on 03-Dec-2019 (Tue)

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For example, time series analysis is frequently used to do demand forecasting for corporate planning, which requires an understanding of seasonality and trend, as well as quantifying the impact of known business drivers. But herein lies the problem: you rarely have sufficient historical data to estimate these components with good precision. And, to make matters worse, validation is more difficult for time series models than it is for classifiers and your audience may not be comfortable with the embedded uncertainty.

A different approach would be to use a Bayesian structural time series model with unobserved components. This technique is more transparent than ARIMA models and deals with uncertainty in a more elegant manner. It is more transparent because its representation does not rely on differencing, lags and moving averages. You can visually inspect the underlying components of the model. It handles uncertainty in a better way because you can quantify the posterior uncertainty of the individual components, control the variance of the components, and impose prior beliefs on the model. Last, but not least, any ARIMA model can be recast as a structural model.

Note that the regressor coefficients, seasonality and trend are estimated simultaneously, which helps avoid strange coefficient estimates due to spurious relationships

General BSTS model. x_t denotes a set of regressors, S_t represents seasonality, and μ_t is the local level term. The local level term defines how the latent state evolves over time and is often referred to as the unobserved trend .

Another advantage of Bayesian structural models is the ability to use spike-and-slab priors. This provides a powerful way of reducing a large set of correlated variables into a parsimonious model, while also imposing prior beliefs on the model. Furthermore, by using priors on the regressor coefficients, the model incorporates uncertainties of the coefficient estimates when producing the credible interval for the forecasts.

spike and slab priors consist of two parts: the spike part and the slab part. The spike part governs the probability of a given variable being chosen for the model (i.e., having a non-zero coefficient). The slab part shrinks the non-zero coefficients toward prior expectations (often zero)

Bayesian structural time series models possess three key features for modeling time series data:

• Ability to incorporate uncertainty into our forecasts so we quantify future risk
• Transparency, so we can truly understand how the model works
• Ability to incorporate outside information for known business drivers when we cannot extract the relationships from the data at hand

Bayesian structural time series models are a widely useful class of time series models, known in various literatures as "structural time series," "state space models," "Kalman filter models," and "dynamic linear models," among others.

bsts can also be configured for specific tasks by an analyst who knows whether the goal is short term or long term forecasting, whether or not the data are likely to contain one or more seasonal effects, and whether the goal is actually to fit an explanatory model, and not primarily to do forecasting at all.

A structural time series model is defined by two equations. The observation equation relates the observed data to a vector of latent variables known as the "state." The transition equation describes how the latent state evolves through time.

The local linear trend is a better model than the local level model if you think the time series is trending in a particular direction and you want future forecasts to reflect a continued increase (or decrease) seen in recent observations. Whereas the local level model bases forecasts around the average value of recent observations, the local linear trend model adds in recent upward or downward slopes as well. As with most statistical models, the extra flexibility comes at the price of extra volatility.

the uncertainty surrounding assumptions about the distributions of the latent and infectious periods may be greater than any uncertainty that arises from noise in the data

consider the evaluation of probabilistic forecasts, or predictive distributions, for count data

Our focus is on the low count situation in which continuum approximations fail; however, our results apply to high counts and rates as well

Gneiting, Balabdaoui, and Raftery (2007) contend that the goal of probabilistic forecasting is to maximize the sharpness of the predictive distributions subject to calibration.

Calibration refers to the statistical consistency between the probabilistic forecasts and the observations, and is a joint property of the predictive distributions and the observations.

Sharpness refers to the concentration of the predictive distributions, and is a property of the forecasts only

For count data, a probabilistic forecast is a predictive probability distribution, P, on the set of the nonnegative integers.

Simplicity, generality, and interpretability are attractive features of the tools discussed by Czado et al. (2009); they apply in Bayesian or classical and parametric or nonparametric settings, and do not require models to be nested, nor be related in any way.

When different models are based on different data, a comparison based on the AIC or related model fit criteria is not feasible here.

In Section 2, we introduce tools for calibration and sharp- ness checks, among them a nonrandomized version of the probability integral transform (PIT) that is tailored to count data, and the marginal calibration diagram. Section 3 dis- cusses the use of scoring rules as omnibus performance mea- sures. We stress the importance of propriety (Gneiting and Raftery, 2007), relate to classical measures of predictive per- formance, and identify the predictive deviance as a variant of the proper logarithmic score. Section 4 turns to a crossvali- dation study, in which we apply these tools to critique count regression models for pharmaceutical and biomedical patents. The epidemiological case study in Section 5 evaluates the pre- dictive performance of Bayesian age-period-cohort models for larynx cancer counts in Germany. The article closes with a discussion in Section 6.

Dawid (1984) proposed the use of the PIT for calibration checks. This is simply the value that the predictive CDF attains at the observation. If the observation is drawn from the predictive distribution—an ideal and desirable situation—and the predictive distribution is continuous, the PIT has a standard uniform distribution

The PIT histogram is typically used informally as a diagnostic tool; formal tests can also be employed though they require care in interpretation (Hamill, 2001; Jolliffe, 2007).

Deviations from uniformity in the PIT hint at reasons for forecast failures and model deficiencies. U-shaped histograms indicate underdispersed predictive distributions, hump or inverse-U shaped histograms point at overdispersion, and skewed histograms occur when central tendencies are biased

In the case of count data, the predictive distribution is discrete which means the PIT is no longer uniform under the hypothesis of an ideal forecast

pick the number of bins, J, compute below equation for equally spaced bins j =1,..., J , plot a histogram with height f j for bin j, and check for uniformity. Under the hypothesis of calibration, that is, if x(i) ∼ P(i) for all forecast cases i =1,..., n, it is straightforward to verify that F(u) has expectation u, so that we expect uniformity. Principled guidelines for the selection of the number of bins remain to be developed; however, J =10 or J = 20 are typical choices that lead to visually informative displays

We now consider what Gneiting et al. (2007) refer to as marginal calibration. The idea is straightforward: If each observed count is a random draw from the respective probabilistic forecast, and if we aggregate over the individual predictive distributions, P⁽ⁱ⁾ , we expect the resulting composite distribution and the histogram of the observed counts to be statistically compatible.

Sharpness refers to the concentration of the predictive dis- tributions. In the context of prediction intervals, this can be rephrased simply: The shorter the intervals, the sharper, and the sharper the better, subject to calibration.

Prediction intervals for continuous predictive distributions are uniquely defined, and Gneiting et al. (2007) suggest to tabulate their average width, or to plot sharpness diagrams, which can be used as a diagnostic tool.

Sharpness continues to be critical for count data; however, we have found these tools to be less useful for discrete predictive distributions, for the ambiguities in specifying prediction intervals. Our preferred way of addressing sharpness is indirectly, via proper scoring rules; see below.

U-shaped PIT histograms indicate underdispersed predictive distributions

Hump or inverse-U shaped PIT histograms indicate overdispersed predictive distributions

Skewed PIT histograms indicate biased central tendencies of predictive distributions

Deviations from uniformity in the PIT hint at reasons for forecast failures and model deficiencies.

The 68000 is no more difficult to program than any of the 8-bit chips. It just has more depth and more capability than those chips. Sure, to get the full use of the 68000 you must understand ad- vanced concepts such as frame pointers, supervisor mode, and memory management, but you don't have to use them.

The 68000 can be used for simple programs just as an 8-bit chip can. But if you have a complicated data structure or routine to program, the 68000 will make your life easier because it provides you with more tools for implementing such things. An 8-bit chip makes you do all the work with long sequences of simple instructions: the 68000 lets you use just a few, more powerful instructions. Many functions that took planning and programming on an 8-bit chip are reduced to a single, automatic operation on the 68000. This process appears in many technologies and is particularly strong in microelec- tronics. While the chips become more powerful, us- ing them doesn't get more difficult (which is nice because we aren't getting smarter). The elemental functions of the chips just keep advancing. Multiply- ing I6-bit numbers or setting up a portion of the stack for a subroutine was a major programming task on a 4-bit microprocessor and required a

Scoring rules provide summary measures in the evaluation of probabilistic forecasts, by assigning a numerical score based on the predictive distribution and the observation.

Suppose, then, that the forecaster’s best judgment is the predictive distribution Q. The forecaster has no incentive to predict any P = Q, and is encouraged to quote her true belief, P = Q, if s(Q, Q) ≤ s(P, Q), (4) with equality if and only if P = Q. A scoring rule with this property is said to be strictly proper. If s(Q, Q) ≤ s(P , Q) for all P and Q, the scoring rule is said to be proper.

Propriety is an essential property of a scoring rule that encourages honest and coherent predictions (Br¨ocker and Smith, 2007; Gneiting and Raftery, 2007).

Strict propriety ensures that both calibration and sharpness are being addressed by the prediction (Winkler, 1996).

In fact, the 68000 is more than a 16-bit microprocessor. Many of its features handle 32-bits at a time. The 68000 family-which is also de- scribed in this book-is a set of chips that includes the 68008 (found in inexpensive home computers) and the 68020 (a full 32-bit chip that is found in super-minicomputers). Learn about the 68000 and you will know most of the details of these chips, too.

The logarithmic score, logs, is defined as logs(P, x) = −log p_x. This is the only proper scoring rule that depends on the pre- dictive distribution P only through the probability mass p x at the observed count (Good, 1952).

The Brier score, qs, is defined as qs(P, x) = −2p_x + ||p||2

The spherical score is defined as sphs(P, x) =−p_x/||p||

Definition of the ranked probabiilty score [Epstein 1969]

The ranked probability score generalizes the absolute error, to which it reduces if P is a point forecast.

The scores introduced in this section are strictly proper, except that the ranked prob- ability score requires Q to have finite first moment for strict inequality in equation (4) to hold

there is a distinct difference between the ranked probability score and the other scores discussed in this section, in that the former blows up score differentials between competing forecasters in cases in which predicted and/or observed counts are unusually high (Candille and Talagrand, 2005)

Viewed as a scoring rule for probabilistic forecasts, the mean squared error of the predictive distribution's mean is proper, but not strictly proper (Gneiting and Raftery, 2007).

It has frequently been argued that the squared Pearson residual or normalized squared error score ought be approximately one when averaged over the predictions (Carroll and Cressie, 1997; Liesenfeld et al., 2006).

There are some advanced details of the 68000 that this book doesn't attempt to cover, including such things as the exact timing of instructions, the interfacing of peripherals, and the algorithms for assembly language subroutines. Those subjects are better left to the original literature from the chip- maker or a book dedicated to this subject.

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