*ANSWER* HW4 #4: graphical hint (ok, more than a hint)
Based on 1,000 runs. Solid straight line is the resulting average hypothesis.
http://www.venturephilosopher.com/?attachment_id=136 Dirk 
Re: *ANSWER* HW4 #4: graphical hint (ok, more than a hint)
Very nice plot. You can clearly see that the hypotheses are restricted to rotation about the origin. If you used R or Matlab (or Octave), could you please post the code to generate the plot?
Thank you. Juan 
Re: *ANSWER* HW4 #4: graphical hint (ok, more than a hint)
Quote:

Re: *ANSWER* HW4 #4: graphical hint (ok, more than a hint)
Juan et al.,
Here is the R code. It plots the graph, prints slope and intercept, and also calculates bias and variance. You will see three code sections  one for each of the hypothesis classes discussed in class and in this homework: h(x) = b (class) h(x) = ax+b (class) h(x) = ax (HW) Dirk ############################################# # Caltech Machine Learning April 2013 # HW4  Bias and variance (problems #47) # Clear workspace rm(list = ls()) # Set seed for random number generation set.seed(2013) # Set sample size N_sample = 1000 # Create sample and estimate expected value for the hypothesis using h(x) = b (same as in lecture example 1) # Initialize variable to store regression parameter for regression h(x) = ax+b (slope a will be forced to zero) a < NA b < NA variance < NA # Plot f(x) plot.new() # Plot f(x) = sin(pi*x) curve(sin(pi*x), 1, 1, main='h(x) = b', col="black") for (i in 1:N_sample) { x_values < runif(2, 1.0, 1.0) data_set < data.frame(x = x_values, y = sin(pi * x_values)) a[i] < 0.0 b[i] < 0.5 * (data_set[1,"y"] + data_set[2,"y"]) # Calculate variance for the specific hypothesis for this particular data set using g_bar(x) = 0.0 integrand < function(x) {(0.0a[i]*x+b[i])^2} # Dividing definitive integral by 2 to calculate exepcted value relative to x since range of x is [1,1] variance[i] < 0.5 * (integrate(integrand,1,1)$value) # Plot current h(x) abline(b[i], a[i], col="grey75") } # Plot expected value for hypothesis g abline(mean(b), mean(a), col="black") # Plot f(x) = sin(pi*x) again to overlay on top of chart par(new=T) curve(sin(pi*x), 1, 1, main='h(x) = b', col="black") # Print average regression parameters as exepcted value for hypothesis g mean(a) mean(b) # Calculate bias using g_bar(x) = 0.0 integrand < function(x) {(0.0sin(pi*x))^2} # Dividing definitive integral by 2 to calculate exepcted value relative to x since range of x is [1,1] bias < 0.5 * (integrate(integrand,1,1)$value) bias # Calculate variance using g_bar(x) = 0.77*x # Average over data set specific variances mean(variance) # Create sample and estimate expected value for the hypothesis using h(x) = ax+b (same as in lecture example 2) # Initialize variable to store regression parameter for regression h(x) = ax+b a < NA b < NA variance < NA # Plot f(x) plot.new() # Plot f(x) = sin(pi*x) curve(sin(pi*x), 1, 1, main='h(x) = ax+b', col="black") for (i in 1:N_sample) { x_values < runif(2, 1.0, 1.0) data_set < data.frame(x = x_values, y = sin(pi * x_values)) regression_params < lm(formula = data_set$y ~ data_set$x) a[i] < regression_params$coefficients[2] b[i] < regression_params$coefficients[1] # Calculate variance for the specific hypothesis for this particular data set using g_bar(x) = 0.77*x integrand < function(x) {(0.77*xa[i]*x+b[i])^2} # Dividing definitive integral by 2 to calculate exepcted value relative to x since range of x is [1,1] variance[i] < 0.5 * (integrate(integrand,1,1)$value) # Plot current h(x) abline(b[i], a[i], col="grey75") } # Plot expected value for hypothesis g abline(mean(b), mean(a), col="black") # Plot f(x) = sin(pi*x) again to overlay on top of chart par(new=T) curve(sin(pi*x), 1, 1, main='h(x) = ax+b', col="black") # Print average regression parameters as exepcted value for hypothesis g mean(a) mean(b) # Calculate bias using g_bar(x) = 0.77*x integrand < function(x) {(0.77*xsin(pi*x))^2} # Dividing definitive integral by 2 to calculate exepcted value relative to x since range of x is [1,1] bias < 0.5 * (integrate(integrand,1,1)$value) bias # Calculate variance using g_bar(x) = 0.77*x # Average over data set specific variances mean(variance) # Create sample and estimate expected value for the hypothesis using h(x) = ax (HW problem) # Initialize variable to store regression parameter for regression h(x) = ax+b (intercept b will be forced to zero) a < NA b < NA variance < NA # Plot f(x) plot.new() # Plot f(x) = sin(pi*x) curve(sin(pi*x), 1, 1, main='h(x) = ax', col="black") for (i in 1:N_sample) { x_values < runif(2, 1.0, 1.0) data_set < data.frame(x = x_values, y = sin(pi * x_values)) # Force intercept to zero through y ~ 0 + x model term regression_params < lm(formula = data_set$y ~ 0 + data_set$x) a[i] < regression_params$coefficients b[i] < 0.0 # Calculate variance for the specific hypothesis for this particular data set using g_bar(x) = 1.40*x integrand < function(x) {(1.40*xa[i]*x+b[i])^2} # Dividing definitive integral by 2 to calculate exepcted value relative to x since range of x is [1,1] variance[i] < 0.5 * (integrate(integrand,1,1)$value) # Plot current h(x) abline(b[i], a[i], col="grey75") } # Plot expected value for hypothesis g abline(mean(b), mean(a), col="black") # Plot f(x) = sin(pi*x) again to overlay on top of chart par(new=T) curve(sin(pi*x), 1, 1, main='h(x) = ax', col="black") # Print average regression parameters as exepcted value for hypothesis g mean(a) mean(b) # Calculate bias using g_bar(x) = 1.40*x integrand < function(x) {(1.40*xsin(pi*x))^2} # Dividing definitive integral by 2 to calculate exepcted value relative to x since range of x is [1,1] bias < 0.5 * (integrate(integrand,1,1)$value) bias # Calculate variance using g_bar(x) = 1.40*x # Average over data set specific variances mean(variance) 
Re: *ANSWER* HW4 #4: graphical hint (ok, more than a hint)
Dirk,
Thanks very much! It works like a charm. Juan 
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