# 预测目标、正则化与模型复杂度 从损失函数推导 ridge，重建相同信息下的完整基线比较，分清两类单位变换。 Entry: zh-qt20 | Node: QT20 | Language: zh | Editorial revision: 2026-09-21 ## Teaching instructions 请以具有微积分、线性代数与基本概率背景的高年级本科至研究生为对象，围绕“预测目标、正则化与模型复杂度”完成一次学习。先实际取得并读完 required_readings 的完整指定单元，记录实际版本/定位/假设/支持范围；目录、摘要或入口不抵扣阅读。可先读取本篇所有必读以备课，但不要把它们全变成读者额外作业。来源打不开时先尝试同版作者可读入口；仍缺承重单元应说明具体缺口，不伪称已读，也不要求无关新批准。先让读者重建2010-01的一列ridge预测，再推导未标准化x×10与仅y×100的区别。反馈必须核矩阵维度、nλ、截距、满秩边界以及总冠军/回归冠军。 先让读者尝试，再根据本文完整解析反馈；引用范围内解释，不把模拟当市场事实、不把最优目标当收益保证。只在读者选择分支时启用 optional_readings。完成后用一项新输入迁移检验，明确其能独立完成什么。 Before substantive teaching, actually retrieve every required reading unit for the selected scope. Read its complete designated section, including necessary assumptions, tables and footnotes. A working URL or an editorial access date is not a runtime reading receipt. Record the actual version, location, scope and what it supports. If unavailable, use a previously verified equivalent source; if the required unit remains unavailable, identify that gap rather than teach it from memory. Start runtime_reading_log empty. Once reading is complete, use a substantive diagnostic or follow the reader's request for direct explanation. Advance one complete reasoning task at a time; skip mastered basics. Distinguish original facts, supplied teaching assumptions and inference. ## Required readings and runtime protocol ```json { "export_mode": "public", "required_readings": [ { "source_id": "QTDE-ISLP", "access": { "kind": "pdf_full_text", "uri": "https://drive.google.com/uc?export=download&id=1ajFkHO6zjrdGNqhqW1jKBZdiNGh_8YQ1", "author_gateway": "https://hastie.su.domains/ISLP/ISLP_website.pdf.download.html", "verified_access_at": "2026-09-21" }, "required_unit": { "locator": "§6.2.1 pp.240–244；§6.2.3 pp.252–253", "scope": "Ridge、截距、尺度与调参完整子节", "purpose": "核目标、单位变换与验证选择，不以书目页抵扣" }, "version": "First Printing July 5, 2023", "supports": "Ridge、截距、尺度与调参完整子节", "title": "An Introduction to Statistical Learning with Applications in Python", "authors": [ "James", "Witten", "Hastie", "Tibshirani", "Taylor" ] }, { "source_id": "QTDE-ACCURACY", "access": { "kind": "html_full_text", "uri": "https://otexts.com/fpp3/accuracy.html", "verified_access_at": "2026-09-21" }, "required_unit": { "locator": "§5.8 全节", "scope": "预测误差与RMSE/MAE", "purpose": "选择与目标匹配的指标" }, "version": "3e online; accessed 2026-09-21", "supports": "预测误差与RMSE/MAE", "title": "Forecasting: Principles and Practice, §5.8", "authors": [ "Rob J Hyndman", "George Athanasopoulos" ] }, { "source_id": "QTDE-FRENCH", "access": { "kind": "html_full_text", "uri": "https://mba.tuck.dartmouth.edu/pages/Faculty/ken.french/Data_Library/det_30_ind_port.html", "verified_access_at": "2026-09-21" }, "required_unit": { "locator": "30 Industry Portfolios完整构造说明；随包432行冻结CSV", "scope": "原始组合、频率、单位", "purpose": "识别重建快照" }, "version": "frozen CRSP database vintage 202607; CIZ reconstruction", "supports": "原始组合、频率、单位", "title": "30 Industry Portfolios", "authors": [ "Kenneth R. French" ] }, { "source_id": "QTDE-FRENCH-REGIME", "access": { "kind": "html_full_text", "uri": "https://mba.tuck.dartmouth.edu/pages/faculty/ken.french/data_library.html", "verified_access_at": "2026-09-21" }, "required_unit": { "locator": "页面FIZ/CIZ生产说明完整单元", "scope": "版本与生产方法", "purpose": "防止将当前历史重建当PIT" }, "version": "accessed 2026-09-21; source transition 2025-01", "supports": "版本与生产方法", "title": "French Data Library：FIZ/CIZ methodology", "authors": [ "Kenneth R. French" ] } ], "optional_readings": [ { "source_id": "QTDE-KMZ", "access": { "kind": "pdf_full_text", "uri": "https://economics.yale.edu/sites/default/files/2024-01/The%20Journal%20of%20Finance%20-%202023%20-%20KELLY%20-%20The%20Virtue%20of%20Complexity%20in%20Return%20Prediction%20%281%29.pdf", "verified_access_at": "2026-09-21" }, "required_unit": { "locator": "§V.A–C pp.487–493，含脚注33、38–41及图", "scope": "实证设置及评价口径", "purpose": "具名现代方法，不等于复现" }, "version": "Journal of Finance 79(1), 459–503, February 2024", "supports": "实证设置及评价口径", "branch": "复杂度研究" }, { "source_id": "QTDE-NAGEL", "access": { "kind": "pdf_full_text", "uri": "https://www.nber.org/system/files/working_papers/w34104/w34104.pdf", "verified_access_at": "2026-09-21" }, "required_unit": { "locator": "§II.A–D pp.6–15；§II.G.1 pp.23–25", "scope": "kernel权重与人工反转机制", "purpose": "与KMZ配对、不作普遍胜负判断" }, "version": "NBER Working Paper 34104, August 2025 frozen version", "supports": "kernel权重与人工反转机制", "branch": "复杂度研究" } ], "runtime_reading_log": [], "supplied_inputs": { "input_identity": "QT-DE-adopted-inputs-20260921-v1", "config": { "content_version": "2026-09-21-QT-DE-review-v2", "input_version": "QT-DE-adopted-inputs-20260921-v1", "dataset": { "file": "https://ou-liu-red-sugar.github.io/notebook/labs/qt-de/data/BusEq-value-weighted-monthly-199001-202512.csv", "origin_input_version": "QT-C-inputs-20260921-v1", "transport_sha256": "72dd0f35d9437d268971de8553827a5c7a4f85ebcf9712c0c1b85ea2ec7cfa74", "source_url": "https://mba.tuck.dartmouth.edu/pages/faculty/ken.french/ftp/30_Industry_Portfolios_CSV.zip", "source_details_url": "https://mba.tuck.dartmouth.edu/pages/Faculty/ken.french/Data_Library/det_30_ind_port.html", "regime_url": "https://mba.tuck.dartmouth.edu/pages/faculty/ken.french/data_library.html", "vintage": "202607 CRSP database; current CIZ reconstruction", "download_date": "2026-09-21", "block": "Average Value Weighted Returns -- Monthly", "column": "BusEq", "sample": [ "199001", "202512" ], "n": 432, "source_lines": [ 775, 1206 ], "missing_removed": 0, "missing_codes": [ -99.99, -999 ], "raw_unit": "percent", "internal_unit": "monthly decimal return", "display_errors": "百分点/月", "annualized": false, "identity": "单一当前重建快照；不是FIZ/CIZ时间拼接，不是逐月当时可见版本。" }, "forecast": { "experiment_id": "EXP-QT-D-FORECAST-01", "p_grid": [ 1, 3, 12 ], "lambda_grid": [ 0.0, 0.1, 1.0, 10.0 ], "default_candidate": "p1-l10", "target": "y_t=r_t", "features": "(r_(t-1),...,r_(t-p))", "warmup": [ "199001", "199012" ], "initial_train": [ "199101", "200912", 228 ], "validation": [ "201001", "201912", 120 ], "historical_evaluation": [ "202001", "202512", 72 ], "train_start_index": 12, "objective": "RSS/n + lambda*||beta||^2", "matrix_penalty": "n*lambda*I", "standardization_ddof": 0, "intercept_penalized": false, "selection": "minimum validation SSE including zero and expanding-mean baselines", "tie_rule": "regression grid: smaller p then smaller lambda", "frozen_hyperparameters_after_validation": true, "refit": "每原点用新成熟标签扩展重拟合；不再次选择p或lambda", "availability": "教学月末可知收益假设；French历史真实发布和接收时点未重建", "future_scaling_control": { "p": 12, "lambdas": [ 0.0, 0.1 ], "scope": "validation only", "identity": "故意信息越界对照" }, "default_clock_month": "201001", "default_clock_lambda": 0.1 }, "selection": { "experiment_id": "EXP-QT-D-SELECT-01", "B": 5000, "m_grid": [ 1, 10, 100, 500 ], "default_m": 100, "seed_all_null": 2201, "seed_mixed": 2202, "rng": "numpy.random.Generator(PCG64)", "n_development": 120, "n_holdout": 120, "sigma": 0.05, "alpha": 0.05, "hypothesis": "事前单侧H0:mu=0 versus H1:mu>0; sigma known", "pvalue": "0.5*erfc(Z/sqrt(2))", "draw_order": "先取(5000,500)开发Z矩阵，再取独立保留Z矩阵；M为共同500列前缀。mixed另重置2202。", "bh_adopted_comparison": "strict <", "source_pilot_comparison": "<=", "empty_BH": "k=0, no rejections, FDP=0", "mixed": { "m": 100, "signals": 10, "signal_mu": 0.01, "null_mu": 0.0 }, "identity": "直接模拟正态样本均值的Z分布；不含逐月路径，不是真实策略面板", "equal_example": { "p": [ 0.025, 0.06 ], "q": 0.05 }, "manual_example": { "p": [ 0.006, 0.012, 0.601, 0.756, 0.918 ], "q": 0.05 } }, "cost": { "experiment_id": "EXP-QT23-COST-01", "rf": 0, "continuous_default": { "mu": 0.015, "rho": 0, "sigma": 0.1, "gamma": 2, "w0": 0.5, "kappa": 0.002, "cap": 0.8, "cash_min": 0.2 }, "integer_default": { "mu": 0.002, "rho": 0, "sigma": 0.02, "gamma": 10, "wealth": 10000, "initial_shares": 0, "mid": 100, "ask": 100.05, "cap": 80, "cash_min": 2000, "mode": "spread_and_commission" }, "fee": { "id": "QTDE-IBKR", "url": "https://www.interactivebrokers.com/en/pricing/commissions-stocks.php?re=amer", "accessed_at": "2026-09-21", "scope": "IBKR Pro US Fixed ordinary whole-share single-order commission component only", "formula": "q=0:0; otherwise min(0.01*execution_price*abs(q),max(1,0.005*abs(q)))" }, "identity": "收益、波动、风险系数、报价、价差、容量为教学设定；佣金是有日期的实际规则组件", "end_value": "期末中价标记；没有退出委托，非往返净利", "excluded_costs": [ "监管/交易所/清算等其他收费", "税", "市场冲击", "持仓融资费", "退出交易费", "零股及改单最低费规则" ] }, "clocks": { "view_id": "EXP-QT-D-CLOCK-01", "shared_experiment": "EXP-QT-D-FORECAST-01", "BEA": [ { "period": "2020Q2", "vintage": "advance", "release_at": "2020-07-30T08:30:00-04:00", "received_at": null, "value_percent_SAAR": -32.9 }, { "period": "2020Q2", "vintage": "second", "release_at": "2020-08-27T08:30:00-04:00", "received_at": null, "value_percent_SAAR": -31.7 } ], "h3_task": { "formula": "prod(1+r_(s+j),j=1..3)-1", "cutoff": "202003", "latest_mature_origin": "201912", "performance_run": false } }, "scale_examples": { "raw_x": [ -1, 1 ], "y": [ -1, 1 ], "lambda": 1.0, "default_feature_scale": 10.0, "response_scale": 100.0 }, "source_identities": { "d_pilot": { "file": "qt-d-experiment-pilot.py", "sha256": "b8a23b01294d543b3b74c6a153168529db74ab7c84837ea3724a55f5071fd3fb", "read_scope": "1–193" }, "ef_pilot": { "file": "qt-ef-calculations.py", "sha256": "28eff60cf87400805af145fabd22d4fe17e2c25bac3708cd9e71506d939acb20", "adopted_scope": "commission, continuous_case, integer_case, decision_calculation" }, "approval": { "file": "qt-de-draft-approved.md", "read_scope": "1–7", "sha256": "80e92333d057772de9463d786a9be435c6dc1208744bf828d5eb2a4d2a5e9a48" }, "plan_review": { "file": "qt-def-plan-review.md", "read_scope": "1–20", "sha256": "56fddfa847ca0f83ec4822652c506f847e9da4e45f7223c039b31428f3aedfd4" } } }, "static_equivalent_html": "

对象：BusEq 1990—2025，432月；前12月预热；228/120/72个训练、验证、历史评价目标。每个原点单独拟合训练均值与ddof=0尺度，截距不惩罚。

RSS/n + λ‖β‖²；(ZcᵀZc+nλI)β=Zcᵀyc；R²OS=1−SSEmodel/SSEexpanding_mean。
全部候选验证RMSE pp/月验证MAE pp/月R²OS
p1-l05.045915644.15173855-0.00653461
p1-l0.15.044344934.15050355-0.00590808
p1-l15.037472534.14494608-0.00316905
p1-l105.030921474.13938860-0.00056157
p3-l05.061215324.10642026-0.01264768
p3-l0.15.057073274.10796478-0.01099088
p3-l15.041518914.11931979-0.00478130
p3-l105.031109724.13445130-0.00063645
p12-l05.135414964.16633893-0.04255704
p12-l0.15.120765734.16109078-0.03661755
p12-l15.068830374.14662389-0.01569721
p12-l105.034939384.13836039-0.00216039
zero5.213875724.31950000-0.07465755
expanding_mean5.029509454.138153610.00000000
历史评价2020—2025RMSE pp/月
selected_ridge6.82421636
expanding_mean6.82382346
zero7.12353747

总冠军是扩展均值，p1-λ10只是回归组冠军。第一个验证原点：ȳ=0.011622807018，z0=0.655343745416，β=0.000242269126，预测1.1781576574%。此历史已被研究者查看，不是首次揭开的独立保留期。

单位迁移：原始x×10、λ仍1时预测从1/2变成100/101；λ改100恢复1/2。仅y×100时预测×100，换回不变。

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"return_decimal": "0.0353", "loss_decimal": "-0.0353", "source_line": "776" }, { "month": "199003", "BusEq_percent": "4.10", "return_decimal": "0.0410", "loss_decimal": "-0.0410", "source_line": "777" }, { "month": "199004", "BusEq_percent": "-1.46", "return_decimal": "-0.0146", "loss_decimal": "0.0146", "source_line": "778" }, { "month": "199005", "BusEq_percent": "11.64", "return_decimal": "0.1164", "loss_decimal": "-0.1164", "source_line": "779" }, { "month": "199006", "BusEq_percent": "-2.00", "return_decimal": "-0.0200", "loss_decimal": "0.0200", "source_line": "780" }, { "month": "199007", "BusEq_percent": "-6.52", "return_decimal": "-0.0652", "loss_decimal": "0.0652", "source_line": "781" }, { "month": "199008", "BusEq_percent": "-12.22", "return_decimal": "-0.1222", "loss_decimal": "0.1222", "source_line": "782" }, { "month": "199009", "BusEq_percent": "-6.16", "return_decimal": "-0.0616", "loss_decimal": "0.0616", "source_line": "783" }, { "month": "199010", "BusEq_percent": 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"BusEq_percent": "-9.19", "return_decimal": "-0.0919", "loss_decimal": "0.0919", "source_line": "792" }, { "month": "199107", "BusEq_percent": "4.15", "return_decimal": "0.0415", "loss_decimal": "-0.0415", "source_line": "793" }, { "month": "199108", "BusEq_percent": "1.37", "return_decimal": "0.0137", "loss_decimal": "-0.0137", "source_line": "794" }, { "month": "199109", "BusEq_percent": "-2.13", "return_decimal": "-0.0213", "loss_decimal": "0.0213", "source_line": "795" }, { "month": "199110", "BusEq_percent": "0.13", "return_decimal": "0.0013", "loss_decimal": "-0.0013", "source_line": "796" }, { "month": "199111", "BusEq_percent": "-5.01", "return_decimal": "-0.0501", "loss_decimal": "0.0501", "source_line": "797" }, { "month": "199112", "BusEq_percent": "7.05", "return_decimal": "0.0705", "loss_decimal": "-0.0705", "source_line": "798" }, { "month": "199201", "BusEq_percent": "7.35", "return_decimal": "0.0735", "loss_decimal": "-0.0735", "source_line": "799" }, { "month": 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"expanding_mean": 0.01442296650717703, "train_target_end": "202510" }, { "month": "202512", "observed": -0.0066, "prediction": 0.014259135296209122, "expanding_mean": 0.014319809069212411, "train_target_end": "202511" } ], "evaluation_scores": { "selected_ridge": { "n": 72, "rmse_decimal": 0.0682421635986639, "mae_decimal": 0.05686840331968504, "sse_decimal_squared": 0.33530348826913026, "benchmark_sse_decimal_squared": 0.3352648795935407, "r2_os_vs_expanding_mean": -0.00011515872356326895 }, "expanding_mean": { "n": 72, "rmse_decimal": 0.06823823459777809, "mae_decimal": 0.05687392806968959, "sse_decimal_squared": 0.3352648795935407, "benchmark_sse_decimal_squared": 0.3352648795935407, "r2_os_vs_expanding_mean": 0.0 }, "zero": { "n": 72, "rmse_decimal": 0.07123537471727871, "mae_decimal": 0.0597638888888889, "sse_decimal_squared": 0.36536246000000006, "benchmark_sse_decimal_squared": 0.3352648795935407, "r2_os_vs_expanding_mean": -0.08977254176741756 } }, "first_prediction_fit": { "prediction": 0.011781576573696191, "n_train": 228, "train_mean_y": 0.011622807017543858, "scaler_mean": [ 0.011504385964912277 ], "scaler_sd": [ 0.0836135454383039 ], "train_z_mean": [ 1.5582077538598687e-17 ], "beta_standardized": [ 0.00024226912557384313 ], "beta_norm": 0.00024226912557384313, "intercept_on_scaled_features": 0.011622807017543858, "new_features": [ 0.0663 ], "new_z": [ 0.6553437454164632 ], "train_rmse": 0.08366456189134369, "train_objective": 0.007000345859762541, "gram_over_n": [ [ 1.0000000000000002 ] ], "cross_over_n": [ 0.0026649603813122744 ] }, "reading_base": "公开同源包；按完整 URL 取得冻结输入与复算文件。来源网站后续更新不覆盖本份 202607 快照。" }, "entry_id": "zh-qt20", "content_version": "2026-09-21-QT-DE-review-v2", "audience": "高年级本科至研究生；默认微积分、线性代数与基本概率", "learning_task": "从损失函数推导 ridge，重建相同信息下的完整基线比较，分清两类单位变换。" } ``` ## Supplied entry 我们已经能给历史收益算出均值和误差，但“把过去描述得更细”不等于“把下个月预测得更准”。本篇做一件具体的事：从一个明确的预测目标出发，推导 ridge 回归，按相同的信息和评价规则比较它与简单基线。最后，你应能解释收缩改变了什么，也能接受一次实验的最好选择仍然是历史均值。 ## 1. 先固定数据、目标和比较对象 我们使用 French 30 Industry Portfolios 的 BusEq、按市值加权的月收益。样本为 1990-01 至 2025-12，共 432 个连续月，原文件缺失标记在此窗口没有出现。内部把百分数除以 100，例如 6.63% 记作 0.0663。这里每个月的观测权重相同；“市值加权”指来源行业组合内部的股票权重，不是对 432 个月再按市值加权。[^french] 数据来自 202607 CRSP 数据库 vintage 的当前 CIZ 重建快照。这是一份用同一生产链重建的历史，不是把 2024 年以前的 FIZ 与以后 CIZ 拼起来，也不保证这些版本在历史预测原点已经可得。下面明确采用“月末已知当月收益”的教学信息假设，研究的是预测运算，不把它称为已重建真实数据接收时点的交易回测。[^french-version] 令 $r_t$ 为月份 $t$ 的简单收益。我们在 $t-1$ 月末预测 $y_t=r_t$，特征是 $x_t=(r_{t-1},\ldots,r_{t-p})^T$。这与用当月市场收益解释当月行业暴露不同：后者可以研究同期关系，却不能直接当作下一期预测。回归系数也不是因果效应。 | 阶段 | 目标月份 | 数量 | 用途 | |---|---|---:|---| | 预热 | 1990-01—1990-12 | 12 | 提供最大 12 期滞后，不进入各模型的目标行 | | 初始训练 | 1991-01—2009-12 | 228 | 拟合第一个验证原点 | | 参数选择 | 2010-01—2019-12 | 120 | 每月扩展训练、预测，再累计验证误差 | | 历史评价 | 2020-01—2025-12 | 72 | 固定选择规则后评价；系数仍按相同扩展规则更新 | 候选为 $p\in\{1,3,12\}$、$\lambda\in\{0,0.1,1,10\}$ 的 12 个回归，另外保留零预测和扩展历史均值两条基线。每个模型从同一目标行开始、预测同一批月份。完整候选账本与逐月结果随实验提供。[^compute] ## 2. 损失函数先决定我们想预测什么 令 $Y\in L^2$，$\mathcal G$ 是预测时已知的信息，$m=E[Y\mid\mathcal G]$。对任意 $\mathcal G$ 可测的平方可积预测量 $g$，条件期望的正交性质给出 $$ E[(Y-g)^2]=E[(Y-m)^2]+E[(m-g)^2]. $$ 推导并不神秘：把 $Y-g=(Y-m)+(m-g)$ 平方展开，交叉项 $E[(Y-m)(m-g)]$ 为零。因此，平方损失对应的总体目标是条件均值；这一步使用了二阶可积性，也限定了允许使用的信息。[条件期望与投影](https://ou-liu-red-sugar.github.io/zh/notebook/conditional-expectation-projection/) 已完整建立这条正交关系。 这并不说线性回归就等于真正的 $m$。我们只是用某一函数族逼近它，再用有限样本估计参数。函数族的限制和估计误差都还存在。换成绝对损失，最优目标一般是条件中位数；因此选 RMSE 或 MAE，不只是换一种报表格式。[^accuracy] 本实验事先采用验证平方误差和 $SSE=\sum_t(r_t-\widehat r_t)^2$ 选模型，同时报告 $RMSE=\sqrt{SSE/n}$ 和 $MAE=n^{-1}\sum_t|r_t-\widehat r_t|$。它们都按月收益的百分点展示。这里没有把收益再除以真实收益去算 MAPE；真实收益接近零时，那样的分母会造成另一种不稳定。 ## 3. 从标准化到 ridge：每个矩阵都要有对象 固定一个预测原点，训练目标有 $n$ 行、特征有 $p$ 列。只在这 $n$ 行上求第 $j$ 列的均值 $\bar x_j$ 与标准差 $s_j=\sqrt{n^{-1}\sum_i(x_{ij}-\bar x_j)^2}$，然后令 $Z_{ij}=(x_{ij}-\bar x_j)/s_j$。本实验各列都非恒定；若 $s_j=0$，须另定删除或保留规则，不能直接除零。未来输入也使用这同一组训练参数。 Ridge 回归在本篇是以下优化问题： $$ \min_{a\in\mathbb R,\;\beta\in\mathbb R^p} \frac1n\|y-a\mathbf1-Z\beta\|_2^2+\lambda\|\beta\|_2^2, \qquad \lambda\ge0. $$ $Z$ 为 $n\times p$，$y$ 为 $n\times1$；截距 $a$ 不受惩罚。惩罚只限制斜率，避免把整体平均收益也机械压向零。ISLP 使用的 RSS 版本与这里相差一个 $n$：把我们的目标整体乘 $n$，惩罚就成为 $n\lambda\|\beta\|^2$，不能不换尺度便移植另一实现的参数值。[^ridge] 对 $a$ 求导得 $a=\bar y-\bar z^T\beta$。设 $Z_c=Z-\mathbf1\bar z^T$、$y_c=y-\bar y\mathbf1$，再对 $\beta$ 求导： $$ -\frac2nZ_c^T(y_c-Z_c\beta)+2\lambda\beta=0, $$ 所以 $$ (Z_c^TZ_c+n\lambda I_p)\widehat\beta=Z_c^Ty_c. $$ 当 $\lambda>0$，任意非零 $v$ 都满足 $v^T(Z_c^TZ_c+n\lambda I)v=\|Z_cv\|^2+n\lambda\|v\|^2>0$，故解唯一。实际计算解线性方程，不显式求逆。$\lambda=0$ 是 OLS；若设计矩阵秩不足，系数可以不唯一，不能顺便宣称每个样本外输入的预测都唯一。 收缩是在训练拟合与系数规模间作取舍。它可能减少对样本扰动的敏感度，也会引入偏差；是否改善新数据预测，仍要评价。随 $\lambda$ 增大，系数向量的整体范数不增，不意味着每一项系数都逐项单调下降。[^ridge] ## 4. 走通一次预测，再看完整比较 先取回归组内最终胜者 $p=1,\lambda=10$，但只重建它在第一个验证原点的拟合。预测目标为 2010-01；训练目标截止 2009-12，共 228 行。 | 中间量 | 内部数值 | |---|---:| | 训练目标均值 $\bar y$ | 0.011622807018 | | 滞后特征训练均值 | 0.011504385965 | | 滞后特征训练标准差 | 0.083613545438 | | 新输入：2009-12 收益 | 0.0663 | | 标准化后的新输入 $z_0$ | 0.655343745416 | | $Z_c^Ty_c/n$ | 0.002664960381 | | $Z_c^TZ_c/n$ | 1 | | 斜率 $\widehat\beta$ | 0.000242269126 | 由于只有一列标准化特征，斜率就是 $0.002664960381/(1+10)$。预测为 $\bar y+z_0\widehat\beta\approx0.011781576574$，即 1.17816%。真实 2010-01 收益随后为 −8.05%，所以这次误差约 −9.22816 个百分点；一次大误差既不证明方法无用，也不能被后续重新挑月份删掉。[^compute] 接下来对每个验证原点重复同一流程。训练窗口可以扩展，但每个候选的 $p,\lambda$ 保持固定，验证段只在结束后比较累计成绩。 | 候选 | 验证 RMSE（百分点/月） | MAE（百分点/月） | $R^2_{OS}$ | |---|---:|---:|---:| | $p=1,\lambda=0$ | 5.04591564 | 4.15173855 | -0.00653461 | | $p=1,\lambda=0.1$ | 5.04434493 | 4.15050355 | -0.00590808 | | $p=1,\lambda=1$ | 5.03747253 | 4.14494608 | -0.00316905 | | $p=1,\lambda=10$ | 5.03092147 | 4.13938860 | -0.00056157 | | $p=3,\lambda=0$ | 5.06121532 | 4.10642026 | -0.01264768 | | $p=3,\lambda=0.1$ | 5.05707327 | 4.10796478 | -0.01099088 | | $p=3,\lambda=1$ | 5.04151891 | 4.11931979 | -0.00478130 | | $p=3,\lambda=10$ | 5.03110972 | 4.13445130 | -0.00063645 | | $p=12,\lambda=0$ | 5.13541496 | 4.16633893 | -0.04255704 | | $p=12,\lambda=0.1$ | 5.12076573 | 4.16109078 | -0.03661755 | | $p=12,\lambda=1$ | 5.06883037 | 4.14662389 | -0.01569721 | | $p=12,\lambda=10$ | 5.03493938 | 4.13836039 | -0.00216039 | | 零预测 | 5.21387572 | 4.31950000 | -0.07465755 | | 扩展历史均值 | 5.02950945 | 4.13815361 | 0.00000000 | 表中 $R^2_{OS}=1-SSE_{model}/SSE_{expanding\ mean}$，分母来自同一批原点上可用历史均值的预测误差，不是全部收益减去一个事后样本均值的方差。负值表示不如这条基线。12 个回归候选内的冠军是 $p=1,\lambda=10$；把两条基线也纳入原定比较，总冠军却是扩展历史均值。 | 2020—2025 历史评价 | RMSE（百分点/月） | 相对均值基线 $R^2_{OS}$ | |---|---:|---:| | 扩展历史均值：验证总冠军 | 6.82382346 | 0 | | $p=1,\lambda=10$：回归组冠军 | 6.82421636 | −0.0001151587 | | 零预测 | 7.12353747 | −0.0897725418 | 这段历史已在研究过程中被查看，不能称为研究者从未见过的保留样本；它仍然能检查一个明示算法的历史运算。微小的 RMSE 差异也没有在这里被证明具有统计显著性。[^compute]
先选一个 $p,\lambda$，说出你期待训练误差、系数范数和验证误差怎样变化，再看完整结果。界面提供实际计算的离散网格；它不会把两个网格点之间插值后称作新拟合。 ## 5. 改单位，到底改变了什么？ 这一点值得单独推导。取 $c>0$，设未经标准化的一列特征从 $x$ 改成 $x'=c x$，响应不变。为了保持同一预测函数，对应系数必须是 $\beta'=\beta/c$。若还施加相同的 $\lambda(\beta')^2$，它在原系数坐标下变成 $\lambda\beta^2/c^2$：预测函数虽然能对应，惩罚强度却改变了。若要维持原惩罚，应改为 $\lambda'=c^2\lambda$。[^ridge] 对照一下：只把响应 $y$ 乘以 $c$，同时允许 $a,\beta$ 乘以 $c$，整个 ridge 目标恰好乘以 $c^2$；$\lambda$ 不变时，解与预测随之乘 $c$，换回原单位仍相同。本实验若把原始滞后收益 $X$ 与 $y$ 同时由小数改成百分数，重新按训练窗标准化后 $Z$ 不变，经济预测也不因此改变。 “尺度影响 ridge”指的是特征坐标与系数惩罚之间的关系，不是看到任何单位变换就宣布模型变了。 ### 选读：复杂模型为什么仍需拆开看 Kelly、Malamud、Zhou 的 2024 年研究在 15 个原始预测量上构造随机 Fourier 特征，改变特征数、训练窗和收缩参数，报告其特定设计下的预测与策略结果。Nagel 的 2025 年工作论文则把相关 ridgeless 预测重写为过去收益的线性权重，研究短训练窗和持久预测量的作用，并用人工反转收益检验机制。[^kmz][^nagel] 这两项研究给我们的任务不是记住“复杂更好”或“复杂无用”，而是问：函数族、信息日期、权重、评价分母分别是什么？例如 $k^TK^{-1}y$ 的权重通常不保证非负或和为 1，不能直接解释成凸平均；矩阵求逆也需要相应秩条件。本篇的小型线性实验没有复现这两篇研究。 ## 6. 独立重建与解析 解释题。 某回归在 12 个回归候选里最好，却比预先保留的均值基线差。应选择谁？换用 MAE 会自动得到同一选择吗？ 解析。 沿事先约定的验证 SSE，应选择均值基线。回归组内胜出不是完整集合胜出。MAE 对应另一种损失与总体目标，排序可能不同；可以同时报告，但不能看完结果才换准则并继续称原先规则。要证明未来优势，还需适合时间依赖的评价与不确定性分析。 迁移题一：原始特征换单位。 两行教学数据为 $x=(-1,1)^T$、$y=(-1,1)^T$，无须额外标准化，含不惩罚截距，$\lambda=1$。先求 ridge；再把 $x$ 乘 10、保持同一惩罚，计算在原来 $x=1$ 处的预测。 解析。 两列都居中，截距为零。原目标为 $(1-\beta)^2+\beta^2$，故 $\beta=1/2$、预测为 $1/2$。新目标为 $(1-10b)^2+b^2$，故 $b=10/101$，在 $x'=10$ 的预测是 $100/101$，不是 $1/2$。改用 $\lambda'=100$ 后，$b=0.05$，才恢复原预测。 迁移题二：响应换单位。 保持原 $x$，只把 $y$ 乘 100，$\lambda=1$。解和换回原单位的预测是什么？ 解析。 目标为 $(100-b)^2+b^2$，解 $b=50$；预测 50 除以 100，仍是 $1/2$。一般情形的结论来自目标函数整体乘 $c^2$，而不是这个两点例的巧合。 现在我们已有一个可重算的预测方法。下一篇把注意力移到它的每个原点：不仅问“算得对不对”，还问“这些输入与拟合参数当时是否可得”。[时间序列验证与信息泄漏](https://ou-liu-red-sugar.github.io/zh/notebook/time-series-validation-leakage/) [^french]: Kenneth French，30 Industry Portfolios：[来源构造说明](https://mba.tuck.dartmouth.edu/pages/Faculty/ken.french/Data_Library/det_30_ind_port.html)。本篇数值使用随包冻结 CSV，而非重新下载后的潜在修订值。 [^french-version]: French [Data Library](https://mba.tuck.dartmouth.edu/pages/faculty/ken.french/data_library.html)，FIZ/CIZ 生产方式说明；本包原文件首行明确为 202607 CRSP database。 [^accuracy]: Hyndman、Athanasopoulos，FPP3，[§5.8 全节](https://otexts.com/fpp3/accuracy.html)，误差度量、训练与新数据评价。 [^ridge]: James 等，ISLP，2023 首印，[作者全文下载入口](https://hastie.su.domains/ISLP/ISLP_website.pdf.download.html)，§6.2.1 pp.240–244、§6.2.3 pp.252–253。这里采用 RSS/n 目标，单位变换计算为本文推导。 [^compute]: [完整实验输入](https://ou-liu-red-sugar.github.io/notebook/labs/qt-de/shared_inputs.json)；[实际计算结果](https://ou-liu-red-sugar.github.io/notebook/labs/qt-de/data/results.json)，forecast 组；[复算程序](https://ou-liu-red-sugar.github.io/notebook/labs/qt-de/compute/reproduce.py)。均值、预测与误差均为这份历史教学实验的输出。 [^kmz]: Kelly、Malamud、Zhou，*The Virtue of Complexity in Return Prediction*，Journal of Finance 79(1), 2024，[发表版全文](https://economics.yale.edu/sites/default/files/2024-01/The%20Journal%20of%20Finance%20-%202023%20-%20KELLY%20-%20The%20Virtue%20of%20Complexity%20in%20Return%20Prediction%20%281%29.pdf)，§V.A–C pp.487–493，特别是脚注33、39–40；本篇不采用其前部理论证明或未读附录。 [^nagel]: Nagel，*Seemingly Virtuous Complexity in Return Prediction*，NBER 34104，2025-08 版，[全文](https://www.nber.org/system/files/working_papers/w34104/w34104.pdf)，§II.A–D 与 §II.G.1；只采用具名方法与机制讨论，不概括为普遍否定复杂模型。 ## Experiment inputs and static equivalents ```json [ { "id": "EXP-QT-D-FORECAST-01", "title": "预测目标、正则化与模型复杂度：可复算实验", "anchor": "qt20-run", "description": "从损失函数推导 ridge，重建相同信息下的完整基线比较，分清两类单位变换。", "inputs": { "identity": "QT-DE-adopted-inputs-20260921-v1", "manifest": "https://ou-liu-red-sugar.github.io/notebook/labs/qt-de/shared_inputs.json" }, "outputs": { "winner": "expanding_mean", "regression_winner": { "key": "p1-l10", "p": 1, "lambda": 10.0 }, "evaluation": { "selected_ridge": { "n": 72, "rmse_decimal": 0.0682421635986639, "mae_decimal": 0.05686840331968504, "sse_decimal_squared": 0.33530348826913026, "benchmark_sse_decimal_squared": 0.3352648795935407, "r2_os_vs_expanding_mean": -0.00011515872356326895 }, "expanding_mean": { "n": 72, "rmse_decimal": 0.06823823459777809, "mae_decimal": 0.05687392806968959, "sse_decimal_squared": 0.3352648795935407, "benchmark_sse_decimal_squared": 0.3352648795935407, "r2_os_vs_expanding_mean": 0.0 }, "zero": { "n": 72, "rmse_decimal": 0.07123537471727871, "mae_decimal": 0.0597638888888889, "sse_decimal_squared": 0.36536246000000006, "benchmark_sse_decimal_squared": 0.3352648795935407, "r2_os_vs_expanding_mean": -0.08977254176741756 } } } } ] ``` ## Sources - [Forecasting: Principles and Practice, §5.8](https://otexts.com/fpp3/accuracy.html): 完整单元支持预测误差不同于残差，RMSE/MAE与mean/median的关系；书中实例不是本站交易绩效。 - [QT-D/E adopted teaching experiments](https://ou-liu-red-sugar.github.io/notebook/labs/qt-de/compute/reproduce.py): 本轮实际沙盒复算，预测遵守冻结算法及数据；选择沿PCG64调用顺序并核strict/inclusive，成本用Fraction。数学演算与假设模型不等于真实投资有效性。 - [30 Industry Portfolios](https://mba.tuck.dartmouth.edu/pages/Faculty/ken.french/Data_Library/det_30_ind_port.html): 冻结原件摘取的432个连续月；百分数除100。来自当前重建历史，不是各月当时可见vintage；无样本期缺失。 - [French Data Library：FIZ/CIZ methodology](https://mba.tuck.dartmouth.edu/pages/faculty/ken.french/data_library.html): 生产链转换与股息复投口径；本批坚持单次202607重建快照，未拼接旧vintage。 - [An Introduction to Statistical Learning with Applications in Python](https://drive.google.com/uc?export=download&id=1ajFkHO6zjrdGNqhqW1jKBZdiNGh_8YQ1): 采用单元的完整镜像原文已在规划轮读取。Ridge 的截距、尺度、参数选择；有效p值、FWER、Bonferroni及BH。BH一般控制定理是引用，原书不提供证明。正文采用严格<；不采用任意相关p值也有效的扩展。 - [The Virtue of Complexity in Return Prediction](https://economics.yale.edu/sites/default/files/2024-01/The%20Journal%20of%20Finance%20-%202023%20-%20KELLY%20-%20The%20Virtue%20of%20Complexity%20in%20Return%20Prediction%20%281%29.pdf): 已读完整实证设置单元，用于配对研究选读：随机特征、训练窗口、信息日期约定、R²分母。未验证前部理论或附录，未复现论文，具体作者结果不普遍化。 - [Seemingly Virtuous Complexity in Return Prediction](https://www.nber.org/system/files/working_papers/w34104/w34104.pdf): 已读方法与人工反转完整单元；线性权重不自动凸权重，短窗机制为具名作者主张。工作论文不作普遍反定理，未采用未读附录。 ## Content relations ```json [ { "from": "zh-qt20", "relation": "part_of", "to": "quant-estimation", "reason": "主要 topic 归属" }, { "from": "zh-qt20", "relation": "requires", "to": "zh-qt04", "required_competence": "读懂收益分布与矩", "reason": "本篇使用该能力，不要求整门随机过程课程。" }, { "from": "zh-qt20", "relation": "requires", "to": "zh-qt19", "required_competence": "分清样本估计与不确定性", "reason": "本篇使用该能力，不要求整门随机过程课程。" }, { "from": "qt20-run", "relation": "illustrated_by", "to": "EXP-QT-D-FORECAST-01", "reason": "完整输入、默认及静态等价支持本篇独立任务。" }, { "from": "zh-qt20", "relation": "supported_by", "to": "QTDE-ISLP", "locator": "§6.2.1 pp.240–244；§6.2.3 pp.252–253", "scope": "Ridge、截距、尺度与调参完整子节", "reason": "核目标、单位变换与验证选择，不以书目页抵扣" }, { "from": "zh-qt20", "relation": "supported_by", "to": "QTDE-ACCURACY", "locator": "§5.8 全节", "scope": "预测误差与RMSE/MAE", "reason": "选择与目标匹配的指标" }, { "from": "zh-qt20", "relation": "supported_by", "to": "QTDE-FRENCH", "locator": "30 Industry Portfolios完整构造说明；随包432行冻结CSV", "scope": "原始组合、频率、单位", "reason": "识别重建快照" }, { "from": "zh-qt20", "relation": "supported_by", "to": "QTDE-FRENCH-REGIME", "locator": "页面FIZ/CIZ生产说明完整单元", "scope": "版本与生产方法", "reason": "防止将当前历史重建当PIT" }, { "from": "zh-qt20", "relation": "supported_by", "to": "QTDE-KMZ", "locator": "§V.A–C pp.487–493，含脚注33、38–41及图", "scope": "实证设置及评价口径", "reason": "具名现代方法，不等于复现" }, { "from": "zh-qt20", "relation": "supported_by", "to": "QTDE-NAGEL", "locator": "§II.A–D pp.6–15；§II.G.1 pp.23–25", "scope": "kernel权重与人工反转机制", "reason": "与KMZ配对、不作普遍胜负判断" }, { "from": "qt20-loss", "relation": "uses_method", "to": "zh-qt11", "reason": "二阶可积平方损失的投影分解。" } ] ``` ## Related entries ## Optional reading path 理解模型并亲手算: step 6/9 推导一轮 ridge，并把全部候选与均值基线放在同一比较中。 下一篇为这些计算逐项检查信息到达和标签成熟时点。 Next: [时间序列验证与信息泄漏](https://ou-liu-red-sugar.github.io/zh/notebook/time-series-validation-leakage/)