José A. AlonsoDadas dos listas de valores
xs = [x(1), x(2), ..., x(n)] ys = [y(1), y(2), ..., y(n)] |
la ecuación de la recta de regresión de ys sobre xs es y = a+bx, donde
b = (nΣx(i)y(i) - Σx(i)Σy(i)) / (nΣx(i)² - (Σx(i))²) a = (Σy(i) - bΣx(i)) / n |
Definir la función
regresionLineal :: [Double] -> [Double] -> (Double,Double) |
tal que (regresionLineal xs ys) es el par (a,b) de los coeficientes de la recta de regresión de ys sobre xs. Por ejemplo, para los valores
ejX, ejY :: [Double] ejX = [5, 7, 10, 12, 16, 20, 23, 27, 19, 14] ejY = [9, 11, 15, 16, 20, 24, 27, 29, 22, 20] |
se tiene
λ> regresionLineal ejX ejY (5.195045748716805,0.9218924347243919) |
Para comprobar la definición, se importa la librería Graphics.Gnuplot.Simple y se define el procedimiento
grafica :: [Double] -> [Double] -> IO ()
grafica xs ys =
plotPathsStyle
[YRange (0,10+mY)]
[(defaultStyle {plotType = Points,
lineSpec = CustomStyle [LineTitle "Datos",
PointType 2,
PointSize 2.5]},
zip xs ys),
(defaultStyle {plotType = Lines,
lineSpec = CustomStyle [LineTitle "Ajuste",
LineWidth 2]},
[(x,a+b*x) | x <- [0..mX]])]
where (a,b) = regresionLineal xs ys
mX = maximum xs
mY = maximum ys |
tal que (grafica xs ys) pinta los puntos correspondientes a las listas de valores xs e ys y su recta de regresión. Por ejemplo, con (grafica ejX ejY) se obtiene el siguiente dibujo
Soluciones
import Data.List (genericLength) import Graphics.Gnuplot.Simple ejX, ejY :: [Double] ejX = [5, 7, 10, 12, 16, 20, 23, 27, 19, 14] ejY = [9, 11, 15, 16, 20, 24, 27, 29, 22, 20] regresionLineal :: [Double] -> [Double] -> (Double,Double) regresionLineal xs ys = (a,b) where n = genericLength xs sumX = sum xs sumY = sum ys sumX2 = sum (zipWith (*) xs xs) sumY2 = sum (zipWith (*) ys ys) sumXY = sum (zipWith (*) xs ys) b = (n*sumXY - sumX*sumY) / (n*sumX2 - sumX^2) a = (sumY - b*sumX) / n grafica :: [Double] -> [Double] -> IO () grafica xs ys = plotPathsStyle [YRange (0,10+mY)] [(defaultStyle {plotType = Points, lineSpec = CustomStyle [LineTitle "Datos", PointType 2, PointSize 2.5]}, zip xs ys), (defaultStyle {plotType = Lines, lineSpec = CustomStyle [LineTitle "Ajuste", LineWidth 2]}, [(x,a+b*x) | x <- [0..mX]])] where (a,b) = regresionLineal xs ys mX = maximum xs mY = maximum ys |