Regresión lineal

PorJosé A. Alonso 22-febrero-201626-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)

Definir el procedimiento

   grafica :: [Double] -> [Double] -> IO ()

1	grafica :: [Double] -> [Double] -> IO ()

tal que (grafica xs ys) pinte 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 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     = fromIntegral (length 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 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 = fromIntegral (length 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

import Data.List

regresionLineal :: [Double] -> [Double] -> (Double,Double)
regresionLineal xs ys = (a,b)
    where  b = (n * sum [x*y | (x,y) <- zip xs ys] - sum xs * sum ys) /
               (n * sum [x^2 | x <- xs] - sum xs ^2)
           a = (sum ys - b * sum xs) / n
           n = genericLength xs

import Data.List

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

regresionLineal xs ys = (a,b)

where b = (n * sum [x*y | (x,y) <- zip xs ys] - sum xs * sum ys) /

(n * sum [x^2 | x <- xs] - sum xs ^2)

a = (sum ys - b * sum xs) / n

n = genericLength xs

Responder

En Maxima

ejX : [5,  7, 10, 12, 16, 20, 23, 27, 19, 14]$
ejY : [9, 11, 15, 16, 20, 24, 27, 29, 22, 20]$

regresionLineal (xs,ys) := 
    block ([n     : length (xs),
            sumX  : lsum (x,x,xs),
            sumY  : lsum (y,y,ys),
            sumX2 : xs . xs,
            sumXY : xs . ys,
            a,b],
           b     : (n*sumXY - sumX*sumY) / (n*sumX2 - sumX^2),
           a     : (sumY - b*sumX) / n,
           float ([a,b]))$ 

grafica (xs,ys) := block ([a,b],
  ( [a,b] : regresionLineal (xs,ys),
    plot2d ([[discrete,xs,ys],a+b*x],
            [x,0,10+lmax(xs)],
            [y,0,10+lmax(ys)],
            [style,[points,4,2,4],[lines,2,3]],
            [title, "Resultados experimentales"],
            [xlabel, "Variable independiente"],
            [ylabel, "Variable dependiente"],
            [legend, "Datos", "Ajuste"]
           )))$

ejX : [5, 7, 10, 12, 16, 20, 23, 27, 19, 14]$

ejY : [9, 11, 15, 16, 20, 24, 27, 29, 22, 20]$

regresionLineal (xs,ys) :=

block ([n : length (xs),

sumX : lsum (x,x,xs),

sumY : lsum (y,y,ys),

sumX2 : xs . xs,

sumXY : xs . ys,

a,b],

b : (n*sumXY - sumX*sumY) / (n*sumX2 - sumX^2),

a : (sumY - b*sumX) / n,

float ([a,b]))$

grafica (xs,ys) := block ([a,b],

( [a,b] : regresionLineal (xs,ys),

plot2d ([[discrete,xs,ys],a+b*x],

[x,0,10+lmax(xs)],

[y,0,10+lmax(ys)],

[style,[points,4,2,4],[lines,2,3]],

[title, "Resultados experimentales"],

[xlabel, "Variable independiente"],

[ylabel, "Variable dependiente"],

[legend, "Datos", "Ajuste"]

)))$

La gráfica se obtiene con grafica(ejX,ejY)

Responder

pendiente :: [Double] -> [Double] -> Double
pendiente xs ys = (n*xiyi - xi*yi) / (n*xi2 - xi^2)
    where n    = genericLength xs
          xiyi = sum (zipWith (*) xs ys)
          xi   = sum xs
          yi   = sum ys
          xi2  = sum (map (^2) xs)
  
ordenada :: [Double] -> [Double] -> Double
ordenada xs ys = (yi - xi * pendiente xs ys)/ n
    where n  = genericLength xs
          xi = sum xs
          yi = sum ys
 
regresionLineal :: [Double] -> [Double] -> (Double,Double)
regresionLineal xs ys= (ordenada xs ys, pendiente xs ys)

pendiente :: [Double] -> [Double] -> Double

pendiente xs ys = (n*xiyi - xi*yi) / (n*xi2 - xi^2)

where n = genericLength xs

xiyi = sum (zipWith (*) xs ys)

xi = sum xs

yi = sum ys

xi2 = sum (map (^2) xs)

ordenada :: [Double] -> [Double] -> Double

ordenada xs ys = (yi - xi * pendiente xs ys)/ n

where n = genericLength xs

xi = sum xs

yi = sum ys

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

regresionLineal xs ys= (ordenada xs ys, pendiente xs ys)

Responder

Gráfico con EasyPlot.

import Graphics.EasyPlot
import Data.List

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)
          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

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)

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

La librería EasyPlot se instala con Cabal ejecutando «cabal install easyplot».

Para ver la gráfica en Windows hay que cambiar X11 por Windows.

Responder

En windows es conveniente añadir la opción Interactive. Asímismo, se debe generalizar más a la hora de indicar el rango:

grafica :: [Double] -> [Double] -> IO Bool
grafica xs ys = do
    let (a,b) = regresionLineal xs ys
    let (mn,mx) = (minimum xs - 5 , maximum xs + 5)
    plot' [Interactive] Windows [
        Data2D [Title "Datos", Color Red, Style Points] [] (zip xs ys),
        Function2D [Title "Ajuste", Color Green, Style Lines] [Range mn mx] (\x -> a + b *x)
        ]

grafica :: [Double] -> [Double] -> IO Bool

grafica xs ys = do

let (a,b) = regresionLineal xs ys

let (mn,mx) = (minimum xs - 5 , maximum xs + 5)

plot' [Interactive] Windows [

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

Function2D [Title "Ajuste", Color Green, Style Lines] [Range mn mx] (\x -> a + b *x)

]

Responder

Con GnuPlot

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

Responder

Regresión lineal

Soluciones

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