smoothPLS_multi_states_03

library(SmoothPLS)
library(pls)
#> Warning: le package 'pls' a été compilé avec la version R 4.4.3
#> 
#> Attachement du package : 'pls'
#> L'objet suivant est masqué depuis 'package:stats':
#> 
#>     loadings

This notebook show the Smooth PLS algorithm for a multi-states Categorical Functional Data (MS-CFD).

Parameters


N_states = 3

nind = 100 # number of individuals (train set)
start = 0 # First time
end = 100 # end time

curve_type = 'cat'

TTRatio = 0.2 # Train Test Ratio means we have floor(nind*TTRatio/(1-TTRatio))
NotS_ratio = 0.2 # noise variance over total variance for Y
beta_0_real=65.4321 # Intercept value for the link between X(t) and Y

nbasis = 10 # number of basis functions
norder = 4 # 4 for cubic splines basis

regul_time = seq(start, end, 5) # regularisation time sequence
regul_time_0 = seq(start, end, 1)

int_mode = 1 # in case of integration errors.

Data generation

lambda_determination

# Initialized the lambdas values
lambdas = lambda_determination(N_states)
lambdas
#> [1] 0.16218960 0.09130628 0.07550633

tranfer_probabilities

# Initialized the transition matrix
transition_df = transfer_probabilities(N_states)
transition_df
#>              dir1      dir2      dir3
#> state_1 0.0000000 0.7863556 0.2136444
#> state_2 0.4314136 0.0000000 0.5685864
#> state_3 0.5306123 0.4693877 0.0000000

df_cfd

df_cfd = generate_X_df_multistates(nind = nind, N_states, start, end,
                              lambdas,  transition_df)
head(df_cfd)
#>   id       time state
#> 1  1  0.0000000     3
#> 2  1  0.4182081     2
#> 3  1  1.5448560     3
#> 4  1  5.7064580     2
#> 5  1 35.5646072     1
#> 6  1 44.2917255     3
plot_CFD_individuals(df_cfd)

plot_CFD_individuals(df_cfd, by_cfda = TRUE)

Basis creation

All the states will share the same basis.

basis = create_bspline_basis(start, end, nbasis, norder)
#basis = fda::create.fourier.basis(c(start,end), nbasis = nbasis)

# All the states will share the same basis.
basis_list = obj_list_creation(N_rep = N_states, obj = basis)

plot(basis, main=paste0(nbasis, " ", basis$type," functions basis"))

Data processing

We have to prepare the data before the FPLS method. For each state, we build its indicator function.

names(df_cfd)
#> [1] "id"    "time"  "state"

cat_data_to_indicator

df_processed = cat_data_to_indicator(df_cfd)

length(df_processed)
#> [1] 3
names(df_processed)
#> [1] "state_1" "state_2" "state_3"
head(df_processed$state_3, 20)
#>    id        time state
#> 1   1   0.0000000     1
#> 2   1   0.4182081     0
#> 3   1   1.5448560     1
#> 4   1   5.7064580     0
#> 6   1  44.2917255     1
#> 7   1  49.3986723     0
#> 12  1  94.5885722     1
#> 13  1 100.0000000     1
#> 14  2   0.0000000     0
#> 16  2  23.5972614     1
#> 17  2  35.2188747     0
#> 19  2  62.3399729     1
#> 20  2  69.0495525     0
#> 23  2 100.0000000     0
#> 24  3   0.0000000     1
#> 25  3   7.7964939     0
#> 27  3  15.5383881     1
#> 28  3  26.7390645     0
#> 29  3  38.7901396     1
#> 30  3  60.9841073     0

beta list

##### beta_real #####
###### beta_0_real ######
beta_0_real
#> [1] 65.4321
beta_func_list = beta_list_generation(N_states = N_states)
for(i in 1:length(beta_func_list)){
  plot(0:end,  beta_func_list[[i]](0:end, end_time = end),
       ylab=paste0("Beta(t) n°=", i), type = 'l')
  title(paste0("Beta(t) n°=", i))
}

Y evaluation

Y generation is based on the following equation : $Y = 0 + {i=1}^K _0^T _i(t) ind_i(t) dt $ with \(ind_i(t) = \{0, 1\}_{t \in [0, T]}\) the indicator function of the state \(i\).

We link \(\beta_i\) with the \(state_i\).

Y_df = generate_Y_df(df = df_processed, curve_type = curve_type,
                     beta_real_func_or_list =  beta_func_list,
                     beta_0_real = beta_0_real,
                     NotS_ratio = NotS_ratio)
Y = Y_df$Y_noised
names(Y_df)
#> [1] "id"       "Y_beta1"  "Y_beta2"  "Y_beta3"  "Y_real"   "Y_noised"
head(Y_df)
#>   id    Y_beta1    Y_beta2     Y_beta3    Y_real  Y_noised
#> 1  1  65.806580  -8.252748  -7.4320011 115.55393 117.36995
#> 2  2  26.027793   3.651140   0.2430665  95.35410  98.68405
#> 3  3  18.239666  -6.248977 -19.0326323  58.39016 102.56355
#> 4  4   5.205643  22.113474 -19.5805494  73.17067  85.04207
#> 5  5 -12.639639  38.985054  -9.9986223  81.77889  49.19584
#> 6  6 -13.391616 -20.803539 -20.0241998  11.21275  -6.47791

Test set

nind_test = floor(nind*TTRatio/(1-TTRatio))
df_test = generate_X_df_multistates(nind = nind_test, N_states, start, end,
                              lambdas,  transition_df)
df_test_processed = cat_data_to_indicator(df_test)

Y_df_test = generate_Y_df(df_test_processed, curve_type = curve_type,
                          beta_real_func_or_list = beta_func_list,
                          beta_0_real = beta_0_real,
                          NotS_ratio= NotS_ratio)

PLS functions

Naive PLS

naivePLS_obj = naivePLS(df_list = df_cfd, Y = Y, regul_time_obj = regul_time,
                        curve_type_obj = 'cat', 
                        id_col_obj = 'id', time_col_obj = 'time',
                        print_steps = TRUE, plot_rmsep = TRUE,
                        print_nbComp = TRUE,plot_reg_curves = TRUE)
#> ### Naive PLS ### 
#> => Input format assertions.
#> => Input format assertions OK.
#> => Data formatting.
#> => PLS model.

#> [1] "Optimal number of PLS components :  2"

Functional PLS

fpls_obj = funcPLS(df_list = df_cfd, Y = Y_df$Y_noised,
                   basis_obj = basis,
                   curve_type_obj = 'cat',
                   regul_time_obj = regul_time,
                   id_col_obj = 'id', time_col_obj = 'time',
                   print_steps = TRUE, plot_rmsep = TRUE, 
                   print_nbComp = TRUE, plot_reg_curves = TRUE)
#> ### Functional PLS ### 
#> => Input format assertions.
#> => Input format assertions OK.
#> => Building alpha matrix.
#> => Building curve names.
#> ==> Evaluating alpha for : CatFD_1_1.
#>  ==> Evaluating alpha for : CatFD_1_2.
#>  ==> Evaluating alpha for : CatFD_1_3.
#> => Evaluate metrix and root_metric.
#> => plsr(Y ~ trans_alphas).

#> Optimal number of PLS components :  2 .
#> => Build Intercept and regression curves for optimal number of components.
#>  ==> Build  regression curve for : CatFD_1_1
#>  ==> Build  regression curve for : CatFD_1_2
#>  ==> Build  regression curve for : CatFD_1_3

Smooth PLS

spls_obj = smoothPLS(df_list = df_cfd, Y = Y_df$Y_noised, basis_obj = basis,
                     orth_obj = TRUE, curve_type_obj = 'cat',
                     int_mode = 1, 
                     id_col_obj ='id', time_col_obj = 'time',
                     print_steps = TRUE, plot_rmsep = TRUE,
                     print_nbComp = TRUE, plot_reg_curves = TRUE)
#> ### Smooth PLS ### 
#> ## Use parallelization in case of heavy computational load. ## 
#> ## Threshold can be manualy adjusted : (default 2500) ## 
#> ## >options(SmoothPLS.parallel_threshold = 500) ## 
#> => Input format assertions.
#> => Input format assertions OK.
#> => Orthonormalize basis.
#> => Data objects formatting.
#> => Evaluate Lambda matrix.
#> ==> Lambda for : CatFD_1_state_1.
#> ==> Lambda for : CatFD_1_state_2.
#> ==> Lambda for : CatFD_1_state_3.
#> => PLSR model.
#> => Optimal number of PLS components : 2

#> => Evaluate SmoothPLS functions and <w_i, p_j> coef.
#> => Build regression functions and intercept.
#> ==> Build regression curve for : CatFD_1_state_1
#> ==> Build regression curve for : CatFD_1_state_2
#> ==> Build regression curve for : CatFD_1_state_3

names(spls_obj$reg_obj)
#> [1] "Intercept"       "CatFD_1_state_1" "CatFD_1_state_2" "CatFD_1_state_3"

Curves comparison

# Warning ms_spls_obj$delta_ms_list[[1]] is the intercept!
cat("curve_1 : smooth PLS regression curve.\n")
#> curve_1 : smooth PLS regression curve.
cat("curve_2 : functional PLS regression curve.\n")
#> curve_2 : functional PLS regression curve.
cat("curve_3 : naive PLS regression coefficients\n")
#> curve_3 : naive PLS regression coefficients

for(i in 1:N_states){
  start = 0
  print(paste0("State_", (i)))
  evaluate_curves_distances(real_f = beta_func_list[[i]],
                          regul_time = regul_time, 
                          fun_fd_list = list(spls_obj$reg_obj[[i+1]], 
                                             fpls_obj$reg_obj[[i+1]],
                                             approxfun(
                                               x = regul_time,
                                               y = naivePLS_obj$opti_reg_coef[
                                                 start:(start+length(regul_time)
                                                        )])
                                             )
                          )
  start = start + length(regul_time)
  
}
#> [1] "State_1"
#> [1] "real_f -> curve_1 / INPROD  : 47.4572000558795 / DIST : 13.2905098633588"
#> [1] "real_f -> curve_2 / INPROD  : 43.6591352214833 / DIST : 13.6320162598076"
#> [1] "real_f -> curve_3 / INPROD  : 13.4690073074092 / DIST : 27.8941820901434"
#> [1] "State_2"
#> [1] "real_f -> curve_1 / INPROD  : 22.9684162701822 / DIST : 14.1561671305604"
#> [1] "real_f -> curve_2 / INPROD  : 25.7051668331251 / DIST : 15.742963397086"
#> [1] "real_f -> curve_3 / INPROD  : 31.4665194389073 / DIST : 38.1269198992419"
#> [1] "State_3"
#> [1] "real_f -> curve_1 / INPROD  : 25.9590236019394 / DIST : 15.1535445344829"
#> [1] "real_f -> curve_2 / INPROD  : 27.0203400870638 / DIST : 11.7454038364192"
#> [1] "real_f -> curve_3 / INPROD  : -64.0495576976778 / DIST : 37.3104708298011"
for(i in 1:N_states){
  start = 0
  
  y_lim = eval_max_min_y(f_list = list(spls_obj$reg_ob[[i+1]],
                                       fpls_obj$reg_ob[[i+1]],
                                       approxfun(
                                         x = regul_time,
                                         y = naivePLS_obj$opti_reg_coef[
                                           start:(start+length(regul_time))]),
                                       beta_func_list[[i]]
                                       ), 
                         regul_time = regul_time_0)
  
  plot(regul_time_0, beta_func_list[[i]](regul_time_0), col = 'black', 
       ylim = y_lim, xlab = 'Time', ylab = 'Value', type = 'l')
  lines(regul_time_0, approxfun(x = regul_time,
                                y = naivePLS_obj$opti_reg_coef[
                                  start:(start+
                                           length(regul_time))])(regul_time_0), 
       col = 'green')
  title(paste0(names(spls_obj$reg_obj)[i+1], " regression curves"))
  plot(spls_obj$reg_obj[[i+1]], col = 'blue', add = TRUE)
  plot(fpls_obj$reg_obj[[i+1]], col = 'red', add = TRUE)
  legend("topleft",
         legend = c("Real curve", "NaivePLS coef", 
                    "SmoothPLS reg curve", "FunctionalPLS reg curve"),
         col = c("black", "green", "blue", "red"),
         lty = 1,
         lwd = 1)
  
  start = start + length(regul_time)
}

Results

train_results = data.frame(matrix(ncol = 5, nrow = 3))
colnames(train_results) = c("PRESS", "RMSE", "MAE", "R2", "var_Y")
rownames(train_results) = c("NaivePLS", "FPLS", "SmoothPLS")

test_results = train_results
print(paste0("There is ", 100*NotS_ratio, "% of noised in Y"))
#> [1] "There is 20% of noised in Y"

Train set

Y_train = Y_df$Y_noised

# Naive
Y_hat = predict(naivePLS_obj$plsr_model, 
                ncomp = naivePLS_obj$nbCP_opti, 
                newdata = naivePLS_obj$plsr_model$model$`as.matrix(df_mod_wide)`)
train_results["NaivePLS", ] = evaluate_results(Y_train, Y_hat)


# FPLS
Y_hat_fpls = (predict(fpls_obj$plsr_model, ncomp = fpls_obj$nbCP_opti,
                newdata = fpls_obj$trans_alphas) 
              + fpls_obj$reg_obj$Intercept
              + mean(Y))

Y_hat_fpls = smoothPLS_predict(df_predict_list = df_cfd,
                               delta_list = fpls_obj$reg_obj, 
                               curve_type_obj = curve_type,
                               int_mode = int_mode,
                               regul_time_obj = regul_time)

train_results["FPLS", ] = evaluate_results(Y_train, Y_hat_fpls)

# Smooth PLS
Y_hat_spls = smoothPLS_predict(df_predict_list = df_cfd,
                               delta_list = spls_obj$reg_obj, 
                               curve_type_obj = curve_type,
                               int_mode = int_mode,
                               regul_time_obj = regul_time)

train_results["SmoothPLS", ] = evaluate_results(Y_train, Y_hat_spls)

train_results["NaivePLS", "nb_cp"] = naivePLS_obj$nbCP_opti
train_results["FPLS", "nb_cp"] = fpls_obj$nbCP_opti
train_results["SmoothPLS", "nb_cp"] = spls_obj$nbCP_opti
train_results
#>              PRESS     RMSE      MAE        R2     var_Y nb_cp
#> NaivePLS  56851.26 23.84350 18.74728 0.8052915  80.52915     2
#> FPLS      51006.48 22.58462 17.48461 0.8253091  77.95544     2
#> SmoothPLS 54449.52 23.33442 18.54678 0.8135171 111.55514     2

Test set

Y_test = Y_df_test$Y_noised

# Naive
df_test_wide = naivePLS_formatting(df_list = df_test,
                                  regul_time_obj = regul_time,
                                  curve_type_obj = curve_type, 
                                  id_col_obj = 'id', time_col_obj = 'time')

Y_hat = predict(naivePLS_obj$plsr_model,
                ncomp = naivePLS_obj$nbCP_opti, 
                newdata = as.matrix(df_test_wide))
test_results["NaivePLS", ] = evaluate_results(Y_test, Y_hat)

# FPLS
Y_hat_fpls = smoothPLS_predict(df_predict_list = df_test,
                               delta_list = fpls_obj$reg_obj,
                               curve_type_obj = curve_type,
                               int_mode = int_mode, 
                               regul_time_obj = regul_time) 

test_results["FPLS", ] = evaluate_results(Y_test, Y_hat_fpls)

# Smooth PLS
Y_hat_spls = smoothPLS_predict(df_predict_list = df_test,
                               delta_list = spls_obj$reg_obj,
                               curve_type_obj = curve_type,
                               int_mode = int_mode, 
                               regul_time_obj = regul_time)

test_results["SmoothPLS", ] = evaluate_results(Y_test, Y_hat_spls)

test_results["NaivePLS", "nb_cp"] = naivePLS_obj$nbCP_opti
test_results["FPLS", "nb_cp"] = fpls_obj$nbCP_opti
test_results["SmoothPLS", "nb_cp"] = spls_obj$nbCP_opti
test_results
#>              PRESS     RMSE      MAE        R2     var_Y nb_cp
#> NaivePLS  12668.95 22.51128 17.71806 0.7918992  96.46796     2
#> FPLS      13999.53 23.66392 17.75687 0.7700429  91.67738     2
#> SmoothPLS 17130.76 26.17691 20.60632 0.7186092 135.21697     2

Plot results

train_results
#>              PRESS     RMSE      MAE        R2     var_Y nb_cp
#> NaivePLS  56851.26 23.84350 18.74728 0.8052915  80.52915     2
#> FPLS      51006.48 22.58462 17.48461 0.8253091  77.95544     2
#> SmoothPLS 54449.52 23.33442 18.54678 0.8135171 111.55514     2

test_results
#>              PRESS     RMSE      MAE        R2     var_Y nb_cp
#> NaivePLS  12668.95 22.51128 17.71806 0.7918992  96.46796     2
#> FPLS      13999.53 23.66392 17.75687 0.7700429  91.67738     2
#> SmoothPLS 17130.76 26.17691 20.60632 0.7186092 135.21697     2
plot_model_metrics_base(train_results, test_results)

plot_model_metrics_base(train_results, test_results,
                        models_to_plot = c('FPLS', 'SmoothPLS'))