The process of reducing the model is the subject of another blog, but now I will show you why the final model is a good one. During that process, I specifically compared including height, weight, and their squared terms in the model to including BMI and its squared term in the model.
Model interpretability is a necessity for inference
I included the squared terms in order to model the curvature. Both approaches produced nearly identical results. Because we have one predictor BMI and one response Body Fat Percentage , we can use a fitted line plot to display the relationship. The fitted line follows the data very nicely because the observations fall randomly around it for the entire range. The R-squared is It reflects the imperfect nature of BMIs.
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We'll assess the residual plots below to really check the model. A side note about the implications of the curved relationship for the raw BMI scores of this population: We tend to think of BMI scores in a linear sense. That is, if you start at either 16 or 30 and increase BMI by 1, we assume that it represents the same increase in fat mass. The curved relationship shown above suggests that this is not true. The change in fat mass varies depending on the specific BMI value you start at. Middle-of-the-range BMI scores tend to underestimate fat mass. Regression analysis actually improves upon raw BMI scores for this population by correctly modeling the curvature.
Practical Statistics for Data Scientists by Andrew Bruce, Peter Bruce
Article information Source Statist. Export citation. Export Cancel. References Afshartous, D. Prediction in multilevel models. Aitchison, J. Statistical Prediction Analysis.
Predictive Analytics in Today's World
Cambridge Univ. The regression model results are used to argue that the lack of skill is not because the dynamical models do not capture realistic relations with SST but rather because the NMME models could not accurately predict the relevant SST patterns. We conclude with a summary and discussion of our results.
We consider the problem of predicting a variable y given a spatial field X.
For the type of problems considered in this paper, the number of predictors S far exceeds the sample size N , in which case the minimization problem is grossly underdetermined [i. A common regularization in seasonal prediction is principal component regression PCR , in which the predictors are represented by a small number of principal components of the predictor data Barnston and Smith The most common regularizations in regression are the L 1 and L 2 norms. The L 1 norm is the sum of the absolute values of the weights: Minimizing 3 based on the L 1 norm 4 is called least absolute shrinkage and selection operator LASSO Tibshirani , LASSO tends to set certain elements of w exactly to zero, which facilitates interpretation by indicating that the corresponding predictors can be discarded.
We propose a new regularization constraint based on the hypothesis that large-scale SST structures provide the most robust predictive information for seasonal weather. This hypothesis is motivated mostly by practical considerations: small-scale SST structures are difficult to observe and not robust across climate models, so they should be filtered out. This principle can be expressed equivalently by saying that if the predictors are represented in a basis set ordered by spatial scale, then most of the amplitudes of the basis vectors are zero or close to zero.
A natural basis set for filtering out short spatial scales is the eigenvectors of the Laplace operator. On a global domain, Laplacian eigenvectors are the spherical harmonics and easily computable. Over an ocean basin, these eigenvectors are difficult to compute using standard boundary conditions. Recent advances in signal processing Saito have led to efficient algorithms for computing these eigenvectors in arbitrary domains.
These eigenvectors typically satisfy unconventional boundary conditions, but the precise boundary conditions are irrelevant if the vectors are used merely as a basis set. The resulting eigenfunctions are orthogonal with respect to an area-weighted norm and normalized to unit-area-weighted norm. The leading Laplacian eigenvectors in the North Pacific are shown in Fig.
The first eigenfunction, not shown, is merely a constant and corresponds to the mean over the North Pacific basin. The second and third eigenfunctions measure the east—west and north—south gradients across the Pacific, respectively. Subsequent eigenfunctions are characterized by tripoles, quadrupoles, etc. The maximum number of Laplacian eigenvectors is set to 50; our results are not sensitive to the choice of the upper limit.
In general, the regression model 1 should include an intercept term, but the intercept term should not be included in the penalty functions of 4 or 5 ; otherwise the solution would depend on the arbitrary choice of origin. It can be shown that minimizing 3 with the intercept is equivalent to minimizing 3 without the intercept, provided all variables are centered. Accordingly, all predictors and predictands were centered before finding the ridge and LASSO solutions. In contrast, OLS is invariant to nonsingular linear transformation of the predictors and thus does not depend on the scale of the predictors.
In regularized regression, it is customary to rescale the predictors to have identical variances, which effectively penalizes all Laplacian functions equally. However, to be consistent with the smoothness hypothesis, small-scale patterns should be penalized more strongly than large-scale patterns.
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We explored a wide variety of penalty functions that increase monotonically with wavenumber and found that the final predictions are not sensitive to the choice of penalty function. Accordingly, a convenient approach is to simply use the time series of the Laplacian eigenvectors with no rescaling. This approach effectively imposes a scale-dependent penalty function because, as is well known in geophysical fluid dynamics, large-scale patterns tend to have larger variance than small-scale patterns Charney ; Nastrom and Gage Consequently, the standard deviation of the time series decreases with wavenumber, and it can be shown that this corresponds to a penalty function that increases with wavenumber if the time series were normalized to the same variance.
A common approach is based on cross validation, in which one sample is withheld and the remaining sample is used to estimate a regression model, after which the resulting regression model is used to predict the withheld sample. The procedure is repeated for each sample in turn until all samples have been used at least once for validation. In the case of PCR, we emphasize that the empirical orthogonal functions EOFs and centering are recomputed for each new training set in the cross-validation procedure. In practice, though, the results are essentially unchanged if the EOFs are computed once for the whole period.
However, we found that leave-one-out cross validation yielded unrealistically high skill scores e. To explore this situation further, we applied tenfold cross validation and found that the regression models estimated from observations had no significant skill for any choice of regularization. These conflicting results demonstrate the danger in estimating regression models from observations of the size considered here i.
However, different cross-validation methods yielded similar results when applied to dynamical model output. The results presented in section 4 are based on tenfold cross validation, which is a generally recommended method in the statistics literature Hastie et al. In contrast, Hastie et al. However, our data are correlated with each other because we pool ensemble members and hindcasts initialized one month apart.
Therefore, the sample standard deviation probably overestimates the true standard deviation. Nevertheless, the standard deviation of the skill score will be shown in the results to follow as a reference. We first attempt to identify relations between SST and land variables in observational data. This area is chosen because the drought or heat wave in that region during raised critical questions about the role of ocean temperatures and the extent to which such events can be predicted in the future Hoerling et al.
The present paper extends previous studies by examining the extent to which such events can be predicted on seasonal time scales by dynamical models and by regression models using SSTs as predictors. Observational estimates of summer land temperature are from the dataset of Fan and Van den Dool , which is a combination of data from the Global Historical Climatology Network, version 2, and the Climate Anomaly Monitoring System.
Inference vs Prediction
We also derive SST—land relations from climate model simulations. A list of models is given in Table 1. A hindcast refers to a dynamical model prediction of historical data in which the verification is available at initialization time. Each model generates an ensemble forecast in which multiple predictions are generated from slightly different initial conditions, each of which are plausible realizations of the state of the system given the available observations.
The reason for choosing this dataset is that the associated dynamical models have been designed and validated specifically for seasonal prediction, so these models are likely to capture realistic seasonal relations between SST and land variables. Unfortunately, seasonal prediction datasets are relatively short e. To increase the sample size, we pool individual ensemble members initialized in the months preceding the June—August verification period.
To be clear, we do not use ensemble averages, but rather we attempt to find the relation between summer land and SST in individual ensemble members. To avoid differences due to different ensemble sizes, we use an equal number of ensemble members per model. Specifically, we use six members, which is the smallest ensemble size among the models in this dataset.
For each model, then, there are 33 years, five initial months January—May , and six ensemble members, giving a total of samples per model. Table 1. The variable we want to predict is summer June—August Texas-area temperature. We first consider predictions based on concurrent SST i. Although predictions based on concurrent SST are not true predictions, they nevertheless are investigated extensively in seasonal prediction studies because they define teleconnection patterns and define an upper bound on predictability.
Later in section 4d we consider time-lagged relations between SST and Texas-area temperature.
Both SST and Texas-area temperature exhibit a significant trend during the period under investigation. The trend presumably reflects the global warming signal and does not reflect a causal relation between land temperature and SST. If these trends are not removed, then all regression models have apparent skill even at large lags.
The observed anomaly time series, relative to the — mean, is shown in Fig. The Texas heat wave of is evident. To gain insight into the Pacific SST pattern relevant to this prediction, it is customary to construct a regression map, which shows the least squares regression coefficient between Texas-area temperature T and local SST from the regression model: where, as in section 2 , s and n denote the spatial location and time step, respectively.
The least squares solution is equivalent to estimating p s independently and individually at each grid point. The resulting coefficients can be collected and displayed on a single map. The regression map p s derived from observations, shown at the bottom of Fig. This regression pattern is quite similar to the actual SST anomaly that occurred in in association with the extreme Texas heat wave Hoerling et al.