Regresión lineal

PorJosé A. Alonso 20-febrero-201726-marzo-2022

Dadas dos listas de valores

   xs = [x(1), x(2), ..., x(n)]
   ys = [y(1), y(2), ..., y(n)]

1 2	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

1 2	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)

1	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]

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)

1 2	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

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

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

6 Comentarios

regresionLineal :: [Double] -> [Double] -> (Double,Double)
regresionLineal xs ys = (a,b)
  where
    sxy = (sum . zipWith (*) xs) ys
    sx  = sum xs
    sx2 = (sum . zipWith (*) xs) xs
    sy  = sum ys
    n   = genericLength xs
    b   = (n * sxy - sx * sy) / (n * sx2 - sx^2)
    a   = (sy - b * sx) / n

regresionLineal :: [Double] -> [Double] -> (Double,Double)

regresionLineal xs ys = (a,b)

where

sxy = (sum . zipWith (*) xs) ys

sx = sum xs

sx2 = (sum . zipWith (*) xs) xs

sy = sum ys

n = genericLength xs

b = (n * sxy - sx * sy) / (n * sx2 - sx^2)

a = (sy - b * sx) / n

Responder

regresionLineal :: [Double] -> [Double] -> (Double,Double)
regresionLineal xs ys = ((d-b*c)/n,b)
  where n = genericLength xs
        d = sum ys
        c = sum xs
        b = (n*sum (zipWith (*) xs ys) - d*c)/(n*q - c^2)
        q = sum (map (^2) xs)

regresionLineal :: [Double] -> [Double] -> (Double,Double)

regresionLineal xs ys = ((d-b*c)/n,b)

where n = genericLength xs

d = sum ys

c = sum xs

b = (n*sum (zipWith (*) xs ys) - d*c)/(n*q - c^2)

q = sum (map (^2) xs)

Responder

import Data.List

regresionLineal :: [Double] -> [Double] -> (Double,Double)
regresionLineal xs ys = (a , b)
  where n  = genericLength xs
        x  = sum xs
        y  = sum ys
        x2 = sum $ map (^2) xs
        xy = sum $ zipWith (*) xs ys
        b  = (n*xy - x*y) / (n*x2 - x^2)
        a  =  (y - b*x) / n

import Data.List

regresionLineal :: [Double] -> [Double] -> (Double,Double)

regresionLineal xs ys = (a , b)

where n = genericLength xs

x = sum xs

y = sum ys

x2 = sum $ map (^2) xs

xy = sum $ zipWith (*) xs ys

b = (n*xy - x*y) / (n*x2 - x^2)

a = (y - b*x) / n

Responder

Usando la librería EasyPlot para las gráficas.

import Graphics.EasyPlot
import Data.List (genericLength)

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 Bool
grafica xs ys = 
    plot X11
         [Data2D     [Title "Datos", Color Red]   []          (zip xs ys),
          Function2D [Title "Ajuste", Color Blue] [Range 0 m] (\x -> a+b*x) ]
    where (a,b) = regresionLineal xs ys                      
          m     = 10 + maximum xs

import Graphics.EasyPlot

import Data.List (genericLength)

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 Bool

grafica xs ys =

plot X11

[Data2D [Title "Datos", Color Red] [] (zip xs ys),

Function2D [Title "Ajuste", Color Blue] [Range 0 m] (\x -> a+b*x) ]

where (a,b) = regresionLineal xs ys

m = 10 + maximum xs

Responder

regresionLineal :: [Double] -> [Double] -> (Double,Double)
regresionLineal xs ys = (a,b)
  where b = covarianza xs ys / varianza xs
        a = media ys - b * media xs

media :: [Double] -> Double
media xs = sum xs / genericLength xs

varianza :: [Double] -> Double
varianza xs = media ys - media xs ^ 2
  where ys = map (^2) xs 
            
covarianza :: [Double] -> [Double] -> Double
covarianza xs ys = media zs - media xs * media ys
  where zs = zipWith (*) xs ys

regresionLineal :: [Double] -> [Double] -> (Double,Double)

regresionLineal xs ys = (a,b)

where b = covarianza xs ys / varianza xs

a = media ys - b * media xs

media :: [Double] -> Double

media xs = sum xs / genericLength xs

varianza :: [Double] -> Double

varianza xs = media ys - media xs ^ 2

where ys = map (^2) xs

covarianza :: [Double] -> [Double] -> Double

covarianza xs ys = media zs - media xs * media ys

where zs = zipWith (*) xs ys

Responder

Usando la librería Statistics.LinearRegression

import Statistics.LinearRegression
import qualified Data.Vector as V

regresionLineal :: [Double] -> [Double] -> (Double,Double)
regresionLineal xs ys =
  linearRegression (V.fromList xs) (V.fromList ys)

import Statistics.LinearRegression

import qualified Data.Vector as V

regresionLineal :: [Double] -> [Double] -> (Double,Double)

regresionLineal xs ys =

linearRegression (V.fromList xs) (V.fromList ys)

Responder

Regresión lineal

Soluciones

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