En estadística , una distribución bimodal es una distribución de probabilidad con dos modos diferentes , que también puede denominarse distribución bimodal. Estos aparecen como picos distintos (máximos locales) en la función de densidad de probabilidad , como se muestra en las Figuras 1 y 2. Los datos categóricos, continuos y discretos pueden formar distribuciones bimodales [ cita requerida ] .
De manera más general, una distribución multimodal es una distribución de probabilidad con dos o más modos, como se ilustra en la Figura 3.
Terminología
Cuando los dos modos son desiguales, el modo más grande se conoce como modo mayor y el otro como modo menor. El valor menos frecuente entre los modos se conoce como antimodo . La diferencia entre los modos mayor y menor se conoce como amplitud . En series de tiempo, el modo principal se llama acrofase y el antimodo batifase . [ cita requerida ]
Clasificación de Galtung
Galtung introdujo un sistema de clasificación (AJUS) para distribuciones: [1]
- A: distribución unimodal - pico en el medio
- J: unimodal - pico en cualquier extremo
- U: bimodal - picos en ambos extremos
- S: bimodal o multimodal - múltiples picos
Desde entonces, esta clasificación se ha modificado ligeramente:
- J: (modificado) - pico a la derecha
- L: unimodal - pico a la izquierda
- F: sin pico (plano)
Bajo esta clasificación, las distribuciones bimodales se clasifican como tipo S o U.
Ejemplos de
Las distribuciones bimodales ocurren tanto en matemáticas como en ciencias naturales.
Distribuciones de probabilidad
Las distribuciones bimodales importantes incluyen la distribución de arcoseno y la distribución beta . Otros incluyen la distribución U-cuadrática .
La relación de dos distribuciones normales también se distribuye bimodalmente. Dejar
donde un y b son constantes y x y y son distribuidas como variables normales con una media de 0 y una desviación estándar de 1. R tiene una densidad conocida que puede ser expresado como una función hipergeométrica confluente . [2]
La distribución del recíproco de una variable aleatoria distribuida en t es bimodal cuando los grados de libertad son más de uno. De manera similar, el recíproco de una variable distribuida normalmente también se distribuye bimodalmente.
Una estadística t generada a partir de un conjunto de datos extraídos de una distribución de Cauchy es bimodal. [3]
Ocurrencias en la naturaleza
Ejemplos de variables con distribuciones bimodales incluyen el tiempo entre erupciones de ciertos géiseres , el color de las galaxias , el tamaño de las hormigas tejedoras obreras , la edad de incidencia del linfoma de Hodgkin , la velocidad de inactivación del fármaco isoniazida en adultos estadounidenses, la magnitud absoluta de novas , y los patrones de actividad circadiana de esos animales crepusculares que están activos tanto en el crepúsculo matutino como vespertino. En la ciencia pesquera, las distribuciones de tallas multimodales reflejan las diferentes clases de años y, por lo tanto, se pueden utilizar para la distribución por edades y las estimaciones de crecimiento de la población de peces. [4] Los sedimentos generalmente se distribuyen de forma bimodal. Al muestrear las galerías mineras que cruzan la roca huésped y las vetas mineralizadas, la distribución de las variables geoquímicas sería bimodal. Las distribuciones bimodales también se ven en el análisis de tráfico, donde el tráfico alcanza su punto máximo durante la hora pico de la mañana y luego nuevamente en la hora pico de la tarde. Este fenómeno también se observa en la distribución diaria del agua, ya que la demanda de agua, en forma de duchas, cocina y uso del baño, generalmente alcanza su punto máximo en los períodos matutino y vespertino.
Econometría
En los modelos econométricos , los parámetros pueden estar distribuidos bimodalmente. [5]
Orígenes
Matemático
Una distribución bimodal surge más comúnmente como una mezcla de dos distribuciones unimodales diferentes (es decir, distribuciones que tienen un solo modo). En otras palabras, la variable aleatoria X distribuida bimodalmente se define como con probabilidad o con probabilidad donde Y y Z son variables aleatorias unimodales y es un coeficiente de mezcla.
No es necesario que las mezclas con dos componentes distintos sean bimodales y las mezclas de dos componentes con densidades de componentes unimodales pueden tener más de dos modos. No existe una conexión inmediata entre el número de componentes en una mezcla y el número de modos de la densidad resultante.
Distribuciones particulares
Las distribuciones bimodales, a pesar de su frecuente aparición en conjuntos de datos, sólo se han estudiado en raras ocasiones [ cita requerida ] . Esto puede deberse a las dificultades para estimar sus parámetros con métodos frecuentistas o bayesianos. Entre los que se han estudiado se encuentran
- Distribución exponencial bimodal. [6]
- Distribución alfa-sesgada-normal. [7]
- Distribución normal bimodal asimétrica asimétrica. [8]
- Se ha ajustado una mezcla de distribuciones de Conway-Maxwell-Poisson a los datos de recuento bimodal. [9]
La bimodalidad también surge naturalmente en la distribución de catástrofes de cúspide .
Biología
En biología, se sabe que cinco factores contribuyen a distribuciones bimodales de tamaños de población [ cita requerida ] :
- la distribución inicial de tamaños individuales
- la distribución de las tasas de crecimiento entre los individuos
- el tamaño y la dependencia del tiempo de la tasa de crecimiento de cada individuo
- tasas de mortalidad que pueden afectar a cada clase de tamaño de manera diferente
- la metilación del ADN en el genoma humano y de ratón.
La distribución bimodal de tamaños de los trabajadores de la hormiga tejedora surge debido a la existencia de dos clases distintas de trabajadores, a saber, los trabajadores mayores y los trabajadores menores. [10]
La distribución de los efectos de aptitud de las mutaciones tanto para los genomas completos [11] [12] como para los genes individuales [13] también se encuentra frecuentemente como bimodal, siendo la mayoría de las mutaciones neutrales o letales, con relativamente pocas de efecto intermedio.
Propiedades generales
Una mezcla de dos distribuciones unimodales con diferentes medias no es necesariamente bimodal. La distribución combinada de alturas de hombres y mujeres se utiliza a veces como un ejemplo de distribución bimodal, pero de hecho la diferencia en las alturas medias de hombres y mujeres es demasiado pequeña en relación con sus desviaciones estándar para producir bimodalidad. [14]
Las distribuciones bimodales tienen la propiedad peculiar de que, a diferencia de las distribuciones unimodales, la media puede ser un estimador muestral más robusto que la mediana. [15] Este es claramente el caso cuando la distribución tiene forma de U como la distribución de arcoseno. Puede que no sea cierto cuando la distribución tiene una o más colas largas.
Momentos de mezclas
Dejar
donde g i es una distribución de probabilidad yp es el parámetro de mezcla.
Los momentos de f ( x ) son [16]
dónde
y S i y K i son la asimetría y la curtosis de la i- ésima distribución.
Mezcla de dos distribuciones normales
No es raro encontrar situaciones en las que un investigador crea que los datos provienen de una mezcla de dos distribuciones normales. Debido a esto, esta mezcla se ha estudiado con cierto detalle. [17]
Una mezcla de dos distribuciones normales tiene cinco parámetros para estimar: las dos medias, las dos varianzas y el parámetro de mezcla. Una mezcla de dos distribuciones normales con desviaciones estándar iguales es bimodal solo si sus medias difieren en al menos el doble de la desviación estándar común. [14] Las estimaciones de los parámetros se simplifican si se puede suponer que las varianzas son iguales (el caso homoscedástico ).
Si las medias de las dos distribuciones normales son iguales, entonces la distribución combinada es unimodal. Eisenberger derivó las condiciones para la unimodalidad de la distribución combinada. [18] Ray y Lindsay han identificado las condiciones necesarias y suficientes para que una mezcla de distribuciones normales sea bimodal. [19]
A mixture of two approximately equal mass normal distributions has a negative kurtosis since the two modes on either side of the center of mass effectively reduces the tails of the distribution.
A mixture of two normal distributions with highly unequal mass has a positive kurtosis since the smaller distribution lengthens the tail of the more dominant normal distribution.
Mixtures of other distributions require additional parameters to be estimated.
Tests for unimodality
- The mixture is unimodal if and only if[20]
or
where p is the mixing parameter and
and where μ1 and μ2 are the means of the two normal distributions and σ1 and σ2 are their standard deviations.
- The following test for the case p = 1/2 was described by Schilling et al.[14] Let
The separation factor (S) is
If the variances are equal then S = 1. The mixture density is unimodal if and only if
- A sufficient condition for unimodality is[21]
- If the two normal distributions have equal standard deviations a sufficient condition for unimodality is[21]
Resumen estadístico
Bimodal distributions are a commonly used example of how summary statistics such as the mean, median, and standard deviation can be deceptive when used on an arbitrary distribution. For example, in the distribution in Figure 1, the mean and median would be about zero, even though zero is not a typical value. The standard deviation is also larger than deviation of each normal distribution.
Although several have been suggested, there is no presently generally agreed summary statistic (or set of statistics) to quantify the parameters of a general bimodal distribution. For a mixture of two normal distributions the means and standard deviations along with the mixing parameter (the weight for the combination) are usually used – a total of five parameters.
Ashman's D
A statistic that may be useful is Ashman's D:[22]
where μ1, μ2 are the means and σ1 σ2 are the standard deviations.
For a mixture of two normal distributions D > 2 is required for a clean separation of the distributions.
van der Eijk's A
This measure is a weighted average of the degree of agreement the frequency distribution.[23] A ranges from -1 (perfect bimodality) to +1 (perfect unimodality). It is defined as
where U is the unimodality of the distribution, S the number of categories that have nonzero frequencies and K the total number of categories.
The value of U is 1 if the distribution has any of the three following characteristics:
- all responses are in a single category
- the responses are evenly distributed among all the categories
- the responses are evenly distributed among two or more contiguous categories, with the other categories with zero responses
With distributions other than these the data must be divided into 'layers'. Within a layer the responses are either equal or zero. The categories do not have to be contiguous. A value for A for each layer (Ai) is calculated and a weighted average for the distribution is determined. The weights (wi) for each layer are the number of responses in that layer. In symbols
A uniform distribution has A = 0: when all the responses fall into one category A = +1.
One theoretical problem with this index is that it assumes that the intervals are equally spaced. This may limit its applicability.
Bimodal separation
This index assumes that the distribution is a mixture of two normal distributions with means (μ1 and μ2) and standard deviations (σ1 and σ2):[24]
Bimodality coefficient
Sarle's bimodality coefficient b is[25]
where γ is the skewness and κ is the kurtosis. The kurtosis is here defined to be the standardised fourth moment around the mean. The value of b lies between 0 and 1.[26] The logic behind this coefficient is that a bimodal distribution with light tails will have very low kurtosis, an asymmetric character, or both – all of which increase this coefficient.
The formula for a finite sample is[27]
where n is the number of items in the sample, g is the sample skewness and k is the sample excess kurtosis.
The value of b for the uniform distribution is 5/9. This is also its value for the exponential distribution. Values greater than 5/9 may indicate a bimodal or multimodal distribution, though corresponding values can also result for heavily skewed unimodal distributions.[28] The maximum value (1.0) is reached only by a Bernoulli distribution with only two distinct values or the sum of two different Dirac delta functions (a bi-delta distribution).
The distribution of this statistic is unknown. It is related to a statistic proposed earlier by Pearson – the difference between the kurtosis and the square of the skewness (vide infra).
Bimodality amplitude
This is defined as[24]
where A1 is the amplitude of the smaller peak and Aan is the amplitude of the antimode.
AB is always < 1. Larger values indicate more distinct peaks.
Bimodal ratio
This is the ratio of the left and right peaks.[24] Mathematically
where Al and Ar are the amplitudes of the left and right peaks respectively.
Bimodality parameter
This parameter (B) is due to Wilcock.[29]
where Al and Ar are the amplitudes of the left and right peaks respectively and Pi is the logarithm taken to the base 2 of the proportion of the distribution in the ith interval. The maximal value of the ΣP is 1 but the value of B may be greater than this.
To use this index, the log of the values are taken. The data is then divided into interval of width Φ whose value is log 2. The width of the peaks are taken to be four times 1/4Φ centered on their maximum values.
Bimodality indices
- Wang's index
The bimodality index proposed by Wang et al assumes that the distribution is a sum of two normal distributions with equal variances but differing means.[30] It is defined as follows:
where μ1, μ2 are the means and σ is the common standard deviation.
where p is the mixing parameter.
- Sturrock's index
A different bimodality index has been proposed by Sturrock.[31]
This index (B) is defined as
When m = 2 and γ is uniformly distributed, B is exponentially distributed.[32]
This statistic is a form of periodogram. It suffers from the usual problems of estimation and spectral leakage common to this form of statistic.
- de Michele and Accatino's index
Another bimodality index has been proposed by de Michele and Accatino.[33] Their index (B) is
where μ is the arithmetic mean of the sample and
where mi is number of data points in the ith bin, xi is the center of the ith bin and L is the number of bins.
The authors suggested a cut off value of 0.1 for B to distinguish between a bimodal (B > 0.1)and unimodal (B < 0.1) distribution. No statistical justification was offered for this value.
- Sambrook Smith's index
A further index (B) has been proposed by Sambrook Smith et al[34]
where p1 and p2 are the proportion contained in the primary (that with the greater amplitude) and secondary (that with the lesser amplitude) mode and φ1 and φ2 are the φ-sizes of the primary and secondary mode. The φ-size is defined as minus one times the log of the data size taken to the base 2. This transformation is commonly used in the study of sediments.
The authors recommended a cut off value of 1.5 with B being greater than 1.5 for a bimodal distribution and less than 1.5 for a unimodal distribution. No statistical justification for this value was given.
- Chaudhuri and Agrawal index
Another bimodality parameter has been proposed by Chaudhuri and Agrawal.[35] This parameter requires knowledge of the variances of the two subpopulations that make up the bimodal distribution. It is defined as
where ni is the number of data points in the ith subpopulation, σi2 is the variance of the ith subpopulation, m is the total size of the sample and σ2 is the sample variance.
It is a weighted average of the variance. The authors suggest that this parameter can be used as the optimisation target to divide a sample into two subpopulations. No statistical justification for this suggestion was given.
Pruebas estadísticas
A number of tests are available to determine if a data set is distributed in a bimodal (or multimodal) fashion.
Graphical methods
In the study of sediments, particle size is frequently bimodal. Empirically, it has been found useful to plot the frequency against the log( size ) of the particles.[36][37] This usually gives a clear separation of the particles into a bimodal distribution. In geological applications the logarithm is normally taken to the base 2. The log transformed values are referred to as phi (Φ) units. This system is known as the Krumbein (or phi) scale.
An alternative method is to plot the log of the particle size against the cumulative frequency. This graph will usually consist two reasonably straight lines with a connecting line corresponding to the antimode.
- Statistics
Approximate values for several statistics can be derived from the graphic plots.[36]
where Mean is the mean, StdDev is the standard deviation, Skew is the skewness, Kurt is the kurtosis and φx is the value of the variate φ at the xth percentage of the distribution.
Unimodal vs. bimodal distribution
Pearson in 1894 was the first to devise a procedure to test whether a distribution could be resolved into two normal distributions.[38] This method required the solution of a ninth order polynomial. In a subsequent paper Pearson reported that for any distribution skewness2 + 1 < kurtosis.[26] Later Pearson showed that[39]
where b2 is the kurtosis and b1 is the square of the skewness. Equality holds only for the two point Bernoulli distribution or the sum of two different Dirac delta functions. These are the most extreme cases of bimodality possible. The kurtosis in both these cases is 1. Since they are both symmetrical their skewness is 0 and the difference is 1.
Baker proposed a transformation to convert a bimodal to a unimodal distribution.[40]
Several tests of unimodality versus bimodality have been proposed: Haldane suggested one based on second central differences.[41] Larkin later introduced a test based on the F test;[42] Benett created one based on Fisher's G test.[43] Tokeshi has proposed a fourth test.[44][45] A test based on a likelihood ratio has been proposed by Holzmann and Vollmer.[20]
A method based on the score and Wald tests has been proposed.[46] This method can distinguish between unimodal and bimodal distributions when the underlying distributions are known.
Antimode tests
Statistical tests for the antimode are known.[47]
- Otsu's method
Otsu's method is commonly employed in computer graphics to determine the optimal separation between two distributions.
General tests
To test if a distribution is other than unimodal, several additional tests have been devised: the bandwidth test,[48] the dip test,[49] the excess mass test,[50] the MAP test,[51] the mode existence test,[52] the runt test,[53][54] the span test,[55] and the saddle test.
An implementation of the dip test is available for the R programming language.[56] The p-values for the dip statistic values range between 0 and 1. P-values less than 0.05 indicate significant multimodality and p-values greater than 0.05 but less than 0.10 suggest multimodality with marginal significance.[57]
Silverman's test
Silverman introduced a bootstrap method for the number of modes.[48] The test uses a fixed bandwidth which reduces the power of the test and its interpretability. Under smoothed densities may have an excessive number of modes whose count during bootstrapping is unstable.
Bajgier-Aggarwal test
Bajgier and Aggarwal have proposed a test based on the kurtosis of the distribution.[58]
Special cases
Additional tests are available for a number of special cases:
- Mixture of two normal distributions
A study of a mixture density of two normal distributions data found that separation into the two normal distributions was difficult unless the means were separated by 4–6 standard deviations.[59]
In astronomy the Kernel Mean Matching algorithm is used to decide if a data set belongs to a single normal distribution or to a mixture of two normal distributions.
- Beta-normal distribution
This distribution is bimodal for certain values of is parameters. A test for these values has been described.[60]
Estimación de parámetros y curvas de ajuste
Assuming that the distribution is known to be bimodal or has been shown to be bimodal by one or more of the tests above, it is frequently desirable to fit a curve to the data. This may be difficult.
Bayesian methods may be useful in difficult cases.
Software
- Two normal distributions
A package for R is available for testing for bimodality.[61] This package assumes that the data are distributed as a sum of two normal distributions. If this assumption is not correct the results may not be reliable. It also includes functions for fitting a sum of two normal distributions to the data.
Assuming that the distribution is a mixture of two normal distributions then the expectation-maximization algorithm may be used to determine the parameters. Several programmes are available for this including Cluster,[62] and the R package nor1mix.[63]
- Other distributions
The mixtools package available for R can test for and estimate the parameters of a number of different distributions.[64] A package for a mixture of two right-tailed gamma distributions is available.[65]
Several other packages for R are available to fit mixture models; these include flexmix,[66] mcclust,[67] agrmt,[68] and mixdist.[69]
The statistical programming language SAS can also fit a variety of mixed distributions with the PROC FREQ procedure.
Ver también
- Overdispersion
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