Weeks 9-12

Question 1

What is a Time Series?

Answer 1

A set of observations y1, y2, …, yt ordered in time (i.e., weekly sales, daily production levels, annual temperature).

Question 2

What components make up a Time Series?

Answer 2

Yt = f(Tt, St, Ct, Et)

• Trend Tt: The systematic long-term increase or decrease in the series.
• Season St: A systematic change in the mean of the seres due to seasonal factors.
• Cyclic Ct: A cyclic pattern exists when there are medium or long run fluctuations in the time series that are not of a fixed period.
• Irregular Et: Short-term fluctuations and noise.

Question 3

What is the Random Walk forecasting method?

Answer 3

Forecast’s the series using the last available observation:

y-hat(t+h) = yt.

Question 4

What is the Seasonal Random Walk method?

Answer 4

Forecasting the series with the last available observation in the same season:

y-hat(t+h) = y(t+h-m) (if h <= m).

Question 5

What is the Drift Forecasting method?

Answer 5

The sum of the most recent value and the average change over time:

y-hat(t+h) = yt + h*sum( yi - y(i-1) / (t-1) )

Question 6

What is the formula for the variance of a point forecast generated using the Random Walk method?

Answer 6

Var( Y(t+h) | y(1:t) ) = h*ó^2

Question 7

What is the equation for the forecast interval for a point forecast generated using the Random Walk method?

Answer 7

Forecast interval = yt +- z(alpha/2) * sqrt( h*ó^2 )

Question 8

What is the concept of an in-sample forecast?

Answer 8

Fitting a model to historical observations and then using that model to make predictions for the same time period that the model was trained on.

Question 9

Explain the Simple Exponential Smoothing method, and include the recursive formulation.

Answer 9

Forecasts are weighted averages of past observations. A higher alpha gives larger weight to recent observations, while lower alpha makes the forecast smoother.

y-hat(t) + 1 = łt
łt = ayt + (1 - a)ł(t-1)
= ayt + (1-a)ay(t-1)+(1-a)^2ay(t-2)+…+(1-a)^(t-1)ay1+(1-a)^t*ł0

Question 10

What is Trend Corrected/Holt Exponential Smoothing?

Answer 10

An adaption of SES to account for a trend in the data.

y-hat(t+1) = łt + bt (forecast equation)
łt = ayt + (1 - a)ł(t-1) (smoothing equation)
bt = B(łt - ł(t-1)) + (1-B)b(t-1) (trend equation)

for initial values ł0 and b0, 0<= a,B <= 1

Question 11

What is Holt Winters Exponential Smoothing?

Answer 11

It extends the trend corrected method to seasonal data, with both additive and multiplicative seasonality.

Question 12

What is the Additive Holt Winters model and what is its formulation?

Answer 12

When the variance of the seasonality component does not change over time:

y-hat(t+1) = łt + bt + s(t+1-L) (forecast equation)
łt = ayt + (1 - a)ł(t-1) (smoothing equation)
bt = B(łt - ł(t-1)) + (1-B)b(t-1) (trend equation)
st = ø(yt - łt) + (1 - ø)s(t-1) (seasonal indices)

for initial values ł0, b0, and s(i-L), 0<= a,B,ø <= 1

Question 13

What is the point forecast for SES?

Answer 13

y-hat(t+h) = = ayt + (1-a)ay(t-1)+(1-a)^2ay(t-2)+…+(1-a)^(t-1)ay1+(1-a)^t*ł0

Question 14

What is Temporal Holdout/Crossvalidation? And why do we need it for Time Series?

Answer 14

Temporal Validation involves splitting training and testing sets into chronological order and forecasting on the next value(s) not included in the training set. We need it to evaluate how well a model can forecast future values while accounting for temporal dependencies.

Question 15

Explain the difference between the concept of Autoregression and the concept of a time series as a function of time.

Answer 15

Autoregression is a function based on past values and approximates time-based functions. Time Series as a function of time is how a variable changes with respect to time.

Question 16

What is the concept of Stationarity?

Answer 16

A time series whose properties do not depend on time (i.e., no trend, no seasonality).

Strictly stationary: when the joint distribution of yt, yt-1,…, yt-k does not depend on t.

Weak/Covariance stationarity: If the TS mean, variance, and autocovariances do not change over time.

Question 17

To figure out the order of an AR model, which graph do you need to look at and how do you determine the order?

Answer 17

PACF

How far is the furthest lag in the PACF that is outside the 95% confidence interval (excluding the lag 1, which is just the points correlation with itself) to get the order

Question 18

To figure out the order of an MA model, which graph do you need to look at and how do you determine the order?

Answer 18

ACF

How far is the furthest lag in the ACF that is outside the 95% confidence interval (excluding the lag 1, which is just the points correlation with itself) to get the order

Question 19

What should the PACF and ACF graphs look like for an AR model?

Answer 19

ACF should decay exponentially

PACF should drop quickly after lag 1

Question 20

What should the PACF and ACF graphs look like for an MA model?

Answer 20

PACF should decay exponentially

ACF should drop quickly after lag 1

Question 21

What is differencing in a Time Series context? Also mention first and second differencing.

Answer 21

A transform to achieve stationarity.

First difference: change in Yt = yt - y(t-1)

Second difference: change^2 in Yt = (yt - y(t-1)) - (y(t-1) - y(t-2))

Question 22

What is the AR model?

Answer 22

The AR model is a linear function of the lags in the TS:

yt = c + ø1y(t-1) + ø2y(t-2) +…+ øp*y(t-p) + Ęt

Question 23

What is Seasonal Differencing?

Answer 23

To address non stationarity caused by seasonality:

change in m*yt = yt - y(t-m)

Question 24

What is the MA model?

Answer 24

A linear function of the errors of the TS:

yt = c + Ęt + ø1Ę(t-1) + ø2Ę(t-2) +…+ øq*Ę(t-q)

Question 25

What is the ARMA model?

Answer 25

Combines the MA and AR models together.

yt = c + ø1y(t-1) +… + ØĘ(t-1) +… + Ęt

Question 26

What is the ARIMA model?

Answer 26

An extension of the ARMA model but including differencing the model to make it stationary (integrated).

ARIMA(p,d,q), where p is the AR order, d is the degree of differencing, and q is the MA order.

Question 27

What does the intercept do in the ARIMA model?

Answer 27

It introduced a linear trend in an ARIMA(p,1,q) model.

Question 28

What is Seasonal ARIMA?

Answer 28

A combination of an ARIMA on the seasonal period and standard ARIMA: ARIMA(p,d,q)(P,D,Q)m.

The seasonal components are capitalised, and m is the season.

Question 28

What is Seasonal ARIMA?

Answer 28

A combination of an ARIMA on the seasonal period and standard ARIMA: ARIMA(p,d,q)(P,D,Q)m.

The seasonal components are capitalised, and m is the season.