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# bivariate density: discussion section week 5

# contour and 3d perspective plots of the function

#   f( x, y ) = 5 * ( x^2 + y ) / 4

# in the region 

#   0 < y < 1 - x^2

n.grid <- 500

x.grid <- seq( -1, 1, length = n.grid )

y.grid <- seq( 0, 1, length = n.grid )

f.grid <- matrix( NA, n.grid, n.grid )

for ( i in 1:n.grid ) {

  for ( j in 1:n.grid ) {

    f.grid[ i, j ] <- ifelse( y.grid[ j ] < 1 - x.grid[ i ]^2, 
      5 * ( x.grid[ i ]^2 + y.grid[ j ] ) / 4, NA )

  }

}

contour( x.grid, y.grid, f.grid, nlevels = 50, col = 'red',
  xlab = 'x', ylab = 'y' )

lines( x.grid, 1 - x.grid^2, lty = 2, lwd = 2, col = 'blue' )

library( plotly )

f.grid.t <- t( f.grid )

interactive.bivariate.density.perspective.plot <- 
  plot_ly( x = x.grid, y = y.grid, z = f.grid.t ) %>% 
  add_surface( contours = list( z = list( show = T, usecolormap = T,
  highlightcolor = '#ff0000', project = list( z = T ) ) ) ) %>%
  layout( title = 'Perspective plot of the bivariate density',
  scene = list( xaxis = list( title = 'x' ), 
  yaxis = list( title = 'y' ), zaxis = list( title = 'f' ),
  camera = list( eye = list( x = 1.5, y = 0.5, z = -0.5 ) ) ) ) %>% 
  colorbar( title = 'Bivariate Density' )

interactive.bivariate.density.perspective.plot

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# bivariate density: extra notes page 98

# contour and 3d perspective plots of the function

#   f( x, y ) = 21 * ( x^2 * y ) / 4 

# in the region 

#   0 < x^2 < y < 1

n.grid <- 500

x.grid <- seq( -1, 1, length = n.grid )

y.grid <- seq( 0, 1, length = n.grid )

f.grid <- matrix( NA, n.grid, n.grid )

for ( i in 1:n.grid ) {

  for ( j in 1:n.grid ) {

    f.grid[ i, j ] <- ifelse( x.grid[ i ]^2 < y.grid[ j ], 
      21 * x.grid[ i ]^2 * y.grid[ j ] / 4, NA )

  }

}

contour( x.grid, y.grid, f.grid, nlevels = 50, col = 'red',
  xlab = 'x', ylab = 'y' )

lines( x.grid, x.grid^2, lty = 2, lwd = 2, col = 'blue' )

library( plotly )

f.grid.t <- t( f.grid )

interactive.bivariate.density.perspective.plot <- 
  plot_ly( x = x.grid, y = y.grid, z = f.grid.t ) %>% 
  add_surface( contours = list( z = list( show = T, usecolormap = T,
  highlightcolor = '#ff0000', project = list( z = T ) ) ) ) %>%
  layout( title = 'Perspective plot of the bivariate density',
  scene = list( xaxis = list( title = 'x' ), 
  yaxis = list( title = 'y' ), zaxis = list( title = 'f' ),
  camera = list( eye = list( x = 1.5, y = 0.5, z = -0.5 ) ) ) ) %>% 
  colorbar( title = 'Bivariate Density' )

interactive.bivariate.density.perspective.plot

##########################################################################

# bivariate density: extra notes page 113

# contour and 3d perspective plots of the function

#   f( x, y ) = 24 * ( x^2 * y^2 ) / pi 

# in the region 

#   0 < x^2 + y^2 < 1

n.grid <- 500

x.grid <- seq( -1, 1, length = n.grid )

y.grid <- seq( -1, 1, length = n.grid )

f.grid <- matrix( NA, n.grid, n.grid )

for ( i in 1:n.grid ) {

  for ( j in 1:n.grid ) {

    f.grid[ i, j ] <- ifelse( x.grid[ i ]^2 + y.grid[ j ]^2 < 1, 
      24 * x.grid[ i ]^2 * y.grid[ j ]^2 / pi, NA )

  }

}

contour( x.grid, y.grid, f.grid, nlevels = 50, col = 'red',
  xlab = 'x', ylab = 'y' )

lines( x.grid, sqrt( 1 - x.grid^2 ), lty = 2, lwd = 2, col = 'blue' )

lines( x.grid, - sqrt( 1 - x.grid^2 ), lty = 2, lwd = 2, col = 'blue' )

library( plotly )

f.grid.t <- t( f.grid )

interactive.bivariate.density.perspective.plot <- 
  plot_ly( x = x.grid, y = y.grid, z = f.grid.t ) %>% 
  add_surface( contours = list( z = list( show = T, usecolormap = T,
  highlightcolor = '#ff0000', project = list( z = T ) ) ) ) %>%
  layout( title = 'Perspective plot of the bivariate density',
  scene = list( xaxis = list( title = 'x' ), 
  yaxis = list( title = 'y' ), zaxis = list( title = 'f' ),
  camera = list( eye = list( x = 1.5, y = 0.5, z = -0.5 ) ) ) ) %>% 
  colorbar( title = 'Bivariate Density' )

interactive.bivariate.density.perspective.plot

##########################################################################

# bivariate density extra notes page 114

# contour and 3d perspective plots of the function

#   f( x, y ) = 2 * exp( - ( x + 2 * y ) ) 

# in the region 

#   0 < x < 3 and 0 < y < 3

n.grid <- 500

x.grid <- seq( 0, 3, length = n.grid )

y.grid <- seq( 0, 3, length = n.grid )

f.grid <- matrix( NA, n.grid, n.grid )

for ( i in 1:n.grid ) {

  for ( j in 1:n.grid ) {

    f.grid[ i, j ] <- 2 * exp( - ( x.grid[ i ] + 2 * y.grid[ j ] ) )

  }

}

contour( x.grid, y.grid, f.grid, nlevels = 50, col = 'red',
  xlab = 'x', ylab = 'y' )

library( plotly )

f.grid.t <- t( f.grid )

interactive.bivariate.density.perspective.plot <- 
  plot_ly( x = x.grid, y = y.grid, z = f.grid.t ) %>% 
  add_surface( contours = list( z = list( show = T, usecolormap = T,
  highlightcolor = '#ff0000', project = list( z = T ) ) ) ) %>%
  layout( title = 'Perspective plot of the bivariate density',
  scene = list( xaxis = list( title = 'x' ), 
  yaxis = list( title = 'y' ), zaxis = list( title = 'f' ),
  camera = list( eye = list( x = 1.5, y = 0.5, z = -0.5 ) ) ) ) %>% 
  colorbar( title = 'Bivariate Density' )

interactive.bivariate.density.perspective.plot

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