Plotting
The spfcICOMP package implements Shrinkage Principal
Fitted Components (SPFC) methodology for sufficient dimension reduction
under high-dimensional settings.
The package combines
within a unified framework.
fit <- spfc_fit(
X = X,
y = y,
d = 1,
ytype = "continuous",
cov_method = "mec",
nslices = 5,
poly_degree = 2
)
fit##
## Shrinkage Principal Fitted Components Fit
## =========================================
##
## Response type: continuous
## Covariance method: mec
## Structural dimension: 1
## Shrinkage rho: 0.334172
## Slices used: 5
##
## Leading eigenvalues:
## [1] 1.756594 0.444314 0.000000 0.000000 0.000000 0.000000
##
## Reduced score matrix dimension:
## [1] 100 1##
## Summary of Shrinkage Principal Fitted Components Fit
## ====================================================
##
## Response type: continuous
## Covariance method: mec
## Structural dimension: 1
## Shrinkage rho: 0.334172
## Slices used: 5
##
## Preprocessing:
## Centred: TRUE
## Scaled: FALSE
##
## Score matrix dimension:
## [1] 100 1
##
## Eigenvalue summary:
## component eigenvalue proportion cumulative_proportion
## 1 1 1.7565935 0.7981225 0.7981225
## 2 2 0.4443135 0.2018775 1.0000000
## 3 3 0.0000000 0.0000000 1.0000000
## 4 4 0.0000000 0.0000000 1.0000000
## 5 5 0.0000000 0.0000000 1.0000000
## 6 6 0.0000000 0.0000000 1.0000000
## 7 7 0.0000000 0.0000000 1.0000000
## 8 8 0.0000000 0.0000000 1.0000000
## 9 9 0.0000000 0.0000000 1.0000000
## 10 10 0.0000000 0.0000000 1.0000000## dir1
## [1,] -0.144789015
## [2,] 0.066982646
## [3,] 0.755598441
## [4,] 0.437298165
## [5,] 0.161266594
## [6,] 0.078488719
## [7,] 0.040066485
## [8,] -0.027000408
## [9,] -0.038766088
## [10,] -0.048790194
## [11,] 0.039893765
## [12,] -0.063439753
## [13,] -0.045789728
## [14,] -0.107824894
## [15,] -0.151024192
## [16,] 0.015384677
## [17,] 0.063066373
## [18,] -0.051192235
## [19,] -0.051227957
## [20,] -0.032227983
## [21,] 0.010966359
## [22,] -0.092314116
## [23,] -0.009047916
## [24,] -0.033750824
## [25,] 0.056240919
## [26,] -0.026041093
## [27,] 0.007381842
## [28,] 0.076870322
## [29,] -0.033637149
## [30,] -0.075260220
## [31,] 0.022255242
## [32,] -0.030489694
## [33,] 0.075816476
## [34,] -0.031958361
## [35,] -0.047534473
## [36,] -0.038778821
## [37,] -0.055427849
## [38,] -0.020983452
## [39,] 0.124279900
## [40,] 0.120473767
## [41,] 0.104758094
## [42,] 0.051068370
## [43,] -0.065226539
## [44,] -0.081000382
## [45,] -0.105136881
## [46,] 0.040933147
## [47,] 0.042819366
## [48,] 0.023815682
## [49,] 0.099651957
## [50,] -0.003062986## SPFC1
## [1,] 1.85053361
## [2,] 2.00267298
## [3,] -2.43877726
## [4,] 0.68275423
## [5,] -0.07949886
## [6,] 2.65260788## SPFC1
## [1,] 1.85053361
## [2,] 2.00267298
## [3,] -2.43877726
## [4,] 0.68275423
## [5,] -0.07949886dsel <- spfc_select_dimension(
X = X,
y = y,
d_grid = 1:3,
cov_method = "mec",
ytype = "continuous"
)## Scoring d = 1 using covariance method: mec## Scoring d = 2 using covariance method: mec## Scoring d = 3 using covariance method: mec## cov_method ytype d rho nslices_used reduced_model loglik npar
## 1 mec continuous 1 0.3341716 5 lm -81.91911 3
## 2 mec continuous 2 0.3341716 5 lm -74.30525 4
## 3 mec continuous 3 0.3341716 5 lm -73.17670 5
## AIC BIC CAIC ICOMP_IFIM ICOMP_MISSPEC CICOMP
## 1 169.8382 177.6537 180.6537 164.2588 164.1571 180.9971
## 2 156.6105 167.0312 171.0312 149.1793 148.9210 171.3878
## 3 156.3534 169.3793 174.3793 147.4786 147.1968 175.3587## criterion selected_d minimum_value
## 1 AIC 3 156.3534
## 2 BIC 2 167.0312
## 3 CAIC 2 171.0312
## 4 ICOMP_IFIM 3 147.4786
## 5 ICOMP_MISSPEC 3 147.1968
## 6 CICOMP 2 171.3878The default C1F-calibrated feature-screening rule uses the covariance of the fitted downstream reduced model. The reduced-space model is therefore fitted before C1F-based screening is applied.
reduced_model <- fit_reduced_model(
Z = scores,
y = y,
ytype = "continuous"
)
vsel <- spfc_select_variables(
fit = fit,
method = "adaptive_weighted_l1",
selection_rule = "c1f",
reduced_model = reduced_model
)
head(vsel)## variable importance weight penalty shrunk_importance selected
## 1 3 0.7555984 1.013777e-05 7.514317e-05 0.7555233 TRUE
## 2 4 0.4372982 3.026699e-05 2.243450e-04 0.4370738 TRUE
## 3 5 0.1612666 2.225542e-04 1.649616e-03 0.1596170 TRUE
## 4 15 0.1510242 2.537649e-04 1.880956e-03 0.1491432 TRUE
## 5 1 0.1447890 2.760917e-04 2.046447e-03 0.1427426 TRUE
## 6 39 0.1242799 3.747337e-04 2.777601e-03 0.1215023 TRUEbench <- benchmark_spfc(
X = X,
y = y,
d = 1,
methods = c(
"mec",
"oas",
"sre",
"sde",
"cse"
),
verbose = FALSE
)
bench##
## SPFC Benchmark
## ==============
##
## Response type: continuous
## Structural dimension: 1
## Variable method: adaptive_weighted_l1
## Validation: resubstitution
## Methods compared: mec, oas, sre, sde, cse
##
## Summary:
## cov_method validation ytype d rho nslices_used reduced_model
## 1 mec resubstitution continuous 1 0.3341716 5 lm
## 2 oas resubstitution continuous 1 0.1898839 NA lm
## 3 sre resubstitution continuous 1 0.1000000 NA lm
## 4 sde resubstitution continuous 1 NA NA lm
## 5 cse resubstitution continuous 1 0.5000000 NA lm
## runtime_sec rmse mae mean_rmse sd_rmse mean_mae sd_mae accuracy
## 1 0.020030975 0.5489503 0.4491404 NA NA NA NA NA
## 2 0.009188890 0.5614532 0.4595925 NA NA NA NA NA
## 3 0.007362843 0.3764402 0.3051238 NA NA NA NA NA
## 4 0.006926060 0.6179009 0.5107200 NA NA NA NA NA
## 5 0.013350010 0.4674027 0.3837767 NA NA NA NA NA
## sensitivity specificity precision f1 balanced_accuracy mean_accuracy
## 1 NA NA NA NA NA NA
## 2 NA NA NA NA NA NA
## 3 NA NA NA NA NA NA
## 4 NA NA NA NA NA NA
## 5 NA NA NA NA NA NA
## mean_sensitivity mean_specificity mean_precision mean_f1
## 1 NA NA NA NA
## 2 NA NA NA NA
## 3 NA NA NA NA
## 4 NA NA NA NA
## 5 NA NA NA NA
## mean_balanced_accuracy n_selected
## 1 NA 36
## 2 NA 32
## 3 NA 31
## 4 NA 27
## 5 NA 32##
## Summary of SPFC Benchmark
## =========================
##
## Response type: continuous
## Structural dimension: 1
## Variable method: adaptive_weighted_l1
## Validation: resubstitution
## Selection metric: rmse
## Smaller is better: TRUE
## Best method: sre
##
## Benchmark table:
## cov_method validation ytype d rho nslices_used reduced_model
## 1 mec resubstitution continuous 1 0.3341716 5 lm
## 2 oas resubstitution continuous 1 0.1898839 NA lm
## 3 sre resubstitution continuous 1 0.1000000 NA lm
## 4 sde resubstitution continuous 1 NA NA lm
## 5 cse resubstitution continuous 1 0.5000000 NA lm
## runtime_sec rmse mae mean_rmse sd_rmse mean_mae sd_mae accuracy
## 1 0.020030975 0.5489503 0.4491404 NA NA NA NA NA
## 2 0.009188890 0.5614532 0.4595925 NA NA NA NA NA
## 3 0.007362843 0.3764402 0.3051238 NA NA NA NA NA
## 4 0.006926060 0.6179009 0.5107200 NA NA NA NA NA
## 5 0.013350010 0.4674027 0.3837767 NA NA NA NA NA
## sensitivity specificity precision f1 balanced_accuracy mean_accuracy
## 1 NA NA NA NA NA NA
## 2 NA NA NA NA NA NA
## 3 NA NA NA NA NA NA
## 4 NA NA NA NA NA NA
## 5 NA NA NA NA NA NA
## mean_sensitivity mean_specificity mean_precision mean_f1
## 1 NA NA NA NA
## 2 NA NA NA NA
## 3 NA NA NA NA
## 4 NA NA NA NA
## 5 NA NA NA NA
## mean_balanced_accuracy n_selected
## 1 NA 36
## 2 NA 32
## 3 NA 31
## 4 NA 27
## 5 NA 32results <- run_spfc_simulation(
response_type = "continuous",
nrep = 5,
n = 100,
p = 50,
d = 1,
s = 5,
rho_x = 0.5,
snr = 2,
cov_methods = c(
"mec",
"oas"
)
)
summary_results <-
summarise_spfc_simulation(
results
)
summary_results## ytype cov_method criterion selection_rule variable_method n p
## 1 continuous mec KNOWN_D c1f adaptive_weighted_l1 100 50
## 2 continuous mec KNOWN_D c1f c1f_extension 100 50
## 3 continuous oas KNOWN_D c1f adaptive_weighted_l1 100 50
## 4 continuous oas KNOWN_D c1f c1f_extension 100 50
## true_d s rho_x snr nrep dimension_recovery_rate mean_selected_d sd_selected_d
## 1 1 5 0.5 2 5 1 1 0
## 2 1 5 0.5 2 5 1 1 0
## 3 1 5 0.5 2 5 1 1 0
## 4 1 5 0.5 2 5 1 1 0
## mean_runtime_sec sd_runtime_sec mean_subspace_distance sd_subspace_distance
## 1 0.018012571 0.002230873 0.6208041 0.1117431
## 2 0.021338415 0.003860143 0.6208041 0.1117431
## 3 0.009530401 0.002788551 0.6260605 0.1097125
## 4 0.008028221 0.002309219 0.6260605 0.1097125
## mean_precision sd_precision mean_recall sd_recall mean_f1_variable
## 1 0.1455734 0.006367509 1 0 0.2541062
## 2 0.1455734 0.006367509 1 0 0.2541062
## 3 0.1545897 0.007176028 1 0 0.2677293
## 4 0.1545897 0.007176028 1 0 0.2677293
## sd_f1_variable mean_n_selected sd_n_selected mean_c1f
## 1 0.009713502 34.4 1.516575 0.195997
## 2 0.009713502 34.4 1.516575 0.195997
## 3 0.010777776 32.4 1.516575 0.202018
## 4 0.010777776 32.4 1.516575 0.202018
## mean_c1f_complexity_fraction mean_c1f_loading_scale mean_c1f_hd_factor
## 1 0.1611789 0.05026529 0.279715
## 2 0.1611789 0.05026529 0.279715
## 3 0.1654130 0.05155130 0.279715
## 4 0.1654130 0.05155130 0.279715
## mean_c1f_complexity_multiplier mean_c1f_global_penalty mean_rmse sd_rmse
## 1 1.161179 0.01619011 0.6444747 0.1153448
## 2 1.161179 0.01619011 0.6444747 0.1153448
## 3 1.165413 0.01661490 0.6465237 0.1171882
## 4 1.165413 0.01661490 0.6465237 0.1171882
## mean_mae sd_mae mean_accuracy sd_accuracy mean_sensitivity
## 1 0.5068063 0.08563456 NA NA NA
## 2 0.5068063 0.08563456 NA NA NA
## 3 0.5083053 0.08434289 NA NA NA
## 4 0.5083053 0.08434289 NA NA NA
## sd_sensitivity mean_specificity sd_specificity mean_f1_classification
## 1 NA NA NA NA
## 2 NA NA NA NA
## 3 NA NA NA NA
## 4 NA NA NA NA
## sd_f1_classification
## 1 NA
## 2 NA
## 3 NA
## 4 NA
Cook, R. D. and Forzani, L. (2008). Principal Fitted Components for dimension reduction in regression. Statistical Science, 23(4), 485–501. doi:10.1214/08-STS275.
Chen, Y., Wiesel, A., Eldar, Y. C. and Hero, A. O. (2010). Shrinkage algorithms for MMSE covariance estimation. IEEE Transactions on Signal Processing, 58(10), 5016–5029. doi:10.1109/TSP.2010.2053029.
Bozdogan, H. (2000). Akaike’s Information Criterion and recent developments in information complexity. Journal of Mathematical Psychology, 44(1), 62–91. doi:10.1006/jmps.1999.1277.
Olorede, K. O. and Yahya, W. B. (2019). A new covariance estimator for sufficient dimension reduction in high-dimensional and undersized sample problems. arXiv. doi:10.48550/arXiv.1909.13017.