The RMSE can be calculated using the following equation:

RMSE=√∑ni=1(yi−ˆyi)2n

While the bias can be calculated:

BIAS=∑ni=1(yi−ˆyi)n

Based on these equations, which of the following statements is TRUE?

Question

The RMSE can be calculated using the following equation:

RMSE=√∑ni=1(yi−ˆyi)2n

While the bias can be calculated:

BIAS=∑ni=1(yi−ˆyi)n

Based on these equations, which of the following statements is TRUE?

BlackTom AI · Accepted Answer

The prompt provides the RMSE and Bias definitions and asks which statement is TRUE.

Option: 'Both RMSE and BIAS are less reliable on small data sets, as the small number of residuals could lead to greater sampling variability.'

Consider the nature of RMSE and Bias:
- RMSE is the square root of the average of squared residuals. Because it squares deviations, RMSE is sensitive to large errors, and its estimate can vary substantially from sample to sample when the data set is small. This introduces sampling variability in RMSE estimates.
- Bias is the average of residuals (y_i - ŷ_i). The sampling distribution of the mean residual has variability that decreases with larger n (approximately sigma / sqrt(n) for independent errors with variance sigma^2). Therefore, in small samples, bias estimates are also quite variable and can be unreliable.

Combining these points, in small data sets, both RMSE and Bias tend to have greater sampling variability, making them less reliable as point estimates of model performance or error characteristics.

Additional considerations:
- RMSE conflates variance and potential bias in a single metric, so its reliability is tied to both the spread of residuals and any systematic error.
- Bias is a measure of systematic error in the average residual; its unreliability in small samples is due to the greater influence of a few residuals on the mean.

Given the statement focuses on the general reliability of both metrics in small samples due to limited residuals and higher sampling variability, it aligns with standard statistical intuition.

Note: There are no other answer options provided to compare against, so the analysis centers solely on evaluating this claim in isolation. If other options existed, we would compare them similarly by examining how sampling variability, sensitivity to outliers, and definitions of RMSE and Bias influence reliability across sample sizes.

ENVX2001 (ND) End of Week 9 Quiz (no quiz for Week 8)

The RMSE can be calculated using the following equation: RMSE=√∑ni=1(yi−ˆyi)2n While the bias can be calculated: BIAS=∑ni=1(yi−ˆyi)n Based on these equations, which of the following statements is TRUE?

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