Métodos Probabilísticos HIDROLOGIA GENERAL
Autor: Valleumbroso Villa Freddy 2017
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- Métodos Probabilísticos
Metodos Probabilisticos PROFESOR: Ing. Dante Salazar Sánchez CURSO: Hidrología General UNIVERSIDAD SAN PEDRO Métodos Probabilísticos
Contenido Distribución de Probabilidades en Hidrología ……………………………..……………… 3 Parámetros Estadísticos……………………………… Estadísticos……………………………………………………… ………………………………... ………... 4 Distribución de Probabilidad para Variables Continuas……………….. 6 Ajuste de Distribuciones ………………………………………..…… ………………………………………..…… 11
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- Métodos Probabilísticos
DISTRIBUCIONES DE PROBABILIDAD EN HIDROLOGÍA El comportamiento de las variables aleatorias discretas o continuas se describe con la ayuda de Distribuciones de Probabilidad. La variable se designa designa por mayúscula y un valor valor específico de ella por minúscula. Por P(x = a) se denota la probabilidad de que un evento asuma el valor a; similarmente P(a ≤ x ≤ b) denota la probabilidad de que un un evento se encuentre encuentre en el intervalo (a, b). Si conocemos conocemos la probabilidad P(a ≤ x ≤ b) para todos los valores de a y b, se dice que conocemos la Distribución de Probabilidades de la variable x. Si x es un número dado y consideramos la probabilidad probabilidad P(X ≤ x):
F(x)= P(X x): y llamamos F(x) la función de distribución acumulada.
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- Métodos Probabilísticos PARAMETROS ESTADISTICOS Los estadísticos extraen información de una muestra, indicando las características de la población. Los principales estadísticos son son los momentos momentos de primer, segundo y tercer orden correspondiente a la media, varianza, y asimetría respectivam r espectivamente. ente.
1.2.1
Media :
Es el valor esperado esperado de la variable variable misma. Primer momento momento respecto al origen. origen. Muestra la tendencia central de la distribución.
El valor estimado de la media a partir de la muestra es:
1.2.2
Varianza ²:
Mide la variabilidad de los datos. Es el segundo momento respecto a la media
El valor estimado de la varianza a partir de la muestra es
En el cual el divisor es n-1 en lugar de n para asegurar que la estadística que no tenga una tendencia, en promedio, a ser ser mayor o menor que el valor verdadero. verdadero. Las unidades de la varianza son la media al cuadrado, la desviación estándar s es una medida de la variabilidad que tiene las mismas dimensiones que la media y simplemente es la raíz cuadrada de la varianza, se estima por s.
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- Métodos Probabilísticos Efectos de la función de densidad de probabilidad causados por cambios en la desviación estándar
Coeficiente de variación
es una medida adimensional de la variabilidad su
estimado es
1.2.3
Coeficiente de asimetría
la distribución de los valores de una distribución alrededor de la media se mide por la asimetría. Se obtiene a partir del tercer momento momento alrededor de la media, dividiéndolo por el cubo de la desviación estándar para que sea adimensional.
tercer momento respecto a la media Un estimativo del coeficiente de asimetría está dado por:
ANALISIS DE FRECUENCIA El análisis de frecuencia es una herramienta utilizada para, predecir el comportamiento futuro de los caudales en un sitio de interés, a partir de la información histórica de caudales. Es un método basado en procedimientos procedimientos estadísticos que permite calcular la magnitud del caudal asociado a un período de retorno. Su confiabilidad depende de la longitud y calidad de la serie histórica, además de la incertidumbre propia de la distribución de probabilidades seleccionada. Para determinar la magnitud de eventos extremos cuando la distribución de probabilidades no es una función fácilmente invertibles se requiere conocer la variación de la variable respecto a la media. media. Chow en 1951 propusó determinar esta variación a partir de un factor fact or de frecuencia KT que puede ser expresado:
y se puede estimar a partir de los datos
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- Métodos Probabilísticos DISTRIBUCIONES DE PROBABILIDAD PARA VARIABLES CONTINUAS 3.1
DISTRIBUCION NORMAL
La distribución normal es una distribución simétrica en forma de campana, también conocida como Campana Campana de Gauss. Aunque muchas veces no se se ajusta a los datos hidrológicos tiene amplia aplicación por ejemplo a los datos transformados que siguen la distribución normal.
3.1.1
Función de densidad:
La función de densidad está dada por
Los dos parámetros de la distribución son la media m y desviación estándar s para los cuales (media) y s (desviación (desviación estándar) son derivados derivados de los datos.
3.1.2
Estimación de parámetros:
3.1.3
Factor de frecuencia: frecuencia:
1.
Si se trabaja con los X sin transformar el K se calcula como
este factor es el mismo de la variable normal estándar
3.1.4
Limites de confianza:
donde a es el nivel de probabilidad es el cuantil de la distribución normal estandarizada para una probabilidad acumulada de 1-a y Se es el error estándar
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- Métodos Probabilísticos 3.2
DISTRIBUCIÓN LOGNORMAL DE DOS PARÁMETROS
Si los logaritmos Y de una variable aleatoria X se distribuyen normalmente se dice que X se distribuye normalmente. Esta distribución es muy usada para el calculo de valores extremos por ejemplo Qmax, Qmínimos, Pmax, Pmínima Pmínima (excelentes resultados resultados en Antioquia). Tiene la ventaja que X>0 y que la transformación Log tiende a reducir la asimetría positiva ya que al sacar logaritmos se reducen en mayor proporción los datos mayores que los menores. Limitaciones: tiene solamente dos parámetros, y requiere que los logaritmos de las variables estén centrados en la media
3.2.1
Función de densidad:
y = ln x donde, my : media media de logaritmos de la población (parámetro escalar), sy : Desviación estándar de los logaritmos de la población, estimado sy.
3.2.2
Estimación de parámetros: parámetros:
3.2.3
Factor de frecuencia:
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- Métodos Probabilísticos 3.3
DISTRIBUCION GUMBEL O EXTREMA TIPO I
Una familia importante de distribuciones usadas en el análisis de frecuencia hidrológico es la distribución general de valores extremos, la cual ha sido ampliamente utilizada para representar el comportamiento de crecientes y sequías (máximos y mínimos).
3.3.1
Función de densidad:
En donde a y b son los parámetros de la distribución.
3.3.2
donde
3.3.3
Estimación de parámetros
son la media y la desviación estándar estimadas con la muestra.
Factor de frecuencia:
Donde Tr es el periodo de retorno. Para la distribución Gumbel Gumbel se tiene que el el caudal para un período de retorno de 2.33 años es igual a la media de los caudales máximos.
3.3.4
Limites de confianza
KT es el factor de frecuencia y t(1-a) es la variable normal estandarizada para una
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- Métodos Probabilísticos 3.4
DISTRIBUCION GAMMA DE TRES PARÁMETROS O PEARSON TIPO 3
Esta distribución ha sido una de las mas utilizadas en en hidrología. Como la mayoría de las las variables hidrológicas son sesgadas, la función Gamma se utiliza para ajustar la distribución de frecuencia de variables tales como crecientes máximas anuales, Caudales mínimos, Volúmenes de flujo anuales y estacionales, valores de precipitaciones extremas y volúmenes de lluvia de corta duración. La función de distribución Gamma Gamma tiene dos o tres parámetros.
3.4.1
Función de densidad:
donde, x0 < x < a para a > 0 a < x < x0 para a < 0 a y b son los parámetros de escala y forma, respectivamente , y x0 es el parámetro de localización.
3.4.2
Estimación de parámetros: parámetros:
3.4.3
Factor de frecuencia:
Este valor de K se encuentra tabulado de acuerdo al valor de Cs calculado con la muestra.
3.4.4
Intervalos de confianza:
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- Métodos Probabilísticos 3.5
DISTRIBUCIÓN LOG GAMMA O LOGPEARSON DE 3 PARÁMETROS
Si los logaritmos Y de una variable aleatoria X se ajustan a una distribución Pearson tipo III, se dice que la variable variable aleatoria X se ajusta ajusta a una distribución Log Pearson Pearson Tipo III. Esta distribución es ampliamente usada en el mundo para el análisis de frecuencia de Caudales máximos. Esta se trabaja igual que para la Pearson Pearson Tipo III pero con Xy y Sy como la media y desviación estándar de los logaritmos de la variable original X.
3.5.1
Función de densidad:
a y b son los parámetros de escala y forma, respectivamente , y y 0 es el parámetro de localización
3.5.2
Estimación de parámetros: parámetros:
Cs es el coeficiente de asimetría, logaritmos de la muestra respectivamente
3.5.3
son la media y la desviación estándar de los
Factor de frecuencia:
donde z es la variable normal estandarizada Este valor de K se encuentra tabulado de acuerdo al valor de Cs calculado con la muestra.
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- Métodos Probabilísticos
AJUSTE DE DISTRIBUCIONES Para la modelación de caudales máximos se utilizan, entre otras, las distribuciones Log Normal, Gumbel Gumbel y Log-Gumbel principalmente. Para seleccionar seleccionar la distribución de probabilidades de la serie histórica se deben tener en cuenta algunas consideraciones. Cuando la información es adecuada el análisis de frecuencia es la metodología más recomendable para la evaluación de eventos extremos, ya que la estimación depende solamente de los caudales máximos anuales que han ocurrido en la cuenca y no da cuenta de los procesos procesos de transformación de la precipitación precipitación en escorrentía. Obviamente tiene algunas limitaciones relacionadas con el comportamiento de la serie histórica y con el tamaño y calidad de los datos de la muestra.
4.1
Plotting Position
Trabaja con la probabilidad de excedencia excedencia asignada a cada valor de la muestra. muestra. Se han propuesto numerosos métodos métodos empíricos. Si n es el total de valores y m es el rango de un valor en una lista ordenada de d e mayor a menor (m=1 para el valor máximo) la probabilidad de excedencia se puede obtener por medio de las l as siguientes expresiones:
California
Weibull
Hazen La expresión más utilizada es la Weibull. Weibull. Con las anteriores expresiones se halla halla lo que se conoce como la distribución empírica de una muestra, esta luego se puede ajustar a una de las distribuciones teóricas presentadas anteriormente. Los resultados pueden ser ser dibujados en el papel de probabilidad; este es diseñado para que los datos se ajusten a una línea recta y se puedan p uedan comparar los datos muestrales con la distribución teórica
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- Métodos Probabilísticos 4.2.1 Prueba Smirnov - Kolmogorov El estadístico Smirnov Kolmogorov D considera la desviación de la función de distribución de probabilidades de la muestra P(x) de la función de probabilidades teórica, escogida Po(x) tal que
La prueba requiere que el valor Dn calculado con la expresión anterior sea menor que el valor tabulado Dn para un nivel de probabilidad requerido. Esta prueba es fácil de realizar y comprende las siguientes etapas:
El estadístico Dn es la máxima diferencia entre la función de distribución acumulada ac umulada de la muestra y la función de distribución acumulada teórica escogida. Se fija el nivel de probabilidad a, valores de 0.05 y 0.01 son los más usuales. El valor crítico Da de la prueba debe ser obtenido de tablas en función de a y n. Si el valor calculado Dn es mayor que el Da, la distribución escogida se debe rechazar.
4.2.2 Prueba Chi Cuadrado Una medida de las discrepancias entre las frecuencias observadas ( f o) y las frecuencias calculadas (f c) por medio de una distribución teórica esta dada por el estadístico χ estadístico χ²²
en donde
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- Métodos Probabilísticos
1. ESTADISTICA DE DATOS HIDROMETRICOS
ANALISIS DE DATOS HIODROMETRICOS EN RIO JEQUETEPQUE
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- Métodos Probabilísticos
CALCULO DE PROBABILIDAD DE DATOS HIDROMETRICOS EN RIO JEQUETEPQUE
ANALISIS DATOS HIDROMETRICOS
PARA LA REALIZACIÓN DEL ESTUDIO HIDROLÓGICO DE LA CUENCA HIDROGRÁFICA SE DISPONE DE MEDICIÓNES DE CAUDALES, CONSIDERANDO LA DISPONIBILIDAD DE ESTOS REGISTROS EN LAS ESTACIONES DE AFORO.
LA DETERMINACION DE LA CURVA DE CALIBRACION - PERIODO DE RETORNO SE REALIZO MEDIANTE EL ANALISIS ESTADISTICO DE AJUSTE DE UNA DISTRIBUCION DE PROBABILIDAD, APLICANDOSE EN ESTE CASO LAS DISTRIBUCIONES: NORMAL, LOGNORMAL, PEARSON, LOGPEARSON, GUMBEL, LOGGUMBEL. ELIGIENDOSE LA MAS
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- Métodos Probabilísticos CALCULO DE PROBABILIDAD DE DATOS HIDROMETRICOS EN RIO JEQUETEPQUE ESTUDIO HIDROLOGICO DE AVENIDAS ANALISI S DE DATOS DATOS HIDRO METRICO METRICO S
ESTACI ESTACI ON HI DROMETRICA DE YONAN.
ORDEN
AÑO
CAUDAL MEDIO
1
2014
11.344414
2
2012
7.519053
3
2008
6.843462
4
1993
6.740720
5
2000
5.896043
6
2011
5.875681
7
1996
4.627101
8
2005
4.470117
9
2007
4.195992
10
1991
4.193391
11
2003
4.178741
12
2009
3.625751
13
2004
3.473280
14
1986
3.469434
15
1994
3.350198
16
1997
2.945584
17
1992
2.626760
18
1972
2.606377
19
1975
2.288143
20
1981
2.162130
21
1984
2.053163
22
1973
1.818151
23
1998
1 804066
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- Métodos Probabilísticos CALCULO DE PROBABILIDAD DE DATOS HIDROMETRICOS EN RIO JEQUETEPQUE ESTUDIO HIDROLOGICO DE AVENIDAS ANALISIS DE DATOS HIDROMETRICOS RESUMEN DEL AJUSTE DE D ISTRIBUCIONES DE PROBABILIDAD PROBABILIDAD A LOS DATOS HIDROM ETRICOS. REGISTRADOS EN LA ESTACION HIDROM ETRICA Y ONAN, JEQUETEPEQUE. JEQUETEPEQUE.
Q MEDIO (m3/s)
DISTRIBUCIONES DE PROBABILIDAD
AÑO
m
P(o P(obs) bs) Tr=(N+1 )/m
2014
1
0.979
47
11.3444
9.1
7.3
9.5
17.6
729.1
566.8
2012 2008
2 3
0.957 0.936
24 16
7.5191 6.8435
7.8 6.9
6.6 6.2
7.7 6.6
11.1 8.5
462.2 364.2
467.5 409.4
1993
4
0.915
12
6.7407
6.4
5.7
5.6
7.0
313.4
368.2
2000
5
0.894
9
5.8960
5.9
5.5
5.2
6.0
282.4
336.3
DP (GUM)
DP (NOR) DP (LNOR) DP (LGUM)
DP (PEAR)
DP (LPEAR)
2011
6
0.872
8
5.8757
5.5
5.2
4.8
5.3
261.5
310.1
1996
7
0.851
7
4.6271
5.2
5.0
4.5
4.7
246.5
288.1
2005
8
0.830
6
4.4701
4.9
4.9
4.2
4.3
235.1
268.9
2007
9
0.809
5
4.1960
4.7
4.7
4.0
3.9
226.3
252.1
1991
10
0.787
5
4.1934
4.4
4.5
3.8
3.6
219.2
237.0
2003
11
0.766
4
4.1787
4.2
4.4
3.6
3.4
213.3
223.3
2009
12
0.745
4
3.6258
4.0
4.1
3.3
3.2
208.5
210.9
2004
13
0.723
4
3.4733
3.8
4.0
3.2
3.0
204.4
199.4
1986
14
0.702
3
3.4694
3.7
3.9
3.0
2.8
200.8
188.8
1994
15
0.681
3
3.3502
3.5
3.8
2.9
2.7
197.7
178.9
1997
16
0.660
3
2.9456
3.4
3.6
2.8
2.5
195.0
169.7
1992
17
0.638
3
2.6268
3.2
3.5
2.7
2.4
192.7
161.0
1972
18
0.617
3
2.6064
3.1
3.4
2.6
2.3
190.5
152.8
1975
19
0.596
2
2.2881
2.9
3.3
2.5
2.2
188.7
145.1
1981
2 0
0.574
2
2.1621
2.8
3.1
2.3
2.1
186.9
137.7
1984
21
0.553
2
2.0532
2.7
3.0
2.2
2.0
185.4
130.7
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- Métodos Probabilísticos
ESTUDIO HIDROLOGICO DE AVENIDAS ANALISIS DE D ATOS HIDROMETRICOS
AJUSTE DE LAS DISTRIBUCIONES GUMBEL SIMPLE Y LOG-GUMBEL A LOS DATOS HIDROMETRICOS DE LA ESTACION YONAN, JEQUETEPEQUE GUM BEL LOG-GUM BEL
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- Métodos Probabilísticos CALCULO DE PROBABILIDAD DE DATOS HIDROMETRICOS EN RIO JEQUETEPQUE ESTUDIO HIDROLOGICO DE AVENIDAS ANALISIS DE DA TOS HIDROMETRICOS HIDROMETRICOS
METODOLOGIA DEAPLICACIÓN DE LA DISTRIBUCION DE PROBABILIDAD GUMBEL DICHA DISTRIBUCION ES DEL TIPO EXPONENCIAL, CASO ESPECIAL DE LA LOG-NORMAL A) DISTR IBUCION GUMBEL
B) DISTR IBUCION LOG-GUM BEL
FUNCION MATEMATICA X = X m + ( ( Y - Y n ) / Tn ) S
FUNCION M ATEMATICA
DONDE: X VALOR BUSCADO Xm, S MEDIA Y DESVIACION DE LA SERIE Yn, Tn CONSTANTES CONSTANTES TEORICAS, TEORICAS, SEG SEG N n (CUADRO 3.6) n NUMERO TOTAL DE DATOS CONSIDERADOS Y 46
W=Wm+((Y-Yn)/Tn)Sw
EL PR OC OCEDIM IE IENTO ES SIMILAR A LA DE GUMBEL, CONSIDER AN ANDO COM O SER IE IE A LOS LOGARITMOS DE LOS DATOS ORIGINALES, ESTO ES Wi = LOG X
T 46 0.5468
1.1538
Y VARIABLE REDUCIDA , FUNCION DE LA PROBABILIDAD X= Xm + (( Y - Y n )/ Tn ) S
Tr 1000 500 200 100 50 25 20 10
p(X<=x)=1-(1 Tr) 0.9990 0.9980 0.9950 0.9900 0.9800 0.9599 0.9500 0 9000
Y
X 6.907 6.214 5.296 4.600 3.902 3.196 2.970 2.250
Tr 1155.1 13.8 13 12.0 12 10.6 9.2 9. 7.9 7. 7.4 7. 60
1000 500 200 100 50 25 20 10
p(X<=x)=1-(1 Tr) 0.9990 0.9980 0.9950 0.9900 0.9800 0.9600 0.9500 0 9000
Y
W 6.907 6.214 5.296 4.600 3.902 3.199 2.970 2.250
X=ANTILOG (W) 2.12 1.92 1.66 1.46 1.26 1.06 1.00 0.79
131.4 83.4 45.7 29.0 18.4 11.6 10.0 6.2
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- Métodos Probabilísticos CALCULO DE PROBABILIDAD DE DATOS HIDROMETRICOS EN RIO JEQUETEPQUE ESTUDIO HIDROLOGICO DE AVENIDAS ANALISIS DE DATOS HIDROMETRICOS
AJUSTE DE LAS DISTRIBUCIONES NORMAL Y LOG-NORMAL A LOS DATOS HIDROMETRICOS DE LA EST. YONAN, JEQUETEPEQUE NORMAL LOG-NORMAL Q MEDIO m P(obs) Tr=(N+1)/m D P ( NO R ) Z F(X) DP (LNOR ) Yi= LOG Xi (Yi-Ym)**3 (m3/s)
AÑ O 20 1 4 20 1 2 2 008 1 993 2 000 2011 1 996 2 005 2 007 1991 2 003 2 009 2 004 1 986
1 2 3 4 5 6 7 8 9 10 11 12 13 14
0.98 0.96 0.94 0.91 0.89 0.87 0.85 0.83 0.81 0.79 0.77 0.74 0.72 0.70
47 24 16 12 9 8 7 6 5 5 4 4 4 3
11.3 7.5 6.8 6.7 5.9 5.9 4.6 4.5 4.2 4.2 4.2 3.6 3.5 3.5
7.3 6.6 6.2 5.7 5.5 5.2 5.0 4.9 4.7 4.5 4.4 4.1 4.0 3.9
2.03 1.75 1.56 1.34 1.23 1.13 1.04 0.96 0.88 0.81 0.74 0.64 0.58 0.53
0.4787 0.46 0.44 0.41 0.39 0.37 0.35 0.33 0.31 0.29 0.27 0.24 0.22 0.20
0. 0 . 978 7 0. 0 . 957 4 0. 0 . 936 2 0.0.9149 0.0.8936 0. 0 . 872 3 0. 0.8511 0.0.8298 0. 0 . 808 5 0. 0 . 787 2 0. 0 . 766 0 0.0.7447 0.0.7234 0. 0 .7021
9.46 7.65 6.63 5.61 5.17 4.79 4.48 4.21 3.97 3.76 3.57 3.31 3.16 3.04
1.1 0.9 0.8 0.8 0.8 0.8 0.7 0.7 0.6 0.6 0.6 0.6 0.5 0.5
0.414 0.182 0.145 0.140 0.098 0.097 0.045 0.040 0.031 0.031 0.030 0.016 0.012 0.012
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- Métodos Probabilísticos METODOLOGIA ETODOLOGIA DEAPLIC DEAPLICAC ACIÓN IÓN DE LA DISTRIBU DISTRIBUCION CION DE PROB PROBAB ABILI ILI
A)
DISTRIBUCION NORMAL
NOR(51.7,15) Z=(Xi-Xm)/ S XN 193.6 182.0 172.5 161.2 149.5 145.5 131.0 114.0
Z 84.90 79.74 75.52 70.49 65.29 63.51 57.06 49.50
F(X)i Tr 0.9980 0.9950 0.9900 0.9798 0.9599 0.9505 0.8997 0.7996
500 200 100 50 25 20 10 5
B) DISTRIBUCION DE PROBABILIDAD LOG-NORMAL
Z=(Yi-Ym)/Sy XLN 396 331
Z 6.97 6 74
F(X)i Tr 0.9980 0 9950
500 200
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- Métodos Probabilísticos CALCULO DE PROBABILIDAD DE DATOS HIDROMETRICOS EN RIO JEQUETEPQUE ESTUDIO HIDROLOGICO DE AVENIDAS ANALISIS DE DATOS HIDROMETRICOS
AJUSTE DE LA DISTRIBUCION PEARSON III ACAUDALES MAXIMOS. ANUALES REGISTRADOS EN LA EST.HIDROLOGICA YONAN, JEQUETEPEQUE A ÑO
OR DEN 2014 2012 2008 1993 2000 2011 1996 2005 2007 1991 2003 2009 2004 1986 1994 1997 1992 1972 1975 1981 1984 1973 1998 1995 1982
Tr=(N+1)/ m 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25
47.00 23.50 15.67 11.75 9.40 7.83 6.71 5.88 5.22 4.70 4.27 3.92 3.62 3.36 3.13 2.94 2.76 2.61 2.47 2.35 2.24 2.14 2.04 1.96 1.88
Q M ED IO (m3/ s)
11.3 7.5 6.8 6.7 5.9 5.9 4.6 4.5 4.2 4.2 4.2 3.6 3.5 3.5 3.4 2.9 2.6 2.6 2.3 2.2 2.1 1.8 1.8 1.6 1.5
D P (PEAR )
566.75 467.48 409.41 368.21 336.25 310.14 288.06 268.94 252.07 236.98 223.33 210.87 199.40 188.79 178.91 169.67 160.98 152.80 145.05 137.71 130.72 124.06 117.69 111.59 105.75
Yi = LOG Xi
1.05 0.88 0.84 0.83 0.77 0.77 0.67 0.65 0.62 0.62 0.62 0.56 0.54 0.54 0.53 0.47 0.42 0.42 0.36 0.33 0.31 0.26 0.26 0.20 0.18
DP (LPEAR )
729.06 462.23 364.15 313.40 282.40 261.51 246.47 235.14 226.28 219.18 213.35 208.48 204.35 200.81 197.73 195.04 192.66 190.55 188.65 186.95 185.40 183.99 182.71 181.53 180.45
LN(Tr) 308.08 3.16 2.75 2.46 2.24 2.06 1.90 1.77 1.65 1.55 1.45 1.37 1.29 1.21 1.14 1.08 1.02 0.96 0.91 0.85 0.81 0.76 0.71 0.67 0.63
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- Métodos Probabilísticos
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- Métodos Probabilísticos COMPARACION ENTRE ENTRE LAS FRECUENCI FRECUENCIAS AS OBSERVADAS Y T EORICAS FRECUENCIAS OBSERVADAS PROBABILIDAD DE NO OCURRENCIA
FRECUENCIAS TEORICAS
VALOR DE LA VARIABLE
DI ST. NORMAL 2P
OBSERVADA
MEDIA
3.43 VALOR DE LA VARIABLE
DESV.EST
3.15
0 . 10
1 . 07
0 . 10
- 0. 60
0. 20
1. 21
0. 20
0. 7 9
0. 30
1. 69
0. 30
1. 7 8
0. 40
1. 95
0. 40
2. 6 4
0. 50
1. 97
0. 50
3. 4 3
0. 60
2. 72
0. 60
4. 2 3
0. 70
3. 33
0. 70
5. 0 8
0. 80
6. 50
0. 80
6. 0 8
0. 90
7. 64
0. 90
7. 4 6
0. 99
12. 33
0. 99
10. 75
DI ST. L OG. NORMAL 2P
MEDIA
0.71 VA VALOR DE LA VARIABLE
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- Métodos Probabilísticos METODO SMIRNOV – KOLMOGOROV KOLMOGOROV PRUEBA DE BONDAD DE AJUSTE SM SMIRNOV IRNOV - KOLMOGOROV KOLMOGORO V ESTACION YONAN - JEQUETEPEQUE "Con Niños" CUADRO N° 03 A m (1) 1 2
Mé todo Gum be l
Q=X m3/s
P (X) m/(n+1)
(2)
(3) 0. 0175 0. 0351
1.08
1.13
Z
X
-
X
=
S
(4) -0.86 -0.86
F( Z) (5) 0. 215 0. 215
D ..........
F ( X ) - P ( X )
(6) 0.1975 0.1799
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- Métodos Probabilísticos PRUEBA DE DE BONDAD BONDAD DE DE AJUSTE AJUSTE SMI RNOV - KOLMOGOROV KOLMOGOROV ESTACION YONAN - JEQUETEPEQUE JEQUETEPEQUE "Con Niños" Ni ños" CUADRO N° 04 A m (1) 1 2 3 4
Mé todo Log P e rson Ti po I II
Q=X m3/s (2)
1.08
1.13
2.61
1.82
Y=LN X
P (X) (X) m/(n+1)
(3) 0.07 0.12 0.96 0 60
(4) 0. 0175 0. 0351 0. 0526 0 0702
Z
X =
-
X
S
(5) -0. 85 -0. 78 0.31 0 16
F( Z) (6) 0. 033 0. 038 0. 053 0 066
.......... D F ( Z ) - P ( X )
(7) 0.0155 0.0029 0.0004 0 0042
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- Métodos Probabilísticos PRUEBA DE DE BONDAD BONDAD DE DE AJUSTE AJUSTE SMIRNOV SMI RNOV - KOLMOGOROV ESTACION YONAN - JEQUETEPEQUE "Sin Niños" CUADRO N° 03 A Mé todo Gum be l m (1) 1 2 3 4
Q=X m3/s
P (X) m/(n+1)
(2)
(3) 0. 0185 0. 0370 0. 0556 0 0741
1.08
1.13
2.61
1.82
Z
X =
-
X
S
(4) -1.01 -1.01 -0.98 -1 00
F( Z) (5) 0. 179 0. 181 0. 187 0 192
.......... D F ( X ) - P ( X )
(6) 0.1605 0.1440 0.1314 0 1179
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- Métodos Probabilísticos PRUEBA DE BONDAD DE AJUSTE SM SMIRNOV IRNOV - KOLMOGOROV KOLMOGO ROV
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- Métodos Probabilísticos