) for the first observation in the series, both estimates of S₀ and T₀ (initial trend) are necessary. These values are computed as:

T₀ = (1/)*M_k-M₁)/[(k-1)*p]

where
     is the smoothing parameter
k       is the number of complete seasonal cycles
M_k    is the mean for the last seasonal cycle
M₁    is the mean for the first seasonal cycle
p       is the length of the seasonal cycle
and S₀ = M₁-p*T₀/2

Multiplicative Season, Exponential Trend. In this Time Series model, the simple exponential smoothing forecasts are "enhanced" both by an exponential trend component (independently smoothed with parameter ) and a multiplicative seasonal component (smoothed with parameter ). For example, suppose we wanted to forecast the monthly revenue for a resort area. Every year, revenue may increase by a certain percentage or factor, resulting in an exponential trend in overall revenue. In addition, there could be an multiplicative seasonal component, that is, given the respective annual revenue, each year 20% of the revenue is produced during the month of December, that is, during Decembers the revenue grows by a particular (multiplicative) factor.

To compute the smoothed values for the first season, initial values for the seasonal components are necessary. Also, to compute the smoothed value (forecast) for the first observation in the series, both estimates of S₀ and T₀ (initial trend) are necessary. By default, these values are computed as:

T₀ = exp{[log(M₂)-log(M₁)]/p}

where
M₂    is the mean for the second seasonal cycle
M₁    is the mean for the first seasonal cycle
p       is the length of the seasonal cycle
and S₀ = exp{log(M₁)-p*log(T₀)/2}

Multiplicative Season, Linear Trend. In this Time Series model, the simple exponential smoothing forecasts are "enhanced" both by a linear trend component (independently smoothed with parameter ) and a multiplicative seasonal component (smoothed with parameter ). For example, suppose we were to predict the monthly budget for snow-removal in a community. There may be a trend component (as the community grows, there is an upward trend for the cost of snow removal from year to year). At the same time, there is obviously a seasonal component, reflecting the differential likelihood of snow during different months of the year. This seasonal component could be multiplicative, meaning that given a respective budget figure, it may increase by a factor of, for example, 1.4 during particular winter months; or it may be additive (see above), that is, a particular fixed additional amount of money is necessary during the winter months. To compute the smoothed values for the first season, initial values for the seasonal components are necessary. Also, to compute the smoothed value (forecast) for the first observation in the series, both estimates of S₀ and T₀ (initial trend) are necessary. By default, these values are computed as:

T₀ = (M_k-M₁)/((k-1)*p)

where
k       is the number of complete seasonal cycles
M_k    is the mean for the last seasonal cycle
M₁    is the mean for the first seasonal cycle
p       is the length of the seasonal cycle
and S₀ = M₁ - T₀/2

Multiplicative Season, No Trend. This Time Series model is partially equivalent to the simple exponential smoothing model; however, in addition, each forecast is "enhanced" by a multiplicative component that is smoothed independently (see The seasonal smoothing parameter ). This model would, for example, be adequate when computing forecasts for monthly expected sales for a particular toy. The level of sales may be stable from year to year, or change only slowly; at the same time, there will be seasonal changes (e.g., greater sales during the December holidays), which again may change slowly from year to year. The seasonal changes may affect the sales in a multiplicative fashion, for example, depending on the respective overall level of sales, December sales may always be greater by a factor of 1.4.

Multivariate Adaptive Regression Splines (MARSplines). Multivariate adaptive regression splines (or MARSplines for short) is a nonparametric regression procedure which makes no assumption about the underlying functional relationship between the dependent and independent variables. Instead MARSplines constructs this relation from a set of coefficients and basis functions that are entirely "driven" from the regression data. The MARSplines technique has become particularly popular in the area of data mining, because it does not assume or impose any particular type or class of relationship (e.g., linear, logistic, and so on) between the predictor variables and the dependent (outcome) variable of interest.

The general MARSplines model equation (see Hastie et al., 2001, equation 9.19) is given as:

where the summation is over the M predictors in the model. To summarize, y is predicted as a function of the predictor variables X (and their interactions); this function consists of an intercept parameter ( and the weighted (by sum of one or more basis functions . You may also think of this model as "selecting" a weighted sum of basis functions from the set of (a large number of) basis functions that span all values of each predictor (i.e., that set would consist of one basis function, and "knot" parameter t, for each distinct value for each predictor variable): The MARSplines algorithm then searches over the space of all inputs and predictor values (knot locations t) as well as interactions between variables. During this search an increasingly larger number of basis functions are added to the model (selected from the set of possible basis functions), to maximize an overall least squares goodness-of-fit criterion.

For more information about this technique, and how it compares to other methods for nonlinear regression (or regression trees), see Hastie, Tishirani, and Friedman (2001).