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#m249 #mathematics #open-university #statistics #time-series

Question

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 or rougher?] the forecasts will be because they are not aﬀected much by recent values.

\(\hat{x}_{n+1}\) = αx

Answer

smoother

status | not learned | measured difficulty | 37% [default] | last interval [days] | |||
---|---|---|---|---|---|---|---|

repetition number in this series | 0 | memorised on | scheduled repetition | ||||

scheduled repetition interval | last repetition or drill |

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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#m249 #mathematics #open-university #statistics #time-series

Question

The 1-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 = [...]

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.

e

The sum of squared errors, or SSE, is given by

SSE = [...]

Given observed values x

Answer

\(\large SSE = \sum_{t=1}^ne_t^2 = \sum_{t=1}^n(x_t-\hat{x}_t)^2\)

status | not learned | measured difficulty | 37% [default] | last interval [days] | |||
---|---|---|---|---|---|---|---|

repetition number in this series | 0 | memorised on | scheduled repetition | ||||

scheduled repetition interval | last repetition or drill |

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.<

Tags

#m249 #mathematics #open-university #statistics #time-series

Question

Give a formula for additive time series model with constant level and no seasonality

Answer

X_{t} = m + W_{t}

status | not learned | measured difficulty | 37% [default] | last interval [days] | |||
---|---|---|---|---|---|---|---|

repetition number in this series | 0 | memorised on | scheduled repetition | ||||

scheduled repetition interval | last repetition or drill |

#m249 #mathematics #open-university #statistics #time-series

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}

X

Note that

X

=(m + bt) + b + W

= m

status | not read | reprioritisations | ||
---|---|---|---|---|

last reprioritisation on | reading queue position [%] | |||

started reading on | finished reading on |

Tags

#m249 #mathematics #open-university #statistics #time-series

Question

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} = [...] , 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}

X

Note that

X

=(m + bt) + b + W

= m

Answer

m + bt + W_{t}

status | not learned | measured difficulty | 37% [default] | last interval [days] | |||
---|---|---|---|---|---|---|---|

repetition number in this series | 0 | memorised on | scheduled repetition | ||||

scheduled repetition interval | last repetition or drill |

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 <

Tags

#m249 #mathematics #open-university #statistics #time-series

Question

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 [...] 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}

X

Note that

X

=(m + bt) + b + W

= m

Answer

slope

status | not learned | measured difficulty | 37% [default] | last interval [days] | |||
---|---|---|---|---|---|---|---|

repetition number in this series | 0 | memorised on | scheduled repetition | ||||

scheduled repetition interval | last repetition or drill |

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

Tags

#m249 #mathematics #open-university #statistics #time-series

Question

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} = [...].

Note that

X_{t+1} = m + b(t +1) + W_{t+1}

=(m + bt) + b + W_{t+1}

= m_{t} + b + W_{t+1}

X

Note that

X

=(m + bt) + b + W

= m

Answer

m + bt

status | not learned | measured difficulty | 37% [default] | last interval [days] | |||
---|---|---|---|---|---|---|---|

repetition number in this series | 0 | memorised on | scheduled repetition | ||||

scheduled repetition interval | last repetition or drill |

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

Tags

#m249 #mathematics #open-university #statistics #time-series

Question

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[...] + W_{t+1}

=(m + bt) + b + W_{t+1}

= m_{t} + b + W_{t+1}

X

Note that

X

=(m + bt) + b + W

= m

Answer

(t +1)

status | not learned | measured difficulty | 37% [default] | last interval [days] | |||
---|---|---|---|---|---|---|---|

repetition number in this series | 0 | memorised on | scheduled repetition | ||||

scheduled repetition interval | last repetition or drill |

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>

Tags

#m249 #mathematics #open-university #statistics #time-series

Question

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_{t} + b + W_{t+1}

X

Note that

X

=[...]

= m

Answer

(m + bt) + b + W_{t+1}

status | not learned | measured difficulty | 37% [default] | last interval [days] | |||
---|---|---|---|---|---|---|---|

repetition number in this series | 0 | memorised on | scheduled repetition | ||||

scheduled repetition interval | last repetition or drill |

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>

Tags

#m249 #mathematics #open-university #statistics #time-series

Question

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}

=_{[...] }

X

Note that

X

=(m + bt) + b + W

=

Answer

m_{t} + b + W_{t+1}

status | not learned | measured difficulty | 37% [default] | last interval [days] | |||
---|---|---|---|---|---|---|---|

repetition number in this series | 0 | memorised on | scheduled repetition | ||||

scheduled repetition interval | last repetition or drill |

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>

#m249 #mathematics #open-university #statistics #time-series

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last reprioritisation on | reading queue position [%] | |||

started reading on | finished reading on |

Tags

#m249 #mathematics #open-university #statistics #time-series

Question

Answer

α(1 − α)^{i}

status | not learned | measured difficulty | 37% [default] | last interval [days] | |||
---|---|---|---|---|---|---|---|

repetition number in this series | 0 | memorised on | scheduled repetition | ||||

scheduled repetition interval | last repetition or drill |

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

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:

\(\hat{x}_{n+1}\)= αx_{n} + (1 − α)\(\hat{x}_n\)

fully recursive is:

\(\hat{x}_{n+1}\)= αx

Answer

expanded m-times is

\(\large \hat{x}_{n+1} = \sum_{i=0}^m\alpha(1-\alpha)^ix_{n-i}+(1-\alpha)^{m+1}\hat{x}_{n-m}\)

\(\large \hat{x}_{n+1} = \sum_{i=0}^m\alpha(1-\alpha)^ix_{n-i}+(1-\alpha)^{m+1}\hat{x}_{n-m}\)

status | not learned | measured difficulty | 37% [default] | last interval [days] | |||
---|---|---|---|---|---|---|---|

repetition number in this series | 0 | memorised on | scheduled repetition | ||||

scheduled repetition interval | last repetition or drill |

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.