# on 17-May-2015 (Sun)

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#### Flashcard 150891134

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#m249 #mathematics #open-university #statistics #time-series
Question
In simple exponential smoothing satisfying the formula:
$$\hat{x}_{n+1}$$ = αxn + (1 − α)$$\hat{x}_n$$, the lower the value of α, the [smoother or rougher?] the forecasts will be because they are not aﬀected much by recent values.
Answer
smoother

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In simple exponential smoothing satisfying the formula: \hat{x}_{n+1} = αx n + (1 − α)\hat{x}_n, the lower the value of α, the smoother the forecasts will be because they are not aﬀected much by recent values.

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#### Flashcard 150891166

Tags
#m249 #mathematics #open-university #statistics #time-series
Question
The 1-step ahead forecast error at time t, which is denoted et, is the diﬀerence between the observed value and the 1-step ahead forecast of Xt:
et = xt - $$\hat{x}_t$$
The sum of squared errors, or SSE, is given by
SSE = [...]
Given observed values x1 ,x2 ,...,xn ,the optimal value of the smoothing parameter α for simple exponential smoothing is the value that minimizes the sum of squared errors.
Answer
$$\large SSE = \sum_{t=1}^ne_t^2 = \sum_{t=1}^n(x_t-\hat{x}_t)^2$$

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step ahead forecast error at time t, which is denoted e t , is the diﬀerence between the observed value and the 1-step ahead forecast of X t : e t = x t - $$\hat{x}_t$$ The sum of squared errors, or SSE, is given by SSE <span>= $$\large \sum_{t=t}^ne_t^2 = \sum_{t=t}^n(x_t-\hat{x}_t)^2$$ Given observed values x 1 ,x 2 ,...,x n ,the optimal value of the smoothing parameter α for simple exponential smoothing is the value that minimizes the sum of squared errors.<

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#### Flashcard 150891282

Tags
#m249 #mathematics #open-university #statistics #time-series
Question
Give a formula for additive time series model with constant level and no seasonality
Answer
Xt = m + Wt

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#### Annotation 150891291

 #m249 #mathematics #open-university #statistics #time-series Suppose that the time series Xt can be described by an additive non-seasonal model with a linear trend component, that is, Xt = m + bt + Wt , where b is the slope of the trend component mt = m + bt. Note that Xt+1 = m + b(t +1) + Wt+1 =(m + bt) + b + Wt+1 = mt + b + Wt+1
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#### Flashcard 150891298

Tags
#m249 #mathematics #open-university #statistics #time-series
Question
Suppose that the time series Xt can be described by an additive non-seasonal model with a linear trend component, that is,
Xt = [...] , where b is the slope of the trend component mt = m + bt.
Note that
Xt+1 = m + b(t +1) + Wt+1
=(m + bt) + b + Wt+1
= mt + b + Wt+1
Answer
m + bt + Wt

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Suppose that the time series X t can be described by an additive non-seasonal model with a linear trend component, that is, X t = m + b t + W t , where b is the slope of the trend component m t = m + bt. Note that X t+1 = m + b(t +1) + W t+1 =(m + bt) + b + W t+1 = m t + b + W t+1 <

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#### Flashcard 150891307

Tags
#m249 #mathematics #open-university #statistics #time-series
Question
Suppose that the time series Xt can be described by an additive non-seasonal model with a linear trend component, that is,
Xt = m + bt + Wt , where b is the [...] of the trend component mt = m + bt.
Note that
Xt+1 = m + b(t +1) + Wt+1
=(m + bt) + b + Wt+1
= mt + b + Wt+1
Answer
slope

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Suppose that the time series X t can be described by an additive non-seasonal model with a linear trend component, that is, X t = m + b t + W t , where b is the slope of the trend component m t = m + bt. Note that X t+1 = m + b(t +1) + W t+1 =(m + bt) + b + W t+1 = m t + b + W t+1

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#### Flashcard 150891319

Tags
#m249 #mathematics #open-university #statistics #time-series
Question
Suppose that the time series Xt can be described by an additive non-seasonal model with a linear trend component, that is,
Xt = m + bt + Wt , where b is the slope of the trend component mt = [...].
Note that
Xt+1 = m + b(t +1) + Wt+1
=(m + bt) + b + Wt+1
= mt + b + Wt+1
Answer
m + bt

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Suppose that the time series X t can be described by an additive non-seasonal model with a linear trend component, that is, X t = m + bt + W t , where b is the slope of the trend component m t = m + bt. Note that X t+1 = m + b(t +1) + W t+1 =(m + bt) + b + W t+1 = m t + b + W t+1

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#### Flashcard 150891325

Tags
#m249 #mathematics #open-university #statistics #time-series
Question
Suppose that the time series Xt can be described by an additive non-seasonal model with a linear trend component, that is,
Xt = m + bt + Wt , where b is the slope of the trend component mt = m + bt.
Note that
Xt+1 = m + b[...] + Wt+1
=(m + bt) + b + Wt+1
= mt + b + Wt+1
Answer
(t +1)

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pose that the time series X t can be described by an additive non-seasonal model with a linear trend component, that is, X t = m + bt + W t , where b is the slope of the trend component m t = m + bt. Note that X t+1 = m + b<span>(t +1) + W t+1 =(m + bt) + b + W t+1 = m t + b + W t+1 <span><body><html>

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#### Flashcard 150891331

Tags
#m249 #mathematics #open-university #statistics #time-series
Question
Suppose that the time series Xt can be described by an additive non-seasonal model with a linear trend component, that is,
Xt = m + bt + Wt , where b is the slope of the trend component mt = m + bt.
Note that
Xt+1 = m + b(t +1) + Wt+1
=[...]
= mt + b + Wt+1
Answer
(m + bt) + b + Wt+1

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can be described by an additive non-seasonal model with a linear trend component, that is, X t = m + bt + W t , where b is the slope of the trend component m t = m + bt. Note that X t+1 = m + b(t +1) + W t+1 =<span>(m + bt) + b + W t+1 = m t + b + W t+1 <span><body><html>

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#### Flashcard 150891340

Tags
#m249 #mathematics #open-university #statistics #time-series
Question
Suppose that the time series Xt can be described by an additive non-seasonal model with a linear trend component, that is,
Xt = m + bt + Wt , where b is the slope of the trend component mt = m + bt.
Note that
Xt+1 = m + b(t +1) + Wt+1
=(m + bt) + b + Wt+1
= [...]
Answer
mt + b + Wt+1

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n-seasonal model with a linear trend component, that is, X t = m + bt + W t , where b is the slope of the trend component m t = m + bt. Note that X t+1 = m + b(t +1) + W t+1 =(m + bt) + b + W t+1 = <span>m t + b + W t+1 <span><body><html>

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#### Annotation 150891346

 #m249 #mathematics #open-university #statistics #time-series simple exponential smoothing: the term exponential refers to the fact that the weights α(1 − α)i lie on an exponential curve.
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#### Flashcard 150891353

Tags
#m249 #mathematics #open-university #statistics #time-series
Question
simple exponential smoothing: the term exponential refers to the fact that the weights [give formula] lie on an exponential curve.
Answer
α(1 − α)i

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simple exponential smoothing: the term exponential refers to the fact that the weights α(1 − α) i lie on an exponential curve.

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#### Flashcard 150891359

Tags
#m249 #mathematics #open-university #statistics #time-series
Question
what does the expanded m-times (i.e. non recursive) simple exponential smoothing formula looks like?
fully recursive is:

$$\hat{x}_{n+1}$$= αxn + (1 − α)$$\hat{x}_n$$
Answer
expanded m-times is
$$\large \hat{x}_{n+1} = \sum_{i=0}^m\alpha(1-\alpha)^ix_{n-i}+(1-\alpha)^{m+1}\hat{x}_{n-m}$$

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If a time series X t is described by an additive model with constant level and no seasonality, 1-step ahead forecasts may be obtained by simple exponential smoothing using the formula $$\hat{x}_{n+1}$$= αx n + (1 − α)$$\hat{x}_n$$ where: x n is the observed value at time n, $$\hat{x}_n$$​and $$\hat{x}_{n+1}$$are the 1-step ahead forecasts of X n and X n+1 , and α is a smoothing parameter, 0 ≤ α ≤ 1.

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