DIDÁCTICA DE LA TECNOLOG Í A

e D₍₂₎ =             1 1 −2 −2 1 1 1 −2 1 1 −2 1 . .. ... ... 1 −2 1             , (4.9)

where empty cells are equal to zero. All together, this leads to the penalized least squares problem

Q(β) = (u−B_cyclicβ)>(u−B_cyclicβ) +λJcyclic(β;d), (4.10)

which is essentially identical to the usual P-spline criterion. Only the basis and the penalty matrices are altered as discussed above.

As discussed in Section 2.3.2, P-splines have a null space, i.e., an unpenalized effect, which depends on the order of the differences in the penalty. Differences of order two lead to a linear effect that remains unpenalized, even in the limit of

λ → ∞. However, cyclic P-splines have a null space that only includes a constant,

irrespective of the order of the difference penalty. Globally seen, i.e., for the com- plete function estimate, the order of the penalty plays no role (even in the limit

λ→∞). Locally, however, the order of the difference penalty has an effect. For ex-

ample, withd =2 the estimated function is penalized for deviations from linearity and hence, locally approaches a straight line (with increasingλ).

As for monotonic effects, we empirically study the properties of cyclic effects in the following section. We use truly periodic effects and evaluate the estimates of models with cyclic constraints and unconstrained models using the MSE, as before. We show that cyclic effects are superior, both with respect to the MSE and regarding the number of violations of cyclicity.

4.3. Empirical Evaluation of Constrained Effect

Estimates

We evaluated the estimation procedure for monotonic smooth and monotonic or- dinal effects, and cyclic effects and compared the resulting effects to unconstrained estimates. In all settings, we used observations xi = (x1i,x2i)>, i = 1, . . . ,n, and

true effects f1(xi)and f2(x2) (see below for the specification of the predictorxi and the effects). The responsey was simulated according to

yi =α+ f1(x1i) + f2(x2i) +εi, i =1, . . . ,n, with intercept α, and random error εi

i.i.d.

∼ N(0,σ2) with σ2 such that fractions of

the explained variance (R2) of approximately 0.3, 0.5, and 0.8 were realized. The number of observations n was chosen to be 100, 200, and 500, respectively. In combination with the different variances, this results in nine scenarios altogether. For each scenario B = 100 data sets were generated. The model fit was evaluated by the mean squared error for N =10, 000 new observations:

MSE = 1 N N

∑

4.3.1. Smooth, Monotonic Effects

The observations x_i = (x1i,x2i)>, i = 1, . . . ,n, were drawn independently from a uniform distribution on [0, 1]. The effect of x1follows a polynomial of degree three

f1(x1) =10(x1−0.5)3,

and the effect of x2 follows a logistic function,

f2(x2) = 1

1+exp{−20(x2−0.5)}.

The intercept was set toα =2. The functions are depicted in Fig. 4.5.

For each of the data sets two models were estimated: one model with monotonic P-splines and one with unconstrained P-splines. The results are given in Table 4.1. One can conclude that the monotonic model outperforms the unconstrained model. Only in cases with many observations (n = 200 andn =500) combined with very little noise in the data (σ2 =0.1), the unconstrained model is marginally better than

the constrained model.

The accuracy of the model fit as measured by the MSE captures only one part of constrained modeling: the overall goodness of fit. However, the task is to estimate monotonic effects. This affects the quality of the model and, in situations with real data, the interpretability of the model. Thus, the resulting effect estimate should be monotonic (or show only minor deviations from monotonicity). Here we

4.3 Empirical Evaluation of Constrained Effect Estimates 79 0.0 0.2 0.4 0.6 0.8 1.0 −1.0 0.0 0.5 1.0 x1 f1 ( x1 ) −0.5 0.0 0.2 0.4 0.6 0.8 1.0 0.0 0.2 0.4 0.6 0.8 1.0 x2 f2 ( x2 )

Figure 4.5.:Empirical evaluation of smooth, monotonic effects: True functions f1(x1) and

f2(x2). Note that both functions are monotonically increasing.

consider models to be monotonic if the differences of coefficients of adjacent knots are positive or only marginally negative. Here, an effect is said to be monotonic if

∆βj >−10−4, for all j. (4.11) Using this boundary to define monotonicity we find that for the monotonically constrained effects, monotonicity is never violated (see Table C.1 in the appendix). Function estimates from models fitted using unconstrained P-splines violate mono- tonicity in approximately 75% of the cases. Violations of monotonicity even in- crease with decreasing noise σ2. Hence, the situations where the unconstrained

Table 4.1.:Smooth, monotonic effects: MSE of monotonicity-constrained and unconstrained models; Mean values and corresponding standard errors (se) estimated from 100 sim- ulation runs.

n σ2 monotonic (se) unconstrained (se) 100 1.0 0.0759 (0.0035) 0.0981 (0.0046) 0.4 0.0349 (0.0015) 0.0431 (0.0019) 0.1 0.0122 (0.0004) 0.0129 (0.0005) 200 1.0 0.0405 (0.0020) 0.0449 (0.0022) 0.4 0.0189 (0.0008) 0.0198 (0.0008) 0.1 0.0076 (0.0002) 0.0063 (0.0002) 500 1.0 0.0155 (0.0008) 0.0171 (0.0007) 0.4 0.0083 (0.0003) 0.0081 (0.0003) 0.1 0.0046 (0.0001) 0.0032 (0.0001)

model outperforms the monotonic model are the worst situations with respect to violations of monotonicity. It seems that the reduced MSE comes at the price of non-monotonicity.

4.3.2. Ordinal, Monotonic Effects

In the second scenario, with ordinal, monotonic effects, the observations xi =

(x1i,x2i)>, i = 1, . . . ,n, were drawn independently from a discrete uniform dis- tribution on {1, 2, . . . , 5}. With dummy coded design matricesX₁ and X₂ (the first category is used as reference category), the true effect functions f1(x1) and f2(x2)

can be written in matrix notation as

fl(xl) =Xlβl, l ∈ {1, 2}, with β1 = (1, 2, 3, 4)> and β2 = _exp(1) 40 , exp(2) 40 , exp(3) 40 , exp(4) 40 >

. The intercept was set to α =4. Both effects are monotonic increasing in with increasing categories.

For each of the data sets two models were estimated: one model with monotonicity- constraint and one with unconstrained effect estimates. The results are given in Table 4.2 . One can conclude that the monotonic model outperforms the uncon- strained model in all given scenarios.

Table 4.2.:Ordinal, monotonic effects: MSE of monotonicity-constrained and unconstrained models; Mean values and corresponding standard errors (se) estimated from 100 sim- ulation runs.

n σ2 monotonic (se) unconstrained (se) 100 1.0 0.3982 (0.0204) 0.5548 (0.0268) 0.4 0.1961 (0.0102) 0.2566 (0.0127) 0.1 0.0533 (0.0028) 0.0657 (0.0033) 200 1.0 0.2098 (0.0104) 0.2569 (0.0115) 0.4 0.0939 (0.0048) 0.1126 (0.0055) 0.1 0.0248 (0.0012) 0.0284 (0.0013) 500 1.0 0.0746 (0.0036) 0.0921 (0.0039) 0.4 0.0326 (0.0016) 0.0388 (0.0017) 0.1 0.0090 (0.0004) 0.0100 (0.0005)

We also assess the frequency of violations of the monotonicity for all estimated models. As before, we consider models to be monotonic if Equation (4.11) holds. The results are given in the appendix (Tab. C.2). Using constrained estimates,

4.3 Empirical Evaluation of Constrained Effect Estimates 81

monotonicity is virtually never violated. In the unconstrained model, much more violations of monotonicity are present. In contrast to the setting with P-splines, the number of violations decreases with increasing sample size and decreasing noise. Furthermore, the overall number of violations is smaller (compare Table C.1 in the appendix). It seems that the signal of categorical covariates is less affected by noise and that the true, monotonic effects are recovered more easily.

4.3.3. Cyclic Effects

For the evaluation of cyclic effects, observations xi = (x1i,x2i)>,i = 1, . . . ,n, were drawn independently from a uniform distribution on[0, 2π]. The true cyclic func-

tion f1(x1) = sin(x1); the second cyclic function f2is a single B-spline basis of order

three, with equidistant inner knots, and boundary knots in 0 and 2π. The intercept α =1 was used. Both functions are depicted in Figure 4.6

0 1 2 3 4 5 6 −1.0 −0.5 0.0 0.5 1.0 x₁ f1 ( x1 ) 0 1 2 3 4 5 6 0.0 0.2 0.4 0.6 x₂ f2 ( x2 )

Figure 4.6.:Empirical evaluation of cyclic effects: True functions f1(x1)and f2(x2). Note that both functions are truly periodic with period 2π.

We fitted the data using a constrained model with cyclic constraints both for x1

andx2and boundary knots in 0 and 2π. As a comparison, an unconstrained model

was fitted. The MSE of the cyclic model is smaller throughout all nine scenarios (see Table 4.3).

To evaluate the ability to recover cyclic effects, we use the absolute difference of the boundaries of the predicted functions, i.e.,

|∆fˆl|:=|fˆl(0)− fˆl(2π)|, l ∈ {1, 2}.

Table 4.3.:Cyclic effects: MSE of models with cyclic constraint and unconstrained models; Mean values and corresponding standard errors (se) estimated from 100 simulation runs.

n σ2 cyclic (se) unconstrained (se) 100 1.0 0.1126 (0.0052) 0.1354 (0.0054) 0.4 0.0539 (0.0025) 0.0679 (0.0027) 0.1 0.0150 (0.0007) 0.0200 (0.0008) 200 1.0 0.0613 (0.0032) 0.0688 (0.0030) 0.4 0.0278 (0.0013) 0.0316 (0.0013) 0.1 0.0074 (0.0003) 0.0090 (0.0003) 500 1.0 0.0232 (0.0013) 0.0263 (0.0013) 0.4 0.0104 (0.0006) 0.0122 (0.0005) 0.1 0.0028 (0.0001) 0.0040 (0.0002)

values at the boundaries, i.e., |∆fˆ| = 0. We considered a function estimate ˆf to be cyclic even if minor deviations occurred, i.e., ˆf is said to be cyclic if |∆fˆ| ≤ 0.1. In all scenarios, the effect estimates from the cyclic model are found to be (truly) cyclic. The effect estimates of the unconstrained model are rarely found to be cyclic. Even more, the unconstrained model shows on average an absolute difference |∆fˆ|

of 0.29 (for details see Table C.3 in the appendix). With ranges of [−1, 1]for f1, and

[0, 0.75] for f2 this absolute difference is remarkable.