> ## Documentation Index
> Fetch the complete documentation index at: https://nixtlaverse.nixtla.io/llms.txt
> Use this file to discover all available pages before exploring further.
# AutoTheta Model
> Step-by-step guide on using the `AutoTheta Model` with
> `Statsforecast`.
During this walkthrough, we will become familiar with the main
`StatsForecast` class and some relevant methods such as
`StatsForecast.plot`, `StatsForecast.forecast` and
`StatsForecast.cross_validation` in other.
The text in this article is copied from and inspired by: 1. [Jose A.
Fiorucci, Tiago R. Pellegrini, Francisco Louzada, Fotios Petropoulos,
Anne B. Koehler (2016). “Models for optimising the theta method and
their relationship to state space models”. International Journal of
Forecasting](https://www.sciencedirect.com/science/article/pii/S0169207016300243).
2\. [Rob J. Hyndman and George Athanasopoulos (2018). “Forecasting
Principles and Practice (3rd ed)”](https://otexts.com/fpp3/tscv.html)
## Table of Contents
* [Introduction](#introduction)
* [Loading libraries and data](#loading)
* [Explore data with the plot method](#plotting)
* [Split the data into training and testing](#splitting)
* [Implementation of AutoTheta with StatsForecast](#implementation)
* [Cross-validation](#cross_validate)
* [Model evaluation](#evaluate)
* [References](#references)
## Introduction
The `AutoTheta` model in `StatsForecast` automatically selects the best
**Theta model** based on the **mean squared error (MSE)**. In this
section, we will discuss each of the models that `AutoTheta` considers
and then explain how it selects the best one.
### 1. Standard Theta Model (STM)
The **Standard Theta Model** is the original version of the Theta model
introduced by Assimakopoulos and Nikolopoulos (2000). It decomposes a
time series into two modified versions of the original series, called
**theta lines**. These lines are created by applying a linear
transformation to the second differences of the original series,
controlled by a parameter called **theta $\theta$**. One theta line
captures the long-term trend, while the other captures short-term
fluctuations. The two theta lines are then combined to produce the final
forecast. The STM assumes that model parameters remain constant over
time.
### 2. Optimized Theta Model (OTM)
The **Optimized Theta Model** extends STM by searching for the best
theta parameters rather than using fixed values. This optimization step
allows the model to better fit series with higher variability.
### 3. Dynamic Standard Theta Model (DSTM)
The **Dynamic Standard Theta Model** allows STM to adapt over time.
Instead of keeping parameters static, it updates them dynamically as new
data becomes available. This dynamic behavior can be useful when
forecasting series with evolving trends or seasonality.
### 4. Dynamic Optimized Theta Model (DOTM)
The **Dynamic Optimized Theta Model** combines features of both OTM and
DSTM. Like OTM, it optimizes the theta parameters. Like DSTM, it updates
the model dynamically with new data.
## How AutoTheta Selects the Best Model
1. `AutoTheta` fits all four variants of the Theta model (STM, OTM,
DSTM, and DOTM) to your data.
2. Each model is evaluated using cross-validation or a hold-out
validation strategy, depending on the configuration.
3. The model that achieves the lowest mean squared error (MSE) is
selected.
4. The selected model is then used to generate the forecast.
## Loading libraries and data
> **Tip**
>
> Statsforecast will be needed. To install, see
> [instructions](../getting-started/installation.html).
Next, we import plotting libraries and configure the plotting style.
```python theme={null}
import pandas as pd
import scipy.stats as stats
```
```python theme={null}
import matplotlib.pyplot as plt
import seaborn as sns
from statsmodels.graphics.tsaplots import plot_acf, plot_pacf
plt.style.use('fivethirtyeight')
plt.rcParams['lines.linewidth'] = 1.5
dark_style = {
'figure.facecolor': '#212946',
'axes.facecolor': '#212946',
'savefig.facecolor':'#212946',
'axes.grid': True,
'axes.grid.which': 'both',
'axes.spines.left': False,
'axes.spines.right': False,
'axes.spines.top': False,
'axes.spines.bottom': False,
'grid.color': '#2A3459',
'grid.linewidth': '1',
'text.color': '0.9',
'axes.labelcolor': '0.9',
'xtick.color': '0.9',
'ytick.color': '0.9',
'font.size': 12 }
plt.rcParams.update(dark_style)
from pylab import rcParams
rcParams['figure.figsize'] = (18,7)
```
### Read Data
```python theme={null}
df = pd.read_csv("https://raw.githubusercontent.com/Naren8520/Serie-de-tiempo-con-Machine-Learning/main/Data/candy_production.csv")
df.head()
```
| | observation\_date | IPG3113N |
| - | ----------------- | -------- |
| 0 | 1972-01-01 | 85.6945 |
| 1 | 1972-02-01 | 71.8200 |
| 2 | 1972-03-01 | 66.0229 |
| 3 | 1972-04-01 | 64.5645 |
| 4 | 1972-05-01 | 65.0100 |
The input to StatsForecast is always a data frame in long format with
three columns: unique\_id, ds and y:
* The `unique_id` (string, int or category) represents an identifier
for the series.
* The `ds` (datestamp) column should be of a format expected by
Pandas, ideally YYYY-MM-DD for a date or YYYY-MM-DD HH:MM:SS for a
timestamp.
* The `y` (numeric) represents the measurement we wish to forecast.
```python theme={null}
df["unique_id"]="1"
df.columns=["ds", "y", "unique_id"]
df.head()
```
| | ds | y | unique\_id |
| - | ---------- | ------- | ---------- |
| 0 | 1972-01-01 | 85.6945 | 1 |
| 1 | 1972-02-01 | 71.8200 | 1 |
| 2 | 1972-03-01 | 66.0229 | 1 |
| 3 | 1972-04-01 | 64.5645 | 1 |
| 4 | 1972-05-01 | 65.0100 | 1 |
```python theme={null}
print(df.dtypes)
```
```text theme={null}
ds object
y float64
unique_id object
dtype: object
```
We can see that our time variable `(ds)` is in an object format, we need
to convert to a date format
```python theme={null}
df["ds"] = pd.to_datetime(df["ds"])
```
## Explore Data with the plot method
Plot some series using the plot method from the StatsForecast class.
This method prints aa random series from the dataset and is useful for
basic EDA.
```python theme={null}
from statsforecast import StatsForecast
StatsForecast.plot(df)
```
### Autocorrelation plots
```python theme={null}
fig, axs = plt.subplots(nrows=1, ncols=2)
plot_acf(df["y"], lags=60, ax=axs[0],color="fuchsia")
axs[0].set_title("Autocorrelation");
plot_pacf(df["y"], lags=60, ax=axs[1],color="lime")
axs[1].set_title('Partial Autocorrelation')
plt.show();
```
## Split the data into training and testing
Let’s divide our data into sets 1. Data to train our `AutoTheta` model
2\. Data to test our model
For the test data we will use the last 12 months to test and evaluate
the performance of our model.
```python theme={null}
train = df[df.ds<='2016-08-01']
test = df[df.ds>'2016-08-01']
```
```python theme={null}
train.shape, test.shape
```
```text theme={null}
((536, 3), (12, 3))
```
Now let’s plot the training data and the test data.
```python theme={null}
sns.lineplot(train,x="ds", y="y", label="Train", linewidth=3, linestyle=":")
sns.lineplot(test, x="ds", y="y", label="Test")
plt.ylabel("Candy Production")
plt.xlabel("Month")
plt.show()
```
## Implementation of AutoTheta with StatsForecast
### Load libraries
```python theme={null}
from statsforecast import StatsForecast
from statsforecast.models import AutoTheta
```
### Instantiate Model
Import and instantiate the models. Setting the argument is sometimes
tricky. This article on [Seasonal
periods](https://robjhyndman.com/hyndsight/seasonal-periods/) by the
master, Rob Hyndmann, can be useful.season\_length.
Automatically selects the best Theta (Standard Theta Model `(‘STM’)`,
Optimized Theta Model `(‘OTM’)`, Dynamic Standard Theta Model
`(‘DSTM’)`, Dynamic Optimized Theta Model `(‘DOTM’))` model using mse.
```python theme={null}
season_length = 12 # Monthly data
horizon = len(test) # number of predictions
# We call the model that we are going to use
models = [AutoTheta(season_length=season_length,
decomposition_type="additive",
model="STM")]
```
We fit the models by instantiating a new StatsForecast object with the
following parameters:
models: a list of models. Select the models you want from models and
import them.
* `freq:` a string indicating the frequency of the data. (See [panda’s
available
frequencies](https://pandas.pydata.org/pandas-docs/stable/user_guide/timeseries.html#offset-aliases).)
* `n_jobs:` n\_jobs: int, number of jobs used in the parallel
processing, use -1 for all cores.
* `fallback_model:` a model to be used if a model fails.
Any settings are passed into the constructor. Then you call its fit
method and pass in the historical data frame.
```python theme={null}
sf = StatsForecast(models=models, freq='MS')
```
### Fit Model
```python theme={null}
sf.fit(df=train)
```
```text theme={null}
StatsForecast(models=[AutoTheta])
```
Let’s see the results of our Theta model. We can observe it with the
following instruction:
```python theme={null}
result=sf.fitted_[0,0].model_
result
```
```text theme={null}
{'mse': 100.57831864069415,
'amse': array([26.13585578, 38.60211513, 44.70605915]),
'fit': results(x=array([258.45064973, 0.7664297 ]), fn=100.57831864069415, nit=32, simplex=array([[250.37338496, 0.76970741],
[232.03915522, 0.76429422],
[258.45064973, 0.7664297 ]])),
'residuals': array([-2.14815337e+02, -6.20562800e+01, -2.13256707e+01, -1.25845480e+01,
-1.19719350e+01, -9.40876632e+00, -8.60141525e+00, -9.00054652e+00,
-1.98778836e+00, 3.14564857e+01, 1.98519673e+01, 2.04962370e+01,
4.98120196e+00, -1.08735375e+01, -1.12328024e+01, -8.08115377e+00,
-9.98197589e+00, -8.39937098e+00, -1.25789505e+01, -1.05952806e+01,
8.47229127e-01, 2.25644616e+01, 2.54401546e+01, 1.73989716e+01,
2.40287275e+00, -2.53475866e+00, -8.00591135e+00, -1.79241479e+01,
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-3.51755220e+00, -1.31441941e+01, -6.13166031e+00, 1.51512150e+00,
-8.05777104e+00, -8.59603388e+00, -1.08617851e+01, -6.72940177e+00,
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-5.16540394e+00, -1.27924628e+01, -9.68434625e+00, -8.76758703e+00,
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2.09363525e+00, 3.13825850e+01, 2.43980063e+01, 1.89899567e+01,
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-1.23296368e+01, -7.43701484e+00, -9.59827267e+00, -6.98198280e+00,
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-3.95418211e+00, 2.43162216e+00, -3.09611231e+00, -1.49779647e+01,
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-4.45625959e+00, -1.37976278e+01, -1.12070229e+01, -1.28293907e+00,
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-1.76399695e+01, -1.50423508e+01, -1.13765532e+01, -9.17973632e+00,
-4.22254178e+00, 2.19090137e+01, 1.90170614e+01, 1.80606278e+01,
4.08981599e+00, 2.02346117e+00, -5.45474659e+00, -1.38725716e+01,
-1.50622791e+01, -1.15367789e+01, -7.55445577e+00, -1.77510788e+00,
9.46335947e+00, 4.88813367e+00, 1.61490895e+01, 1.93212548e+01,
1.03075610e+01, -6.46758291e-01, -5.79530543e-01, -1.35917659e+01,
-1.62148912e+01, -1.29823949e+01, -1.02149087e+01, -3.24211066e+00,
3.05411201e-01, 1.19385090e+01, 2.08979477e+01, 2.19927470e+01,
1.32364223e+00, 1.68626515e+00, -3.52030557e+00, -1.50337436e+01,
-1.75865944e+01, -1.23980840e+01, -1.19670311e+01, -1.59575440e+00,
4.32015112e+00, 1.39461330e+01, 2.63901690e+01, 2.11431667e+01,
1.19960552e+00, 1.22769386e+00, -3.12851420e+00, -1.23388328e+01,
-1.66429432e+01, -9.08277509e+00, -7.92637338e+00, 2.43702321e+00,
-3.53211182e+00, 1.00606776e+01, 1.39608421e+01, 1.44689452e+01,
6.50770562e+00, 3.13940836e+00, -4.89894478e-01, -1.05833296e+01,
-1.34863098e+01, -1.20763793e+01, -1.00738904e+01, -9.39207297e+00]),
'm': 12,
'states': array([[1.24021769e+02, 8.30544193e+01, 8.40569047e+01, 6.14692129e-02,
3.00509837e+02],
[8.50161150e+01, 7.80917592e+01, 8.40569047e+01, 6.14692129e-02,
1.33876280e+02],
[7.65735210e+01, 7.67280499e+01, 8.40569047e+01, 6.14692129e-02,
8.73485707e+01],
...,
[1.12984989e+02, 1.00517846e+02, 8.40569047e+01, 6.14692129e-02,
1.14480779e+02],
[1.14049672e+02, 1.00543746e+02, 8.40569047e+01, 6.14692129e-02,
1.13025090e+02],
[1.10946036e+02, 1.00561388e+02, 8.40569047e+01, 6.14692129e-02,
1.14089773e+02]]),
'par': {'initial_smoothed': 258.45064973324986,
'alpha': 0.7664297044277045,
'theta': 2.0},
'n': 536,
'modeltype': 'STM',
'mean_y': 100.56138830499272,
'decompose': True,
'decomposition_type': 'additive',
'seas_forecast': {'mean': array([ 0.08977811, 18.09442035, 20.24848682, 19.4306462 ,
2.64008067, -1.30909907, -7.97773123, -12.32640613,
-12.02777406, -10.1369666 , -11.42293515, -5.30249992])},
'fitted': array([300.50983667, 133.87628004, 87.34857069, 77.149048 ,
76.98193501, 77.05546632, 77.64431525, 79.83754652,
77.03398836, 75.47241431, 85.74423271, 85.47106297,
86.31849804, 88.14353755, 80.8438024 , 78.37975377,
81.66417589, 83.26287098, 84.62535048, 83.7700806 ,
79.74427087, 80.35553844, 83.81224541, 87.82202841,
86.29562725, 86.14455866, 85.23591135, 85.24504787,
80.98550693, 85.35566468, 88.73347592, 80.13900931,
77.37219166, 71.81797618, 78.98325428, 80.46128324,
70.5292522 , 65.84059407, 56.80056031, 58.2461785 ,
68.88547104, 71.95893388, 73.17068509, 73.63150177,
72.56861641, 67.74141718, 75.823697 , 83.17867619,
82.88182158, 84.77950058, 78.46220336, 71.99792744,
75.71080394, 81.00106285, 78.99654625, 80.35978703,
77.73477595, 77.0624998 , 86.86366484, 90.37877922,
93.59485284, 94.48135717, 92.0871403 , 86.88905458,
88.05659723, 89.54567652, 88.73256441, 87.66000103,
84.49066084, 84.28553517, 89.56282177, 86.79937702,
92.06289264, 88.57657843, 83.39583679, 84.22817596,
88.48908916, 88.70896704, 88.43688493, 84.98139923,
82.12001187, 82.55098375, 85.9525497 , 90.19133861,
93.64043798, 95.47816961, 88.30724672, 88.90190843,
89.38115594, 90.19718831, 91.02162087, 87.36982498,
83.60211476, 81.42239492, 82.66423005, 85.61780804,
86.08812516, 84.73820883, 85.54103724, 83.21124039,
79.16163028, 84.73307206, 86.60047462, 83.45823664,
82.8036783 , 79.0463675 , 81.90332096, 85.42827927,
87.13594255, 92.20371202, 91.92571881, 87.47001337,
89.71173884, 94.08746707, 93.42067364, 91.36829945,
88.10499619, 84.3807048 , 96.69192329, 96.73805538,
94.53625569, 93.65491108, 94.17929385, 91.81488763,
87.30208784, 88.22493606, 88.60917145, 87.97178146,
84.24395894, 81.95085029, 92.78029537, 94.275001 ,
95.43756282, 93.25335889, 89.64588248, 86.84354705,
86.27398255, 87.31900372, 85.50115807, 87.26078723,
85.45187034, 83.45436489, 90.30572204, 87.23685484,
85.22183448, 89.83718668, 88.17711021, 87.21418481,
87.84436043, 89.45885582, 88.22276703, 88.33640489,
86.986106 , 86.10365179, 92.24300578, 94.58859369,
93.69697622, 94.76073782, 89.93749012, 87.8093004 ,
88.3949345 , 89.16392452, 86.74878426, 86.67946215,
85.59093176, 88.57095695, 92.89938688, 93.34684947,
96.69921709, 95.24315643, 95.05395839, 88.76162712,
86.66136449, 88.14065857, 87.23099008, 85.41872572,
85.01380191, 87.60818255, 95.45557821, 98.3240456 ,
96.5726075 , 95.04052437, 95.49116642, 92.53977272,
90.36888696, 90.16393454, 89.5384766 , 88.04727997,
88.67676475, 90.24331499, 100.45849371, 103.6695433 ,
103.36252038, 95.57789385, 96.3997412 , 97.54332409,
94.2091886 , 94.45290287, 96.36634424, 100.85347739,
102.80919411, 102.5472505 , 106.48312008, 104.03628873,
103.29067992, 101.03748421, 103.07742567, 101.82224878,
101.27230355, 100.28657648, 100.63629155, 101.06606288,
101.71464486, 101.14884962, 101.78769257, 102.7950277 ,
105.71545223, 99.33413407, 101.80714675, 102.61832483,
101.21735442, 100.85561542, 102.45166863, 107.05014533,
107.51717356, 108.33874738, 106.85496498, 111.55980808,
114.23636971, 107.06776156, 111.42831624, 112.50186659,
109.36957611, 107.54585583, 111.62426724, 111.62202077,
114.31102418, 111.83959398, 107.39758714, 108.70651652,
106.30953697, 102.78440744, 103.82046147, 103.14437142,
104.11684481, 102.33982409, 102.8723728 , 103.47360882,
102.01677319, 103.62723339, 101.56281457, 101.87268181,
102.03905301, 102.36943582, 103.33817557, 102.95055886,
102.19972468, 100.90281095, 104.03647886, 106.44257555,
108.22178492, 108.49688565, 107.17885267, 105.19959511,
103.80611838, 102.98366417, 102.30386854, 105.52016297,
102.58638056, 99.89919716, 103.02022955, 106.77390074,
107.96263336, 109.29927875, 106.01489901, 103.6864972 ,
105.1475863 , 105.11603549, 101.63327513, 103.61001775,
105.92623683, 105.72561484, 107.82567267, 109.2548828 ,
107.99841184, 107.35109379, 103.5233805 , 103.62365533,
108.13938211, 103.11607784, 106.01381231, 109.78596466,
107.78446605, 108.81079898, 110.17164752, 109.84536969,
112.60070842, 114.23275493, 109.59549173, 112.69372438,
115.81058738, 109.85108519, 110.73448348, 113.67239919,
111.26167729, 109.87022246, 111.31252448, 110.13430105,
114.10103454, 114.77969912, 112.22984999, 114.31642742,
116.09441204, 116.92384863, 118.16082896, 118.67694524,
118.56524723, 119.03853891, 120.55739465, 120.49404273,
122.4297822 , 121.24085099, 116.16013458, 116.63300608,
116.58468172, 115.20069256, 115.84357956, 115.28811168,
117.8563375 , 119.42647419, 118.72610568, 119.14774202,
118.15012563, 116.95870856, 114.38003444, 112.21454843,
114.48341518, 119.11962906, 120.63092958, 121.65618463,
126.75939743, 122.1827986 , 123.8699932 , 123.46128916,
123.29574937, 125.06196634, 120.23216725, 116.54759242,
118.66743152, 119.67090937, 122.40414386, 125.63126387,
126.72874773, 125.40591101, 125.48596965, 125.07720702,
125.03320929, 123.8308448 , 111.2956079 , 109.17137615,
110.01274633, 113.80892938, 115.40216099, 117.64202977,
117.19533719, 114.68331622, 120.59245198, 121.56787494,
122.56854548, 120.16921999, 109.29738449, 108.58309477,
105.67469335, 109.31954714, 111.77844749, 117.49308896,
117.5137965 , 119.17248724, 121.21684679, 115.92686418,
117.89155825, 117.02722838, 109.44298667, 111.84906053,
109.99544591, 112.7496681 , 117.55482364, 113.24182371,
114.25985959, 120.09362779, 117.31872292, 117.51493907,
120.62638451, 120.66695807, 115.74879597, 113.59360961,
111.30546837, 119.47810143, 123.92117003, 117.29474313,
118.73046831, 122.40358785, 118.54651485, 120.94522623,
120.34509238, 119.83191444, 119.28258839, 116.90606293,
119.67953588, 116.61541466, 118.57002916, 116.93539025,
119.24189675, 115.26824869, 112.85500598, 113.09982676,
116.36816979, 118.09076126, 113.10484433, 110.84983175,
112.21846295, 117.52051435, 117.77135585, 119.97014793,
113.71329113, 110.97321598, 106.47875848, 103.69775119,
105.5329286 , 109.03535469, 100.5185778 , 98.7783504 ,
100.72723124, 104.88073539, 103.53983956, 104.83050157,
104.37041809, 101.78207536, 99.79195805, 97.33147199,
94.7051695 , 101.23419515, 97.22698298, 96.03053349,
85.56579529, 86.89526346, 89.52989363, 92.63258395,
92.20654345, 92.29302095, 90.33797118, 92.97269973,
94.06049433, 99.45748428, 104.17945492, 98.57230063,
97.48447184, 97.71075906, 99.74481829, 99.92754582,
101.40703208, 101.89152524, 100.19790135, 106.12158657,
110.71243486, 114.61516239, 109.71600444, 100.36181401,
99.59503236, 100.26066735, 103.05302578, 106.07904849,
109.00286948, 104.73225081, 101.00335324, 101.06963632,
98.12874178, 94.85438633, 97.80873857, 96.89567218,
95.87638401, 97.01823883, 99.60314659, 101.56757157,
100.41327905, 98.11827889, 97.07615577, 100.07180788,
102.80604053, 110.02096633, 99.93001054, 96.81884524,
96.765739 , 102.67305829, 103.21143054, 108.91236591,
107.9732912 , 104.79489485, 102.64480872, 103.60141066,
105.2112888 , 105.40729101, 100.7199523 , 101.24845301,
103.24285777, 102.26463485, 104.59110557, 108.03814361,
105.99389435, 101.76418396, 100.06193105, 99.6756544 ,
102.54734888, 105.82036702, 102.67173097, 107.40963325,
108.75289448, 107.67960614, 109.6546142 , 113.40193278,
113.42314323, 109.91667509, 110.75537338, 113.46597679,
119.42851182, 116.6833224 , 110.55675793, 105.76845483,
101.99639438, 104.99139164, 108.43159448, 114.20122959,
115.56790983, 114.4807793 , 113.0250904 , 114.08977297])}
```
Let us now visualize the residuals of our models.
As we can see, the result obtained above has an output in a dictionary,
to extract each element from the dictionary we are going to use the
`.get()` function to extract the element and then we are going to save
it in a `pd.DataFrame()`.
```python theme={null}
residual=pd.DataFrame(result.get("residuals"), columns=["residual Model"])
residual
```
| | residual Model |
| --- | -------------- |
| 0 | -214.815337 |
| 1 | -62.056280 |
| 2 | -21.325671 |
| ... | ... |
| 533 | -12.076379 |
| 534 | -10.073890 |
| 535 | -9.392073 |
```python theme={null}
fig, axs = plt.subplots(nrows=2, ncols=2)
residual.plot(ax=axs[0,0])
axs[0,0].set_title("Residuals");
sns.distplot(residual, ax=axs[0,1]);
axs[0,1].set_title("Density plot - Residual");
stats.probplot(residual["residual Model"], dist="norm", plot=axs[1,0])
axs[1,0].set_title('Plot Q-Q')
plot_acf(residual, lags=35, ax=axs[1,1],color="fuchsia")
axs[1,1].set_title("Autocorrelation");
plt.show();
```
### Forecast Method
If you want to gain speed in productive settings where you have multiple
series or models we recommend using the `StatsForecast.forecast` method
instead of `.fit` and `.predict`.
The main difference is that the `.forecast` doest not store the fitted
values and is highly scalable in distributed environments.
The forecast method takes two arguments: forecasts next `h` (horizon)
and `level`.
* `h (int):` represents the forecast h steps into the future. In this
case, 12 months ahead.
* `level (list of floats):` this optional parameter is used for
probabilistic forecasting. Set the level (or confidence percentile)
of your prediction interval. For example, `level=[90]` means that
the model expects the real value to be inside that interval 90% of
the times.
The forecast object here is a new data frame that includes a column with
the name of the model and the y hat values, as well as columns for the
uncertainty intervals. Depending on your computer, this step should take
around 1min. (If you want to speed things up to a couple of seconds,
remove the AutoModels like `ARIMA` and `Theta`)
```python theme={null}
# Prediction
Y_hat = sf.forecast(df=train, h=horizon, fitted=True)
Y_hat
```
| | unique\_id | ds | AutoTheta |
| --- | ---------- | ---------- | ---------- |
| 0 | 1 | 2016-09-01 | 111.075915 |
| 1 | 1 | 2016-10-01 | 129.111292 |
| 2 | 1 | 2016-11-01 | 131.296093 |
| ... | ... | ... | ... |
| 9 | 1 | 2017-06-01 | 101.125782 |
| 10 | 1 | 2017-07-01 | 99.870548 |
| 11 | 1 | 2017-08-01 | 106.021718 |
```python theme={null}
values=sf.forecast_fitted_values()
values.head()
```
| | unique\_id | ds | y | AutoTheta |
| - | ---------- | ---------- | ------- | ---------- |
| 0 | 1 | 1972-01-01 | 85.6945 | 300.509837 |
| 1 | 1 | 1972-02-01 | 71.8200 | 133.876280 |
| 2 | 1 | 1972-03-01 | 66.0229 | 87.348571 |
| 3 | 1 | 1972-04-01 | 64.5645 | 77.149048 |
| 4 | 1 | 1972-05-01 | 65.0100 | 76.981935 |
```python theme={null}
StatsForecast.plot(values)
```
Adding 95% confidence interval with the forecast method
```python theme={null}
sf.forecast(df=train, h=horizon, level=[95])
```
| | unique\_id | ds | AutoTheta | AutoTheta-lo-95 | AutoTheta-hi-95 |
| --- | ---------- | ---------- | ---------- | --------------- | --------------- |
| 0 | 1 | 2016-09-01 | 111.075915 | 90.139234 | 136.011109 |
| 1 | 1 | 2016-10-01 | 129.111292 | 94.795409 | 160.387128 |
| 2 | 1 | 2016-11-01 | 131.296093 | 90.579813 | 168.268538 |
| ... | ... | ... | ... | ... | ... |
| 9 | 1 | 2017-06-01 | 101.125782 | 41.186268 | 159.159903 |
| 10 | 1 | 2017-07-01 | 99.870548 | 35.144354 | 152.867267 |
| 11 | 1 | 2017-08-01 | 106.021718 | 38.753454 | 166.048584 |
```python theme={null}
# Merge the forecasts with the true values
Y_hat1 = test.merge(Y_hat, how='left', on=['unique_id', 'ds'])
Y_hat1
```
| | ds | y | unique\_id | AutoTheta |
| --- | ---------- | -------- | ---------- | ---------- |
| 0 | 2016-09-01 | 109.3191 | 1 | 111.075915 |
| 1 | 2016-10-01 | 119.0502 | 1 | 129.111292 |
| 2 | 2016-11-01 | 116.8431 | 1 | 131.296093 |
| ... | ... | ... | ... | ... |
| 9 | 2017-06-01 | 104.2022 | 1 | 101.125782 |
| 10 | 2017-07-01 | 102.5861 | 1 | 99.870548 |
| 11 | 2017-08-01 | 114.0613 | 1 | 106.021718 |
```python theme={null}
sf.plot(train, Y_hat1)
```
### Predict method with confidence interval
To generate forecasts use the predict method.
The predict method takes two arguments: forecasts the next `h` (for
horizon) and `level`.
* `h (int):` represents the forecast h steps into the future. In this
case, 12 months ahead.
* `level (list of floats):` this optional parameter is used for
probabilistic forecasting. Set the level (or confidence percentile)
of your prediction interval. For example, `level=[95]` means that
the model expects the real value to be inside that interval 95% of
the times.
The forecast object here is a new data frame that includes a column with
the name of the model and the y hat values, as well as columns for the
uncertainty intervals.
This step should take less than 1 second.
```python theme={null}
sf.predict(h=horizon)
```
| | unique\_id | ds | AutoTheta |
| --- | ---------- | ---------- | ---------- |
| 0 | 1 | 2016-09-01 | 111.075915 |
| 1 | 1 | 2016-10-01 | 129.111292 |
| 2 | 1 | 2016-11-01 | 131.296093 |
| ... | ... | ... | ... |
| 9 | 1 | 2017-06-01 | 101.125782 |
| 10 | 1 | 2017-07-01 | 99.870548 |
| 11 | 1 | 2017-08-01 | 106.021718 |
```python theme={null}
forecast_df = sf.predict(h=horizon, level=[95])
forecast_df
```
| | unique\_id | ds | AutoTheta | AutoTheta-lo-95 | AutoTheta-hi-95 |
| --- | ---------- | ---------- | ---------- | --------------- | --------------- |
| 0 | 1 | 2016-09-01 | 111.075915 | 90.139234 | 136.011109 |
| 1 | 1 | 2016-10-01 | 129.111292 | 94.795409 | 160.387128 |
| 2 | 1 | 2016-11-01 | 131.296093 | 90.579813 | 168.268538 |
| ... | ... | ... | ... | ... | ... |
| 9 | 1 | 2017-06-01 | 101.125782 | 41.186268 | 159.159903 |
| 10 | 1 | 2017-07-01 | 99.870548 | 35.144354 | 152.867267 |
| 11 | 1 | 2017-08-01 | 106.021718 | 38.753454 | 166.048584 |
```python theme={null}
sf.plot(train, test.merge(forecast_df), level=[95])
```
## Cross-validation
In previous steps, we’ve taken our historical data to predict the
future. However, to asses its accuracy we would also like to know how
the model would have performed in the past. To assess the accuracy and
robustness of your models on your data perform Cross-Validation.
With time series data, Cross Validation is done by defining a sliding
window across the historical data and predicting the period following
it. This form of cross-validation allows us to arrive at a better
estimation of our model’s predictive abilities across a wider range of
temporal instances while also keeping the data in the training set
contiguous as is required by our models.
The following graph depicts such a Cross Validation Strategy:
### Perform time series cross-validation
Cross-validation of time series models is considered a best practice but
most implementations are very slow. The statsforecast library implements
cross-validation as a distributed operation, making the process less
time-consuming to perform. If you have big datasets you can also perform
Cross Validation in a distributed cluster using Ray, Dask or Spark.
In this case, we want to evaluate the performance of each model for the
last 5 months `(n_windows=5)`, forecasting every second months
`(step_size=12)`. Depending on your computer, this step should take
around 1 min.
The cross\_validation method from the StatsForecast class takes the
following arguments.
* `df:` training data frame
* `h (int):` represents h steps into the future that are being
forecasted. In this case, 12 months ahead.
* `step_size (int):` step size between each window. In other words:
how often do you want to run the forecasting processes.
* `n_windows(int):` number of windows used for cross validation. In
other words: what number of forecasting processes in the past do you
want to evaluate.
```python theme={null}
crossvalidation_df = sf.cross_validation(
df=train,
h=horizon,
step_size=12,
n_windows=5
)
```
The crossvaldation\_df object is a new data frame that includes the
following columns:
* `unique_id:` series identifier.
* `ds:` datestamp or temporal index
* `cutoff:` the last datestamp or temporal index for the n\_windows.
* `y:` true value
* `"model":` columns with the model’s name and fitted value.
```python theme={null}
crossvalidation_df
```
| | unique\_id | ds | cutoff | y | AutoTheta |
| --- | ---------- | ---------- | ---------- | -------- | ---------- |
| 0 | 1 | 2011-09-01 | 2011-08-01 | 93.9062 | 98.167469 |
| 1 | 1 | 2011-10-01 | 2011-08-01 | 116.7634 | 116.969932 |
| 2 | 1 | 2011-11-01 | 2011-08-01 | 116.8258 | 119.135142 |
| ... | ... | ... | ... | ... | ... |
| 57 | 1 | 2016-06-01 | 2015-08-01 | 102.4044 | 109.600469 |
| 58 | 1 | 2016-07-01 | 2015-08-01 | 102.9512 | 108.260160 |
| 59 | 1 | 2016-08-01 | 2015-08-01 | 104.6977 | 114.248270 |
## Model Evaluation
Now we are going to evaluate our model with the results of the
predictions, we will use different types of metrics MAE, MAPE, MASE,
RMSE, SMAPE to evaluate the accuracy.
```python theme={null}
from functools import partial
import utilsforecast.losses as ufl
from utilsforecast.evaluation import evaluate
```
```python theme={null}
evaluate(
test.merge(Y_hat),
metrics=[ufl.mae, ufl.mape, partial(ufl.mase, seasonality=season_length), ufl.rmse, ufl.smape],
train_df=train,
)
```
| | unique\_id | metric | AutoTheta |
| - | ---------- | ------ | --------- |
| 0 | 1 | mae | 6.281513 |
| 1 | 1 | mape | 0.055683 |
| 2 | 1 | mase | 1.212473 |
| 3 | 1 | rmse | 7.683669 |
| 4 | 1 | smape | 0.027399 |
## References
1. [Jose A. Fiorucci, Tiago R. Pellegrini, Francisco Louzada, Fotios
Petropoulos, Anne B. Koehler (2016). “Models for optimising the
theta method and their relationship to state space models”.
International Journal of
Forecasting](https://www.sciencedirect.com/science/article/pii/S0169207016300243).
2. [Nixtla AutoTheta API](../../src/core/models.html#autotheta)
3. [Pandas available
frequencies](https://pandas.pydata.org/pandas-docs/stable/user_guide/timeseries.html#offset-aliases).
4. [Rob J. Hyndman and George Athanasopoulos (2018). “Forecasting
Principles and Practice (3rd
ed)”](https://otexts.com/fpp3/tscv.html)
5. [Seasonal periods- Rob J
Hyndman](https://robjhyndman.com/hyndsight/seasonal-periods/).