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TimeSeriesForecast
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TimeSeriesForecast

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TimeSeriesForecast[tproc,data,k]

gives the k-step-ahead forecast beyond data according to the time series process tproc.

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gives the k-step-ahead forecast for TimeSeriesModel tsmod.

Details and Options

  • TimeSeriesForecast[tproc,{x0,…,xm},k] will give Expectation[x[m+k]x[0]x0∧…∧x[m]xm], where xtproc, the expected value of the process given data.
  • TimeSeriesForecast allows tproc to be a time series process such as ARProcess, ARMAProcess, SARIMAProcess, etc.
  • The data can be a list of numeric values {x1,x2,…}, a list of time-value pairs {{t1,x1},{t2,x2},…}, or TemporalData.
  • The following forecast specifications can be given:
  • kat the k^(th) step ahead
    {kmax}at 1, …, kmax steps ahead
    {kmin,kmax}at kmin, …, kmax steps ahead
    {{k1,k2,…}}use explicit {k1,k2,…} steps ahead
  • TimeSeriesForecast returns the forecasted value if k is an integer and TemporalData otherwise.
  • The default for k is 1.
  • TimeSeriesForecast supports a Method option with the following settings:
  • Automaticautomatically determine the method
    "AR"approximate with a large-order AR process
    "Covariance"exact covariance function-based
    "Kalman"use Kalman filter
  • The mean squared errors of the prediction are the compounded noise errors and are given as MetaInformation in the TemporalData output. For forecast=TimeSeriesForecast[tproc,data,k], the mean squared errors can be accessed by forecast["MeanSquaredErrors"].

Examples

open allclose all

Basic Examples  (3)Summary of the most common use cases

Forecast three steps ahead for an ARProcess:

In[1]:=1
✖
https://wolfram.com/xid/0jz9y9fp6f2gie-oyzkai
In[2]:=2
✖
https://wolfram.com/xid/0jz9y9fp6f2gie-fc5vyq
Out[2]=2

An ARMAProcess:

In[3]:=3
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https://wolfram.com/xid/0jz9y9fp6f2gie-dodo4
Out[3]=3

Predict the seventh value from TimeSeriesModel:

In[1]:=1
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https://wolfram.com/xid/0jz9y9fp6f2gie-kpk3yi
In[2]:=2
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https://wolfram.com/xid/0jz9y9fp6f2gie-fckpgv
Out[2]=2

Mean squared error of the forecast:

In[3]:=3
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https://wolfram.com/xid/0jz9y9fp6f2gie-cnhyyb
Out[3]=3

Forecast a vector-valued time series process:

In[1]:=1
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https://wolfram.com/xid/0jz9y9fp6f2gie-z2srib

Find the forecast for the next 10 steps:

In[2]:=2
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https://wolfram.com/xid/0jz9y9fp6f2gie-uxaf8m
Out[2]=2

Plot the data and the forecast for each component:

In[3]:=3
✖
https://wolfram.com/xid/0jz9y9fp6f2gie-f5ibl8
Out[3]=3

Scope  (7)Survey of the scope of standard use cases

Step  (4)

Forecast one step ahead:

In[1]:=1
✖
https://wolfram.com/xid/0jz9y9fp6f2gie-sf1dzs
In[3]:=3
✖
https://wolfram.com/xid/0jz9y9fp6f2gie-1r6zyl
Out[3]=3

Forecast the third step ahead:

In[4]:=4
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https://wolfram.com/xid/0jz9y9fp6f2gie-fjbrvl
Out[4]=4

Find forecast for all the steps ahead up to the fifth:

In[1]:=1
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https://wolfram.com/xid/0jz9y9fp6f2gie-pyg5ri
In[2]:=2
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https://wolfram.com/xid/0jz9y9fp6f2gie-qh00x6
Out[2]=2

Show the path:

In[3]:=3
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https://wolfram.com/xid/0jz9y9fp6f2gie-y1m35l
Out[3]=3

Forecast all the steps ahead in the range from third to fifth:

In[1]:=1
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https://wolfram.com/xid/0jz9y9fp6f2gie-jbz2ft
In[2]:=2
✖
https://wolfram.com/xid/0jz9y9fp6f2gie-wubp48
Out[2]=2

Show the path:

In[3]:=3
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https://wolfram.com/xid/0jz9y9fp6f2gie-1s4z8k
Out[3]=3

Forecast for a list of steps:

In[1]:=1
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https://wolfram.com/xid/0jz9y9fp6f2gie-wvlg5a
In[2]:=2
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https://wolfram.com/xid/0jz9y9fp6f2gie-hr5d7t
Out[2]=2

Path:

In[3]:=3
✖
https://wolfram.com/xid/0jz9y9fp6f2gie-gxwpmw
Out[3]=3

Mean Squared Errors  (3)

Forecast one step ahead:

In[1]:=1
✖
https://wolfram.com/xid/0jz9y9fp6f2gie-qdv7mz
In[3]:=3
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https://wolfram.com/xid/0jz9y9fp6f2gie-ezbgro
Out[3]=3

Return the forecast as TemporalData to extract mean squared errors:

In[4]:=4
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https://wolfram.com/xid/0jz9y9fp6f2gie-pm7akx
Out[4]=4
In[5]:=5
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https://wolfram.com/xid/0jz9y9fp6f2gie-vy2sta
Out[5]=5
In[6]:=6
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https://wolfram.com/xid/0jz9y9fp6f2gie-bd3ye0
Out[6]=6

Find the forecast with mean squared errors:

In[1]:=1
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https://wolfram.com/xid/0jz9y9fp6f2gie-ecot7l

Calculate the forecast:

In[2]:=2
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https://wolfram.com/xid/0jz9y9fp6f2gie-s0dtlb

Standard errors:

In[3]:=3
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https://wolfram.com/xid/0jz9y9fp6f2gie-h3pqyi

Define the error bands:

In[4]:=4
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https://wolfram.com/xid/0jz9y9fp6f2gie-cu49hu
In[5]:=5
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https://wolfram.com/xid/0jz9y9fp6f2gie-0hg2yf

Plot the data and forecast with mean error bands:

In[6]:=6
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https://wolfram.com/xid/0jz9y9fp6f2gie-i2hoip
Out[6]=6

Find the forecast with 95% confidence intervals:

In[1]:=1
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https://wolfram.com/xid/0jz9y9fp6f2gie-ki9f6k

Find the forecast for the next 10 steps:

In[2]:=2
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https://wolfram.com/xid/0jz9y9fp6f2gie-x57kie

Find mean squared errors and confidence intervals:

In[3]:=3
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https://wolfram.com/xid/0jz9y9fp6f2gie-pdviaz
In[4]:=4
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https://wolfram.com/xid/0jz9y9fp6f2gie-f5fojr
In[5]:=5
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https://wolfram.com/xid/0jz9y9fp6f2gie-wkr3wc

Plot data, forecast, and the forecast limits:

In[6]:=6
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https://wolfram.com/xid/0jz9y9fp6f2gie-4adu48
Out[6]=6

Options  (4)Common values & functionality for each option

Method  (4)

Find the forecast using the covariance-based method:

In[1]:=1
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https://wolfram.com/xid/0jz9y9fp6f2gie-zc5umt
Out[3]=3
In[4]:=4
✖
https://wolfram.com/xid/0jz9y9fp6f2gie-mdf3gq
Out[4]=4

Find the forecast using the autoregressive method:

In[1]:=1
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https://wolfram.com/xid/0jz9y9fp6f2gie-fyqj6x
Out[1]=1
In[2]:=2
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https://wolfram.com/xid/0jz9y9fp6f2gie-z5oan8
Out[2]=2

Find the forecast using the Kalman filter method:

In[1]:=1
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https://wolfram.com/xid/0jz9y9fp6f2gie-t8k8zl
Out[1]=1
In[2]:=2
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https://wolfram.com/xid/0jz9y9fp6f2gie-06t7na
Out[2]=2

Compare exact and approximate methods for an MAProcess:

In[1]:=1
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https://wolfram.com/xid/0jz9y9fp6f2gie-52s586
In[2]:=2
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https://wolfram.com/xid/0jz9y9fp6f2gie-2d20n7
Out[2]=2
In[3]:=3
✖
https://wolfram.com/xid/0jz9y9fp6f2gie-oat5jb
Out[3]=3

Both methods agree for the autoregressive processes:

In[4]:=4
✖
https://wolfram.com/xid/0jz9y9fp6f2gie-5kgd4d
In[5]:=5
✖
https://wolfram.com/xid/0jz9y9fp6f2gie-w8vvbn
Out[5]=5
In[6]:=6
✖
https://wolfram.com/xid/0jz9y9fp6f2gie-rgulqm
Out[6]=6

Applications  (3)Sample problems that can be solved with this function

The daily exchange rates of the euro to the dollar from May 2012 through September 2012:

In[1]:=1
✖
https://wolfram.com/xid/0jz9y9fp6f2gie-iqliwg
In[2]:=2
✖
https://wolfram.com/xid/0jz9y9fp6f2gie-ksytjw
Out[2]=2
In[3]:=3
✖
https://wolfram.com/xid/0jz9y9fp6f2gie-jm5uvi
Out[3]=3

Fit an AR process to the exchange rates:

In[4]:=4
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https://wolfram.com/xid/0jz9y9fp6f2gie-1lkzx3
Out[4]=4

Forecast for 20 business days ahead:

In[5]:=5
✖
https://wolfram.com/xid/0jz9y9fp6f2gie-3da2v4
Out[5]=5

Plot the forecast with original data:

In[6]:=6
✖
https://wolfram.com/xid/0jz9y9fp6f2gie-x82zye
Out[6]=6

Consider hourly temperature readings for September 9, 2012, near your location:

In[1]:=1
✖
https://wolfram.com/xid/0jz9y9fp6f2gie-2wl7zr
Out[1]=1

The data contains missing values:

In[2]:=2
✖
https://wolfram.com/xid/0jz9y9fp6f2gie-3nxtyb
Out[2]=2

Redefine time series with MissingDataMethod to fill in missing data with interpolated values:

In[3]:=3
✖
https://wolfram.com/xid/0jz9y9fp6f2gie-plysjd
Out[3]=3

Check if the time stamps are regularly spaced:

In[4]:=4
✖
https://wolfram.com/xid/0jz9y9fp6f2gie-i472ls
Out[4]=4

Resample by an hour:

In[5]:=5
✖
https://wolfram.com/xid/0jz9y9fp6f2gie-umr0td
Out[5]=5
In[6]:=6
✖
https://wolfram.com/xid/0jz9y9fp6f2gie-3nubr8
Out[6]=6

Estimate an ARProcess:

In[7]:=7
✖
https://wolfram.com/xid/0jz9y9fp6f2gie-xyaiaj
Out[7]=7

Calculate prediction for the next 12 hours:

In[8]:=8
✖
https://wolfram.com/xid/0jz9y9fp6f2gie-p8j6el
Out[8]=8

Plot forecast with original data:

In[9]:=9
✖
https://wolfram.com/xid/0jz9y9fp6f2gie-vga9xk
Out[9]=9

Retail monthly sales in United States:

In[1]:=1
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https://wolfram.com/xid/0jz9y9fp6f2gie-qjusom
In[2]:=2
✖
https://wolfram.com/xid/0jz9y9fp6f2gie-uczmaw
Out[2]=2

Create TimeSeries from the selection:

In[3]:=3
✖
https://wolfram.com/xid/0jz9y9fp6f2gie-oen04n
Out[3]=3

Plot the sales with grid lines at December peaks:

In[4]:=4
✖
https://wolfram.com/xid/0jz9y9fp6f2gie-n5gcnj
Out[4]=4

Fit a seasonal model:

In[5]:=5
✖
https://wolfram.com/xid/0jz9y9fp6f2gie-lfflei
Out[5]=5

The process parameters:

In[6]:=6
✖
https://wolfram.com/xid/0jz9y9fp6f2gie-y101ce
Out[6]=6

Find forecast for the next 7 years:

In[7]:=7
✖
https://wolfram.com/xid/0jz9y9fp6f2gie-9o4zwf
Out[7]=7

Calculate 95% confidence bands for the forecast:

In[8]:=8
✖
https://wolfram.com/xid/0jz9y9fp6f2gie-q4uytd
Out[8]=8

There is an upper and a lower band:

In[9]:=9
✖
https://wolfram.com/xid/0jz9y9fp6f2gie-jy9kj7
Out[9]=9

Plot the forecast within the 95% confidence region:

In[10]:=10
✖
https://wolfram.com/xid/0jz9y9fp6f2gie-66trwe
In[11]:=11
✖
https://wolfram.com/xid/0jz9y9fp6f2gie-mrioe4
Out[11]=11

Properties & Relations  (3)Properties of the function, and connections to other functions

Forecasting with ARProcess using the exact or approximate method gives the same result:

In[1]:=1
✖
https://wolfram.com/xid/0jz9y9fp6f2gie-hiusje
Out[1]=1
In[2]:=2
✖
https://wolfram.com/xid/0jz9y9fp6f2gie-lwbd3n
Out[2]=2
In[3]:=3
✖
https://wolfram.com/xid/0jz9y9fp6f2gie-66q78i
Out[3]=3

Forecast is the same for a time series process and its invertible representation:

In[1]:=1
✖
https://wolfram.com/xid/0jz9y9fp6f2gie-y93h5i

This process is not invertible:

In[2]:=2
✖
https://wolfram.com/xid/0jz9y9fp6f2gie-e5am82
Out[2]=2

Find its invertible representation:

In[3]:=3
✖
https://wolfram.com/xid/0jz9y9fp6f2gie-ik4d2e
Out[3]=3

Compare forecasts:

In[4]:=4
✖
https://wolfram.com/xid/0jz9y9fp6f2gie-w1v9ni
In[5]:=5
✖
https://wolfram.com/xid/0jz9y9fp6f2gie-1dl7yq
Out[5]=5
In[6]:=6
✖
https://wolfram.com/xid/0jz9y9fp6f2gie-5vjd1v
Out[6]=6
In[7]:=7
✖
https://wolfram.com/xid/0jz9y9fp6f2gie-zxt6qz
Out[7]=7
In[8]:=8
✖
https://wolfram.com/xid/0jz9y9fp6f2gie-ox1f7m
Out[8]=8

Use TimeSeriesModel to forecast:

In[1]:=1
✖
https://wolfram.com/xid/0jz9y9fp6f2gie-7pdlz2
In[2]:=2
✖
https://wolfram.com/xid/0jz9y9fp6f2gie-jgcsub
Out[2]=2

Compute forecast for 20 steps:

In[3]:=3
✖
https://wolfram.com/xid/0jz9y9fp6f2gie-elat0r
Out[3]=3

Use process and data explicitly:

In[4]:=4
✖
https://wolfram.com/xid/0jz9y9fp6f2gie-f56h3h
Out[4]=4

Compare paths:

In[5]:=5
✖
https://wolfram.com/xid/0jz9y9fp6f2gie-6xjb4r
Out[5]=5

Possible Issues  (2)Common pitfalls and unexpected behavior

"Kalman" method requires the parameters of the process to be numeric:

In[1]:=1
✖
https://wolfram.com/xid/0jz9y9fp6f2gie-ysiqus
Out[1]=1

If the invertible representation does not exist, the forecast may not be reliable:

In[1]:=1
✖
https://wolfram.com/xid/0jz9y9fp6f2gie-18s83k
Out[1]=1
In[2]:=2
✖
https://wolfram.com/xid/0jz9y9fp6f2gie-qozxo
Out[2]=2
Wolfram Research (2012), TimeSeriesForecast, Wolfram Language function, https://reference.wolfram.com/language/ref/TimeSeriesForecast.html (updated 2014).
✖
Wolfram Research (2012), TimeSeriesForecast, Wolfram Language function, https://reference.wolfram.com/language/ref/TimeSeriesForecast.html (updated 2014).

Text

Wolfram Research (2012), TimeSeriesForecast, Wolfram Language function, https://reference.wolfram.com/language/ref/TimeSeriesForecast.html (updated 2014).

✖
Wolfram Research (2012), TimeSeriesForecast, Wolfram Language function, https://reference.wolfram.com/language/ref/TimeSeriesForecast.html (updated 2014).

CMS

Wolfram Language. 2012. "TimeSeriesForecast." Wolfram Language & System Documentation Center. Wolfram Research. Last Modified 2014. https://reference.wolfram.com/language/ref/TimeSeriesForecast.html.

✖
Wolfram Language. 2012. "TimeSeriesForecast." Wolfram Language & System Documentation Center. Wolfram Research. Last Modified 2014. https://reference.wolfram.com/language/ref/TimeSeriesForecast.html.

APA

Wolfram Language. (2012). TimeSeriesForecast. Wolfram Language & System Documentation Center. Retrieved from https://reference.wolfram.com/language/ref/TimeSeriesForecast.html

✖
Wolfram Language. (2012). TimeSeriesForecast. Wolfram Language & System Documentation Center. Retrieved from https://reference.wolfram.com/language/ref/TimeSeriesForecast.html

BibTeX

@misc{reference.wolfram_2025_timeseriesforecast, author="Wolfram Research", title="{TimeSeriesForecast}", year="2014", howpublished="\url{https://reference.wolfram.com/language/ref/TimeSeriesForecast.html}", note=[Accessed: 11-May-2025 ]}

✖
@misc{reference.wolfram_2025_timeseriesforecast, author="Wolfram Research", title="{TimeSeriesForecast}", year="2014", howpublished="\url{https://reference.wolfram.com/language/ref/TimeSeriesForecast.html}", note=[Accessed: 11-May-2025 ]}

BibLaTeX

@online{reference.wolfram_2025_timeseriesforecast, organization={Wolfram Research}, title={TimeSeriesForecast}, year={2014}, url={https://reference.wolfram.com/language/ref/TimeSeriesForecast.html}, note=[Accessed: 11-May-2025 ]}

✖
@online{reference.wolfram_2025_timeseriesforecast, organization={Wolfram Research}, title={TimeSeriesForecast}, year={2014}, url={https://reference.wolfram.com/language/ref/TimeSeriesForecast.html}, note=[Accessed: 11-May-2025 ]}