From bbbdd398c82f73ecb054917aece0e39c6fd422af Mon Sep 17 00:00:00 2001 From: "Documenter.jl" Date: Sun, 18 Feb 2024 04:58:58 +0000 Subject: [PATCH] build based on 00cc547 --- dev/api/index.html | 6 +- dev/examples/conditions/index.html | 41716 +++++++++++++------------- dev/examples/diff_sort/index.html | 198 +- dev/examples/model_G.pt | Bin 97728 -> 97728 bytes dev/examples/monoci_mlp/index.html | 366 +- dev/examples/monofit/index.html | 5006 ++-- dev/examples/monofit_mlp/index.html | 382 +- dev/examples/ph/index.html | 2490 +- dev/index.html | 2 +- dev/search/index.html | 2 +- 10 files changed, 25083 insertions(+), 25085 deletions(-) diff --git a/dev/api/index.html b/dev/api/index.html index 738f3db..591ce3a 100644 --- a/dev/api/index.html +++ b/dev/api/index.html @@ -1,7 +1,7 @@ -API · MonotoneSplines.jl
Edit on GitHub

API

MonotoneSplines.Spl — Type

A Spl object.

Fields

  • H: an RObject generated by splines::bs()
  • β: the coefficients for the B-spline.
source
MonotoneSplines.build_model — Method
build_model(x::AbstractVector{T}; <keyword arguments>)

Construct design matrix and other internal variables for smoothing spline.

Arguments

  • all_knots = false: whether to use all knots. If false, use the same rule as in R's smooth.spline.
  • prop_nknots = 1.0: a proportion for using fewer knots. Suppose the number of knots is nknots, then the final number of knots is prop_nknots * nknots. Currently, it is only effective when all_knots = false.
  • ε = 6.06e-6: a small number added to the diagonal of matrix Ω to ensure it is positive definite.

Returns

  • B-spline design matrix B at x for cubic splines
  • L: cholesky decomposition of Ω = LL'
  • J: number of basis functions, which does not change for cubic splines, so it is only intended for smoothing splines

the above four are shared with the method for cubic splines, but for smoothing splines, it also returns

  • mx, rx, idx, idx0: only for smoothing splines
source
MonotoneSplines.build_model — Method
build_model(x::AbstractVector{T}, J::Int; <keyword arguments>)

Construct design matrix and other internal variables for cubic spline with J basis functions.

Returns

  • B: B-spline design matrix B at x for cubic splines
  • rB: raw RObject of B
source
MonotoneSplines.check_CI — Method
check_CI(; <keyword arguments>)

Conduct repeated experiments to check the overlap of confidence bands (default, check_acc = false) or accuracy of fitting curves (check_acc = true) between MLP generator and OPT solution.

Arguments

  • n = 100: sample size
  • σ = 0.1: noise level
  • f::Function = exp: the truth curve
  • seed = 1234: random seed for the simulated data
  • check_acc = false: check overlap of confidence bands (default: false) or accuracy of fitting curves (true)
  • nepoch0 = 5: number of epoch in the first step to fit the curve
  • nepoch = 50: number of epoch in the second step to obtain the confidence band
  • niter_per_epoch = 100: number of iterations in each epoch
  • η0 = 1e-4: learning rate in step 1
  • η = 1e-4: learning rate in step 2 (NOTE: lr did not make much difference, unify these two)
  • K0 = 32: Monte Carlo size for averaging λ in step 2
  • K = 32: Monte Carlo size for averaging λ in step 1 and for averaging y in step 2. (NOTE: unify these two Monte Carlo size)
  • nB = 2000: number of bootstrap replications
  • nrep = 5: number of repeated experiments
  • fig = true: whether to plot
  • figfolder = ~: folder for saving the figures if fig = true
  • λs = exp.(range(-8, -2, length = 10)): region of continuous λ
  • nhidden = 1000: number of hidden layers
  • depth = 2: depth of MLP
  • demo = false: whether to save internal results for demo purpose
  • model_file = nothing: if not nothing, load the model from the file.
  • gpu_id = 0: specify the id of GPU, -1 for CPU.
  • prop_nknots = 0.2: proportion of number of knots in B-spline basis.
  • backend = "flux": train MLP generator with Flux or PyTorch
source
MonotoneSplines.ci_mono_ss_mlp — Method
ci_mono_ss_mlp(x::AbstractVector{T}, y::AbstractVector{T}, λs::AbstractVector{T}; )

Fit data x, y at each λs with confidence bands.

Arguments

  • prop_nknots = 0.2: proportion of number of knots
  • backend = "flux": flux or pytorch
  • model_file: path for saving trained model
  • nepoch0 = 3: number of epoch in training step 1
  • nepoch = 3: number of epoch in training step 2
  • niter_per_epoch = 100: number of iterations in each epoch
  • M = 10: Monte Carlo size
  • nhidden = 100: number of hidden units
  • disable_progressbar = false: set true if generating documentation
  • device = :cpu: train using :cpu or :gpu
  • sort_in_nn = true: (only for backend = "flux") whether put sort in MLP
  • eval_in_batch = false: (only for backend = "flux") Currently, Flux does not support sort in batch mode. A workaround with customized Zygote.batch_sort needs further verifications.
source
MonotoneSplines.coverage_prob — Method
coverage_prob(CIs::AbstractMatrix, y0::AbstractVector)

Calculate coverage probability given n x 2 CI matrix CIs and true vector y0 of size n.

source
MonotoneSplines.cv_mono_ss — Method
cv_mono_ss(x::AbstractVector{T}, y::AbstractVector{T}, λs::AbstractVector{T})

Cross-validation for monotone fitting with smoothing spline on y ~ x among parameters λs.

source
MonotoneSplines.div_into_folds — Method
div_into_folds(N::Int; K = 10, seed = 1234)

Equally divide 1:N into K folds with random seed seed. If seed is negative, it is a non-random division, where the i-th fold would be the i-th equidistant range.

source
MonotoneSplines.eval_penalty — Method
eval_penalty(model::Spl{T}, x::AbstractVector{T})

Evaluate the penalty matrix by R's fda::eval.penalty. To make sure the corresponding design matrix contructed by fda::eval.basis is the same as model.H, it asserts the norm difference should be smaller than sqrt(eps()).

source
MonotoneSplines.fit — Function
fit(X, y, paras, method)

paras is either the number of basis functions, or the sequence of interior knots. Return a Spl object.

n = 100
+API · MonotoneSplines.jl
Edit on GitHub
API
MonotoneSplines.Spl — Type
A Spl object.
Fields
  • H: an RObject generated by splines::bs()
  • β: the coefficients for the B-spline.
source
MonotoneSplines.build_model — Method
build_model(x::AbstractVector{T}; <keyword arguments>)
Construct design matrix and other internal variables for smoothing spline.
Arguments
  • all_knots = false: whether to use all knots. If false, use the same rule as in R's smooth.spline.
  • prop_nknots = 1.0: a proportion for using fewer knots. Suppose the number of knots is nknots, then the final number of knots is prop_nknots * nknots. Currently, it is only effective when all_knots = false.
  • ε = 6.06e-6: a small number added to the diagonal of matrix Ω to ensure it is positive definite.
Returns
  • B-spline design matrix B at x for cubic splines
  • L: cholesky decomposition of Ω = LL'
  • J: number of basis functions, which does not change for cubic splines, so it is only intended for smoothing splines 
the above four are shared with the method for cubic splines, but for smoothing splines, it also returns 
  • mx, rx, idx, idx0: only for smoothing splines
source
MonotoneSplines.build_model — Method
build_model(x::AbstractVector{T}, J::Int; <keyword arguments>)
Construct design matrix and other internal variables for cubic spline with J basis functions.
Returns
  • B: B-spline design matrix B at x for cubic splines
  • rB: raw RObject of B
source
MonotoneSplines.check_CI — Method
check_CI(; <keyword arguments>)
Conduct repeated experiments to check the overlap of confidence bands (default, check_acc = false) or accuracy of fitting curves (check_acc = true) between MLP generator and OPT solution. 
Arguments
  • n = 100: sample size
  • σ = 0.1: noise level
  • f::Function = exp: the truth curve
  • seed = 1234: random seed for the simulated data
  • check_acc = false: check overlap of confidence bands (default: false) or accuracy of fitting curves (true)
  • nepoch0 = 5: number of epoch in the first step to fit the curve
  • nepoch = 50: number of epoch in the second step to obtain the confidence band
  • niter_per_epoch = 100: number of iterations in each epoch
  • η0 = 1e-4: learning rate in step 1
  • η = 1e-4: learning rate in step 2 (NOTE: lr did not make much difference, unify these two)
  • K0 = 32: Monte Carlo size for averaging λ in step 2
  • K = 32: Monte Carlo size for averaging λ in step 1 and for averaging y in step 2. (NOTE: unify these two Monte Carlo size)
  • nB = 2000: number of bootstrap replications
  • nrep = 5: number of repeated experiments
  • fig = true: whether to plot
  • figfolder = ~: folder for saving the figures if fig = true
  • λs = exp.(range(-8, -2, length = 10)): region of continuous λ
  • nhidden = 1000: number of hidden layers
  • depth = 2: depth of MLP
  • demo = false: whether to save internal results for demo purpose
  • model_file = nothing: if not nothing, load the model from the file.
  • gpu_id = 0: specify the id of GPU, -1 for CPU.
  • prop_nknots = 0.2: proportion of number of knots in B-spline basis. 
  • backend = "flux": train MLP generator with Flux or PyTorch
source
MonotoneSplines.ci_mono_ss_mlp — Method
ci_mono_ss_mlp(x::AbstractVector{T}, y::AbstractVector{T}, λs::AbstractVector{T}; )
Fit data x, y at each λs with confidence bands.
Arguments
  • prop_nknots = 0.2: proportion of number of knots
  • backend = "flux": flux or pytorch
  • model_file: path for saving trained model
  • nepoch0 = 3: number of epoch in training step 1
  • nepoch = 3: number of epoch in training step 2
  • niter_per_epoch = 100: number of iterations in each epoch
  • M = 10: Monte Carlo size
  • nhidden = 100: number of hidden units
  • disable_progressbar = false: set true if generating documentation
  • device = :cpu: train using :cpu or :gpu
  • sort_in_nn = true: (only for backend = "flux") whether put sort in MLP 
  • eval_in_batch = false: (only for backend = "flux") Currently, Flux does not support sort in batch mode. A workaround with customized Zygote.batch_sort needs further verifications. 
source
MonotoneSplines.coverage_prob — Method
coverage_prob(CIs::AbstractMatrix, y0::AbstractVector)
Calculate coverage probability given n x 2 CI matrix CIs and true vector y0 of size n.
source
MonotoneSplines.cv_mono_ss — Method
cv_mono_ss(x::AbstractVector{T}, y::AbstractVector{T}, λs::AbstractVector{T})
Cross-validation for monotone fitting with smoothing spline on y ~ x among parameters λs.
source
MonotoneSplines.div_into_folds — Method
div_into_folds(N::Int; K = 10, seed = 1234)
Equally divide 1:N into K folds with random seed seed. If seed is negative, it is a non-random division, where the i-th fold would be the i-th equidistant range.
source
MonotoneSplines.eval_penalty — Method
eval_penalty(model::Spl{T}, x::AbstractVector{T})
Evaluate the penalty matrix by R's fda::eval.penalty. To make sure the corresponding design matrix contructed by fda::eval.basis is the same as model.H, it asserts the norm difference should be smaller than sqrt(eps()).
source
MonotoneSplines.fit — Function
fit(X, y, paras, method)
paras is either the number of basis functions, or the sequence of interior knots. Return a Spl object.
n = 100
 x = rand(n) * 2 .- 1
 y = x .^3 + randn(n) * 0.01
-res = fit(x, y, 10, "monotone")
source
MonotoneSplines.gen_data — Method
gen_data(n, σ, f::Union{Function, String}; xmin = -1, xmax = 1, k = 10)
Generate n data points (xi, yi) from curve f with noise level σ, i.e., yi = f(xi) + N(0, σ^2).
It returns four vectors, x, y, x0, y0, where
  • x, y: pair points of length n.
  • x0, y0: true curve without noise, represented by k*n points.
source
MonotoneSplines.jaccard_index — Method
jaccard_index(a::AbstractVector, b::AbstractVector)
Calculate Jaccard Index for two confidence intervals a and b
jaccard_index(a::AbstractMatrix, b::AbstractMatrix)
Calculate Jaccard Index for two confidence intervals a[i, :] and b[i, :]
source
MonotoneSplines.load_model — Method
load_model(n::Int, J::Int, nhidden::Int, model_file::String; dim_lam = 8, gpu_id = 3)
Load trained model from model_file.
source
MonotoneSplines.mono_cs — Function
mono_cs(x::AbstractVector, y::AbstractVector, J::Int = 4; increasing::Bool = true)
Monotone splines with cubic splines.
source
MonotoneSplines.mono_ss — Function
mono_ss(x::AbstractVector, y::AbstractVector, λ = 1.0; prop_nknots = 1.0)
Monotone splines with smoothing splines, return a MonotoneSS object.
source
MonotoneSplines.mono_ss — Function
mono_ss(B::AbstractMatrix, y::AbstractVector, L::AbstractMatrix, J::Int, λ::AbstractFloat)
Monotone Fitting with Smoothing Splines given design matrix B and cholesky-decomposed matrix L.
Returns
  • βhat: estimated coefficient
  • yhat: fitted values
  • (optional) B and L
source
MonotoneSplines.mono_ss_mlp — Method
mono_ss_mlp(x::AbstractVector, y::AbstractVector; λl, λu)
Fit monotone smoothing spline by training a MLP generator.
Arguments
  • prop_nknots = 0.2: proportion of number of knots
  • backend = flux: use flux or pytorch
  • device = :cpu: use :cpu or :gpu
  • nhidden = 100: number of hidden units
  • disable_progressbar = false: disable progressbar (useful in Documenter.jl)
source
MonotoneSplines.py_train_G_lambda — Method
py_train_G_lambda(y::AbstractVector, B::AbstractMatrix, L::AbstractMatrix; <keyword arguments>)
Wrapper for training MLP generator using PyTorch.
Arguments
  • η0, η: learning rate
  • K0, K: Monte Carlo size
  • nepoch0, nepoch: number of epoch
  • nhidden, depth: size of MLP
  • λl, λu: range of λ
  • use_torchsort = false: torch.sort (default: false) or torchsort.soft_sort (true)
  • sort_reg_strength = 0.1: tuning parameter when use_torchsort = true.
  • model_file: path for saving trained model
  • gpu_id = 0: use specified GPU
  • niter_per_epoch = 100: number of iterations in each epoch
  • disable_tqdm = false: set true when generating documentation
source
MonotoneSplines.smooth_spline — Method
smooth_spline(x::AbstractVector, y::AbstractVector, xnew::AbstractVector)
Perform smoothing spline on (x, y), and make predictions on xnew.
Returns: yhat, ynewhat,....
source
MonotoneSplines.train_Gyλ — Method
train_Gyλ(rawy::AbstractVector, rawB::AbstractMatrix, rawL::AbstractMatrix, model_file::String)
Train MLP generator G(y, λ) for λ ∈ [λl, λu] and y ~ N(f, σ²)
source
MonotoneSplines.train_Gλ — Method
train_Gλ(rawy::AbstractVector, rawB::AbstractMatrix, rawL::AbstractMatrix; λl, λu)
Train MLP generator G(λ) for λ ∈ [λl, λu].
source
RCall.rcopy — Method
rcopy(s::Spl)
Convert RObject s.H as a Julia matrix, and s.β keeps the same.
source
StatsAPI.predict — Method
predict(model::Spl{T}, xs::AbstractVector{T})
+res = fit(x, y, 10, "monotone")
source
MonotoneSplines.gen_data — Method
gen_data(n, σ, f::Union{Function, String}; xmin = -1, xmax = 1, k = 10)
Generate n data points (xi, yi) from curve f with noise level σ, i.e., yi = f(xi) + N(0, σ^2).
It returns four vectors, x, y, x0, y0, where
  • x, y: pair points of length n.
  • x0, y0: true curve without noise, represented by k*n points.
source
MonotoneSplines.jaccard_index — Method
jaccard_index(a::AbstractVector, b::AbstractVector)
Calculate Jaccard Index for two confidence intervals a and b
jaccard_index(a::AbstractMatrix, b::AbstractMatrix)
Calculate Jaccard Index for two confidence intervals a[i, :] and b[i, :]
source
MonotoneSplines.load_model — Method
load_model(n::Int, J::Int, nhidden::Int, model_file::String; dim_lam = 8, gpu_id = 3)
Load trained model from model_file.
source
MonotoneSplines.mono_cs — Function
mono_cs(x::AbstractVector, y::AbstractVector, J::Int = 4; increasing::Bool = true)
Monotone splines with cubic splines.
source
MonotoneSplines.mono_ss — Function
mono_ss(x::AbstractVector, y::AbstractVector, λ = 1.0; prop_nknots = 1.0)
Monotone splines with smoothing splines, return a MonotoneSS object.
source
MonotoneSplines.mono_ss — Function
mono_ss(B::AbstractMatrix, y::AbstractVector, L::AbstractMatrix, J::Int, λ::AbstractFloat)
Monotone Fitting with Smoothing Splines given design matrix B and cholesky-decomposed matrix L.
Returns
  • βhat: estimated coefficient
  • yhat: fitted values
  • (optional) B and L
source
MonotoneSplines.mono_ss_mlp — Method
mono_ss_mlp(x::AbstractVector, y::AbstractVector; λl, λu)
Fit monotone smoothing spline by training a MLP generator.
Arguments
  • prop_nknots = 0.2: proportion of number of knots
  • backend = flux: use flux or pytorch
  • device = :cpu: use :cpu or :gpu
  • nhidden = 100: number of hidden units
  • disable_progressbar = false: disable progressbar (useful in Documenter.jl)
source
MonotoneSplines.py_train_G_lambda — Method
py_train_G_lambda(y::AbstractVector, B::AbstractMatrix, L::AbstractMatrix; <keyword arguments>)
Wrapper for training MLP generator using PyTorch.
Arguments
  • η0, η: learning rate
  • K0, K: Monte Carlo size
  • nepoch0, nepoch: number of epoch
  • nhidden, depth: size of MLP
  • λl, λu: range of λ
  • use_torchsort = false: torch.sort (default: false) or torchsort.soft_sort (true)
  • sort_reg_strength = 0.1: tuning parameter when use_torchsort = true.
  • model_file: path for saving trained model
  • gpu_id = 0: use specified GPU
  • niter_per_epoch = 100: number of iterations in each epoch
  • disable_tqdm = false: set true when generating documentation
source
MonotoneSplines.smooth_spline — Method
smooth_spline(x::AbstractVector, y::AbstractVector, xnew::AbstractVector)
Perform smoothing spline on (x, y), and make predictions on xnew.
Returns: yhat, ynewhat,....
source
MonotoneSplines.train_Gyλ — Method
train_Gyλ(rawy::AbstractVector, rawB::AbstractMatrix, rawL::AbstractMatrix, model_file::String)
Train MLP generator G(y, λ) for λ ∈ [λl, λu] and y ~ N(f, σ²)
source
MonotoneSplines.train_Gλ — Method
train_Gλ(rawy::AbstractVector, rawB::AbstractMatrix, rawL::AbstractMatrix; λl, λu)
Train MLP generator G(λ) for λ ∈ [λl, λu].
source
RCall.rcopy — Method
rcopy(s::Spl)
Convert RObject s.H as a Julia matrix, and s.β keeps the same.
source
StatsAPI.predict — Method
predict(model::Spl{T}, xs::AbstractVector{T})
 predict(X::Vector{Float64}, y::Vector{Float64}, J::Int, Xnew::AbstractVector{Float64}, ynew::AbstractVector{Float64}
-predict(X::Vector{Float64}, y::Vector{Float64}, J::Int, Xnew::Vector{Float64}, ynew::Vector{Float64}, σ::Vector{Float64}
Make prediction based on fitted Spl on new points xs. If Xnew is provided, then also returns the prediction error ‖yhat - ynew‖_2^2.
source

+predict(X::Vector{Float64}, y::Vector{Float64}, J::Int, Xnew::Vector{Float64}, ynew::Vector{Float64}, σ::Vector{Float64}

Make prediction based on fitted Spl on new points xs. If Xnew is provided, then also returns the prediction error ‖yhat - ynew‖_2^2.

source
diff --git a/dev/examples/conditions/index.html b/dev/examples/conditions/index.html index f87ea10..33fe4e5 100644 --- a/dev/examples/conditions/index.html +++ b/dev/examples/conditions/index.html @@ -60,20872 +60,20872 @@ end
plot_intervals (generic function with 1 method)

reproduce the figure in the paper

plot_intervals()
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+ + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + diff --git a/dev/examples/diff_sort/index.html b/dev/examples/diff_sort/index.html index 71ad74a..685cff3 100644 --- a/dev/examples/diff_sort/index.html +++ b/dev/examples/diff_sort/index.html @@ -15,10 +15,10 @@ use_torchsort=true, sort_reg_strength=1.0, disable_progressbar = true); @time Ghat4, loss4 = mono_ss_mlp(x, y, λl = λl, λu = λu, device = :cpu, prop_nknots = 0.2, backend = "pytorch", - use_torchsort=false, sort_reg_strength=1.0, disable_progressbar = true);
  8.256786 seconds (19.08 M allocations: 1.023 GiB, 5.06% gc time, 79.10% compilation time: 5% of which was recompilation)
-  0.453283 seconds (1.91 k allocations: 68.078 KiB)
-  0.454391 seconds (1.91 k allocations: 68.078 KiB)
-  0.401884 seconds (1.92 k allocations: 69.328 KiB)

Evaluate the fitted curve,

λ = λl
+                                    use_torchsort=false, sort_reg_strength=1.0, disable_progressbar = true);
  7.781853 seconds (19.38 M allocations: 1.041 GiB, 3.59% gc time, 76.54% compilation time: 4% of which was recompilation)
+  0.474219 seconds (1.94 k allocations: 69.328 KiB)
+  0.471298 seconds (1.94 k allocations: 69.328 KiB)
+  0.420955 seconds (1.94 k allocations: 70.578 KiB)

Evaluate the fitted curve,

λ = λl
 yhat1 = Ghat1(y, λ)
 yhat2 = Ghat2(y, λ)
 yhat3 = Ghat3(y, λ)
@@ -29,126 +29,126 @@
 plot!(x, yhat4, label = "no")
- + - + - + - + - + - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + +

And the traing loss is

plot(loss1[1:100], label = "1e-4", xlab = "iter", ylab = "loss")
 plot!(loss2[1:100], label = "1e-1")
 plot!(loss3[1:100], label = "1")
 plot!(loss4[1:100], label = "no")
- + - + - + - + - + - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + diff --git a/dev/examples/model_G.pt b/dev/examples/model_G.pt index 159a5eac314c145fc7a29ef188d134ea389f39d0..cad158c9e416162f5002c541e8cbcdf54dd827bd 100644 GIT binary patch literal 97728 zcmb5V2{e}9_cx5p^B9@uq0BS)wNFxMAX6zSk)cvjhUQVE2$5NZR7y#cg!|e@ktmIn z3KeP6s7a!D?|$F^|G(bv^Q`B2*L$yZE$cq+>vKNWIeVY8&)%QQ+kKoMAD^fw-~aZJ z<&)=Iy(TO?GH8ZH*wVPA=4)3)y17d8E&SJKr$A`vn#joT(71>-QL*A)F=62`;bGGv zLgU1v`AmGH`HlU>dC zQL$@cg4SD$M~^c$_7~yZ70dhO(&gdNg8$qi?iRUbY24g@8pWf9{P}smv5Xe>AIJL} zx^`W($eaQ3xM)!ycRu$pclD%bF?V%${-kK}ImMl1Qt^6rm}hzegC6Kihy z-%mCEUpUs$$}2?X$Qs2(tNcSIEGc>dj}woK>OWv=anb6&(Hj2J{{s6DCCz_9*+gso zLrMF;DCzvAr27wwUR<=kZ?wU`C|UhCCBuK=*hU-u7bWBWp=9z8rHTK5O^S;)^^G?B z7bUBIM#=nNPlLdoiy`dJ8z85D*f{@#!-@9kF6zT2Yb{?pEL zw*L%E^t^4+e*ZN6^^c`I|5?BEuMb7f-xj^#KmS-48)EM5E+TrR|DAde&&2;;-v38; zmE+U>A9oGe6c@hcuP3^>YW^?&`Y-3@eTj^Si;E2Z+s$t1n;^`i^7r~bcO%V5{->M2 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0.01:0.09:0.1

Run the optimization toolbox to fit the monotone spline, and conduct (parametric) bootstrap to obtain the confidence band of the fitted curve.

@time RES0 = [ci_mono_ss(x, y, λ, prop_nknots = 0.2) for λ in λs]
 Yhat0 = hcat([RES0[i][1] for i=1:2]...)
 YCIs0 = [RES0[i][2] for i = 1:2]
2-element Vector{LinearAlgebra.Adjoint{Float64, Matrix{Float64}}}:
- [0.17550133381777902 0.3561331547960619; 0.22691771847055056 0.38791522735467926; … ; 2.272412076137895 2.435187713899899; 2.4782545426362135 2.682879045715268]
- [-0.03637363524958645 0.22031271625165838; 0.048160750845113114 0.29085815145127897; … ; 2.09153852288662 2.334946940092237; 2.23676312068496 2.5138153336583]

Estimate the confidence band with the Flux backend

@time Yhat, YCIs, LOSS = ci_mono_ss_mlp(x, y, λs, prop_nknots = 0.2, device = :cpu, backend = "flux", nepoch0 = 5, nepoch = 5, disable_progressbar = true);
┌ Warning: Layer with Float32 parameters got Float64 input.
+ [0.17858290436268556 0.3540184674400672; 0.2308478035883113 0.3867083077647926; … ; 2.2746707007432705 2.44111602923719; 2.483094463213197 2.6858812986042784]
+ [-0.030958049671792942 0.22735188418026964; 0.05033624873038706 0.29366626648300687; … ; 2.098645447180695 2.334031554437537; 2.2488866258423057 2.5136065052986813]

Estimate the confidence band with the Flux backend

@time Yhat, YCIs, LOSS = ci_mono_ss_mlp(x, y, λs, prop_nknots = 0.2, device = :cpu, backend = "flux", nepoch0 = 5, nepoch = 5, disable_progressbar = true);
┌ Warning: Layer with Float32 parameters got Float64 input.
 │   The input will be converted, but any earlier layers may be very slow.
 │   layer = Dense(28 => 100, gelu)  # 2_900 parameters
 │   summary(x) = "28-element Vector{Float64}"
@@ -19,62 +19,60 @@
 │   layer = Dense(28 => 100, gelu)  # 2_900 parameters
 │   summary(x) = "28-element Vector{Float64}"
 └ @ Flux ~/.julia/packages/Flux/n3cOc/src/layers/stateless.jl:60
- 22.668316 seconds (74.96 M allocations: 26.869 GiB, 8.81% gc time)

Alternatively, we can also estimate it with the PyTorch backend

@time Yhat2, YCIs2, LOSS2 = ci_mono_ss_mlp(x, y, λs, prop_nknots = 0.2, device = :cpu, backend = "pytorch", nepoch0 = 5, nepoch = 5, disable_progressbar = true);
  1.891592 seconds (397.15 k allocations: 27.987 MiB, 8.92% compilation time)

plot the traceplot of training loss

plot(log.(LOSS), label = "MLP generator (Flux)")
+ 23.311207 seconds (75.55 M allocations: 26.887 GiB, 8.05% gc time)

Alternatively, we can also estimate it with the PyTorch backend

@time Yhat2, YCIs2, LOSS2 = ci_mono_ss_mlp(x, y, λs, prop_nknots = 0.2, device = :cpu, backend = "pytorch", nepoch0 = 5, nepoch = 5, disable_progressbar = true);
  1.956923 seconds (398.86 k allocations: 28.107 MiB, 9.00% compilation time)

plot the traceplot of training loss

plot(log.(LOSS), label = "MLP generator (Flux)")
 plot!(log.(LOSS2), label = "MLP generator (PyTorch)")
- + - + - + - + - + - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - + + + + + + + + + + + + + + + + + + + + + + + + + + + + +

Calculate the jaccard index OPT solution vs MLP generator (Flux)

[MonotoneSplines.jaccard_index(YCIs[i], YCIs0[i]) for i = 1:2]
2-element Vector{Float64}:
- 0.7565022593343843
- 0.8498920339670173

OPT solution vs MLP generator (PyTorch)

[MonotoneSplines.jaccard_index(YCIs2[i], YCIs0[i]) for i = 1:2]
2-element Vector{Float64}:
- 0.7613000652539665
- 0.8196980821837483
Note

For simple demonstration, the training might not be sufficient, so the Jaccard index might not be good enough. For a better performance, please train it with a larger nepoch and nepoch0.

Plot the fitted curves and their confidence bands.

OPT solution vs MLP generator (Flux)

scatter(x, y, label = "")
+ 0.46654970338047674
+ 0.5292903256026571

OPT solution vs MLP generator (PyTorch)

[MonotoneSplines.jaccard_index(YCIs2[i], YCIs0[i]) for i = 1:2]
2-element Vector{Float64}:
+ 0.6712830919662872
+ 0.7983503521824987
Note

For simple demonstration, the training might not be sufficient, so the Jaccard index might not be good enough. For a better performance, please train it with a larger nepoch and nepoch0.

Plot the fitted curves and their confidence bands.

OPT solution vs MLP generator (Flux)

scatter(x, y, label = "")
 plot!(x0, y0, label = "truth", legend = :topleft, ls = :dot)
 plot!(x, Yhat0[:, 1], label = "OPT solution")
 plot!(x, Yhat0[:, 2], label = "OPT solution")
@@ -86,87 +84,87 @@
 plot!(x, YCIs[2][:, 1], fillrange = YCIs[2][:, 2], linealpha = 0, label = "", fillalpha = 0.5)
- + - + - + - + - + - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + +

OPT solution vs MLP generator (PyTorch)

scatter(x, y, label = "")
 plot!(x0, y0, label = "truth", legend = :topleft, ls = :dot)
 plot!(x, Yhat0[:, 1], label = "OPT solution")
@@ -179,85 +177,85 @@
 plot!(x, YCIs2[2][:, 1], fillrange = YCIs2[2][:, 2], linealpha = 0, label = "", fillalpha = 0.5)
- + - + - + - + - + - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + diff --git a/dev/examples/monofit/index.html b/dev/examples/monofit/index.html index 7ee54a8..02e1335 100644 --- a/dev/examples/monofit/index.html +++ b/dev/examples/monofit/index.html @@ -11,150 +11,150 @@ λs = exp.(range(-10, -4, length = 100));

Perform cross-validation for monotone fitting with smoothing splines,

@time errs, B, L, J = MonotoneSplines.cv_mono_ss(x, y, λs)
([0.03836494968310219, 0.038356054988443196, 0.038346860326416564, 0.03833810480828669, 0.03833030336201492, 0.0383181815212615, 0.038306086526776, 0.03829277370690645, 0.03827569988291665, 0.03825256780043086  …  0.041377653962167346, 0.04164026736341345, 0.041913261668632816, 0.04219672525984907, 0.042490701739591435, 0.04279519591471907, 0.04311015485047943, 0.04343543081198881, 0.0437708461692481, 0.04411615119567598], [0.0 0.0 … 0.0 0.0; 0.0 0.0 … 0.0 0.0; … ; 0.0 0.0 … 0.0 0.0; 0.0 0.0 … 0.0 0.0], [3722.3861277938377 0.0 … 0.0 0.0; -5484.443813834549 2256.9009327110152 … 0.0 0.0; … ; 0.0 0.0 … 5.503905144035493 0.0; 0.0 0.0 … -5.4879608135793845 0.011543876496072695], 64)

Then plot the CV curve

scatter(log.(λs), errs, title = "seed = $seed")
- + - + - + - + - + - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + +

Then we can choose λ which minimized the CV error.

idx = argmin(errs)
 λopt = λs[idx]
0.0008846636600765526

Fit with λopt

βhat, yhat = MonotoneSplines.mono_ss(B, y, L, J, λopt);

Alternatively,

res = MonotoneSplines.mono_ss(x, y, λopt);
 yhat = res.fitted
@@ -182,251 +182,251 @@
 scatter!(x, yhat)
- + - + - + - + - + - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + +

We can also compare it with smooth.spline,

spl = R"smooth.spline($x, $y)"
RCall.RObject{RCall.VecSxp}
 Call:
 smooth.spline(x = `#JL`$x, y = `#JL`$y)
@@ -439,352 +439,352 @@
 scatter!(x, yhat_ss)
- + - + - + - + - + - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + +

For ease of demonstrating other examples, we wrap up the above procedures as a function

function demo_mono_ss(x, y, λs)
     errs, B, L, J = MonotoneSplines.cv_mono_ss(x, y, λs)
     fig1 = plot(log.(λs), errs, xlab = "λ", ylab = "CV error", legend=false)
@@ -809,375 +809,375 @@
 demo_mono_ss(x, y, λs)
- + - + - + - + - + - - - - - - - - - - - - - - - - - - - - - - + + + + + + + + + + + + + + + + + + + + + + - + - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + +

σ = 2.0

σ = 2.0
 Random.seed!(seed)
 x, y, x0, y0 = MonotoneSplines.gen_data(n, σ, z->1/(1-0.42log(z)), xmin = 0, xmax = 10)
@@ -1187,381 +1187,381 @@ 
σ = 2.0
- + - + - + - + - + - - - - - - - - - - - - - - - - - - - - - - - - - - - - + + + + + + + + + + + + + + + + + + + + + + + + + + + + - + - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + +

σ = 0.5

σ = 0.5
 Random.seed!(seed)
 x, y, x0, y0 = MonotoneSplines.gen_data(n, σ, z->1/(1-0.42log(z)), xmin = 0, xmax = 10)
@@ -1571,379 +1571,379 @@ 
σ = 0.5
- + - + - + - + - + - - - - - - - - - - - - - - - - - - - - - - - - - - + + + + + + + + + + + + + + + + + + + + + + + + + + - + - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + +

Logistic Curve

λs = exp.(range(-10, 0, length = 100));

σ = 0.2

σ = 0.2
 Random.seed!(seed)
 x, y, x0, y0 = MonotoneSplines.gen_data(n, σ, z->exp(z)/(1+exp(z)), xmin = -5, xmax = 5)
@@ -1953,387 +1953,387 @@ 
Lo
 demo_mono_ss(x, y, λs)
- + - + - + - + - + - - - - - - - - - - - - - - - - - - - - - - - - - - - - + + + + + + + + + + + + + + + + + + + + + + + + + + + + - + - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + +

σ = 1.0

σ = 1.0
 Random.seed!(seed)
 x, y, x0, y0 = MonotoneSplines.gen_data(n, σ, z->exp(z)/(1+exp(z)), xmin = -5, xmax = 5)
@@ -2343,381 +2343,381 @@ 
σ = 1.0
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we also support the well-known PyTorch backend with the help of PyCall.jl,

@time Ghat2, loss2 = mono_ss_mlp(x, y, λl = λ, λu = λ, device = :cpu, backend = "pytorch", disable_progressbar = true);
  0.431089 seconds (36.92 k allocations: 2.254 MiB, 4.60% compilation time)
Note

Showing the progressbar is quite useful in practice, but here in the documenter environment, it cannot display properly, so currently I simply disable it via disable_progressbar = true.

plot the log training loss

plot(log.(loss), label = "Flux")
+  2.258574 seconds (7.45 M allocations: 2.582 GiB, 8.46% gc time)

we also support the well-known PyTorch backend with the help of PyCall.jl,

@time Ghat2, loss2 = mono_ss_mlp(x, y, λl = λ, λu = λ, device = :cpu, backend = "pytorch", disable_progressbar = true);
  0.456176 seconds (39.88 k allocations: 2.470 MiB, 3.64% compilation time)
Note

Showing the progressbar is quite useful in practice, but here in the documenter environment, it cannot display properly, so currently I simply disable it via disable_progressbar = true.

plot the log training loss

plot(log.(loss), label = "Flux")
 plot!(log.(loss2), label = "Pytorch")
- + - + - + - + - + - - - - - - - - - - - - - - - - - - - - - - - - - - - - - + + + + + + + + + + + + + + + + + + + + + + + + + + + + +

The fitting can be obtained via evaluating at $λ$,

yhat = Ghat(y, λ);
 yhat2 = Ghat2(y, λ);
┌ Warning: Layer with Float32 parameters got Float64 input.
 │   The input will be converted, but any earlier layers may be very slow.
@@ -81,75 +81,75 @@
 plot!(x, yhat2, label = "MLP generator (Pytorch)", ls = :dash, lw = 2)
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The fitting curves obtained from optimization solution and MLP generator overlap quite well.

continus $λ$

Here we train a generator $G(\lambda), \lambda\in [\lambda_l, \lambda_u]$,

λl = 1e-2
 λu = 1e-1
 @time Ghat, loss = mono_ss_mlp(x, y, λl = λl, λu = λu, prop_nknots = 0.2, device = :cpu);
┌ Warning: Layer with Float32 parameters got Float64 input.
@@ -157,52 +157,52 @@
 │   layer = Dense(28 => 100, gelu)  # 2_900 parameters
 │   summary(x) = "28-element Vector{Float64}"
 └ @ Flux ~/.julia/packages/Flux/n3cOc/src/layers/stateless.jl:60
-  2.294526 seconds (7.41 M allocations: 2.581 GiB, 8.39% gc time, 0.43% compilation time)

Plot the training losses along with the iterations.

plot(loss)
+ 2.244706 seconds (7.46 M allocations: 2.583 GiB, 8.82% gc time, 0.42% compilation time)

Plot the training losses along with the iterations.

plot(loss)
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Evaluate the generator at $\lambda_l$, $\lambda_u$ and their middle $\lambda_m$

λm = (λl + λu) / 2
 yhat_l = Ghat(y, λl)
 yhat_u = Ghat(y, λu)
@@ -233,79 +233,79 @@
 plot!(x, yhat_u, label = "MLP generator (λ = $λu)", ls = :dash, lw = 2)
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-9.14043389977092:0.018317502805152146:0.0

Then we can check how the data looks like

scatter(x, y, label = "")
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Monotone Cubic Splines

Perform the monotone cubic splines with different number of basis functions $J=4, 10$

fit_mcs4 = mono_cs(x, y, 4, increasing = false)
 plot!(x0, predict(fit_mcs4, x0), label = "J = 4", legend = :bottomleft)
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Monotone Smoothing Splines
Perform smoothing splines
yhat_ss, yhatnew_ss, _, λ = MonotoneSplines.smooth_spline(x, y, x0);
use the same $\lambda$,
fit_mss = mono_ss(x, y, λ, increasing  = false)
MonotoneSplines.MonotoneSS(-9.14043389977092, 9.14043389977092, [0.0, 1.454834583480431e-5, 0.040255767815961296, 0.040275046566675785, 0.053120406912553514, 0.07158510316354796, 0.07160151251650909, 0.08108010267692153, 0.08880327201348134, 0.08881246768312859  …  0.7735049160513184, 0.7735584982043695, 0.7900313672860578, 0.8038873718450078, 0.8041381144717783, 0.8438188519580327, 0.8519292828788234, 0.8646148315616505, 0.9417901105035358, 1.0], [0.6734819458337752 0.32650221063496454 … 0.0 0.0; 0.0 0.0 … 0.0 0.0; … ; 0.0 0.0 … 0.0 0.0; 0.0 0.9991961711349227 … 0.0 0.0], [0.9999999999999999 0.0 … 0.0 0.0; 0.0 0.0 … 0.0 1.0], [-206209.0105682694 206209.0105682694 … 0.0 0.0; 0.0 0.0 … -51.53763434273872 51.53763434273872], [6.242667422805045e7 0.0 … 0.0 0.0; -6.24492310139133e7 428.8793536648739 … 0.0 0.0; … ; 0.0 0.0 … 71.43886038880825 0.0; 0.0 0.0 … -70.05267748833886 0.011036672741150104], [-0.00010540172462759652, -0.00010545870572024371, -0.00024925570697206567, -0.0003172203651837067, -0.0003179371183418965, -0.00031810004755924455, -0.0007954049885768426, -0.0007954689356193098, -0.0007955053849623059, -0.0007955411056740728  …  -0.4730285521501108, -0.5926980987393097, -0.5926980988779484, -0.5926980993447567, -1.0019660275247402, -1.0019660276423883, -1.001966028018416, -1.3849584959992833, -1.3849584970545323, -1.3849584977221743], [-0.00010542260825206995, -0.0003981393749492754, -0.0007955739445121291, -0.0018508856543992124, -0.002793771958923699, -0.002981867803659064, -0.005703884110722602, -0.006235107369037761, -0.006235107375595559, -0.00570701833081863  …  -0.005702326746152812, -0.006235107359301814, -0.006235107367597545, -0.005705108724269978, -0.002981868441137281, -0.0027937789477241034, -0.001850908765655973, -0.0007955763502580652, -0.00039833929655984357, -0.00010557430491573791])
then plot it
plot!(x0, yhatnew_ss, ls = :dot, label = "Smoothing Spline (λ = $(round(λ, sigdigits = 3)))")
 plot!(x0, predict(fit_mss, x0), ls = :solid, label = "Monotone Smoothing Spline (λ = $(round(λ, sigdigits = 3)))")

 
 
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Monotone smoothing splines with cross-validation
Alternatively, we can find the optimal tuning parameter $\lambda$ by cross-validation,
λs = exp.(-10:0.2:1)
 errs, B, L, J = cv_mono_ss(x, y, λs, increasing = false)
 λopt = λs[argmin(errs)]
4.5399929762484854e-5
Fit with the optimal tuning parameter
fit_mss2 = mono_ss(x, y, λopt, increasing = false)
 plot!(x0, predict(fit_mss2, x0), label = "Monotone Smoothing Spline (λ = $(round(λopt, sigdigits = 3)))")

 
 
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where the cross-validation error curve is as follows,
scatter(log.(λs), errs, label = "")

 
 
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diff --git a/dev/index.html b/dev/index.html
index 32ad81c..771c778 100644
--- a/dev/index.html
+++ b/dev/index.html
@@ -2,4 +2,4 @@
 Home · MonotoneSplines.jl
Edit on GitHub
MonotoneSplines.jl Documentation
Wang, L., Fan, X., Li, H., & Liu, J. S. (2023). Monotone Cubic B-Splines (arXiv:2307.01748). arXiv. https://doi.org/10.48550/arXiv.2307.01748
MonotoneSplines.jl is a Julia package for monotone splines, which impose a monotonicity constraint on the smoothing splines. 
\[\underset{\color{red}{f\textbf{ is monotonic}}}{\arg\min} \sum_{i=1}^n\left\{y_i-f(x_i)\right\}^2 + \lambda \int \left\{f''(t)\right\}^2dt\,,\]
where $f$ is formed with B-spline basis $f(x) = \sum_{j=1}^J\gamma_j B_j(x)$. A sufficient condition for $f$ to be monotonic is $\gamma_1,\ldots,\gamma_J$ is monotonic. With matrix notation ${\mathbf y} = [y_1,\ldots, y_n], {\mathbf B}_{ij} = B_j(x_i), {\boldsymbol\Omega}_{ij} = \int B_i''(s)B_j''(s)ds$, the problem can be rewritten as
\[\begin{aligned}
 \underset{\gamma}{\arg\min} & \Vert {\mathbf y} - {\mathbf B} \gamma\Vert_2^2 + \lambda \gamma^T\boldsymbol\Omega\gamma\\
 \text{subject to } & \alpha \gamma_1 \le \alpha \gamma_2\le \cdots \le \alpha\gamma_J\,,
-\end{aligned}\]
where $\alpha=1$ implies non-decreasing and $\alpha=-1$ indicates non-increasing.
The package provides two algorithms (frameworks) for fitting the monotone splines.
  • Convert the problem into a classical convex second-order cone optimization problem. There are many mature existing optimization toolboxes can be used, such as ECOS.jl.
  • Approximate the solution with an Multi-Layer Perceptrons (MLP) generator, using the powerful representation ability of neural network.
Particularly, the second approach can achieve good approximations and it can save much time by avoiding repeating to run the optimization problems of the first approach when we conduct bootstrap to estimate the confidence band. 
We do not reinvent the wheel. Instead, we fully take advantage of the existing widely-used implementations in other programming languages with the help of the flexible integration feature of Julia. For example, the package adopts the calculation of B-splines from R's splines package via RCall.jl, and provides the PyTorch deep learning backend via PyCall.jl as an alternative to the pure-Julia deep learning framework Flux.jl.

+\end{aligned}\]
where $\alpha=1$ implies non-decreasing and $\alpha=-1$ indicates non-increasing.
The package provides two algorithms (frameworks) for fitting the monotone splines.
  • Convert the problem into a classical convex second-order cone optimization problem. There are many mature existing optimization toolboxes can be used, such as ECOS.jl.
  • Approximate the solution with an Multi-Layer Perceptrons (MLP) generator, using the powerful representation ability of neural network.
Particularly, the second approach can achieve good approximations and it can save much time by avoiding repeating to run the optimization problems of the first approach when we conduct bootstrap to estimate the confidence band. 
We do not reinvent the wheel. Instead, we fully take advantage of the existing widely-used implementations in other programming languages with the help of the flexible integration feature of Julia. For example, the package adopts the calculation of B-splines from R's splines package via RCall.jl, and provides the PyTorch deep learning backend via PyCall.jl as an alternative to the pure-Julia deep learning framework Flux.jl.

diff --git a/dev/search/index.html b/dev/search/index.html
index be3e892..a6f5206 100644
--- a/dev/search/index.html
+++ b/dev/search/index.html
@@ -1,2 +1,2 @@
 
-Search · MonotoneSplines.jl
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+Search · MonotoneSplines.jl
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