pushing after approximate counting chapter

Author: leios

contents/IFS/IFS.html

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+using Test 
+
+# This function takes 
+#     - v: value in register
+#     - a: a  scaling value for the logarithm based on Morris's paper
+# It returns n(v,a), the approximate count
+function n(v, a)
+    a*((1+1/a)^v-1)  
+end
+
+# This function takes
+#    - v: value in register
+#    - a: a scaling value for the logarithm based on Morris's paper
+# It returns a new value for v
+function increment(v, a)
+    # delta is the probability of incrementing our counter
+    delta = 1/(n(v+1, a)-n(v, a))
+
+    if rand() <= delta
+        return v += 1
+    else
+        return v
+    end
+end
+
+# This simulates counting and takes
+#     - n_items: number of items to count and loop over
+#     - a: a scaling value for the logarithm based on Morris's paper
+# It returns n(v,a), the approximate count
+function approximate_count(n_items, a)
+    v = 0
+    for i = 1:n_items
+        v = increment(v, a)
+    end
+
+    return n(v, a)
+end
+
+# This function takes
+#     - n_trials: the number of counting trials
+#     - n_items: the number of items to count to
+#     - a: a scaling value for the logarithm based on Morris's paper
+#     - threshold: the maximum percent error allowed
+# It returns a true / false test value
+function test_approximate_count(n_trials, n_items, a, threshold)
+    samples = [approximate_count(n_items, a) for i = 1:n_trials]
+
+    avg = sum(samples)/n_trials
+
+    @test ((avg - n_items) / n_items < threshold)
+end
+
+@testset "Counting Tests, 100 trials" begin
+    println("testing 1,000, a = 30, 1% error")
+        test_approximate_count(0.1, 100, 1000, 30)
+    println("testing 12,345, a = 10, 1% error")
+        test_approximate_count(0.1, 100, 12345, 10)
+    # Note: with a lower a, we need more trials, so a higher % error here.
+    println("testing 222,222, a = 0.5, 10% error")
+        test_approximate_count(0.1, 100, 222222, 0.5)
+end

contents/affine_transformations/affine_transformations.html

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+using DelimitedFiles
+
+# n = number of events
+# prob = probability of incrementing counter
+function random_count(n, prob; a=30, prob_calc=false)
+    v = 0
+    base = 1+1/a
+    output = zeros(n)
+
+    if prob_calc
+        prob = 1
+        v = 1
+    end
+
+    if rand() < prob
+        output[1] = 1/prob
+    end
+
+    for i = 2:n
+        if prob_calc
+            prob = 1/((a*(base^(v+1)-1))-output[i-1])
+
+            if rand() <= prob
+                v += 1
+            end
+            output[i] = a*(base^v-1)
+        else
+            if rand() <= prob
+                output[i] = output[i-1]+1/prob
+            else
+                output[i] = output[i-1]
+            end
+        end
+    end
+
+    return output
+end
+
+# m = number of counting trials
+# l = number of saved trials
+function multi_count(n, m, l, prob; a=30, prob_calc=false, file_mod="",
+                     stops = [100000, 500000, 1000000])
+    out = zeros(n, l)
+    extremes = zeros(n, 2)
+
+    chosen_set = zeros(Int, l)
+    
+    for i = 1:l
+        chosen_number = rand(1:m)
+        while chosen_number in chosen_set
+            chosen_number = rand(1:m)
+        end
+        chosen_set[i] = chosen_number
+    end
+
+    histograms = zeros(Float64, m, length(stops))
+
+    out_count = 1
+    for i = 1:m
+        current_dist = random_count(n, prob; a, prob_calc)
+        if i == 1
+            extremes[:,1] .= current_dist
+            extremes[:,2] .= current_dist
+        else
+            for j = 1:n
+                if current_dist[j] < extremes[j,1]
+                    extremes[j,1] = current_dist[j]
+                end
+                if current_dist[j] > extremes[j,2]
+                    extremes[j,2] = current_dist[j]
+                end
+            
+            end
+        end
+
+        for j = 1:length(stops)
+            histograms[i,j] = current_dist[stops[j]]
+        end
+
+        if i in chosen_set
+            out[:,out_count] = current_dist
+            out_count += 1
+        end
+    end
+
+    output_file = open("out"*file_mod*".dat", "w")
+    writedlm(output_file, out)
+    close(output_file)
+
+    extreme_output_file = open("extremes"*file_mod*".dat", "w")
+    writedlm(extreme_output_file, extremes)
+    close(extreme_output_file)
+
+    formatted_histograms = [[],[],[]]
+    # Going through the histogram data to put it into the right format
+    for j = 1:length(stops)
+
+        max = floor(Int,maximum(histograms[:,j]))
+        min = floor(Int,minimum(histograms[:,j]))
+
+        println(min, '\t', max, '\t', sum(histograms[:,j])/m)
+
+        temp_array = zeros(max - min+1)
+
+        for i = 1:m
+
+            temp_array[floor(Int,histograms[i,j])-min+1] += 1
+
+        end
+
+        formatted_histograms[j] = temp_array
+    end
+
+    # output histograms into different files for each one
+
+    for i = 1:length(stops)
+        histogram_file = open("histogram_" * string(i)*file_mod* ".dat", "w")
+        writedlm(histogram_file, formatted_histograms[i])
+        close(histogram_file)
+    end
+    return formatted_histograms
+end
+

contents/approximate_counting/approximate_counting.html

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+set terminal epslatex standalone color
+
+#set output "histexp_1.tex"
+#set output "histexp_2.tex"
+set output "histexp_3.tex"
+
+set size square
+
+#set title "True count of 10,000"
+#set title "True count of 500,000"
+set title "True count of 1,000,000"
+
+set xlabel ''
+#set xrange [-255:9745]
+#set xtics ("6000" -255, "10,000" 3745, "16,000" 9745)
+
+#set xrange [352000:644000]
+#set xrange[-1458:538542]
+#set xtics ("320,000" -1458, "500,000" 178542, "860,000" 538542)
+
+#set xrange [808000:1240000]
+set xrange[-19374:1100626]
+set xtics ("600,0000" -19374, "1,000,000" 380626, "1,720,000" 1100626)
+
+#set ylabel 'Approximate count $\left( \times 10^{5} \right)$'
+#set ytics ("0" 0, "2" 200000, "4" 400000, "6" 600000, "8" 800000, "10" 1000000)
+
+#plot "histogram_1exp.dat" w l lw 10 title ""
+#plot "histogram_2exp.dat" w l lw 10 title ""
+plot "histogram_3exp.dat" w l lw 10 title ""
+

contents/approximate_counting/code/julia/approximate_counting.jl

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+set terminal epslatex standalone color
+
+#set output "hist_1.tex"
+#set output "hist_2.tex"
+set output "hist_3.tex"
+
+set size square
+
+#set title "True count of 10,000"
+#set title "True count of 500,000"
+set title "True count of 1,000,000"
+
+#set xlabel ''
+#set xrange [0:40000]
+#set xtics ("0" 0, "20,000" 20000, "40,000" 40000)
+
+#set xrange [352000:644000]
+#set xrange[10000:310000]
+#set xtics ("350,000" 10000, "500,000" 160000, "650,000" 310000)
+
+#set xrange [808000:1240000]
+set xrange[-10000:490000]
+set xtics ("750,0000" -10000, "1,000,000" 240000, "1,250,000" 490000)
+
+#set ylabel 'Approximate count $\left( \times 10^{5} \right)$'
+#set ytics ("0" 0, "2" 200000, "4" 400000, "6" 600000, "8" 800000, "10" 1000000)
+
+#plot "histogram_1.dat" w l lw 10 title ""
+#plot "histogram_2.dat" w l lw 10 title ""
+plot "histogram_3.dat" w l lw 10 title ""
+

contents/approximate_counting/res/a_change.png

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+set terminal epslatex standalone color
+#set terminal pdf enhanced
+#set style fill transparent solid 0.3
+
+#set output "check.tex"
+set output "checkexp.tex"
+#set size square
+
+set title "Approximate counting of 1,000,000 items"
+
+set xlabel 'True number of items $\left( \times 10^{5} \right)$'
+set xtics ("0" 0, "2" 200000, "4" 400000, "6" 600000, "8" 800000, "10" 1000000)
+
+set ylabel 'Approximate count $\left( \times 10^{5} \right)$'
+set ytics ("0" 0, "2" 200000, "4" 400000, "6" 600000, "8" 800000, "10" 1000000, "12" 1200000, "14" 1400000)
+
+#plot  "extremes.dat" u 0:1:2 w filledcu lc "gray" title "", for [i=1:10] "out.dat" u 0:i w l lw 3 title "", x w l lw 3 dt 3 lc "black" title ""
+plot  "extremesexp.dat" u 0:1:2 w filledcu lc "gray" title "", for [i=1:10] "outexp.dat" u 0:i w l lw 3 title "", x w l lw 3 dt 3 lc "black" title ""
+

contents/approximate_counting/res/approximations.png

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+set terminal epslatex standalone color
+
+set output "a_change.tex"
+#set size square
+
+set title '$n_v$ vs $a$ for $v=255$'
+
+set xlabel '$a$'
+set xtics ("25" 0, "27.5" 5, "30" 10, "32.5" 15, "35" 20)
+
+set ylabel '$n_v (\times 10^5)$'
+set ytics ("0" 0, "1" 100000, "2" 200000, "3" 300000, "4" 400000, "5" 500000)
+
+plot for [i=1:10] "a_change.dat" w l lw 3 title ""
+

contents/approximate_counting/res/approximationsexp.png

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+set terminal epslatex standalone color
+
+set output "deltas.tex"
+#set size square
+
+set title '$\Delta$ vs $v$'
+
+set xlabel '$v$'
+
+set ylabel '$\Delta$ (logscale)'
+set logscale y
+
+plot for [i=1:10] "deltas.dat" w l lw 3 title ""
+