statsmodels.tsa.holtwinters.HoltWintersResults.simulate#
- HoltWintersResults.simulate(nsimulations, anchor=None, repetitions=1, error='add', random_errors=None, *, rng=None)[source]#
Random simulations using the state space formulation
- Parameters:
- nsimulations
int The number of simulation steps.
- anchor
int,str,ordatetime,optional First period for simulation. The simulation will be conditional on all existing datapoints prior to the anchor. Type depends on the index of the given endog in the model. Two special cases are the strings ‘start’ and ‘end’. start refers to beginning the simulation at the first period of the sample, and end refers to beginning the simulation at the first period after the sample. Integer values can run from 0 to nobs, or can be negative to apply negative indexing. Finally, if a date/time index was provided to the model, then this argument can be a date string to parse or a datetime type. Default is ‘end’.
- repetitions
int,optional Number of simulated paths to generate. Default is 1 simulated path.
- error{“add”, “mul”, “additive”, “multiplicative”},
optional Error model for state space formulation. Default is
"add".- random_errors
optional Specifies how the random errors should be obtained. Can be one of the following:
None: Random normally distributed values with variance estimated from the fit errors drawn from numpy’s standard RNG (can be seeded with the rng argument). This is the default option.A distribution function from
scipy.stats, e.g.scipy.stats.norm: Fits the distribution function to the fit errors and draws from the fitted distribution. Note the difference betweenscipy.stats.normandscipy.stats.norm(), the latter one is a frozen distribution function. The methodrvsis called on the distribution function to draw the random errors.A frozen distribution function from
scipy.stats, e.g.scipy.stats.norm(scale=2): Draws from the frozen distribution function. The methodrvsis called on the distribution function to draw the random errors.A
np.ndarraywith shape (nsimulations, repetitions): Uses the given values as random errors."bootstrap": Samples the random errors from the fit errors.
- rng{
None,int,numpy.random.Generator,numpy.random.RandomState},optional If rng is None, a new
Generatoris created using fresh entropy from the operating system. If rng is an int, a newRandomStateinstance is created, seeded with rng; this integer-seeding behavior is deprecated and will change to creating aGeneratorin a future release. If rng is already aGeneratororRandomStateinstance, that instance is used. Only used if random_errors isNoneor"bootstrap".- random_state{
None,int, array_like[int],numpy.random.Generator,numpy.random.RandomState},optional Deprecated since version 0.15: random_state has been deprecated. In-line with SPEC-007, use rng for passing a random number generator or seed.
- nsimulations
- Returns:
- sim
pd.Series,pd.DataFrameornp.ndarray An
np.ndarray,pd.Series, orpd.DataFrameof simulated values. If the original data was apd.Seriesorpd.DataFrame, sim will be apd.Seriesif repetitions is 1, and apd.DataFrameof shape (nsimulations, repetitions) else. Otherwise, if repetitions is 1, anp.ndarrayof shape (nsimulations,) is returned, and if repetitions is not 1 anp.ndarrayof shape (nsimulations, repetitions) is returned.
- sim
Notes
The simulation is based on the state space model of the Holt-Winter’s methods. The state space model assumes that the true value at time \(t\) is randomly distributed around the prediction value. If using the additive error model, this means:
\[\begin{split}y_t &= \hat{y}_{t|t-1} + e_t\\ e_t &\sim \mathcal{N}(0, \sigma^2)\end{split}\]Using the multiplicative error model:
\[\begin{split}y_t &= \hat{y}_{t|t-1} \cdot (1 + e_t)\\ e_t &\sim \mathcal{N}(0, \sigma^2)\end{split}\]Inserting these equations into the smoothing equation formulation leads to the state space equations. The notation used here follows [1].
Additionally,
\[\begin{split}B_t &= b_{t-1} \circ_d \phi\\ L_t &= l_{t-1} \circ_b B_t\\ S_t &= s_{t-m}\\ Y_t &= L_t \circ_s S_t,\end{split}\]where \(\circ_d\) is the operation linking trend and damping parameter (multiplication if the trend is additive, power if the trend is multiplicative), \(\circ_b\) is the operation linking level and trend (addition if the trend is additive, multiplication if the trend is multiplicative), and \(\circ_s\) is the operation linking seasonality to the rest.
The state space equations can then be formulated as
\[\begin{split}y_t &= Y_t + \eta \cdot e_t\\ l_t &= L_t + \alpha \cdot (M_e \cdot L_t + \kappa_l) \cdot e_t\\ b_t &= B_t + \beta \cdot (M_e \cdot B_t + \kappa_b) \cdot e_t\\ s_t &= S_t + \gamma \cdot (M_e \cdot S_t + \kappa_s) \cdot e_t\\\end{split}\]with
\[\begin{split}\eta &= \begin{cases} Y_t\quad\text{if error is multiplicative}\\ 1\quad\text{else} \end{cases}\\ M_e &= \begin{cases} 1\quad\text{if error is multiplicative}\\ 0\quad\text{else} \end{cases}\\\end{split}\]and, when using the additive error model,
\[\begin{split}\kappa_l &= \begin{cases} \frac{1}{S_t}\quad \text{if seasonality is multiplicative}\\ 1\quad\text{else} \end{cases}\\ \kappa_b &= \begin{cases} \frac{\kappa_l}{l_{t-1}}\quad \text{if trend is multiplicative}\\ \kappa_l\quad\text{else} \end{cases}\\ \kappa_s &= \begin{cases} \frac{1}{L_t}\quad\text{if seasonality is multiplicative}\\ 1\quad\text{else} \end{cases}\end{split}\]When using the multiplicative error model
\[\begin{split}\kappa_l &= \begin{cases} 0\quad \text{if seasonality is multiplicative}\\ S_t\quad\text{else} \end{cases}\\ \kappa_b &= \begin{cases} \frac{\kappa_l}{l_{t-1}}\quad \text{if trend is multiplicative}\\ \kappa_l + l_{t-1}\quad\text{else} \end{cases}\\ \kappa_s &= \begin{cases} 0\quad\text{if seasonality is multiplicative}\\ L_t\quad\text{else} \end{cases}\end{split}\]References
[1]Hyndman, R.J., & Athanasopoulos, G. (2018) Forecasting: principles and practice, 2nd edition, OTexts: Melbourne, Australia. OTexts.com/fpp2. Accessed on February 28th 2020.