(pdf) (cdf) Matlab χ ( ) F t

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1 (, ) (univariate) (bivariate) (multi-variate) Matlab Octave Matlab Matlab/Octave

2 (pdf) (cdf) Matlab χ ( ) F t

3 --... (population) (sample) (parameter) population mean µ ( population variance)σ ˆµ (statistics) (sample mean)x (sample variance)s statistics statistics.. (discrete) (continuous) (trial) (event) (elementary event) Ω trial, event (stochastic variable) stochastic

4 -3-0~ (probability) /6.3. (pdf) (cdf) X x (probability function) i PX ( = xi ( )) = W = Wi ( ) (.) xi () P W xi,,...,6 P()=P()=...=P(6)=/6 X X x x+ x (probability density) (probability distribution function, PDF) Px ( < X x+ x) = Wx ( ) x W X x F( x) = P( X x) (.) (distribution function) (cumulative distribution function, CDF) (cumulative probabilities) PDF CDF x F( x) = W( x) dx (.3) d W( x) = F( x) (.4) dx PDF (, ) x CDF CDF α x x= W ( α) CDF PDF CDF 0 0 0

5 -4- (inverse of cumulative distribution function, ICDF) PDF CDF.4. (expected value) EX ( ) = Pxi ( ( )) xi ( ) (.5) i EX ( ) = Pxxdx ( ) (.6) Ω Ω X.5. xi ( ), yi ( ),,..., y () i = ax () i + b (.7) a b x, y ( x i) x() i x, y () i y() i y a, b (residual) ε = ( ax + b y ) i i { a x i b y i abx i ax iy i by i} { a x i b y i ax iy i} = = + + i i ) a ( x = y = 0 b ( axi xiyi) a ε = b b ε = b=0

6 -5- x () i y () i ( x() i x)( y() i y) a = = x () i ( x() i x) a (regression coefficient) x, y ( ) (.8) ye( i) y = a x() i x (.9) e (estimated value) ( / x ) (.8) (.8) r = ( xi ( ) x)( yi ( ) y) ( xi ( ) x) ( yi ( ) y) (correlation coefficient) r ρ x y S, S x y (.0) ( xi () x)( yi () y) r = = x() i y() i xy S S S S (.) ( ) x y x y S y a = r (.) S x (covariance) Cxy = ( x() i x)( y() i y) (.3) (normalize) ± (residual)

7 -6-, y () i, r y () i = y() i y () i r e (.9) Sy Sy [ yr() i ] [ ye() i y] = y() i r ( x() i x) y r ( x() i x) + y y Sx Sx Sy Sy = ( yi () y) r ( xi () x) r ( xi () x) S S = rs = x x y rsy 0 Sy = [ y() i y] = [ ye() i + yr() i y] = + = + ( ye() i y) ( yr() i ) ( ye() i y) yr() i S y ( ye() i y) = r ( x() i x) = r Sy x S r - r 0.5, 0.6, 0.7 /4, /3, / 0.7 (dominant)

8 (Monte-Carlo method) (Monte-Carlo simulation) Matlab randn rand (0,). (-, ) n m (PDF) n=00 m=00, 000, 0000 hist v x=-0.3:0.0:0.3;(x=-0.3, -0.8, -0.6 ) bin vh vh=hist(v,x)

9 Matlab ( ) t F t F t F Matlab Statistical Toolbox Octave statistics/distribution. Matlab statistics Toolbox Octave statistics/distribution PDF CDF ICDF x CDF normpdf [,mu,sigma] 0 (PDF) (CDF) (ICDF) normpdf(x[,mu,sigma])) normal_pdf chipdf chi_pdf F fpdf(x,,) f_pdf normcdf normal_cdf chicdf chi_cdf fcdf f_cdf norminv normal_inv chiinv chi_inv finv f_inv

10 -9- t tpdf(x,) tcdf tinv t_pdf t_cdf t_inv.7.. (normal distribution) (Gaussian distribution) (pdf) ( x µ ) W( x) = exp πσ σ (.4) µ σ σ µ (, ) (µ, σ ) 0 (0, ) (standard normal distribution) x(), x(),, x() 3 bi-modal x (), x (), x () (µ, σ ) x = x()/ i n (µ, σ /) n Q. n A. (µ, σ /n) σ / n 3 996

11 -0- (log normal distribution) (x) (log(x)).7.. χ ( ) (0, ) x (), x (), x () Y = x() i i = χ 4 (PDF) ( /) y/ W ( y, ) = y e (.5) χ / Γ( /) (Gamma function) λ x Γ ( λ) = x e dx (.6) 0 Γ ( λ+ ) = λγ ( λ) (.7) Γ () = (.8) λ Γ ( n+ ) = n! / Γ ( /) = π (.9) (µ, σ ) x Y = ( x ( i ) µ ) / σ Y χ Y = ( x ( i ) x ) / σ - ( 996, p. ) , p , 973 p , p

12 --. χ = = 0.5, 3 = (ICDF) Matlab chiinv chiinv chiinv ICDF ( ) x a b wγ ( x a, b) = x e a b Γ( a) (.5) PDF a = /, b= ICDF Matlab gaminv(p, a, b) a,b chiinv(p,v) chiinv_nint(p,v) chiinv chiinv_nint function icdf = chiinv_nint(p,v); % v chi % % p: (0<=p<=) % v: (0<v) % % icdf: inverse of cumulative distribution function

13 -- % statistical toolbox chiinv v % a % 00//0 if isempty(p); error('*chiinv_nint* p is empty'); end if isempty(v); error('*chiinv_nint* v is empty'); end if sum(p<0); error('*chiinv_nint* p '); end if sum(p>); error('*chiinv_nint* p '); end if sum(v<0); error('*chiinv_nint* v '); end if sum(v==0); error('*chiinv_nint* v '); end if (ndims(p)~=ndims(v)); error('p v '); end if (sum(size(p)~=size(v)); error('p v '); end % icdf = gaminv(p,v/,);.7.4. F Y, Y, Y = ( Y/ )/( Y / ) (, )F (F-distribution) F, F (Snedecor's F-distribution), (Fisher distribution) (, )F PDF 6 ( /) ( /) + Γ ( /) y W( y,, ) = ( + )/ Γ( / ) Γ ( / ) ( y+ ) (.0) ( /) ( /) ( /) y = ( + )/ ( y + ) B, y>0 y 0 W=0 B a 0 b B( a, b) = B( b, a) = t ( t) dt Γ( a) Γ( b) Bab (, ) = (.) Γ ( a + b) 6 996, p , p.76-79

14 =5 F PDF.7.5. t- (0,) Z Y / T = Z Y / t (t-distribution) t t (Student s t-distribution) 7 8 t PDF ( 996, p. 89; 973, p. 8) t = + B, + + Γ (( + )/) t WT (, t ) = + π Γ( /) Γ( α) Γ( β) B ( α, β) = Γ ( α + β) Γ(/ ) Γ( / ) Γ( / ) B, = = π Γ (( + )/) Γ (( + )/) t- F t t- t =F F (,)F 9 t- F- (.) 7 Student W. L. Gossett (Thiebaux, p. 55) 8 t- 9 (973), p. 83

15 -4-.4.,,0,00 t PDF.8. (estimation) (estimator) (point estimation) 95% (interval estimation) 95% ( θl, θu) l,u lower, upper 95%0.95 (confidence level) (confidence coefficient) 0 (confidence interval), θl θ u (confidence limits).8.. (consistency) 00 0 Emery and Thomson 997 p. 6 confidence level significance level 95%=-

16 (unbiased estimator) biased.8.. s = ( xi µ ) i xi i = x = ( ) (.3) V = xi x i = ( ) V = ( xi ( ) x) = [ ( x µ ) ( x µ )] = ( xi ( ) µ ) ( xi ( ) µ )( x µ ) + ( x µ ) = σ ( x µ ) σ / V = σ

17 -6- V /(-) S = ( x( i) x) (.4) (.3) (biased variance)(.4) (unbiased variance).8.3. σ (30 ) µ - σ σ x z( α /) < µ < x + z( α/) (.5) P σ σ P x z( α / ) < µ < x + z( α/ ) = α (.6) z( α /) PDF W zα / z α / zα / α + zα / W ( x) dx= W ( x) dx= /, W ( z) dz = α (.7) CDF F F ( z( α/)) = α/, F ( z( α/)) = α/ (.8) (.7)(.8) PDFCDF.5 PDF PDF z=0 z z -z z PDF (, z ) ( z, ) / - z z α / CDF F ( ( /)) z α PDF α / F ( ( /)) z α PDF α /

18 PDF CDF α= 0.05 (.7)(.8) α ICDF F z α / z( α /) F ( α /) = F ( α /) (.9) Matlab norminv Octave normal_inv α α = 0.05 α α = 0.0 z α / (.6) x = x()/ i ( µσ, / ) x µ µ x z = (0,) P z( α/ ) < < z( α/ ) = α σ / σ / (.6)

19 t- s s P x t( α /, ) < µ < x + t( α /, ) = α (.30) S t( α /, ν) t- ICDF F t t( α /, ν) = F t ( α /, ν).8.5. σ S ( ) S ( ) P σ F α α χ, F χ, < < = (.3) χ = S ( )/ σ = ( xi ( ) x ) / σ α (.3) - χ α α χ P n < χ < χ n = α χ = S ( )/ σ.8.6. r ln + r z = r z (Fisher's z-transform) z ln + r, r n 3 z Z Z zˆ z zˆ, σ < < + = α/ α/ σ σ n 3 (.3) tanh( r = z) tanh( zˆ Z / σ ) < r < tanh( zˆ+ Z / σ ) (.33) α/ α/ 996 p p.77

20 -9- Emery and Thomson (997)p. 53 Storch and Zwiers (999)p (statistical test) H 0 (reject) (null hypothesis) null-hypothesis test (significance level, level of significance)α( -α) 5% 95% H 0 t-, F- P (P-value) P 5% 0% 5% 0% P.9.. (error of the first kind, type I error) H 0 (error of the second kind, type II error) H 0 H 0 - (power of the statistical test) (Emery and Thomson, 998, p. 49) II I

21 x(), x(),, x( x )y(), y(),, y( y ) S, S x y S / S x -, y - F x y S / S < F ( α/,, ) F ( α/,, ) < S / S (.34) x y F x y F x y x y σ = σ x y.9.3. x(), x(),, x( x )y(), y(),, y( y ) µ = µ x y T = x y ( nx ) Sx + ( ny ) Sy + n + n n n x y x y (.35) x+y+ t- T T = F ( α /, + ) t x y Welch x y T = x y S n x x S + n y y (.36) k t- k

22 -- Sx c ( c ) n, x = + c = k nx n S S x + n n (.36) t- (996) x y y.9.4. r (Degree of Freedom) t r ( ) = T r r ( ) t- F- F = (,-)F 3 r t- - -α/ ( T = F α /, ) T r = + T α T = F ( α /, ) T(r) F- F F = F F ( α,, ) t- -α/ F- -α F α F r = + F t t.. t- F- 40 5% t- F- 3 (958), 958: ( ) / [ ],, p. 6.

23 icholls (00) 0 5% , , % (996) icholls (00) 0.6 (996) , 0.6, 0.7 /4, /3, / icholls (00) Gardner and Altman (989) icholls (00) icholls (00) icholls (00)

24 -3- Monte-Carlo.0. Emery, W. J., and Thomson, R. E., 997: Data analysis methods in physical oceanography. Pergamon press, pp Gardner,M. J., and D. G. Altman, 989: Statistics with confidence intervals and statistical guideline. British Medical Journal, pp :,, pp # (973) , 996: 30, p. 38. ISB icholls,. 00: The insignificance of significance testing. Bull, Am. Met. Soc., 8, , 996:,, pp.7. Trenberth, K., E., 984: Some effects of finite sample size and persistence on meteorological statistics. Mon. Wea. Rev., (Dec), von Stroch, H. and F. W. Zwiers, 999: Statistical analysis in climate research. Cambridge University Press, pp. 483 ISB:

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untitled 2 : n =1, 2,, 10000 0.5125 0.51 0.5075 0.505 0.5025 0.5 0.4975 0.495 0 2000 4000 6000 8000 10000 2 weak law of large numbers 1. X 1,X 2,,X n 2. µ = E(X i ),i=1, 2,,n 3. σi 2 = V (X i ) σ 2,i=1, 2,,n ɛ>0