Weeks 9-12 Flashcards
What is a Time Series?
A set of observations y1, y2, …, yt ordered in time (i.e., weekly sales, daily production levels, annual temperature).
What components make up a Time Series?
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.
What is the Random Walk forecasting method?
Forecast’s the series using the last available observation:
y-hat(t+h) = yt.
What is the Seasonal Random Walk method?
Forecasting the series with the last available observation in the same season:
y-hat(t+h) = y(t+h-m) (if h <= m).
What is the Drift Forecasting method?
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) )
What is the formula for the variance of a point forecast generated using the Random Walk method?
Var( Y(t+h) | y(1:t) ) = h*ó^2
What is the equation for the forecast interval for a point forecast generated using the Random Walk method?
Forecast interval = yt +- z(alpha/2) * sqrt( h*ó^2 )
What is the concept of an in-sample forecast?
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.
Explain the Simple Exponential Smoothing method, and include the recursive formulation.
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
What is Trend Corrected/Holt Exponential Smoothing?
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
What is Holt Winters Exponential Smoothing?
It extends the trend corrected method to seasonal data, with both additive and multiplicative seasonality.
What is the Additive Holt Winters model and what is its formulation?
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
What is the point forecast for SES?
y-hat(t+h) = = ayt + (1-a)ay(t-1)+(1-a)^2ay(t-2)+…+(1-a)^(t-1)ay1+(1-a)^t*ł0
What is Temporal Holdout/Crossvalidation? And why do we need it for Time Series?
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.
Explain the difference between the concept of Autoregression and the concept of a time series as a function of time.
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.
What is the concept of Stationarity?
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.
To figure out the order of an AR model, which graph do you need to look at and how do you determine the order?
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
To figure out the order of an MA model, which graph do you need to look at and how do you determine the order?
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
What should the PACF and ACF graphs look like for an AR model?
ACF should decay exponentially
PACF should drop quickly after lag 1
What should the PACF and ACF graphs look like for an MA model?
PACF should decay exponentially
ACF should drop quickly after lag 1
What is differencing in a Time Series context? Also mention first and second differencing.
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))
What is the AR model?
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
What is Seasonal Differencing?
To address non stationarity caused by seasonality:
change in m*yt = yt - y(t-m)
What is the MA model?
A linear function of the errors of the TS:
yt = c + Ęt + ø1Ę(t-1) + ø2Ę(t-2) +…+ øq*Ę(t-q)
What is the ARMA model?
Combines the MA and AR models together.
yt = c + ø1y(t-1) +… + ØĘ(t-1) +… + Ęt
What is the ARIMA model?
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.
What does the intercept do in the ARIMA model?
It introduced a linear trend in an ARIMA(p,1,q) model.
What is Seasonal ARIMA?
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.
What is Seasonal ARIMA?
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.
