The slide presents a title on nonparametric functional data regression, focusing on handling incomplete observations. It is authored by Ibnu Abdul Hadi (student ID H062242008) from the Mathematics Program at [University Name].

On Nonparametric Functional Data Regression with Incomplete Observations

Penyusun: Ibnu Abdul Hadi (H062242008), Program Studi Matematika, Universitas [Nama Universitas]

Source: Review Artikel oleh Ibnu Abdul Hadi

Kernel regression on functional data often encounters incomplete observations, particularly in MNAR cases where missing data occurs non-randomly. This leads to major challenges like estimation bias and information loss, necessitating robust methods to handle data incompleteness.

Latar Belakang Masalah

• Regresi kernel pada data fungsional menghadapi observasi tidak lengkap.
• Kasus MNAR menyebabkan missing data yang tidak acak.
• Tantangan utama: bias estimasi dan kehilangan informasi.
• Diperlukan metode robust untuk menangani ketidaklengkapan data.

Source: On Nonparametric Functional Data Regression with Incomplete Observations, Ibnu Abdul Hadi (H062242008)

This slide outlines the objectives and contributions of a research study focused on developing a nonparametric kernel estimator for regression with incomplete functional data, while providing convergence theory, proving classification optimality for partially labeled data, and applying the method to functional classification tasks. It also highlights improvements to the literature on nonparametric regression under missing-not-at-random (MNAR) assumptions.

Tujuan dan Kontribusi Penelitian

• Mengembangkan estimator kernel nonparametrik untuk regresi data fungsional tidak lengkap.
• Menyediakan teori konvergensi dan tingkat konvergensi estimator.
• Membuktikan optimalitas klasifikasi pada data berlabel sebagian.
• Menerapkan metode pada aplikasi klasifikasi fungsional.
• Meningkatkan literatur regresi nonparametrik di bawah asumsi MNAR.

Source: On Nonparametric Functional Data Regression with Incomplete Observations, Ibnu Abdul Hadi (H062242008)

The slide outlines general methodology for functional regression, with the left column explaining the training-validation split for parameter estimation, which enables robust model evaluation, avoids overfitting, and ensures good generalization on incomplete functional data. The right column describes the kernel estimator applied to follow-up subsamples to handle missing observations, using an iterative process to reduce bias and achieve convergence for improved nonparametric regression accuracy.

Metodologi Umum

The slide outlines key assumptions (A0–A8) for a regression model, starting with A0 assuming continuous and smooth functional data, followed by A1–A3 requiring positive, symmetric kernels that satisfy integration conditions. Assumptions A4–A6 impose finite upper bounds on density for stability, while A7–A8 ensure high-degree regularity for estimator convergence, collectively guaranteeing the estimator's stability and consistency.

Asumsi Model (A0–A8)

• A0: Data fungsional diasumsikan kontinu dan lancar.
• A1-A3: Kernel positif, simetris, dan memenuhi kondisi integrasi.
• A4-A6: Densitas memiliki batas atas terhingga untuk kestabilan.
• A7-A8: Regularitas derajat tinggi memastikan konvergensi estimator.
• Asumsi ini menjamin kestabilan dan konsistensi estimator regresi.

Source: On Nonparametric Functional Data Regression with Incomplete Observations, Ibnu Abdul Hadi (H062242008)

The slide, titled "Teorema Utama," features a quote explaining that Theorem 1 proves the convergence of the kernel estimator to the true value under assumptions A0-A8, while Theorem 2 demonstrates the optimality of classification rates for incomplete data based on precise functional approximations. It is attributed to Ibnu Abdul Hadi, a statistics student at Universitas X (student ID H062242008).

Teorema Utama

> Teorema 1 membuktikan konvergensi estimator kernel ke nilai sejati di bawah asumsi A0-A8, sementara Teorema 2 menunjukkan optimalitas tingkat klasifikasi untuk data tidak lengkap, didasarkan pada aproksimasi fungsional yang presisi.

— Ibnu Abdul Hadi (H062242008, Mahasiswa Program Studi Statistika, Universitas X)

Source: On Nonparametric Functional Data Regression with Incomplete Observations

This slide discusses applications of classification on partially labeled data using kernel estimators. It highlights how subsampling methods effectively complete missing labels, leading to significant accuracy improvements over conventional approaches, with successful results on incomplete functional datasets.

Aplikasi pada Klasifikasi

• Klasifikasi data berlabel sebagian menggunakan estimator kernel.
• Metode subsample melengkapi label hilang secara efektif.
• Peningkatan akurasi signifikan dibanding metode konvensional.
• Aplikasi berhasil pada dataset fungsional tidak lengkap.

Source: Ibnu Abdul Hadi (H062242008)

This slide outlines a numerical study on simulation design, focusing on generating synthetic functional data with MNAR proportions ranging from 20% to 50%. It compares complete versus incomplete estimators and Gaussian versus Epanechnikov kernels, while evaluating performance through Mean Squared Error (MSE) and classification error rates.

Studi Numerik: Rancangan Simulasi

• Simulasi data fungsional sintetis dengan proporsi MNAR 20-50%.
• Perbandingan estimator lengkap versus tidak lengkap.
• Perbandingan kernel Gaussian versus Epanechnikov.
• Pengukuran kinerja menggunakan Mean Squared Error (MSE).
• Pengukuran kinerja menggunakan tingkat kesalahan klasifikasi.

Source: On Nonparametric Functional Data Regression with Incomplete Observations, Ibnu Abdul Hadi (H062242008)

The simulation results highlight an average MSE of 0.05 for the subsample estimator, which is significantly lower than the 0.12 MSE of the comparison method. Additionally, the classification accuracy for MNAR data reaches 85% even with 30% incomplete data.

Hasil Utama Simulasi

• 0.05: MSE Rata-rata Estimator

Subsample vs. 0.12 metode lain

• 85%: Akurasi Klasifikasi MNAR

Pada data 30% tidak lengkap

• 0.12: MSE Metode Perbandingan

Lebih tinggi dari subsample Source: Studi Numerik Artikel

The new method proves effective for incomplete functional regression, offering theoretical contributions and numerical evidence specifically for MNAR cases while enriching nonparametric statistical literature. It suggests further applications to real data, ending with thanks for attention and an invitation to discuss practical implementations.

Kesimpulan Utama

- Metode baru efektif untuk regresi fungsional tidak lengkap.

• Kontribusi: Teori dan bukti numerik untuk kasus MNAR.
• Saran: Aplikasi lebih lanjut pada data real.
• Memperkaya literatur statistik nonparametrik.

Pesan Penutup: Terima kasih atas perhatiannya.

Ajakan Bertindak: Mari diskusikan aplikasi lebih lanjut pada data nyata.

Source: On Nonparametric Functional Data Regression with Incomplete Observations - Ibnu Abdul Hadi (H062242008)