Resolution dependence of the information-preserving compression¶

In [1]:

using BitInformation, PyPlot

In [2]:

N = 100_000
x = zeros(N)
y = zeros(N)

x[1] = randn()
y[1] = randn()

cx = 0.95
cy = 0.999

for i in 2:N
    x[i] = cx*x[i-1] + sqrt(1-cx^2)*randn()
    y[i] = cy*y[i-1] + sqrt(1-cy^2)*randn()
end

In [3]:

# subsample and m last time steps to plot
sub = 10
m = 1000

Out[3]:

In [4]:

hcat(bitinformation(x)[12:20], bitinformation(y)[12:20])

Out[4]:

9×2 Matrix{Float64}:
 0.0266057   0.404852
 0.00152374  0.216059
 0.0         0.0862215
 0.0         0.0162929
 0.0         0.000569383
 0.0         5.96659e-5
 0.0         0.0
 0.0         0.0
 0.0         0.0

In [5]:

hcat(bitinformation(x[1:sub:end])[12:20], bitinformation(y[1:sub:end])[12:20])

Out[5]:

9×2 Matrix{Float64}:
 0.0  0.124283
 0.0  0.0311626
 0.0  0.00262315
 0.0  0.000871424
 0.0  0.0
 0.0  0.0
 0.0  0.0
 0.0  0.0
 0.0  0.0

In [6]:

bitinformation(randn(N))[1:12]

Out[6]:

12-element Vector{Float64}:
 0.0
 0.0
 0.0
 0.0
 0.0
 0.0
 0.0
 0.0
 0.0
 0.0
 0.0
 0.0

In [11]:

fig,axs = subplots(2,3,sharex=true,sharey=true,figsize=(9,4)) 

c = "teal"

axs[1,3].plot(randn(m),c,lw=3)
axs[1,2].plot(x[1:m],c,lw=3)
axs[1,1].plot(y[1:m],c,lw=3,label="full precision")

axs[2,3].plot(1:sub:m,randn(m÷sub),c,lw=3)
axs[2,2].plot(1:sub:m,x[1:sub:m],c,lw=3)
axs[2,1].plot(1:sub:m,y[1:sub:m],c,lw=3)

axs[1,3].plot(round(randn(m),-11),"C1",lw=2)
axs[1,2].plot(round(x[1:m],1),"C1")
axs[1,1].plot(round(y[1:m],4),"C1",label="round to\n99% information")

axs[2,3].plot(1:sub:m,round(randn(m÷sub),-11),"C1",lw=2)
axs[2,2].plot(1:sub:m,round(x[1:sub:m],-1),"C1")
axs[2,1].plot(1:sub:m,round(y[1:sub:m],2),"C1")

axs[1,3].set_title("High resolution, noise",loc="left")
axs[1,2].set_title("High resolution, rough",loc="left")
axs[1,1].set_title("High resolution, smooth",loc="left")

axs[2,3].set_title("Low resolution, noise",loc="left")
axs[2,2].set_title("Low resolution, rough",loc="left")
axs[2,1].set_title("Low resolution, smooth",loc="left")

for (iax,ax) in enumerate(axs)
    ax.set_title(string(Char(iax+96)),loc="right",fontweight="bold")
end

for ax in axs[2,:]
    ax.set_xlabel("dimension to compress")
end

for ax in axs[:,1]
    ax.set_ylabel("values")
    ax.set_yticks([-2,0,2])
end

axs[1,1].legend(loc=3,fontsize=8)
axs[1,1].set_xlim(1,m-1)

tight_layout()
savefig("resolution_information.png",dpi=200)
savefig("resolution_information.pdf")

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