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In general, y is an (n 1) vector of the observations, X is an (n p) matrix of the levels of the independent variables, is a ( p 1) vector of the regression coefficients, and is a (n 1) vector of random errors. We wish to find the vector of least squares estimators, , that minimizes

I _____ C _____ LNS PC _____ Cd _____ BD _____ BD _____ MR _____ MR _____ PA _____ OA _____ OA _____ SS _____ OA _____

We will not give the details of taking the derivatives above; however, the resulting equations that must be solved are

PA _____

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(12-12)

_____ (1.2) _____ +40 _____ _____

Equations 12-12 are the least squares normal equations in matrix form. They are identical to the scalar form of the normal equations given earlier in Equations 12-10. To solve the normal equations, multiply both sides of Equations 12-12 by the inverse of X X. Therefore, the least squares estimate of is

(X X)

[Mic00] [Mof88] [MSDN] [NG98] [NIST800-12]

_____ (1.1) _____ +34 _____ _____

(12-13)

_____ (1.5) _____ +40 _____ _____

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Note that there are p k 1 normal equations in p k 1 unknowns (the values of , , p , 2. Furthermore, the matrix X X is always nonsingular, as was assumed above, 0 1 k so the methods described in textbooks on determinants and matrices for inverting these matrices can be used to find 1X X2 1. In practice, multiple regression calculations are almost always performed using a computer. It is easy to see that the matrix form of the normal equations is identical to the scalar form. Writing out Equation 12-12 in detail, we obtain

Microsoft Corporation: Windows 2000 Security Technical Reference, Microsoft Press, 2000 J. D. Moffett, M. S. Sloman: The Source of Authority for Commercial Access Control, Computer, Vol. 21, No. 2, IEEE, pp. 59 69, February 1988 Microsoft Developer Network: http://msdn.microsoft.com/ F. Das Neves, A. Garrido: Bodyguard, published in [PLoPD3], 1997 National Institute of Standards and Technology (NIST): An Introduction to Computer Security: The NIST Handbook, NIST Special Publication 800-12, Information Technology Laboratory, 25 August 2003

If the indicated matrix multiplication is performed, the scalar form of the normal equations (that is, Equation 12-10) will result. In this form it is easy to see that X X is a ( p p) symmetric matrix and X y is a ( p 1) column vector. Note the special structure of the X X matrix. The diagonal elements of X X are the sums of squares of the elements in the columns of X, and the off-diagonal elements are the sums of cross-products of the elements in the columns of X. Furthermore, note that the elements of X y are the sums of cross-products of the columns of X and the observations 5 yi 6. The fitted regression model is

_____ (1.9) _____ +43 _____ _____ _____ (Continues)

(12-14)

(Continued)

[NIST800-30]

The difference between the observation yi and the fitted value yi is a residual, say, i. The (n 1) vector of residuals is denoted by ei yi y

(12-15)

Instructions for Calculating Strengths and Weaknesses of Horn Standard Scores Step 1. Enter the person s standard score for each of the four Horn categories in the appropriate column and transfer the Working Memory Index (WMI) to the Horn Short-Term Memory standard score. Step 2. Compute the mean of the five standard scores, rounding the mean to the nearest whole number. Step 3. Subtract the person s mean score from each of the four standard scores. Enter the difference scores in the column headed Difference. Use negative signs to denote scores below the person s mean. Step 4. Determine whether each difference is significant at the .05 level using the values provided. Differences that equal or exceed these values are significant strengths or weaknesses for the person. Step 5. Enter an S or W in the column provided to denote significance. Examinee s Name _________________________________ Date: ________________________ Size of Difference Needed for Significance Difference ______ ______ ______ ______ ______ ( p < .05) 7 8 9 11 8 (p < .01) 9 9 10 13 10 S or W ______ ______ ______ ______ ______

EXAMPLE 12-2

G. Stoneburner, A. Goguen, A. Feringa: Risk Management Guide for Information Technology Systems, NIST Special Publication SP800-30, National Institute of Standards and Technology (NIST), 2001 G. Stoneburner: Underlying Technical Models for Information Technology Security: Recommendations of the National Institute of Standards and Technology {USA), NIST Special Publication SP800-33, National Institute of Standards and Technology (NIST), December 2001 T. Grance et al: Guide to Information Technology Security Services, NIST Special Publication NIST SP800-35, National Institute of Standards and Technology (NIST), 2003 W. E. Burr, D. F. Dodson, W. T. Polk: Electronic Authentication Guideline, NIST Special Publication NIST SP800-63, National Institute of Standards and Technology (NIST), 2004

Horn Category Crystallized Fluid Broad Visualization Broad Speediness Short-Term Memory (= WMI)

0 1 x1

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