distributions are probably the most generally used, loads of real-world information sadly will not be regular. When confronted with extraordinarily skewed information, it’s tempting for us to make the most of log transformations to normalize the distribution and stabilize the variance. I just lately labored on a undertaking analyzing the vitality consumption of coaching AI fashions, utilizing information from Epoch AI [1]. There is no such thing as a official information on vitality utilization of every mannequin, so I calculated it by multiplying every mannequin’s energy draw with its coaching time. The brand new variable, Vitality (in kWh), was extremely right-skewed, together with some excessive and overdispersed outliers (Fig. 1).
To deal with this skewness and heteroskedasticity, my first intuition was to use a log transformation to the Vitality variable. The distribution of log(Vitality) appeared rather more regular (Fig. 2), and a Shapiro-Wilk take a look at confirmed the borderline normality (p ≈ 0.5).
Modeling Dilemma: Log Transformation vs Log Hyperlink
The visualization appeared good, however once I moved on to modeling, I confronted a dilemma: Ought to I mannequin the log-transformed response variable (log(Y) ~ X), or ought to I mannequin the unique response variable utilizing a log hyperlink operate (Y ~ X, hyperlink = “log")? I additionally thought-about two distributions — Gaussian (regular) and Gamma distributions — and mixed every distribution with each log approaches. This gave me 4 totally different fashions as under, all fitted utilizing R’s Generalized Linear Fashions (GLM):
all_gaussian_log_link Mannequin Comparability: AIC and Diagnostic Plots
I in contrast the 4 fashions utilizing Akaike Info Criterion (AIC), which is an estimator of prediction error. Usually, the decrease the AIC, the higher the mannequin suits.
AIC(all_gaussian_log_link, all_gaussian_log_transform, all_gamma_log_link, all_gamma_log_transform)
df AIC
all_gaussian_log_link 25 2005.8263
all_gaussian_log_transform 25 311.5963
all_gamma_log_link 25 1780.8524
all_gamma_log_transform 25 352.5450Among the many 4 fashions, fashions utilizing log-transformed outcomes have a lot decrease AIC values than those utilizing log hyperlinks. For the reason that distinction in AIC between log-transformed and log-link fashions was substantial (311 and 352 vs 1780 and 2005), I additionally examined the diagnostics plots to additional validate that log-transformed fashions match higher:
Primarily based on the AIC values and diagnostic plots, I made a decision to maneuver ahead with the log-transformed Gamma mannequin, because it had the second-lowest AIC worth and its Residuals vs Fitted plot appears higher than that of the log-transformed Gaussian mannequin.
I proceeded to discover which explanatory variables have been helpful and which interactions could have been vital. The ultimate mannequin I chosen was:
glm(system = log(Energy_kWh) ~ Training_time_hour * Hardware_quantity +
Training_hardware + 0, household = Gamma(), information = df)Decoding Coefficients
Nevertheless, once I began deciphering the mannequin’s coefficients, one thing felt off. Since solely the response variable was log-transformed, the consequences of the predictors are multiplicative, and we have to exponentiate the coefficients to transform them again to the unique scale. A one-unit improve in 𝓍 multiplies the end result 𝓎 by exp(β), or every further unit in 𝓍 results in a (exp(β) — 1) × 100 % change in 𝓎 [2].
Wanting on the outcomes desk of the mannequin under, we’ve Training_time_hour, Hardware_quantity, and their interplay time period Training_time_hour:Hardware_quantity are steady variables, so their coefficients characterize slopes. In the meantime, since I specified +0 within the mannequin system, all ranges of the explicit Training_hardware act as intercepts, which means that every {hardware} sort acted because the intercept β₀ when its corresponding dummy variable was energetic.
> glm(system = log(Energy_kWh) ~ Training_time_hour * Hardware_quantity +
Training_hardware + 0, household = Gamma(), information = df)
Coefficients:
Estimate Std. Error t worth Pr(>|t|)
Training_time_hour -1.587e-05 3.112e-06 -5.098 5.76e-06 ***
Hardware_quantity -5.121e-06 1.564e-06 -3.275 0.00196 **
Training_hardwareGoogle TPU v2 1.396e-01 2.297e-02 6.079 1.90e-07 ***
Training_hardwareGoogle TPU v3 1.106e-01 7.048e-03 15.696 When changing the slopes to % change in response variable, the impact of every steady variable was nearly zero, even barely adverse:
All of the intercepts have been additionally transformed again to simply round 1 kWh on the unique scale. The outcomes didn’t make any sense as at the very least one of many slopes ought to develop together with the big vitality consumption. I questioned if utilizing the log-linked mannequin with the identical predictors could yield totally different outcomes, so I match the mannequin once more:
glm(system = Energy_kWh ~ Training_time_hour * Hardware_quantity +
Training_hardware + 0, household = Gamma(hyperlink = "log"), information = df)
Coefficients:
Estimate Std. Error t worth Pr(>|t|)
Training_time_hour 1.818e-03 1.640e-04 11.088 7.74e-15 ***
Hardware_quantity 7.373e-04 1.008e-04 7.315 2.42e-09 ***
Training_hardwareGoogle TPU v2 7.136e+00 7.379e-01 9.670 7.51e-13 ***
Training_hardwareGoogle TPU v3 1.004e+01 3.156e-01 31.808 This time, Training_time and Hardware_quantity would improve the entire vitality consumption by 0.18% per further hour and 0.07% per further chip, respectively. In the meantime, their interplay would lower the vitality use by 2 × 10⁵%. These outcomes made extra sense as Training_time can attain as much as 7000 hours and Hardware_quantity as much as 16000 items.
To visualise the variations higher, I created two plots evaluating the predictions (proven as dashed traces) from each fashions. The left panel used the log-transformed Gamma GLM mannequin, the place the dashed traces have been almost flat and near zero, nowhere close to the fitted stable traces of uncooked information. Then again, the appropriate panel used log-linked Gamma GLM mannequin, the place the dashed traces aligned rather more intently with the precise fitted traces.
test_data %
mutate(
pred_energy1 = exp(predict(glm3, newdata = test_data)),
pred_energy2 = predict(glm3_alt, newdata = test_data, sort = "response"),
)
y_limits 
Why Log Transformation Fails
To grasp the rationale why the log-transformed mannequin can’t seize the underlying results because the log-linked one, let’s stroll by means of what occurs after we apply a log transformation to the response variable:
Let’s say Y is the same as some operate of X plus the error time period:
Once we apply a log remodeling to Y, we are literally compressing each f(X) and the error:
Which means we’re modeling an entire new response variable, log(Y). Once we plug in our personal operate g(X)— in my case g(X) = Training_time_hour*Hardware_quantity + Training_hardware — it’s attempting to seize the mixed results of each the “shrunk” f(X) and error time period.
In distinction, after we use a log hyperlink, we’re nonetheless modeling the unique Y, not the remodeled model. As an alternative, the mannequin exponentiates our personal operate g(X) to foretell Y.
The mannequin then minimizes the distinction between the precise Y and the expected Y. That means, the error phrases stays intact on the unique scale:
Conclusion
Log-transforming a variable will not be the identical as utilizing a log hyperlink, and it might not at all times yield dependable outcomes. Below the hood, a log transformation alters the variable itself and distorts each the variation and noise. Understanding this delicate mathematical distinction behind your fashions is simply as essential as looking for the best-fitting mannequin.
[1] Epoch AI. Knowledge on Notable AI Fashions. Retrieved from https://epoch.ai/data/notable-ai-models
[2] College of Virginia Library. Decoding Log Transformations in a Linear Mannequin. Retrieved from https://library.virginia.edu/data/articles/interpreting-log-transformations-in-a-linear-model
