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V x, y,..., z, w, y x y = f(x) +, (1) f(x) y f(x) f(x) f(x) D() = D(y - f(x)) D(y - f(x)) min.

f(x) = M(y/x), (2) M(y/x) y x f(x) y x f(x) (x, y) f(x) y = a0 + a1x + a2x2 + a0, a1, a2 y x y u = (x1, x2,..., xk) y = f(b, u) +, (3) b = (b1, b2,..., b ) f(b, u) y u y x1, x2,..., xk f(b, u) N y x1 x2 xk 1 y1 x11 x21 xk1 2 y2 x12 x22 xk2 i yi x1i x2i xki n yn x1n x2n xkn (yi, x1i, x2i,..., xki) 1 i n n b L(b) y u n 2 L(b) = yi - f(b, u[i]), (4) i=1 u[i] = (x1i, x2i,..., xki) 1 i n L(b) b L(b) min. (5) f(x) f(b, u) y x1 (x1i, yi) 1 i n x1 y x1 y x1 x1 x1 y y = b0 + b u + = b0 + b1 x1 +... + bi xi +... + bk xk +. (6) b0, b1,..., bi,..., bk 1 i k bi = 0 xi y bi = 0 y xi 1 i k bi = 0 y y xi M() = 0, D() = 2 = const > 0. (7) i j i = j M(i j) = 0 i = yi - b0 - b ui j = yj - b0 - b uj yi, yj, ui, uj y x1, x2,..., xk y = b0 + b1 x1 +. (8) y x1 n n 1 1 x1 = x1i, y = yi, (9 a) n n i=1 i=1 n n Qx x1 = x2 - n (x1)2, Qx y = (x1i yi) - n x1 y, (9 b) 1 1i 1 i=1 i=1 b0 b1 Qx y 1 b1 =, b0 = y - b1 x1. (10) Qx x1 1 e1, e2,..., ei,..., en, (11) yi ei = yi - b0 - b1 x1i, 1 i n. (12) n Qee = e2, (13) i i=1 b0 b1 b0 b0 b1 b1 b1 b1 = 0 y x1 b1 n b1 1 b1 Qee 1 =. (14) (n - 2) Qx x1 1 x1 = x 1 = b0 + b1 x (15) 1 y y y y y Qee 1 (x - x1)2 1 y = 1 + +. (16) n - 2 n Qx x1 1 y x1 H0 b1 = 0 100% H0 H0 b2 Qx x1 (n - 2) 1 1 F =. (17) Qee F f1 = 1 f2 = n - 2 F F H0 F > F H0 x1 y b1 b1 I1 = ( - t 1, b1 + t 1), (18) b1 t n - 2 I1 b1 p = 1 - I1 b1 = 0 H I b1 = 0 H x1 y x1 = x y y Iy = ( - t y, + t y), (19) y p = 1 - t, y c1 = - t y c2 = + t y x y x x y c cy x1 x y y x1 xy = b0 + b1 x1 + b2 x2 +. (20) x1 x2 x2 = a1 x1+a2 a1 = y x1, xn n n 1 1 x1 = x1i, x2 = x2i, y = yi, (21 a) n n n i=1 i=1 i=n = Qx x1 Qx x2 - Q2, Qx x1 = x2 - n (x1)2, (21 b) 1 2 x1x2 1 1i i=n n Qx x2 = x2 - n (x2)2, Qyy = yi - n (y)2, (21 c) 2 2i i=1 i=n n Qx y = (x1i yi) - n x1 y, Qx y = (x2i yi) - n x2 y, (21 d) 1 i=1 i=n Qx x2 = (x1i x2i) - n x1 x2. (21 e) i=Qx x2 Qx y - Qx x2 Qx y 2 1 1 b1 =, (22 a) Qx x1 Qx y - Qx x2 Qx y 1 2 1 b2 =, (22 b) b0 = y - b1 x1 - b2 x2. (22 c) e1, e2,..., ei,..., en, (23) yi ei = yi - b0 - b1 x1i - b2 x2i, 1 i n. (24) n Qee = e2, (25) i i= Qee Qee = Qyy - ( Qx y + b2 Qx y). (26) b1 b0 b1 b b0 b0, b1 b1, b2 b2.

b1 b2 b1 = b2 = y x1 x2 bb2 n b1 b2 1 2 b1 b Qee Qx x2 Qee Qx x2 1 = , 2 = . (27) n - 3 n - x1 = x x2 = x 1 = b0 + b1 x + b2 x (28) 1 y y y y y Qee 1 y = 1 + + y, (29) n - 3 n y y = Qx x2(x - x1)2 - 2 Qx x2(x - x1)(x - x2) + Qx x1(x - x2)2. (30) 2 1 1 1 2 1 y x1 x2 H b1 = 0 b2 = 100% H0 H ( Qx y + b2 Qx y) (n - 3) b1 F = (31) 2 Qee F f1 = f2 = n - F F H F > F H y x1, x x1 x2 y 100% b1 b b1 I1 = ( - t 1, b1 + t 1), b2 I2 = ( - t 2, b2 + t 2), (32) b1 b t n - 1 I1 I2 b1 b2 p = 1 - I b1 = 0 y xI2 b2 = y x2 I1 I y b1 > 0 x1 y b1 < y x2 b y x1 x x1 = x x2 = x y 1 y Iy = ( - t y, + t y), (33) y p = 1 - t, y c1 = - t y c2 = + t y x x 1 y (x, x) (x1, x2) 1 H b0 b1 b2 b0 b1 b y H 100% H dL, dU n k k = 1 k = n (ei - ei-1)i=d =, (34) Qee [0, 4] Qee d < dL H d > 4-dL H dU d 4 - dU H0 dL d dU 4 - dU d 4 - dL y H 100% H H y n n n n 1 = ei, s2 = (ei - )2, (35) e n n - i=1 i= se Qee - n ()s2 =, (36) e n - Qee se se = se (-, - se), [ - se, - 0.5 se), [ - 0.5 se, ), [, + 0.5 se), [ + 0.5 se, + se), [ + se, +).

n1 n2 n3 n4 n5 n ni = n. (37) i= n > n2 + n2 n2 + n2 n2 + n1 6 2 5 3 N2 = + + - n. (38) 0.1587 n 0.1498 n 0.1915 n N2 7. N2 > 11. H 25 n (n1 + n2)2 + (n5 + n6)2 n2 + n3 M2 = + - n. (39) 0.3085 n 0.1915 n M2 3. M2 > 6. H 7.81 < N2 11.3 3.84 < M2 6. H0 = 0. se Ae Ee n n 1 1 1 Ae = (ei - )3, Ee = (ei - )4 - 3. (40) s3 n s4 n e e i=1 i=6 (n - 1) 24 n (n - 2) (n - 3) D(Ae) =, D(Ee) =. (41) (n + 1) (n + 3) (n + 1)2 (n + 3) (n + 5) |Ae| 3 D(Ae), |Ee| 5 D(Ee), (42) n 1 2,... n y1 y2,... yn D(i) = 2 = const > 0 1 i n D(i) i i yi, x1i, x2i b0 b1 b H 100% H H e x1 x2 y (e, x1) (e, x2) (e, x1) xx11, x12,..., x1i,..., x1n, (43) e1, e2,..., ei,..., en, (44) x1i 1, 2,..., j,..., n, (45) j ei m n/m n G1 = (j)2, G2 = (j)2, (46) j=1 j=n-m+ Gmax Gmin Gmax G = (47) Gmin F f1 = f2 = m-k k k = 1 k = G F H G > F H n x|e1|, |e2|,..., |ei|,..., |en|. (48) x1i x1i x1i 1i |ei| 1i wi = 1i - 1i 1 i n n Rs = 1 - wi. (49) n3 - n i= Rs n - Ts = |Rs| , (50) 1 - Rs t n - Ts t H Ts > t H H = 0.05, 0.01, 0.001.

H0 = 0.05 = 0. = 0. = 0. H0 = 0.05 = 0. = 0.01 = 0. Hy = f(b, u) + = f(b, x1, x2,..., xk) +, (51) f y = b0 + b1 x + b2 x2 +... + bk xk + x1 = x, x2 = x2,..., xk = xk y = b0 + b1 ln x + x1 = ln x y = b0 + b1 exp(x) + x1 = exp(x) y = b0 + b1/x + x1 = 1/x y = x x+ + > 1 ln y = ln + ln x1 + ln x2 + ln + f y x1, x2,..., xk Qyy = Qrr + Qee, (52) n Qyy = (yi - y) i= n Qrr = (yi - f( u[i]))b, i= Qee Qrr Qee R2 = = 1 -. (53) Qyy Qyy 0 R2 1 R Qyy y R RF R2 n - k - F = , (54) 1 - R2 k k k = 1 k = F Fp R y x1 (x1, y) xx1 28 29 32 35 40 44 45 51 y 5.3 9.2 15.2 20.7 21.7 36.5 39.3 52.7 55.x1 58 64 65 73 75 80 83 y 64.3 76 79.1 94.8 101 89.5 114.8 137. x y y x 5% n = x1 = 55.76, y = 59.55, Qx x1 = 6557.06, Qx y = 12714.79, 1 Qee = 27.09, b0 = -48.4705, b1 = 1.9377.

b0 b1 e1, e2,..., e ee2 = y2 - ( + b1 x12) = 9.2 - (-48.4705 + 1.9377 29) = 1.4772 1.48.

b H = 5% k = 1 n = dL = 1.13, dU = 1.38 d d = 1.67 dU d 4 - dU H H Ae = -2.65 Ee = 8. |Ae| = 2.65 > D(Ae) = 3 0.266 = 1.55, |Ee| = 8.05 > 5 D(Ee) = 5 0.601 = 3.88.

b y H0 = 5% x1i x1i 1i = i 1 i 11 = 4, 12 = 8, 13 = 10, 14 = 7, 15 = 16, 16 = 1, 17 = 5, 18 = 12, 19 = 6, 110 = 2, 111 = 3, 112 = 9, 113 = 11, 114 = 14, 115 = 17, 116 = 13, 117 = 15.

Rs = 0.45 Ts = 1.952 Ts t n - 2 = Ts = 1.952 < t = 2.H y xy -48.47+1.94 x y xx1 3.5 5.0 6.5 10.5 13.y 16.4 15.2 14.6 20.8 26.x1 4.0 7.5 8.5 6.0 12.y 12.7 15.5 17.0 14.2 25. y x 5% n = b0 = 7.72, b1 = 1.32 y x y 7.72 + 1.32 x b0 b1 e1, e2,..., e4.06, 0.88, -1.71, -0.79, 1.71, -0.30, -2.13, -1.95, -1.45, 1.67.

Ae = 0.86 Ee = 0. |Ae| = 0.86 < D(Ae) = 3 0.377 = 1.84, |Ee| = 0.003 < 5 D(Ee) = 5 0.569 = 3.77.

H = 5% k = 1 n = dL = 0.88, dU = 1.32 d d = 1.12 dL d dU H Hx11 = 3.5, x12 = 5.0, x13 = 6.5, x14 = 10.5, x15 = 13.0, x16 = 4.0, x17 = 7.5, x18 = 8.5, x19 = 6.0, x110 = 12.5.

xx11, x16, x12, x19, x13, x17, x18, x14, x110, x15.

e1, e6, e2, e9, e3, e7, e8, e4, e10, e5, 4.06, -0.30, 0.88, -1.45, -1.71, -2.13, -1.95, -0.79, 1.67, 1.71.

m n/3 = 10/ m = 3 n - m + 1 = 8 G G GG1 = (4.06)2 + (-0.30)2 + (0.88)2 = 17.348, G2 = (-0.79)2 + (1.67)2 + (1.71)2 = 6.3371, 17.Gmax = 17.348, Gmin = 6.3371, G = = 2.74.

6. F = 5% f1 = f2 = m - k = 3 - 1 = 2 F = 19. G F H 5% F F = 38. F f1 = f2 = 8 = 0. F0.05 = 5.32 F > F0. x1 y b1 5% t = t0.05 = 2.306 b1 I1 = (0.84, 1.8) I x1 y y 7.72 + 1.32 x y x1 = x d b0 = 7. b1 = 1. y x1 x n = x1 31 34 35 41 38 32 29 x2 29.5 14.2 18.0 21.3 47.5 10.0 21.0 36.y 22.0 14.0 23.0 43.0 66.0 7.6 12.0 36. y x1 x 5% i = yi - b0 - b1 x1i - b2 x2i, 1 i 8 8 x1i = 274, x2i = 198, yi = 223.6, i=1 i=1 i=8 8 x2 = 9488, x2 = 5979.08, yi = 8911.76, 1i 2i i=1 i=1 i=8 8 (x1i x2i) = 6875.6, (x1i yi) = 8049.2, (x2i yi) = 6954.7.

i=1 i=1 i= b0 = -94.55, b1 = 2.80, b2 = 1.07.

H F = 151.7 f1 = f2 = 5 F0.05 = 5.79 F > F0.05 H0 y x1 x b1, b x1 x2 y I1 = (2.03, 3.57) I2 = (0.83, 1.31) y y -94.55 + 2.80 x1 + 1.07 xy(t) = f(t, x(t)) + (t), t = 0, 1, 2,..., T. (55) t y(t) x(t) (t) f(t, x(t)) M((t)) = y(t) y(t1), y(t2), y(t3),..., y(tn), (56) V t = t1 < t2 < t3 <... < tn. (57) y(t) y(t) t > T = tn y(t) t > T y(t) y(t) y(t) t = t y(t) F(x) = P {y(t) x}. (58) y(t) y(t) tt2,... tk F (x1, x2,..., xk) = P {y(t1) x1, y(t2) x2,..., y(tk) xk}. (59) y(tk) y(t1), y(t2),..., y(tk-1), t1 < t2 < < tk-1 < tk. (60) {y(t1), y(t2),..., y(tk)} {y(t1), y(t2),..., y(tk)} y(t) V {y(t1), y(t2),..., y(tk)} y(t) T < t T y(t) y(t) = g(t) + (t), (61) g(t) t t1 < t2 <... < tn g(t) a + b t, a + b t + c t2, ea+bt, a tb, a + b ln t. (62) a, b, c g(t) = a + b t y(t) y(t) - g(t) (t) M((t)) = 0, D((t)) = 2 = const > 0, (63) (t1), (t2),..., (tn) (64) y(t) t y(t) y(t) b) a) 0 T t y(t) a) b) y(t) = a + b t + c t2 + (t), (65) b0 = a b1 = b b2 = c y = y(t) x1 = t x2 = t2 = (t) y = b0 + b1 x1 + b2 x2 +. (66) y(1) = 1.5, y(2) = 2.4, y(3) = 0.8, y(4) = 2.7, y(5) = 3.0 (67) y(t) t = 1, 2, 3, 4, x11 = 1, x12 = 2, x13 = 3, x14 = 4, x15 = 5, (68 a) x21 = 1, x22 = 4, x23 = 9, x24 = 16, x25 = 25, (68 b) y1 = 1.5, y2 = 2.4, y3 = 0.8, y4 = 2.7, y5 = 3.0. (68 c) x1 x2 y b0, b1, b y(t) (t = 1, 2,..., 48) 1984 1985 1986 5.50 5.73 5.82 5. 5.46 5.70 5.79 5. 5.48 5.73 5.80 5. 5.49 5.74 5.81 5. 5.48 5.69 5.78 5. 5.50 5.70 5.79 5. 5.53 5.70 5.79 5. 5.55 5.69 5.83 5. 5.63 5.75 5.91 5. 5.61 5.74 5.87 5. 5.61 5.75 5.87 5. 5.68 5.80 5.90 5. y(t) n = g(t) = a + b t + c t = 5.434, b = 0.019, c = -0.0002 d d = 0.922 = 5% k = dL = 1.35, dU = 1.59 d < dL y(t) x(t) t t y(t) = b0 + b1 x(t) + (t) (69) y(t) x(t) t y(t) x(t) y(t) = g1(t) + 1(t), (70) x(t) = g2(t) + 2(t). (71) g1(t) g2(t) x1 = x(t) - g2(t), y = y(t) - g1(t). (72) t y = b0 + b1 x1 +. (73) y(t) x(t) y(t) = b0 + b1 x(t) + b2 t + (t). (74) t y = y(t) x = x(t) t x(t) y(t) t x(t) y(t) t x(t) y(t) t x(t) y(t) 1 127.2 15.12 7 143.2 17.66 13 158.2 20.2 129.0 16.62 8 144.3 18.40 14 159.3 23.3 131.2 17.16 9 145.6 19.97 15 159.6 21.4 134.8 17.63 10 148.7 21.35 16 163.6 23.5 136.6 18.14 11 150.4 20.16 17 164.9 23.6 138.6 16.92 12 154.7 21.04 18 169.4 23. 1(t) 2(t) 5% g1(t) = a0 + a1 t, g2(t) = c0 + c1 t.

0 = 15.16 1 = 0.c0 = 124.4 c1 = 2.46 x1 y x15 yy5 = y(5) - 1(5) = 18.14 - (15.16 + 0.49 5) = 0.53, x15 = x(5) - 2(5) = 136.6 - (124.4 + 2.46 5) = -0.1.

x11, x12,..., x1 18; y1, y2,..., y18.

x1 y b0 = -0.032 b1 = -0. F F = 0.781 F f1 = 1 f2 = 16 = 0. F0.05 = 4.F < F0. x1 y y(t) x(t) y(t) y(t) x(t) y(t) x(t) y(t) = + 0 x(t) + 1 x(t - 1) +... + k x(t - k) + (t). (75) y(t) x(t - 1), x(t - 2),..., x(t - k) (t), 0, 1,..., k y(t) y(t) +2 x(t-2)+3 x(t-3) x(t) = x2, x(t - 1) = xy(t + 2) + 2x2 + 3x k i x(t) y(t) i y(t) x(t) y(t) x(t) t x(t) y(t) t x(t) y(t) t x(t) y(t) 1 1.52 17.955 7 1.34 15.029 13 1.24 16.2 1.23 16.842 8 1.38 15.626 14 1.35 16.3 1.78 18.448 9 1.45 16.241 15 1.68 17.4 1.45 18.286 10 1.14 16.100 16 1.65 18.5 1.48 17.388 11 1.58 17.446 17 1.56 18.6 1.12 16.345 12 1.45 17.351 18 1.50 19.y(t) x(t) y(t) x(t) k = y(t) = + 0 x(t) + 1 x(t - 1) + 2 x(t - 2) + 3 x(t - 3) + (t). (76) y = y(t), x1 = x(t), x2 = x(t - 1), x3 = x(t - 2), x4 = x(t - 3), = (t). (77) b0 =, b1 = 0, b2 = 1, b3 = 2, b4 = 3. (78) y = b0 + b1 x1 + b2 x2 + b3 x3 + b4 x4 +. (79) t = y1 = y(4) = 18.286, x11 = x(4) = 1.45, x21 = x(3) = 1.78, x31 = x(2) = 1.23, x41 = x(1) = 1.52.

0 = 4.68, 1 = 3.72, 2 = 2.52, 3 = 1.08, = -0. y(t) t - 1 t - t - 3 t F f1 f = 0.f2 f1 = 1 f1 = 2 f1 = 3 f1 = 4 f1 = 6.61 5.79 5.41 5.19 5. 5.99 5.14 4.76 4.53 4. 5.59 4.74 4.35 4.12 3. 5.32 4.46 4.07 3.84 3. 5.12 4.26 3.86 3.63 3. 4.96 4.10 3.71 3.43 3. 4.84 3.98 3.59 3.36 3. 4.75 3.89 3.49 3.26 3. 4.67 3.81 3.41 3.18 3. 4.60 3.74 3.34 3.11 2. 4.54 3.68 3.29 3.06 2. 4.49 3.63 3.24 3.01 2. 4.45 3.59 3.20 2.96 2. 4.41 3.55 3.16 2.93 2. 4.38 3.52 3.13 2.90 2. 4.35 3.49 3.10 2.87 2. = 0.f2 f1 = 1 f1 = 2 f1 = 3 f1 = 4 f1 = 16.26 13.27 12.06 11.39 10. 13.75 10.92 9.78 9.15 8. 12.25 9.55 8.45 7.85 7. 11.26 8.65 7.59 7.01 6. 10.56 8.02 6.99 6.42 6. 10.04 7.56 6.55 5.99 5. 9.65 7.21 6.22 5.67 5. 9.33 6.93 5.95 5.41 5. 9.07 6.70 5.74 5.21 4. 8.86 6.51 5.56 5.04 4. 8.68 6.36 5.42 4.89 4. 8.53 6.23 5.29 4.77 4. 8.40 6.11 5.18 4.67 4. 8.29 6.01 5.09 4.58 4. 8.18 5.93 5.01 4.50 4. 8.10 5.85 4.94 4.43 4. t f f = 0.1 = 0.05 = 0. 6.314 12.706 63. 2.920 4.303 9. 2.353 3.182 5. 2.132 2.776 4. 2.015 2.571 4. 1.943 2.447 3. 1.895 2.365 3. 1.860 2.306 3. 1.833 2.262 3. 1.812 2.228 3. 1.796 2.201 3. 1.782 2.179 3. 1.771 2.160 3. 1.761 2.145 2. 1.753 2.131 2. 1.746 2.120 2. dL, dU = 0. k = n dL dU n dL dU n dL dU 8 0.76 1.33 18 1.16 1.39 28 1.33 1.9 0.82 1.32 19 1.18 1.40 29 1.34 1.10 0.82 1.32 20 1.20 1.41 30 1.35 1.11 0.93 1.32 21 1.22 1.42 31 1.36 1.12 0.97 1.33 22 1.24 1.43 32 1.37 1.13 1.01 1.34 23 1.26 1.44 33 1.38 1.14 1.05 1.35 24 1.27 1.45 34 1.39 1.15 1.08 1.36 25 1.29 1.45 35 1.40 1.16 1.10 1.37 26 1.30 1.46 36 1.41 1.17 1.13 1.38 27 1.32 1.47 37 1.42 1.




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