# 8.22 MULTIPLE LINEAR MODEL - REVISION OF MATRICES - ESTIMATION OF THE PARAMETERS OF FOX INTEGRATED MODEL (IFOX)

(7.3)

REVISION OF MATRICES

GROUP I

Consider the matrices A and B:

 A = 2 3 0 1 B = 1 1 0 3 1 1 4 1 1 3 2 5 0 4 2 2 2 1 6 0 1 0 3 3 2 2 1 0

1. Using a spreadsheet, calculate: A + B, A * B, Det(A), Det(B), A-1 e B-1

2. Show that (A.B)-1 = B-1.A-1

3. Show that (A.B)T = BT.AT

GROUP II

Let the Matrices:

 M(4,4) = (1/4) 1 1 1 1 O(4,4) = 0 0 0 0 I(4,4) = 1 0 0 0 1 1 1 1 0 0 0 0 0 1 0 0 1 1 1 1 0 0 0 0 0 0 1 0 1 1 1 1 0 0 0 0 0 0 0 1

1. Verify that the matrix null 0 is idempotent.

2. Verify that the matrix identity I is idempotent.

3. Verify that the matrix M is idempotent.

4. What are the traces of M and I?

5. Calculate the ranks, r, of M and of I.

6. What is the value of the determinant of M and I?

GROUP III

1. Verify that the product Mx, where x is the vector given by xT = (3 4 8 1), is a vector with all elements equal to the arithmetic mean, , of the 4 elements of vector x.

2. Verify that (I-M)x is the vector of the deviation.

3. Verify that the sum of the squares of xi, Σ(xi2) can be written as: xT. x

4. Verify that the sum of the squares of the deviations, , can also be written in a matricial form, as: xT (I-M)x

GROUP IV

 1. Consider the vector x = 2 + θ where θ is an unknown parameter. 3 θ 5 - θ

a) Write the derivative of the vector x
b) Calculate xT x
c) Calculate
d) Show that

 2. Consider the vector x = 2 + 4θ1 - 5θ2 where θ1 and θ2 are two unknown constants. 1+ θ1 + θ2 θ1+4θ2

a) Write the derivative matrix (take θ1 and θ2 as variables)
b) Calculate xT x
c) Transpose
d) Show that the transposed matrix

GROUP V

Consider the following system of 2 equations with 2 unknowns;

5 = 2 A + 3 B
4 = A - 2 B

1. Show that the equation system can be written in matrix form as,

Y(2,1) = X(2,2) θ(2,1)

where Y is the vector of the independent terms (5 e 4) of the system,

θ is the vector of the unknowns A and B
and X is the matrix of the coefficients of the unknowns

2. Verify that the solution of the system can be given as θ = (XTX)-1XTY

3. Show that X is a square, non singular matrix, and then that the solution of the system can be θ = X-1Y

ESTIMATION OF THE PARAMETERS OF THE YOSHIMOTO AND CLARKE MODEL (1993)

4. Estimate the parameters k, q and r, of the Fox integrated model (IFOX) and of Yoshimoto & Clarke (1993) using the following data:

 Year Y (t) CPUE (kg/day) 1983 538 235 1984 638 131 1985 431 63 1986 99 22 1987 37 8 1988 62 21 1989 437 77 1990 146 28 1991 126 26 1992 53 25 1993 91 41 1994 232 66

which represent the total annual catches (in tons) and the respective catches by fishing effort unit (kg/fishing day of the fleet PESCRUL) of the stock of Deepwater rose shrimp, Parapenaeus longirostris of the Algarve during the period 1983 to 1994 (Mattos Silva, 1995).

Comment on the obtained results comparing them with those presented in Section 8.20.