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plot_Dbinary_vector.jl
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plot_Dbinary_vector.jl
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using PyPlot;
using Distributions;
using LaTeXStrings;
### Useful functions
## There are a number of alternative ways to calculate pdf and cdf inverse
dist_pdf(x) = pdf(Normal(0,1), x);
dist_cdf(x) = cdf(Normal(0,1), x);
# Note: inv_cdf(x) != 1.0 / cdf(Normal(0,1), x); #Not 1/fn but inverse function!!
include("inverse_cdf.jl"); #contains invnorm(), consider switching to invphi()
invphi(p) = sqrt(2) * erfinv(2 * p - 1.0) # variance 1 inverse phi function
#invphi(p) = 2.0 * erfinv(2 * p - 1.0) # variance 2 (1+1) inverse phi function
include("plotting_assist_functions.jl");
function setup_plot_D_binary_basic_variables(local_a = 0.5, local_c = -1)
## Plotting over D, D~ (+ve), and p optional
global use_plot_over_D_pos = false :: Bool;
global use_plot_over_D = false :: Bool;
global use_plot_over_p = true :: Bool;
# performance lines can be overlaid on plots in D-space to aid legibility
global use_overlay_performance_on_D = true :: Bool;
# simulation trajectories
global use_add_trajectories_to_plot = false :: Bool;
global sub_task_id_to_plot = 2;
global use_plot_measured_proportion_correct = false ::Bool;
#TODO: forward Euler trajectories not yet implemented for binary outputs
# separate components of flow field
global use_include_learning_term_in_flow = false :: Bool;
global use_include_internal_bias_term_in_flow = true :: Bool;
global use_include_external_bias_term_in_flow = false :: Bool;
## Space over which vector field is calculated / plotted
global no_points = 15; #17; #25;
#no_points = 10;
#no_y_points = no_points - 1;
# The no_y_points is to ensure that I plot the vector field in the right direction,
# julia is column major but matplot lib is row major which causes confusion!
# Set no_y_points = no_points - 1; to check if an error is thrown, no error means
# that the array access is correct.
global epsilon = 1e-7
global no_y_points = no_points;
global p = linspace(0+epsilon, 1-epsilon, no_points);
global p_y = linspace(0+epsilon, 1-epsilon, no_y_points);
global d_a = linspace(-3,3, no_points);
global d_b = linspace(-3,3, no_points);
global D_pos_scale = 20.0 :: Float64;
global D_scale = 20.0 :: Float64;
global p_scale = 0.5 :: Float64;
#debug vars
global Da = zeros(no_points);
global Db = zeros(no_y_points);
## Vector flow field variables
global deriv_p_a = zeros(no_points, no_y_points);
global deriv_p_b = zeros(no_points, no_y_points);
global p_deriv_D_a = zeros(no_points, no_y_points);
global p_deriv_D_b = zeros(no_points, no_y_points);
global deriv_D_a = zeros(no_points, no_y_points);
global deriv_D_b = zeros(no_points, no_y_points);
global deriv_D_a_pos = zeros(no_points, no_y_points);
global deriv_D_b_pos = zeros(no_points, no_y_points);
# Confusion parameter
global critic_dimensions = 4;
# perfect critic (overwritten if any of the following are active)
global C = eye(critic_dimensions)
#=
# equal mix critic
c = 1 / critic_dimensions; # currently equal confusion mix of all true critics
C = ones(critic_dimensions,critic_dimensions)
C *= c
A = eye(critic_dimensions) - C=#
# Probabilistic presentation of individual tasks
global prob_task = ones(1,critic_dimensions);
prob_task /= critic_dimensions;
#prob_task = [1, 0.001, 10, 10]; # manual tweaking
#prob_task /= sum(prob_task); # normalise, so I can use arbitrary units
# this influences Confustion matrix
for k = 1:critic_dimensions
C[k,:] = prob_task;
end
A = eye(critic_dimensions) - C;
if(local_c != -1)
global A = eye(critic_dimensions) - local_c;
end
# Input representation similarity parameter
global a = local_a; #0.5; #0.9;
global S = [1 a; a 1]
S /= S[1,1];
# Output correlation with +ve D
global O = [1; -1];
# Noise and external bias
global sigma = 1;
global R_ext = 1;
end
function calculate_binary_model_flow_vectors()
for i = 1:no_points
for j = 1:(no_y_points)
#####
#
# Calculation of change of difference in outputs
#
# positive association in plotting of D with reward (use d_a as d_a^~)
if (use_plot_over_D_pos)
temp_a = 0;
temp_b = 0;
if (use_include_learning_term_in_flow)
# *2 for R^{true} = (2p-1)
temp_a = 4 * cdf(Normal(0,sigma), (d_a[i]) ) * (1 - cdf(Normal(0,sigma), (d_a[i]) ) )
temp_b = 4 * cdf(Normal(0,sigma), (d_b[j]) ) * (1 - cdf(Normal(0,sigma), (d_b[j]) ) )
end
if (use_include_internal_bias_term_in_flow)
# equations for R^{true} = (2p-1)
temp_a += A[1,1] * (2 * cdf(Normal(0,sigma), (d_a[i]) ) - 1) * (2 * cdf(Normal(0,sigma), (d_a[i]) ) - 1);
temp_a += A[1,2] * (2 * cdf(Normal(0,sigma), (d_b[j]) ) - 1) * (2 * cdf(Normal(0,sigma), (d_a[i]) ) - 1);
temp_b += A[2,1] * (2 * cdf(Normal(0,sigma), (d_a[i]) ) - 1) * (2 * cdf(Normal(0,sigma), (d_b[j]) ) - 1);
temp_b += A[2,2] * (2 * cdf(Normal(0,sigma), (d_b[j]) ) - 1) * (2 * cdf(Normal(0,sigma), (d_b[j]) ) - 1);
end
if (use_include_external_bias_term_in_flow)
# Bias from other tasks
if(critic_dimensions > 2)
# a_multiplier assumes equal for all
a_multiplier = (critic_dimensions - 2) / critic_dimensions
#=temp_a += (2 * cdf(Normal(0,sigma),d_a[i]) - 1) * (-0.5 * R_ext);
temp_b += (2 * cdf(Normal(0,sigma),d_b[j]) - 1) * (-0.5 * R_ext);=#
#=temp_a += (2 * cdf(Normal(0,sigma),d_a[i]) - 1) * (-a_multiplier * R_ext);
temp_b += (2 * cdf(Normal(0,sigma),d_b[j]) - 1) * (-a_multiplier * R_ext);=#
for(k = 3:critic_dimensions)
temp_a += (2 * cdf(Normal(0,sigma), (d_a[i]) ) - 1) * (A[1,k] * R_ext);
temp_b += (2 * cdf(Normal(0,sigma), (d_b[j]) ) - 1) * (A[2,k] * R_ext);
end
end
end
# Multiply by probability of occurence of each task
temp_a *= prob_task[1];
temp_b *= prob_task[2];
# putting it all together
deriv_D_a_pos[i,j] = ( O[1] * S[1,1] * temp_a + O[2] * S[1,2] * temp_b );
deriv_D_b_pos[i,j] = ( O[1] * S[2,1] * temp_a + O[2] * S[2,2] * temp_b );
# multiply again by output encoding to give +ve D for success representation
deriv_D_a_pos[i,j] *= O[1];
deriv_D_b_pos[i,j] *= O[2];
end
######
#
# no correction for -ve association in plotting of D with reward (use d_a as d_a)
if (use_plot_over_D)
temp_a = 0;
temp_b = 0;
if (use_include_learning_term_in_flow)
# *2 for R^{true} = (2p-1)
temp_a = 4 * cdf(Normal(0,sigma), (d_a[i]*O[1]) ) * (1 - cdf(Normal(0,sigma), (d_a[i]*O[1]) ) )
temp_b = 4 * cdf(Normal(0,sigma), (d_b[j]*O[2]) ) * (1 - cdf(Normal(0,sigma), (d_b[j]*O[2]) ) )
end
if (use_include_internal_bias_term_in_flow)
# equations for R^{true} = (2p-1)
temp_a += A[1,1] * (2 * cdf(Normal(0,sigma), (d_a[i]*O[1]) ) - 1) * (2 * cdf(Normal(0,sigma), (d_a[i]*O[1]) ) - 1);
temp_a += A[1,2] * (2 * cdf(Normal(0,sigma), (d_b[j]*O[2]) ) - 1) * (2 * cdf(Normal(0,sigma), (d_a[i]*O[1]) ) - 1);
temp_b += A[2,1] * (2 * cdf(Normal(0,sigma), (d_a[i]*O[1]) ) - 1) * (2 * cdf(Normal(0,sigma), (d_b[j]*O[2]) ) - 1);
temp_b += A[2,2] * (2 * cdf(Normal(0,sigma), (d_b[j]*O[2]) ) - 1) * (2 * cdf(Normal(0,sigma), (d_b[j]*O[2]) ) - 1);
end
if (use_include_external_bias_term_in_flow)
# Bias from other tasks
if(critic_dimensions > 2)
# a_multiplier assumes equal for all
a_multiplier = (critic_dimensions - 2) / critic_dimensions
#=temp_a += (2 * cdf(Normal(0,sigma),d_a[i]) - 1) * (-0.5 * R_ext);
temp_b += (2 * cdf(Normal(0,sigma),d_b[j]) - 1) * (-0.5 * R_ext);=#
#=temp_a += (2 * cdf(Normal(0,sigma),d_a[i]) - 1) * (-a_multiplier * R_ext);
temp_b += (2 * cdf(Normal(0,sigma),d_b[j]) - 1) * (-a_multiplier * R_ext);=#
for(k = 3:critic_dimensions)
temp_a += (2 * cdf(Normal(0,sigma), (d_a[i]*O[1]) ) - 1) * (A[1,k] * R_ext);
temp_b += (2 * cdf(Normal(0,sigma), (d_b[j]*O[2]) ) - 1) * (A[2,k] * R_ext);
end
end
end
# Multiply by probability of occurence of each task
temp_a *= prob_task[1];
temp_b *= prob_task[2];
# putting it all together
deriv_D_a[i,j] = ( O[1] * S[1,1] * temp_a + O[2] * S[1,2] * temp_b );
deriv_D_b[i,j] = ( O[1] * S[2,1] * temp_a + O[2] * S[2,2] * temp_b );
end
#####
#
# Calculation of change of probability of outcome
#
if (use_plot_over_p)
Da[i] = invphi(p[i]);
Db[j] = invphi(p_y[j]);
p_temp_a = 0;
p_temp_b = 0;
if (use_include_learning_term_in_flow)
p_temp_a = 4 * cdf(Normal(0,sigma),Da[i]) * (1 - cdf(Normal(0,sigma),Da[i]))
p_temp_b = 4 * cdf(Normal(0,sigma),Db[j]) * (1 - cdf(Normal(0,sigma),Db[j]))
end
if (use_include_internal_bias_term_in_flow)
# equations for R^{true} = (2p-1)
p_temp_a += A[1,1] * (2 * p[i] - 1) * (2 * p[i] - 1);
p_temp_a += A[1,2] * (2 * p[j] - 1) * (2 * p[i] - 1);
p_temp_b += A[2,1] * (2 * p[i] - 1) * (2 * p[j] - 1);
p_temp_b += A[2,2] * (2 * p[j] - 1) * (2 * p[j] - 1);
end
if (use_include_external_bias_term_in_flow)
# Bias from other tasks
if(critic_dimensions > 2)
a_multiplier = (critic_dimensions - 2) / critic_dimensions
#p_temp_a += (2 * p[i] - 1) * (-0.5 * R_ext);
#p_temp_b += (2 * p[j] - 1) * (-0.5 * R_ext);
#p_temp_a += (2 * p[i] - 1) * (-a_multiplier * R_ext);
#p_temp_b += (2 * p[j] - 1) * (-a_multiplier * R_ext);
for(k = 3:critic_dimensions)
p_temp_a += (2 * p[i] - 1) * (A[1,k] * R_ext);
p_temp_b += (2 * p[j] - 1) * (A[2,k] * R_ext);
end
end
end
# Multiply by probability of occurence of each task
p_temp_a *= prob_task[1];
p_temp_b *= prob_task[2];
# putting it all together
p_deriv_D_a[i,j] = (O[1] * S[1,1] * p_temp_a + O[2] * S[1,2] * p_temp_b);
p_deriv_D_b[i,j] = (O[1] * S[2,1] * p_temp_a + O[2] * S[2,2] * p_temp_b);
# we need to transform derivatives to D_pos space
p_deriv_D_a[i,j] *= O[1];
p_deriv_D_b[i,j] *= O[2];
# and we scale everything by the pdf of the underlying probability
deriv_p_a[i,j] = pdf(Normal(0,sigma), Da[i]) * p_deriv_D_a[i,j];
deriv_p_b[i,j] = pdf(Normal(0,sigma), Db[j]) * p_deriv_D_b[i,j];
end
end
end
end # end function calculate_binary_model_flow_vectors()
function plot_binary_model_flow_vectors()
## Plotting
print("Plotting...\n")
filename_change = "unbounded_post"
filename_change = "binary_new"
#filename_change = "rescaled_new"
file_name_change = "blah"
filename_base = string("vector_field_", filename_change);
filename_quiver = string("quiver_",filename_base,".pdf")
filename_stream = string("stream_",filename_base,".pdf")
#figure();
#quiver(p,p,deriv_p_a', deriv_p_b');
#quiver(p,p_y,deriv_p_a', deriv_p_b');
#savefig(filename_quiver);
#figure();
#streamplot(p,p,deriv_p_a',deriv_p_b');
#streamplot(p,p_y,deriv_p_a',deriv_p_b')
#savefig(filename_stream);
if (use_plot_over_D_pos)
## Difference in positive outputs view
figure();
#streamplot(d_a,d_b,deriv_D_a',deriv_D_b');
quiver(d_a,d_b,deriv_D_a_pos',deriv_D_b_pos', units="width", scale=D_pos_scale);
xtxt = latexstring("D_1^+");
ytxt = latexstring("D_2^+");
xlabel(xtxt)
ylabel(ytxt) # L"D_2"
title("Similarity s=$a");
if (critic_dimensions > 2)
titletxt = latexstring();
title("Similarity s=$a, R_ext = $R_ext, no external processes = $(critic_dimensions-2)");
end
if ( use_add_trajectories_to_plot )
scalar_for_d_pos = 10.0 / sigma; # since sigma in simulation is 10 times sigma here
for j = 1:no_subjects
local_prop_sub_1_correct = zeros(no_blocks_in_experiment);
local_prop_sub_2_correct = zeros(no_blocks_in_experiment);
for i = 1:no_blocks_in_experiment
#scatter(latest_experiment_results.subjects_task[j,sub_task_id].blocks[i].proportion_task_correct[1], latest_experiment_results.subjects_task[j,sub_task_id].blocks[i].proportion_task_correct[2], marker="o", c="c")
local_prop_sub_1_correct[i] = (exp_results[1].subjects_task[j,sub_task_id_to_plot].blocks[i].noise_free_positive_output[sub_task_id_to_plot,1]) / scalar_for_d_pos;
local_prop_sub_2_correct[i] = (exp_results[1].subjects_task[j,sub_task_id_to_plot].blocks[i].noise_free_positive_output[sub_task_id_to_plot,2]) / scalar_for_d_pos;
end
plot(local_prop_sub_1_correct, local_prop_sub_2_correct, "r", zorder=1)
#print("",local_prop_sub_1_correct, local_prop_sub_2_correct, "\n-----\n")
end
for j = 1:no_subjects
for i = 1:no_blocks_in_experiment
# start point
scatter(exp_results[1].subjects_task[j,sub_task_id_to_plot].blocks[1].noise_free_positive_output[sub_task_id_to_plot,1] / scalar_for_d_pos, exp_results[1].subjects_task[j,sub_task_id_to_plot].blocks[1].noise_free_positive_output[sub_task_id_to_plot,2] / scalar_for_d_pos, marker="s", c="r", s=40, zorder=2)
# end point
scatter(exp_results[1].subjects_task[j,sub_task_id_to_plot].blocks[end].noise_free_positive_output[sub_task_id_to_plot,1] /scalar_for_d_pos, exp_results[1].subjects_task[j,sub_task_id_to_plot].blocks[end].noise_free_positive_output[sub_task_id_to_plot,2] / scalar_for_d_pos, marker="D", c="g", s=60, zorder=3)
end
end
end
if ( use_overlay_performance_on_D )
overlay_level_80 = invphi(0.8);
overlay_level_90 = invphi(0.9);
overlay_level_95 = invphi(0.95);
overlay_level_99 = invphi(0.99);
performance_overlay = ones(no_points);
plot(performance_overlay * overlay_level_80, d_a, linewidth=2, c="c", zorder=0);
#plot(performance_overlay * overlay_level_90, d_a, linewidth=2, c="m", zorder=0);
plot(performance_overlay * overlay_level_95, d_a, linewidth=2, c="y", zorder=0);
plot(performance_overlay * overlay_level_99, d_a, linewidth=2, c="g", zorder=0);
plot(-performance_overlay * overlay_level_80, d_a, linewidth=2, c="c", zorder=0);
#plot(-performance_overlay * overlay_level_90, d_a, linewidth=2, c="m", zorder=0);
plot(-performance_overlay * overlay_level_95, d_a, linewidth=2, c="y", zorder=0);
plot(-performance_overlay * overlay_level_99, d_a, linewidth=2, c="g", zorder=0);
plot(d_a, performance_overlay * overlay_level_80, linewidth=2, c="c", zorder=0);
#plot(d_a, performance_overlay * overlay_level_90, linewidth=2, c="m", zorder=0);
plot(d_a, performance_overlay * overlay_level_95, linewidth=2, c="y", zorder=0);
plot(d_a, performance_overlay * overlay_level_99, linewidth=2, c="g", zorder=0);
plot(d_a, -performance_overlay * overlay_level_80, linewidth=2, c="c", zorder=0);
#plot(d_a, -performance_overlay * overlay_level_90, linewidth=2, c="m", zorder=0);
plot(d_a, -performance_overlay * overlay_level_95, linewidth=2, c="y", zorder=0);
plot(d_a, -performance_overlay * overlay_level_99, linewidth=2, c="g", zorder=0);
#plot(d_a, Db_null);
## x=0 and y=0 lines for visual inspection
origin = zeros(no_points);
#origin_space = linspace(-100,100,no_points);
plot(origin, d_a, linewidth=1, c="0.75", zorder=-1);
plot(d_b, origin, linewidth=1, c="0.75", zorder=-1);
end
end
if (use_plot_over_D)
## Difference in outputs view
figure();
#streamplot(d_a,d_b,deriv_D_a',deriv_D_b');
quiver(d_a,d_b,deriv_D_a',deriv_D_b', units="width", scale=D_scale);
xtxt = latexstring("D_1");
ytxt = latexstring("D_2");
xlabel(xtxt)
ylabel(ytxt) # L"D_2"
title("Similarity s=$a");
if (critic_dimensions > 2)
titletxt = latexstring();
title("Similarity s=$a, R_ext = $R_ext, no external processes = $(critic_dimensions-2)");
end
if ( use_overlay_performance_on_D )
overlay_level_80 = invphi(0.8);
overlay_level_90 = invphi(0.9);
overlay_level_95 = invphi(0.95);
overlay_level_99 = invphi(0.99);
performance_overlay = ones(no_points);
plot(performance_overlay * overlay_level_80, d_a, linewidth=2, c="c", zorder=0);
#plot(performance_overlay * overlay_level_90, d_a, linewidth=2, c="m", zorder=0);
plot(performance_overlay * overlay_level_95, d_a, linewidth=2, c="y", zorder=0);
plot(performance_overlay * overlay_level_99, d_a, linewidth=2, c="g", zorder=0);
plot(-performance_overlay * overlay_level_80, d_a, linewidth=2, c="c", zorder=0);
#plot(-performance_overlay * overlay_level_90, d_a, linewidth=2, c="m", zorder=0);
plot(-performance_overlay * overlay_level_95, d_a, linewidth=2, c="y", zorder=0);
plot(-performance_overlay * overlay_level_99, d_a, linewidth=2, c="g", zorder=0);
plot(d_a, performance_overlay * overlay_level_80, linewidth=2, c="c", zorder=0);
#plot(d_a, performance_overlay * overlay_level_90, linewidth=2, c="m", zorder=0);
plot(d_a, performance_overlay * overlay_level_95, linewidth=2, c="y", zorder=0);
plot(d_a, performance_overlay * overlay_level_99, linewidth=2, c="g", zorder=0);
plot(d_a, -performance_overlay * overlay_level_80, linewidth=2, c="c", zorder=0);
#plot(d_a, -performance_overlay * overlay_level_90, linewidth=2, c="m", zorder=0);
plot(d_a, -performance_overlay * overlay_level_95, linewidth=2, c="y", zorder=0);
plot(d_a, -performance_overlay * overlay_level_99, linewidth=2, c="g", zorder=0);
#plot(d_a, Db_null);
## x=0 and y=0 lines for visual inspection
origin = zeros(no_points);
#origin_space = linspace(-100,100,no_points);
plot(origin, d_a, linewidth=1, c="0.75", zorder=-1);
plot(d_b, origin, linewidth=1, c="0.75", zorder=-1);
end
end
if (use_plot_over_p)
## probabilistic view
figure();
##streamplot(d_a,d_b,deriv_D_a',deriv_D_b');
quiver(p,p_y,deriv_p_a',deriv_p_b', units="width", scale=p_scale);
xtxt = latexstring("p_1");
ytxt = latexstring("p_2");
xlabel(xtxt)
ylabel(ytxt) # L"D_2"
aa = abs(a);
title("Similarity s=$aa");
if (critic_dimensions > 2)
titletxt = latexstring();
title("Similarity s=$aa, R_ext = $R_ext, no external processes = $(critic_dimensions-2)");
end
axis([-0.02,1.02,-0.02,1.02]);
end
if (use_plot_over_p && use_add_trajectories_to_plot)
if (critic_dimensions == 2)
add_trajectories_to_linear_p_plot(exp_results[1],sub_task_id_to_plot);
elseif (critic_dimensions == 4)
add_biased_trajectories_to_linear_p_plot(exp_results[1],sub_task_id_to_plot);
end
end
end # end function plot_binary_model_flow_vectors()
function run_binary_model_flow(local_similarity = 0.7)
setup_plot_D_binary_basic_variables(local_similarity)
calculate_binary_model_flow_vectors()
plot_binary_model_flow_vectors()
end