Time-Dependent and Stochastic Models

Introduction and Context

Many observations in artificial intelligence, engineering, finance, logistics, and operations occur sequentially. A customer arrives, a machine fails, a sensor records a temperature, or a user clicks an advertisement. The order and timing of observations matter because the present may depend on the past. Probability theory for such evolving random phenomena is the subject of stochastic processes.

A stochastic process provides a mathematical framework for uncertainty that changes over time. It supports models of queues, inventories, communication networks, financial prices, speech, weather, demand, biological signals, and user behavior. In AI and machine learning, stochastic-process ideas appear in hidden Markov models, reinforcement learning, state-space models, Kalman filters, speech recognition, anomaly detection, and time-series forecasting.

This chapter develops discrete-time and continuous-time models, with particular attention to Markov chains, Poisson processes, stationarity, autocorrelation, time-series decomposition, and forecasting. The emphasis is not only on calculation but also on assumptions, interpretation, model selection, and responsible use.

Learning Objectives

After studying this chapter, you should be able to:

1. Define a stochastic process and distinguish discrete-time from continuous-time processes.

2. Explain the Markov property and analyze finite-state Markov chains.

3. Calculate multi-step transition probabilities and long-run stationary distributions.

4. Describe Poisson processes and use them to model random event arrivals.

5. Distinguish strict stationarity, weak stationarity, and nonstationarity.

6. Interpret autocovariance, autocorrelation, and partial autocorrelation.

7. Decompose time series into trend, seasonal, cyclical, and irregular components.

8. Apply basic forecasting methods and evaluate forecast errors.

9. Identify assumptions, limitations, ethical concerns, and operational implications of stochastic models.

1. Stochastic Processes: The Basic Framework

A stochastic process is a collection of random variables indexed by time, commonly written as

\[

\{X_t:t\in T\}.

\]

Here, $X_t$ is the random quantity observed at time $t$, and $T$ is the index set. If $T=\{0,1,2,\ldots\}$, the process is in discrete time. If $T$ is an interval of real numbers, it is in continuous time. The set of possible values of $X_t$ is the state space.

A realized sequence, such as $(x_0,x_1,x_2,\ldots)$, is called a sample path or realization. The process describes uncertainty over possible sample paths; a time series is usually one observed realization.

Examples include:

• $X_t$: number of calls received during hour $t$;
• $X_t$: daily demand for a product;
• $X(t)$: number of machine failures by continuous time $t$;
• $X_t$: traffic state of a network link;
• $X_t$: hidden speech state in a recognition system.

A process may be classified by its time index and state space. A discrete-time, discrete-state process might describe the number of customers in a queue at the end of each minute. A continuous-time, continuous-state process might describe a physical temperature signal.

The filtration or information history up to time $t$ represents the information available by that time. Informally, a model is causal when predictions at time $t$ use only information available by $t$, not future observations.

Dependence over time

The central issue is temporal dependence. If $X_t$ and $X_{t-1}$ are independent, yesterday's value gives no information about today's value. In many systems this is unrealistic. Demand may persist, machine conditions may deteriorate, and financial volatility may cluster.

Dependence can be summarized using conditional probabilities, covariance, correlation, transition probabilities, or a model for the data-generating mechanism. A useful model should represent important dependence while remaining interpretable and estimable.

2. Markov Chains

A discrete-time process is a Markov chain if the conditional distribution of the next state depends on the past only through the current state:

\[

P(X_{t+1}=j\mid X_t=i,X_{t-1},\ldots,X_0)

=P(X_{t+1}=j\mid X_t=i).

\]

This is the Markov property. It does not say that successive states are independent. It says that the current state is a sufficient summary of the relevant past for predicting the next state.

For a finite state space $\{1,\ldots,m\}$, define the one-step transition matrix $P$ by

\[

p_{ij}=P(X_{t+1}=j\mid X_t=i).

\]

Each $p_{ij}$ is nonnegative and every row sums to one:

\[

p_{ij}\geq 0,\qquad \sum_{j=1}^{m}p_{ij}=1.

\]

The matrix convention used here is row-based: a row vector of state probabilities is multiplied on the right by $P$. Other texts use column vectors, so the convention must always be stated.

Multi-step transitions

The probability of moving from state $i$ to state $j$ in $n$ steps is the $(i,j)$ entry of $P^n$:

\[

p_{ij}^{(n)}=P(X_{t+n}=j\mid X_t=i).

\]

This follows from the law of total probability and matrix multiplication. For two steps,

\[

p_{ij}^{(2)}=\sum_{k=1}^{m}p_{ik}p_{kj}.

\]

If the initial row distribution is $\boldsymbol{\pi}_0$, then the distribution after $n$ steps is

\[

\boldsymbol{\pi}_n=\boldsymbol{\pi}_0P^n.

\]

Worked Example 1: A machine-condition chain

Suppose a machine is in one of three conditions: Good (G), Degraded (D), or Failed (F). The transition matrix is

\[

P=

\begin{pmatrix}

0.80&0.18&0.02\\

0.10&0.70&0.20\\

0&0&1

\end{pmatrix}.

\]

The failed state is absorbing: once failure occurs, the model assumes the machine remains failed. If the machine is initially good, its initial distribution is $(1,0,0)$. After one period it is $(0.80,0.18,0.02)$.

The probability that a good machine fails within two periods can be found by calculating the probability of failure by time two. The probability of remaining nonfailed after two periods is

\[

P(X_2\in\{G,D\}\mid X_0=G)

=0.80(0.80+0.18)+0.18(0.10+0.70)

=0.784+0.144=0.928.

\]

Therefore,

\[

P(X_2=F\mid X_0=G)=1-0.928=0.072.

\]

This calculation assumes fixed transition probabilities and that the current condition captures all relevant information about future failure.

Communicating classes and absorbing chains

A state $i$ can reach state $j$ if some power of $P$ has a positive $(i,j)$ entry. States that can reach each other belong to the same communicating class. A state is absorbing if $p_{ii}=1$. Absorbing chains are useful for modeling completion, failure, default, or termination.

For an absorbing chain, one may calculate expected time to absorption and absorption probabilities. If transient states are ordered first, the transition matrix has a transient block $Q$. The fundamental matrix is

\[

N=(I-Q)^{-1},

\]

provided the inverse exists. The entry $N_{ij}$ is the expected number of visits to transient state $j$ when starting in transient state $i$, including the initial visit. This result applies to a finite absorbing chain in which absorption occurs with probability one.

Stationary distributions and long-run behavior

A stationary distribution is a probability row vector $\boldsymbol{\pi}$ satisfying

\[

\boldsymbol{\pi}=\boldsymbol{\pi}P,\qquad \sum_i\pi_i=1,\qquad \pi_i\geq0.

\]

If a finite chain is irreducible and aperiodic, it has a unique stationary distribution and the distribution of $X_n$ converges to it from any initial distribution. Irreducibility means every state can eventually reach every other state. Aperiodicity rules out deterministic cycles. These conditions are sufficient for convergence; without them, a stationary distribution may still exist, but convergence behavior can differ.

For example, consider

\[

P=\begin{pmatrix}0.8&0.2\\0.3&0.7\end{pmatrix}.

\]

Let $\boldsymbol{\pi}=(\pi_1,\pi_2)$. From $\pi_1=0.8\pi_1+0.3\pi_2$ and $\pi_1+\pi_2=1$, we obtain $0.2\pi_1=0.3\pi_2$, so $\pi_1: \pi_2=3:2$. Hence

\[

\boldsymbol{\pi}=(0.6,0.4).

\]

The long-run proportion of time in state 1 is therefore 0.6, under the model assumptions.

3. Poisson Processes

A Poisson process models counts of events occurring over continuous time. Let $N(t)$ be the number of events in the interval $(0,t]$. A homogeneous Poisson process with rate $\lambda>0$ satisfies:

1. $N(0)=0$;

2. increments over disjoint time intervals are independent;

3. the number of events in an interval of length $t$ has a Poisson distribution;

4. the distribution depends only on interval length, not calendar location.

Thus,

\[

P(N(t)=k)=e^{-\lambda t}\frac{(\lambda t)^k}{k!},\qquad k=0,1,2,\ldots.

\]

The parameter $\lambda$ is the average event rate and must have units of events per unit time. For a time interval of length $t$, both the mean and variance are $\lambda t$:

\[

E[N(t)]=\lambda t,\qquad \operatorname{Var}[N(t)]=\lambda t.

\]

The waiting time $T$ until the first event has an exponential distribution:

\[

P(T>t)=e^{-\lambda t},\qquad f_T(t)=\lambda e^{-\lambda t},\quad t\geq0.

\]

Its mean is $1/\lambda$. The exponential distribution is memoryless:

\[

P(T>s+t\mid T>s)=P(T>t).

\]

Worked Example 2: Help-desk arrivals

A help desk receives an average of 6 calls per hour. Assuming a homogeneous Poisson process, the number of calls in 30 minutes has mean $6(0.5)=3$. The probability of exactly two calls is

\[

P(N(0.5)=2)=e^{-3}\frac{3^2}{2!}=4.5e^{-3}\approx0.2240.

\]

The probability of at least one call in the next 10 minutes is

\[

P(N(1/6)\geq1)=1-P(N(1/6)=0)=1-e^{-6/6}=1-e^{-1}\approx0.6321.

\]

These calculations assume a constant rate, independent increments, and no systematic variation by time of day.

A nonhomogeneous Poisson process allows a time-varying rate $\lambda(t)$. Its mean measure over $(a,b]$ is

\[

\Lambda(a,b)=\int_a^b\lambda(u)\,du,

\]

and, under the process assumptions,

\[

P(N(b)-N(a)=k)=e^{-\Lambda(a,b)}\frac{\Lambda(a,b)^k}{k!}.

\]

This is more appropriate when arrivals vary predictably across hours or days. A Poisson model may be inappropriate when events cluster, inhibit one another, or have overdispersion relative to the Poisson variance assumption.

4. Stationarity and Dependence

Stationarity describes stability of probabilistic behavior over time. A process is strictly stationary if, for every collection of times and every shift $h$, the joint distribution is unchanged:

\[

(X_{t_1},\ldots,X_{t_k})\overset{d}{=}(X_{t_1+h},\ldots,X_{t_k+h}).

\]

This is a strong condition involving all joint distributions.

A process is weakly stationary or covariance stationary if its mean is constant, its variance is finite and constant, and its covariance depends only on lag rather than absolute time:

\[

E[X_t]=\mu,

\]

\[

\operatorname{Cov}(X_t,X_{t+h})=\gamma(h).

\]

The autocorrelation function (ACF) is

\[

\rho(h)=\frac{\gamma(h)}{\gamma(0)},

\]

where $\gamma(0)=\operatorname{Var}(X_t)>0$. The ACF measures linear dependence between observations separated by $h$ time units. Autocorrelation is not proof of causation and does not capture every form of dependence.

The partial autocorrelation at lag $h$ measures the relationship between $X_t$ and $X_{t-h}$ after controlling linearly for intermediate lags. ACF and partial ACF plots are useful for identifying candidate autoregressive and moving-average models, but they should be combined with subject knowledge and diagnostic tests.

A process with a unit root, deterministic trend, changing variance, structural break, or evolving seasonal pattern is generally nonstationary. Differencing may help: the first difference is

\[

\Delta X_t=X_t-X_{t-1}.

\]

Differencing can remove a stochastic trend, but unnecessary differencing can introduce dependence and obscure interpretation. Transformations such as logarithms may stabilize variance when variability increases with the level, provided the data are positive or suitably shifted.

5. Time-Series Decomposition

A time series is often represented as the combination of interpretable components. The additive decomposition is

\[

Y_t=T_t+S_t+C_t+R_t,

\]

where $T_t$ is trend, $S_t$ is seasonal variation, $C_t$ is a longer-term cyclical component, and $R_t$ is an irregular or remainder component.

A multiplicative decomposition is

\[

Y_t=T_t\times S_t\times C_t\times R_t,

\]

which is appropriate when seasonal variation grows proportionally with the level. Taking logarithms converts a multiplicative model into an additive form when all values are positive:

\[

\log Y_t=\log T_t+\log S_t+\log C_t+\log R_t.

\]

Trend is persistent long-term movement. Seasonality is a systematic pattern with a known or stable period, such as weekly retail demand. Cycles are fluctuations whose duration is not necessarily fixed, often associated with economic or operational conditions. The remainder contains variation not explained by the systematic components.

Moving averages can smooth short-term noise. A centered moving average of odd width $2m+1$ is

\[

\widetilde{Y}_t=\frac{1}{2m+1}\sum_{j=-m}^{m}Y_{t+j}.

\]

This formula is most directly applicable away from the boundaries and assumes equal weighting is suitable. Smoothing is descriptive; it should not automatically be interpreted as forecasting.

6. Forecasting Sequential Data

Forecasting estimates future values using historical information. A forecast should be accompanied by an uncertainty statement, because a point prediction alone can create false precision.

Naive and seasonal-naive methods

The naive forecast for the next period is

\[

\hat{Y}_{t+1|t}=Y_t.

\]

For seasonal data with period $m$, a seasonal-naive forecast is

\[

\hat{Y}_{t+h|t}=Y_{t+h-mk},

\]

where $k$ is chosen so that the referenced observation is the most recent value from the same seasonal position. These methods are strong baselines and should rarely be omitted.

Moving-average forecasting

A simple moving-average forecast using the last $k$ observations is

\[

\hat{Y}_{t+1|t}=\frac{1}{k}\sum_{i=0}^{k-1}Y_{t-i}.

\]

A larger $k$ gives more smoothing but reacts more slowly to changes. The method assumes that a recent local average is informative for the next period.

Exponential smoothing

Simple exponential smoothing updates a level estimate by

\[

\ell_t=\alpha Y_t+(1-\alpha)\ell_{t-1},\qquad 0<\alpha\leq1,

\]

with one-step forecast $\hat{Y}_{t+1|t}=\ell_t$. The initial level must be specified, for example as the first observation or an estimated value. Larger $\alpha$ places more weight on recent observations.

Trend and seasonality require extensions such as Holt's method and Holt–Winters methods. These methods are useful when patterns are reasonably stable, but they can fail under abrupt regime changes or unusual interventions.

Autoregressive models

An autoregressive model of order $p$, AR($p$), is

\[

X_t=c+\phi_1X_{t-1}+\cdots+\phi_pX_{t-p}+\varepsilon_t,

\]

where $\varepsilon_t$ is a zero-mean innovation, often assumed uncorrelated over time and sometimes modeled as Gaussian. For an AR(1), stationarity requires $|\phi_1|<1$ when the model is used in its standard stationary form. The stationary mean is $c/(1-\phi_1)$ when the innovation has mean zero.

AR models describe dependence on previous values. Moving-average models describe dependence on previous shocks, and ARIMA models combine autoregression, differencing, and moving-average terms. Model identification, estimation, residual diagnostics, and out-of-sample evaluation are essential; fitting a flexible model without validation can produce overfitting.

Forecast evaluation

For actual values $y_t$ and forecasts $\hat y_t$, common measures include:

\[

\operatorname{MAE}=\frac{1}{n}\sum_{t=1}^{n}|y_t-\hat y_t|,

\]

\[

\operatorname{RMSE}=\sqrt{\frac{1}{n}\sum_{t=1}^{n}(y_t-\hat y_t)^2}.

\]

MAE is in the original units and treats errors linearly. RMSE is also in the original units but penalizes large errors more heavily. Percentage measures can be problematic when actual values are zero or near zero. Evaluation must preserve temporal order: training on future observations and testing on past observations creates leakage.

Forecast residuals should be examined for remaining autocorrelation, changing variance, outliers, and systematic bias. A well-specified model does not require residuals to be exactly normal for every practical purpose, but unmodeled structure indicates that useful predictive information may remain.

7. Applied Case: Predictive Maintenance for a Packaging Line

The following is an illustrative case. A manufacturer operates a packaging line with sensors recorded every 10 minutes. The engineering team wants to reduce unplanned downtime while avoiding unnecessary maintenance.

Background and problem

The line can be in normal, warning, or failed condition. Historical maintenance records show that warning conditions often precede failure, but not every warning leads to failure. The organization also records failure events and vibration measurements.

Analysis

The team first defines a three-state Markov model using condition labels. Transition counts from historical sequences are converted into row probabilities. A transition matrix estimates the probability of entering failure within the next monitoring interval. Because the warning state contains information about imminent failure, maintenance decisions are conditioned on the current state rather than on an overall average failure rate.

Failure events are also analyzed as arrivals. During stable production periods, a Poisson model provides a baseline for the number of minor stoppages per shift. However, the data show higher event rates during product changeovers, so a nonhomogeneous model or separate regime-specific rates are more appropriate than one constant rate.

The vibration series displays a slow upward trend and daily operating cycles. The team decomposes the series into trend, seasonality, and remainder, then uses rolling-origin validation to compare a seasonal-naive model, exponential smoothing, and an autoregressive model. A model is selected based on predictive performance and operational interpretability, not merely in-sample fit.

Approach and solution

The maintenance policy assigns an inspection when the estimated probability of failure before the next planned service exceeds a threshold. The threshold is based on the cost of inspection, the cost of downtime, safety considerations, and the consequences of false alarms. Sensor readings are not used as an automatic basis for disciplinary or employment decisions; they are used to support engineering review.

The system includes monitoring for data drift. If sensor calibration changes or the production process is redesigned, transition probabilities and forecasting relationships may no longer be valid. Model outputs are stored with timestamps and version information so that decisions can be audited.

Outcome and learning points

In this illustrative case, combining state transitions with time-series forecasting gives a more informative system than either approach alone. The Markov model represents discrete machine condition, while forecasting captures gradual sensor changes. The main learning points are:

• state definitions must be operationally meaningful;
• rates should reflect exposure time and operating regime;
• temporal validation is essential;
• predictive accuracy must be weighed against maintenance cost and safety;
• monitoring and human oversight are part of the model design.

8. Challenges, Limitations, and Governance

Stochastic models simplify reality. The Markov assumption may omit relevant history, such as cumulative fatigue or time since the last repair. A Poisson process may fail when events cluster, have changing rates, or influence one another. Stationarity may be violated by policy changes, technological development, seasonality, or crises.

Parameter estimates are uncertain. A transition probability estimated from few observations can be unstable, and rare-event rates can be difficult to estimate reliably. Confidence intervals, predictive intervals, sensitivity analysis, and stress testing should be used where decisions are consequential.

Data quality is particularly important for sequential data. Missing timestamps, irregular sampling, censoring, delayed reporting, sensor drift, and changes in measurement definitions can create apparent patterns. Interpolation may be inappropriate when events are abrupt or missingness is informative.

In AI systems, temporal leakage is a major governance risk. Features created using future information can produce impressive validation results but fail in deployment. Data splitting, feature engineering, and preprocessing must respect the information available at prediction time.

Fairness concerns arise when sequential models influence access to services, insurance, employment, credit, or healthcare. Historical data may encode unequal treatment. A model that predicts future behavior may reinforce feedback loops if its predictions determine the observations later collected. Governance should therefore address transparency, contestability, privacy, security, monitoring, and accountability.

Best practice includes documenting the data-generating process, assumptions, time horizon, state definitions, estimation method, validation design, known failure modes, and responsible owner. Models should be recalibrated when the underlying process changes, and simple baselines should remain available for comparison.

9. Contemporary Developments

Modern forecasting increasingly combines classical statistical models with machine learning. Gradient-boosted trees and neural networks can use lagged values, calendar variables, external covariates, and high-dimensional sensor information. Recurrent neural networks, temporal convolutional networks, and transformers can represent complex dependencies, although greater flexibility increases the need for careful validation and interpretability.

Probabilistic forecasting produces a distribution or prediction interval rather than only a point estimate. Quantile forecasts, Bayesian state-space models, Gaussian processes, and simulation-based approaches are useful when decision-makers need risk estimates. In operations, the probability of exceeding capacity may be more valuable than the expected demand alone.

Hidden Markov models extend Markov chains by distinguishing an unobserved state process from observed emissions. They are widely used in speech recognition, biological sequence analysis, regime detection, and anomaly detection. Reinforcement learning also uses state-transition concepts, but its transition model is embedded in sequential decision-making and reward optimization.

These developments do not eliminate the importance of foundational probability. All advanced models still require clear definitions of time, information, dependence, uncertainty, and evaluation.

Chapter Summary

Stochastic processes model random quantities that evolve over time. Markov chains use the current state as a sufficient summary of the past and are analyzed through transition matrices, multi-step probabilities, communicating classes, absorbing states, and stationary distributions. Poisson processes model event counts in continuous time under assumptions of independent increments and a constant or time-varying rate; their waiting times are exponentially distributed in the homogeneous case.

Stationarity concerns whether probabilistic behavior remains stable under time shifts. Autocovariance, autocorrelation, and partial autocorrelation describe temporal dependence and support model identification. Time-series decomposition separates trend, seasonality, cycles, and irregular variation. Forecasting methods range from naive and seasonal-naive baselines to moving averages, exponential smoothing, autoregressive models, and modern machine-learning systems.

Reliable application requires explicit assumptions, temporal validation, residual diagnostics, uncertainty quantification, monitoring for drift, and attention to privacy, fairness, safety, and accountability. In AI and operations, the best model is not necessarily the most complex; it is the model whose assumptions, performance, and limitations are appropriate for the decision.

Key Terms

Stochastic process: A collection of random variables indexed by time.

Sample path: One realized sequence from a stochastic process.

State space: The set of possible values or conditions of a process.

Markov property: The condition that the future depends on the past only through the present state.

Transition matrix: A matrix containing one-step state-transition probabilities.

Absorbing state: A state that, once entered, cannot be left.

Stationary distribution: A probability distribution unchanged by application of the transition matrix.

Irreducibility: The property that every state can eventually reach every other state.

Aperiodicity: The absence of a forced cyclic return pattern.

Poisson process: A continuous-time event-counting process with specified independent-increment and rate properties.

Intensity or rate: Expected event frequency per unit time.

Exponential distribution: The waiting-time distribution for the first event in a homogeneous Poisson process.

Strict stationarity: Invariance of all joint distributions under time shifts.

Weak stationarity: Constant mean and variance with covariance depending only on lag.

Autocovariance: Covariance between observations separated by a specified lag.

Autocorrelation: Autocovariance divided by the process variance.

Seasonality: A systematic pattern repeating at a known period.

Differencing: Subtracting a prior observation to reduce trend or nonstationarity.

Forecast leakage: Use of information unavailable at prediction time.

Prediction interval: An interval representing uncertainty about a future observation or value.

Review Questions

Short Answer Questions

1. What is a stochastic process, and how does it differ from a single random variable?

2. State the Markov property in words and mathematical notation.

3. What conditions define a weakly stationary process?

4. What is the mean number of events in an interval of length $t$ for a homogeneous Poisson process with rate $\lambda$?

5. Why must temporal order be preserved when evaluating forecasts?

Long Answer Questions

1. Explain transition matrices, multi-step transition probabilities, and stationary distributions for a finite Markov chain.

2. Describe the assumptions of a homogeneous Poisson process and explain situations in which those assumptions may fail.

3. Compare strict stationarity and weak stationarity, including their relevance to time-series modeling.

4. Explain additive and multiplicative time-series decomposition and provide an example of when each is appropriate.

5. Compare naive forecasting, exponential smoothing, and autoregressive forecasting in terms of assumptions and limitations.

Analytical / Application Questions

1. A service center receives an average of 4 calls per hour. Assuming a homogeneous Poisson process, calculate the probability of exactly three calls in a 30-minute interval and the probability of no calls in 15 minutes.

2. For the transition matrix $P=\begin{pmatrix}0.7&0.3\\0.4&0.6\end{pmatrix}$, find the stationary distribution and interpret its components.

3. A demand series has increasing variability as its level rises and a repeating weekly pattern. Recommend a decomposition and forecasting strategy, identify two validation risks, and explain how you would monitor the deployed model.

Multiple Choice Questions with Answers

Question 1

Which statement best expresses the Markov property?

A. The next state is independent of the current state.

B. The future depends on the entire history but not the present.

C. Given the current state, the future is conditionally independent of the past.

D. All states must have equal transition probabilities.

Correct Answer: C. Given the current state, the future is conditionally independent of the past.

Question 2

For a homogeneous Poisson process with rate 6 events per hour, what is the probability of zero events in 30 minutes?

A. e^-6

B. e^-3

C. 1-e^-3

D. 6e^-3

Correct Answer: B. e^-3

Question 3

A stationary distribution for a transition matrix P satisfies which equation?

A. π=P+π

B. π=πP

C. π=PπP

D. π=I-P

Correct Answer: B. π=πP

Question 4

In a homogeneous Poisson process, the expected number of events in an interval of length t is:

A. λ+t

B. λ/t

C. λt

D. t/λ

Correct Answer: C. λt

Question 5

Which condition is required for the standard stationary AR(1) model X_t=c+φX_{t-1}+ε_t?

A. φ=0 only

B. |φ|<1

C. |φ|>1

D. φ must equal 1

Correct Answer: B. |φ|<1

Question 6

What does weak stationarity require regarding the mean?

A. It must increase linearly over time.

B. It must be constant over time.

C. It must equal zero.

D. It must equal the variance.

Correct Answer: B. It must be constant over time.

Question 7

Which forecast uses the most recent observation as the next-period prediction?

A. Naive forecast

B. Seasonal decomposition forecast

C. Poisson forecast

D. Absorbing-state forecast

Correct Answer: A. Naive forecast

Question 8

A multiplicative decomposition is most suitable when seasonal fluctuations:

A. Are proportional to the series level

B. Are always exactly zero

C. Have no repeated pattern

D. Are independent of the series level in absolute size

Correct Answer: A. Are proportional to the series level

Question 9

Which evaluation practice creates temporal leakage?

A. Training on earlier observations and testing on later observations

B. Using rolling-origin validation

C. Using future observations to construct a feature for an earlier prediction

D. Comparing with a naive baseline

Correct Answer: C. Using future observations to construct a feature for an earlier prediction

Question 10

For the transition matrix [[0.8,0.2],[0.3,0.7]], the stationary probability of state 1 is:

A. 0.30

B. 0.40

C. 0.50

D. 0.60

Correct Answer: D. 0.60