为回归任务赋值Y：寻求风险评估系统的最优Y建模方法

Great question! This is a classic problem in time-sensitive risk modeling, and there are several practical, data-driven approaches to figure out the optimal functional form for your risk score Y (0-100) relative to your feature matrix X. Let’s break this down step by step:

Before diving into models, you need to understand the inherent patterns in your data:

• Plot Y against your most critical features (especially time-to-event, since you noted risk rises as you approach a crisis). Look for visual trends: does Y grow linearly, spike exponentially, level off as it hits 100 (sigmoidal), or follow a Gaussian curve?
• Calculate correlation coefficients between Y (and transformed versions of Y) and your features. For example:
  • Correlate log(Y) with X to test for exponential relationships
  • Correlate log(100 - Y) with X to check for sigmoidal/logistic trends
  • Correlate sqrt(Y) with X to spot square-root patterns
    The transformation that gives the strongest linear correlation is a good starting point.

Once you have EDA hints, use models to validate and compare different forms:

• Linear Regression with Transformations: Fit linear models to transformed Y values and compare performance metrics. For example:
  • Basic linear: Y ~ X1 + X2 + ...
  • Exponential: log(Y) ~ X1 + X2 + ... (reverse-transform to get Y = exp(intercept + coeffs*X))
  • Sigmoidal (since Y caps at 100): log(100 - Y) ~ X1 + X2 + ... (solves to Y = 100 - exp(intercept + coeffs*X))
    Use metrics like R², adjusted R², or RMSE to pick the best fit.
• Generalized Linear Models (GLMs): GLMs are perfect for bounded outcomes like your 0-100 Y. Try:
  • Beta Regression: Scale Y to 0-1 first, then use a logit or log link function to model the relationship between X and Y. This naturally handles bounded values and can capture non-linear trends.
  • Tweedie Regression: Useful if you have zero-inflated risk scores (some cases where Y starts at 0) and want to model both linear and non-linear components.
• Non-Linear Least Squares: If your EDA points to a specific non-linear shape (e.g., Gaussian, quadratic), fit a custom non-linear model. For example, an exponential rise toward crisis could look like:

  from scipy.optimize import curve_fit
  import numpy as np
  
  def exponential_model(time, a, b, c):
      return a * np.exp(b * time) + c
  params, _ = curve_fit(exponential_model, X['time_to_event'], Y)
  

• Regularized Regression with Transformed Features: Create a matrix of transformed features (polynomial terms, log terms, exponential terms) then use LASSO or Ridge regression. The regularization will automatically select the most impactful transformations, helping you narrow down the optimal functional form.

Never rely solely on training data performance:

• Split your data into training and test sets, or use k-fold cross-validation, to ensure your chosen functional form generalizes to unseen data.
• Pick metrics that align with your goals:
  • If ranking risk is key (e.g., prioritizing high-risk cases), use AUC-ROC or Spearman correlation.
  • If absolute prediction accuracy matters, use RMSE or MAE.

Don’t ignore real-world context:

• If your domain expects risk to accelerate rapidly as a crisis nears (e.g., equipment failure, credit default), an exponential or sigmoidal form is likely more realistic than linear.
• If interpretability is critical (e.g., explaining risk to stakeholders), prioritize simpler forms like linear or log-linear models over complex non-linear ones.

Start simple, then build complexity only if needed:

• Begin with the most parsimonious model that fits the data well, then add transformations or interaction terms if they improve validation performance.
• After selecting a functional form, fine-tune your model by engineering relevant features (e.g., rolling averages of X variables, squared time-to-event terms) to capture more nuance.

内容的提问来源于stack exchange，提问作者Ernest Montañà Ortiz