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Time Series Analysis using iPython

ipython-resizedIn this example, we will examine ARMA and ARIMA models with Python using the Statsmodels package. This package can be downloaded at http://statsmodels.sourceforge.net/stable/index.html. Autogressive Moving-Average Processes (ARMA) and Auto-Regressive Integrated Moving Average (ARIMA) can be called from the tsa (Time Series) module from the Statamodels package.

Note: I am not as expert in time-series analysis as I am in other areas of Analytics, so if you find errors I would be happy to know about them and correct them.

Introduction

ARIMA models are, in theory, the most general class of models for forecasting a time series, which can be made to be “stationary” by differencing (if necessary), perhaps in conjunction with nonlinear transformations such as logging or deflating (if necessary). A random variable that is a time series is stationary if its statistical properties are all constant over time. A stationary series has no trend, its variations around its mean have a constant amplitude, and it wiggles in a consistent fashion, i.e., its short-term random time patterns always look the same in a statistical sense. The latter condition means that its autocorrelations (correlations with its prior deviations from the mean) remain constant over time, or equivalently, that its power spectrum remains constant over time. A random variable of this form can be viewed (as usual) as a combination of signal and noise, and the signal (if one is apparent) could be a pattern of fast or slow mean reversion, or sinusoidal oscillation, or rapid alternation in sign, and it could also have a seasonal component. An ARIMA model can be viewed as a “filter” that tries to separate the signal from the noise, and the signal is then extrapolated into the future to obtain forecasts.

The Data

The data we will use is annual sunspot data from 1700 – 2008 recording the number of sunspots per year. The file sunspots.csv and can be downloaded from the line below.

Import Packages

In addition to Statsmodels, we will need to import additional packages, including Numpy, Scipy, Pandas, and Matplotlib.

Also, from Statsmodels we will need to import qqplot.

In [1]:
import numpy as np
from scipy import stats
import pandas as pd
import matplotlib.pyplot as plt
import statsmodels.api as sm
In [2]:
from statsmodels.graphics.api import qqplot

Loading the Dataset

We will use pandas to load the .csv file which we downloaded earlier.

In [3]:
dta= pd.read_csv("C:/Users/Strickland/Documents/Python Scripts/sunspots.csv")

Check the Data Upload

Next we check to see that the data uploaded correctly by viewing the dataset Notes.

In [4]:
print sm.datasets.sunspots.NOTE
::

    Number of Observations - 309 (Annual 1700 - 2008)
    Number of Variables - 1
    Variable name definitions::

        SUNACTIVITY - Number of sunspots for each year

    The data file contains a 'YEAR' variable that is not returned by load.

Preparing the Data

Next we need to do a little dataset preparation. Here, an annual date series must be date-times at the end of the year.

In [5]:
dta.index = pd.Index(sm.tsa.datetools.dates_from_range('1700', '2008'))
del dta["YEAR"]

Examine the Data

Now we take a look at the data.

In [6]:
# show plots in the notebook
%matplotlib inline
dta.plot(figsize=(12,8));
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Auto-correlations

Before we decide which model to use, we need to look at auto-correlations.

Autocorrelation correlogram. Seasonal patterns of time series can be examined via correlograms, which display graphically and numerically the autocorrelation function (ACF). Auto-correlation in pandas plotting and statsmodels graphics standardize the data before computing the auto-correlation. These libraries subtract the mean and divide by the standard deviation of the data.

When using standardization, they make an assumption that your data has been generated with a Gaussian law (with a certain mean and standard deviation). This may not be the case in reality.

Correlation is sensitive. Both (matplotlib and pandas plotting) of these functions have their drawbacks. The figure generated by the following code using matplotlib will be identical to figure generated by pandas plotting or statsmodels graphics.

Partial autocorrelations. Another useful method to examine serial dependencies is to examine the partial autocorrelation function (PACF) – an extension of autocorrelation, where the dependence on the intermediate elements (those within the lag) is removed.

Once we determine the nature of the auto-correlations we use the following rules of thumb.

  • Rule 1: If the ACF shows exponential decay, the PACF has a spike at lag 1, and no correlation for other lags, then use one autoregressive (p)parameter
  • Rule 2: If the ACF shows a sine-wave shape pattern or a set of exponential decays, the PACF has spikes at lags 1 and 2, and no correlation for other lags, the use two autoregressive (pparameters
  • Rule 3: If the ACF has a spike at lag 1, no correlation for other lags, and the PACF damps out exponentially, then use one moving average (qparameter.
  • Rule 4: If the ACF has spikes at lags 1 and 2, no correlation for other lags, and the PACF has a sine-wave shape pattern or a set of exponential decays, then use two moving average (qparameter.
  • Rule 5: If the ACF shows exponential decay starting at lag 1, and the PACF shows exponential decay starting at lag 1, then use one autoregressive (pand one moving average (qparameter.

Removing serial dependency. Serial dependency for a particular lag can be removed by differencing the series. There are two major reasons for such transformations.

  • First, we can identify the hidden nature of seasonal dependencies in the series. Autocorrelations for consecutive lags are interdependent, so removing some of the autocorrelations will change other auto correlations, making other seasonalities more apparent.
  • Second, removing serial dependencies will make the series stationary, which is necessary for ARIMA and other techniques.

Another popular test for serial correlation is the Durbin-Watson statistic. The DW statistic will lie in the 0-4 range, with a value near two indicating no first-order serial correlation. Positive serial correlation is associated with DW values below 2 and negative serial correlation with DW values above 2.

In [7]:
sm.stats.durbin_watson(dta)
Out[7]:
array([ 0.13952893])
 The value of Durbin-Watson statistic is close to 2 if the errors are uncorrelated. In our example, it is 0.1395. That means that there is a strong evidence that the variable open has high autocorrelation.
In [8]:
# show plots in the notebook
%matplotlib inline
fig = plt.figure(figsize=(12,8))
ax1 = fig.add_subplot(211)
fig = sm.graphics.tsa.plot_acf(dta.values.squeeze(), lags=40, ax=ax1)
ax2 = fig.add_subplot(212)
fig = sm.graphics.tsa.plot_pacf(dta, lags=40, ax=ax2)
C:\Anaconda\lib\site-packages\matplotlib\collections.py:590: FutureWarning: elementwise comparison failed; returning scalar instead, but in the future will perform elementwise comparison
  if self._edgecolors == str('face'):
ts-02
 The plots also indicate that autocorrelation is present. Another set of plots (shown below) are available using the autocorrelation_plot function from Pandas.
In [9]:
from pandas.tools.plotting import autocorrelation_plot
In [10]:
# show plots in the notebook
%matplotlib inline
dta['SUNACTIVITY_2'] = dta['SUNACTIVITY']
dta['SUNACTIVITY_2'] = (dta['SUNACTIVITY_2'] - dta['SUNACTIVITY_2'].mean()) / (dta['SUNACTIVITY_2'].std())
plt.acorr(dta['SUNACTIVITY_2'],maxlags = len(dta['SUNACTIVITY_2']) -1, linestyle = "solid", usevlines = False, marker='')
plt.show()
autocorrelation_plot(dta['SUNACTIVITY'])
plt.show()
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ts-04
For mixed ARMA processes the Autocorrelation function is a mixture of exponentials and damped sine waves after (q-p) lags. The partial autocorrelation function is a mixture of exponentials and dampened sine waves after (p-q) lags.

Times Series Modeling

We will only explore two methods here. An ARMA model is classified as ARMA(p,q), with no differenceing terms. ARMA models can be described by a series of equations. The equations are somewhat simpler if the time series is first reduced to zero-mean by subtracting the sample mean. Therefore, we will work with the mean-adjusted series

yt; = Yt , t = 1, …N

where Yt is the original time series, is its sample mean, and yt is the mean-adjusted series. One subset of ARMA models are the so-called autoregressive, or AR models. An AR model expresses a time series as a linear function of its past values. The order of the AR model tells how many lagged past values are included. The simplest AR model is the first-order autoregressive, or AR(1), model

yt + a1 yt-1 = et

where yt is the mean-adjusted series in year t, yt-1 is the series in the previous year, at is the lag-1 autoregressive coefficient, and et is the noise. The noise also goes by various other names: the error, the random-shock, and the residual. The residuals et are assumed to be random in time (not autocorrelated), and normally distributed. Be rewriting the equation for the AR(1) model as

yt = a1 yt-1 + et

We see that the AR(1) model has the form of a regression model in which yt is regressed on its previous value. In this form, at is analogous to the negative of the regression coefficient, and et to the regression residuals. The name autoregressive refers to the regression on self (auto).

A nonseasonal ARIMA model is classified as an ARIMA(p,d,q) model, where:

  • p is the number of autoregressive terms,
  • d is the number of nonseasonal differences needed for stationarity, and
  • q is the number of lagged forecast errors in the prediction equation.

The forecasting equation is constructed as follows. First, let y denote the dth difference of Y, which means:

If d=0: yt = Yt

If d=1: yt = Yt – Yt-1

If d=2: yt = (Yt – Yt-1) – (Yt-1 – Yt-2) = Yt – 2Yt-1 + Yt-2

Note that the second difference of Y (the d=2 case) is not the difference from two periods ago. Rather, it is the first-difference-of-the-first difference, which is the discrete analog of a second derivative, i.e., the local acceleration of the series rather than its local trend.

Modeling the Data

Model One: ARMA(2,0)

Using Rule 2 from above, we will first try an ARMA(2,0) model with two autoregressive terms and no moving averages.

In [11]:
arma_mod20 = sm.tsa.ARMA(dta, (2,0)).fit()
print arma_mod20.params
const                49.659389
ar.L1.SUNACTIVITY     1.390656
ar.L2.SUNACTIVITY    -0.688571
dtype: float64
 We now calculate the Akaike Information Criterion (AIC), Schwarz Bayesian Information Criterion (BIC), and Hannan-Quinn Information Criterion (HQIC). Our goalis to choose a model that minimizes (AIC, BIC, HQIC).
In [12]:
print arma_mod20.aic, arma_mod20.bic, arma_mod20.hqic
2622.63633806 2637.56970317 2628.60672591
Does our model obey the theory? We will use the Durbin-Watson test for autocorrelation. The Durbin-Watson statistic ranges in value from 0 to 4. A value near 2 indicates non-autocorrelation; a value toward 0 indicates positive autocorrelation; a value toward 4 indicates negative autocorrelation.
In [13]:
sm.stats.durbin_watson(arma_mod20.resid.values)
Out[13]:
2.1458269461951018
 The Durbin-Watson test shows no autocorrelation.

Plotting the Data

Next we plot and study the data it represents.

In [14]:
# show plots in the notebook
%matplotlib inline
fig = plt.figure(figsize=(12,8))
ax = fig.add_subplot(111)
ax = arma_mod20.resid.plot(ax=ax);
ts-05

Analyzing the Residuals

In the following steps, we calculate the residuals, tests the null hypothesis that the residuals come from a normal distribution, and construct a qq-plot.

In [15]:
resid20 = arma_mod20.resid
In [16]:
stats.normaltest(resid20)
Out[16]:
NormaltestResult(statistic=41.736012879102248, pvalue=8.6524670674533609e-10)
In [17]:
# show plots in the notebook
%matplotlib inline
fig = plt.figure(figsize=(12,8))
ax = fig.add_subplot(111)
fig = qqplot(resid20, line='q', ax=ax, fit=True)
ts-06

Model Autocorrelation

Now we investigate the autocorrelation of our ARMA(2,0) model.

In [18]:
%matplotlib inline
fig = plt.figure(figsize=(12,8))
ax1 = fig.add_subplot(211)
fig = sm.graphics.tsa.plot_acf(resid20.values.squeeze(), lags=40, ax=ax1)
ax2 = fig.add_subplot(212)
fig = sm.graphics.tsa.plot_pacf(resid20, lags=40, ax=ax2)
ts-07

Next, we calculate the lag, autocorrelation (AC), Q statistic and Prob>Q. The Ljung–Box Q test (named for Greta M. Ljung and George E. P. Box) is a type of statistical test of whether any of a group of autocorrelations of a time series are different from zero. The null hypothesis is, H0: The data are independently distributed (i.e. the correlations in the population from which the sample is taken are 0, so that any observed correlations in the data result from randomness of the sampling process).

In [19]:
r,q,p = sm.tsa.acf(resid20.values.squeeze(), qstat=True)
data = np.c_[range(1,41), r[1:], q, p]
table = pd.DataFrame(data, columns=['lag', "AC", "Q", "Prob(>Q)"])
print table.set_index('lag')
           AC          Q  Prob(>Q)
lag                               
1   -0.085220   2.265964  0.132244
2    0.103691   5.631592  0.059857
3   -0.027833   5.874877  0.117859
4    0.091122   8.491069  0.075158
5    0.019010   8.605302  0.125881
6    0.031320   8.916425  0.178333
7    0.044485   9.546121  0.215786
8   -0.034337   9.922552  0.270504
9    0.185690  20.967727  0.012794
10   0.191608  32.767487  0.000298
11   0.190385  44.456231  0.000006
12   0.121693  49.247963  0.000002
13  -0.016219  49.333365  0.000004
14   0.014986  49.406527  0.000008
15  -0.063197  50.711976  0.000009
16   0.039730  51.229689  0.000015
17   0.009577  51.259872  0.000027
18  -0.073645  53.050932  0.000026
19   0.076469  54.988666  0.000023
20  -0.006827  55.004163  0.000041
21   0.088818  57.636429  0.000029
22   0.120485  62.497142  0.000009
23   0.103328  66.084650  0.000005
24  -0.085728  68.562767  0.000004
25   0.013730  68.626556  0.000006
26  -0.036183  69.071127  0.000009
27  -0.150156  76.754545  0.000001
28   0.049680  77.598615  0.000002
29  -0.055467  78.654537  0.000002
30   0.003354  78.658411  0.000003
31  -0.010905  78.699521  0.000005
32   0.120386  83.727446  0.000002
33   0.042680  84.361682  0.000002
34   0.011107  84.404793  0.000004
35   0.024261  84.611235  0.000005
36  -0.125046  90.115463  0.000002
37  -0.036394  90.583412  0.000002
38  -0.060509  91.881735  0.000002
39  -0.024440  92.094325  0.000003
40   0.000581  92.094446  0.000005
Notice that the p-values for the Ljung–Box Q test all are well above .05 for lags 1 through 8, indicating “significance.” This is not a desirable result. However, the p-values for the remaining lags through 40 data values as less than .05. So there is much data not contributing to correlations at high lags.

Predictions

Next, we compute the predictions and analyze their fit against actual values.

In [20]:
predict_sunspots20 = arma_mod20.predict('1990', '2012', dynamic=True)
print predict_sunspots20
1990-12-31    164.966812
1991-12-31    135.687529
1992-12-31     89.897551
1993-12-31     46.380327
1994-12-31     17.392506
1995-12-31      7.045131
1996-12-31     12.615671
1997-12-31     27.487278
1998-12-31     44.332851
1999-12-31     57.519084
2000-12-31     64.257219
2001-12-31     64.547986
2002-12-31     60.312657
2003-12-31     54.222559
2004-12-31     48.669655
2005-12-31     45.140943
2006-12-31     44.057288
2007-12-31     44.980067
2008-12-31     47.009508
2009-12-31     49.196363
2010-12-31     50.840111
2011-12-31     51.620193
2012-12-31     51.573181
Freq: A-DEC, dtype: float64
In [21]:
ax = dta.ix['1950':].plot(figsize=(12,8))
ax = predict_sunspots20.plot(ax=ax, style='r--', label='Dynamic Prediction');
ax.legend();
ax.axis((-20.0, 38.0, -4.0, 200.0));
ts-08

The fit looks good up to about 1998 and underfit the data afterwards.

Calculate Forecast Errors

Mean absolute error: The mean absolute error (MAE) value is computed as the average absolute error value. If this value is 0 (zero), the fit (forecast) is perfect. As compared to the mean squared error value, this measure of fit will “de-emphasize” outliers, that is, unique or rare large error values will affect the MAE less than the MSE value.

Mean Forecast Error (Bias). The mean forecast error (MFE) is the average error in the observations. A large positive MFE means that the forecast is undershooting the actual observations, and a large negative MFE means the forecast is overshooting the actual observations. A value near zero is ideal.

The MAE is a better indicator of fit than the MFE.

In [22]:
def mean_forecast_err(y, yhat):
    return y.sub(yhat).mean()
In [23]:
def mean_absolute_err(y, yhat):
    return np.mean((np.abs(y.sub(yhat).mean()) / yhat)) # or percent error = * 100
In [24]:
print "MFE = ", mean_forecast_err(dta.SUNACTIVITY, predict_sunspots20)
print "MAE = ", mean_absolute_err(dta.SUNACTIVITY, predict_sunspots20)
MFE =  4.73038833847
MAE =  0.134688778061

For MFE > 0, models tends to under-forecast. However, as long as the tracking signal is between –4 and 4, we assume the model is working correctly. The measure of MAE being small would indicate a pretty good fit.

Model Two: ARMA(3,0)

As a second model, we will try an ARMA(3,0) model with three autoregressive terms and no moving averages.

In [25]:
arma_mod30 = sm.tsa.ARMA(dta, (3,0)).fit()
In [26]:
print arma_mod30.params
const                49.749953
ar.L1.SUNACTIVITY     1.300810
ar.L2.SUNACTIVITY    -0.508093
ar.L3.SUNACTIVITY    -0.129650
dtype: float64
In [27]:
print arma_mod30.aic, arma_mod30.bic, arma_mod30.hqic
2619.4036287 2638.07033508 2626.8666135

Does our model obey the theory?

In [28]:
sm.stats.durbin_watson(arma_mod30.resid.values)
Out[28]:
1.9564808156070777

The Durbin-Watson is lose to 2, so the test shows very little if any positive autocorrelation.

In [29]:
# show plots in the notebook
%matplotlib inline
fig = plt.figure(figsize=(12,8))
ax = fig.add_subplot(111)
ax = arma_mod30.resid.plot(ax=ax);
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In [30]:
resid30 = arma_mod30.resid
In [31]:
stats.normaltest(resid30)
Out[31]:
NormaltestResult(statistic=49.845020032909339, pvalue=1.5006915069024666e-11)
In [32]:
# show plots in the notebook
%matplotlib inline
fig = plt.figure(figsize=(12,8))
ax = fig.add_subplot(111)
fig = qqplot(resid30, line='q', ax=ax, fit=True)
ts-10
In [33]:
%matplotlib inline
fig = plt.figure(figsize=(12,8))
ax1 = fig.add_subplot(211)
fig = sm.graphics.tsa.plot_acf(resid30.values.squeeze(), lags=40, ax=ax1)
ax2 = fig.add_subplot(212)
fig = sm.graphics.tsa.plot_pacf(resid30, lags=40, ax=ax2)
ts-11
In [34]:
r,q,p = sm.tsa.acf(resid30.values.squeeze(), qstat=True)
data = np.c_[range(1,41), r[1:], q, p]
table = pd.DataFrame(data, columns=['lag', "AC", "Q", "Prob(>Q)"])
print table.set_index('lag')
           AC          Q      Prob(>Q)
lag                                   
1    0.009179   0.026286  8.712028e-01
2    0.041793   0.573042  7.508715e-01
3   -0.001335   0.573601  9.024483e-01
4    0.136089   6.408921  1.706203e-01
5    0.092468   9.111828  1.046860e-01
6    0.091948  11.793244  6.674345e-02
7    0.068748  13.297201  6.518982e-02
8   -0.015020  13.369229  9.976132e-02
9    0.187592  24.641909  3.393910e-03
10   0.213718  39.321995  2.229474e-05
11   0.201082  52.361139  2.344948e-07
12   0.117182  56.804191  8.574251e-08
13  -0.014055  56.868327  1.893901e-07
14   0.015398  56.945566  3.997655e-07
15  -0.024967  57.149321  7.741463e-07
16   0.080916  59.296772  6.872157e-07
17   0.041138  59.853741  1.110943e-06
18  -0.052021  60.747431  1.548430e-06
19   0.062496  62.041694  1.831641e-06
20  -0.010301  62.076981  3.381239e-06
21   0.074453  63.926656  3.193583e-06
22   0.124955  69.154773  8.978346e-07
23   0.093162  72.071037  5.799777e-07
24  -0.082152  74.346689  4.713011e-07
25   0.015695  74.430045  8.289033e-07
26  -0.025037  74.642904  1.367283e-06
27  -0.125861  80.041150  3.722562e-07
28   0.053225  81.009983  4.716274e-07
29  -0.038693  81.523809  6.916626e-07
30  -0.016904  81.622228  1.151660e-06
31  -0.019296  81.750940  1.868764e-06
32   0.104990  85.575067  8.927949e-07
33   0.040086  86.134568  1.247508e-06
34   0.008829  86.161811  2.047823e-06
35   0.014588  86.236449  3.263804e-06
36  -0.119329  91.248899  1.084453e-06
37  -0.036665  91.723868  1.521921e-06
38  -0.046193  92.480517  1.938732e-06
39  -0.017768  92.592886  2.990676e-06
40  -0.006220  92.606708  4.696979e-06

This could indicate a lack of fit. Notice that the p-values for the Ljung–Box Q test all are well above 0.05 for lags 1 through 8, indicating “significance.” This is not a desirable result. However, the p-values for the remaining lags through 40 data values as less than 0.05. So there is much data not contributing to correlations at high lags.

In-sample dynamic prediction. How well does our model do?

In [35]:
predict_sunspots30 = arma_mod30.predict('1990', '2012', dynamic=True)
print predict_sunspots30
1990-12-31    167.047421
1991-12-31    140.993013
1992-12-31     94.859134
1993-12-31     46.860929
1994-12-31     11.242619
1995-12-31     -4.721260
1996-12-31     -1.166883
1997-12-31     16.185712
1998-12-31     39.021896
1999-12-31     59.449879
2000-12-31     72.170148
2001-12-31     75.376790
2002-12-31     70.436468
2003-12-31     60.731600
2004-12-31     50.201814
2005-12-31     42.076047
2006-12-31     38.114308
2007-12-31     38.454663
2008-12-31     41.963833
2009-12-31     46.869301
2010-12-31     51.423272
2011-12-31     54.399728
2012-12-31     55.321701
Freq: A-DEC, dtype: float64
In [36]:
ax = dta.ix['1950':].plot(figsize=(12,8))
ax = predict_sunspots30.plot(ax=ax, style='r--', label='Dynamic Prediction');
ax.legend();
ax.axis((-20.0, 38.0, -4.0, 200.0));
ts-12
In [37]:
print "MFE = ", mean_forecast_err(dta.SUNACTIVITY, predict_sunspots30)
print "MAE = ", mean_absolute_err(dta.SUNACTIVITY, predict_sunspots30)
MFE =  5.63694046318
MAE =  -0.141136496478

For small MAE the model is a pretty good fit. The MFE being slightly above 4 might indicate the model tends to under-forecast.

Model Three: ARMA(2,1)

As a third model, we will try an ARMA(2,1) model with two autoregressive terms and one moving average term.

In [38]:
arma_mod40 = sm.tsa.ARMA(dta, (2,1)).fit()
In [39]:
res40 = arma_mod40.resid
In [40]:
print arma_mod40.params
const                49.749195
ar.L1.SUNACTIVITY     1.470738
ar.L2.SUNACTIVITY    -0.755121
ma.L1.SUNACTIVITY    -0.153691
dtype: float64
In [41]:
print arma_mod40.aic, arma_mod40.bic, arma_mod40.hqic
2620.27719156 2638.94389794 2627.74017636
In [42]:
sm.stats.durbin_watson(arma_mod40.resid.values)
Out[42]:
1.990341245334406
In [43]:
predict_sunspots40 = arma_mod40.predict('1990', '2012', dynamic=True)
In [44]:
resid40 = arma_mod40.resid
In [45]:
stats.normaltest(resid40)
Out[45]:
NormaltestResult(statistic=48.550090206155012, pvalue=2.8673576316879386e-11)
In [46]:
predict_sunspots40 = arma_mod40.predict('1990', '2012', dynamic=True)
In [47]:
print "Metric for ARMA(2,1):    ",mean_forecast_err(dta.SUNACTIVITY, predict_sunspots40),"MFE"
Metric for ARMA(2,1):     5.65195019619 MFE
In [48]:
print "Metric for ARMA(2,1):    ",mean_absolute_err(dta.SUNACTIVITY, predict_sunspots40),"MAE"
Metric for ARMA(2,1):     4.82202232657 MAE

Model Four: ARMA(2,3)

As a fourth model we will try an ARMA(2,3) model with three autoregressive terms and two moving averages.

In [49]:
arma_mod50 = sm.tsa.ARMA(dta, (2,3)).fit()
In [50]:
sm.stats.durbin_watson(arma_mod50.resid.values)
Out[50]:
1.9674807976736908
In [51]:
print arma_mod50.params
const                49.718535
ar.L1.SUNACTIVITY     1.447478
ar.L2.SUNACTIVITY    -0.749029
ma.L1.SUNACTIVITY    -0.144607
ma.L2.SUNACTIVITY     0.067941
ma.L3.SUNACTIVITY     0.009734
dtype: float64
In [52]:
print arma_mod50.aic, arma_mod50.bic, arma_mod50.hqic
2622.85169078 2648.98507972 2633.29986951
In [53]:
resid50 = arma_mod50.resid
In [54]:
stats.normaltest(resid50)
Out[54]:
NormaltestResult(statistic=48.615275864226881, pvalue=2.7754089086771973e-11)
In [55]:
predict_sunspots50 = arma_mod50.predict('1990', '2012', dynamic=True)
In [56]:
ax = dta.ix['1950':].plot(figsize=(12,8))
ax = predict_sunspots50.plot(ax=ax, style='r--', label='Dynamic Prediction');
ax.legend();
ax.axis((-20.0, 38.0, -4.0, 200.0));
ts-13
In [57]:
print "Metric for ARMA(3,2):    ",mean_forecast_err(dta.SUNACTIVITY, predict_sunspots50),"MLE"
Metric for ARMA(3,2):     5.27758730154 MLE
In [58]:
print "Metric for ARMA(3,2):    ",mean_absolute_err(dta.SUNACTIVITY, predict_sunspots50),"MAE"
Metric for ARMA(3,2):     0.325297875678 MAE

Model Five: ARIMA(3,0,2)

As a fifth model we will try an AIMA(3,0,2) model with three autoregressive terms, no differences, and two moving averages.

In [59]:
arima_mod1 = sm.tsa.ARIMA(dta, (3,0,2)).fit()
In [60]:
print arima_mod1.params
const                48.486959
ar.L1.SUNACTIVITY     2.560637
ar.L2.SUNACTIVITY    -2.471094
ar.L3.SUNACTIVITY     0.892054
ma.L1.SUNACTIVITY    -1.518372
ma.L2.SUNACTIVITY     0.663744
dtype: float64
In [61]:
sm.stats.durbin_watson(arima_mod1.resid.values)
Out[61]:
1.7634439830432345
In [62]:
print arima_mod1.aic, arima_mod1.bic, arima_mod1.hqic
2581.57229897 2607.7056879 2592.0204777
In [63]:
# show plots in the notebook
%matplotlib inline
fig = plt.figure(figsize=(12,8))
ax = fig.add_subplot(111)
ax = arima_mod1.resid.plot(ax=ax);
ts-14
In [64]:
resid1 = arima_mod1.resid
In [65]:
stats.normaltest(resid1)
Out[65]:
NormaltestResult(statistic=45.133708715833087, pvalue=1.5824852780515422e-10)
In [66]:
%matplotlib inline
fig = plt.figure(figsize=(12,8))
ax = fig.add_subplot(111)
fig = qqplot(resid1, line='q', ax=ax, fit=True)
ts-15
In [67]:
%matplotlib inline
fig = plt.figure(figsize=(12,8))
ax1 = fig.add_subplot(211)
fig = sm.graphics.tsa.plot_acf(resid1.values.squeeze(), lags=40, ax=ax1)
ax2 = fig.add_subplot(212)
fig = sm.graphics.tsa.plot_pacf(resid1, lags=40, ax=ax2)
ts-16
In [68]:
r,q,p = sm.tsa.acf(resid1.values.squeeze(), qstat=True)
data = np.c_[range(1,41), r[1:], q, p]
table = pd.DataFrame(data, columns=['lag', "AC", "Q", "Prob(>Q)"])
print table.set_index('lag')
           AC          Q  Prob(>Q)
lag                               
1    0.102089   3.251829  0.071344
2   -0.073750   4.954393  0.083978
3   -0.128911  10.173307  0.017149
4    0.069554  11.697568  0.019748
5    0.095314  14.569406  0.012370
6    0.072379  16.230927  0.012567
7   -0.023626  16.408542  0.021635
8   -0.136402  22.348679  0.004309
9    0.040967  22.886280  0.006457
10   0.100846  26.154915  0.003537
11   0.089601  28.743898  0.002487
12  -0.014426  28.811239  0.004202
13  -0.132360  34.498960  0.001010
14  -0.065597  35.900688  0.001080
15  -0.011676  35.945248  0.001800
16   0.122644  40.878636  0.000579
17   0.064368  42.242196  0.000619
18  -0.075295  44.114441  0.000556
19  -0.051250  44.984809  0.000689
20  -0.115365  49.410363  0.000269
21  -0.011647  49.455629  0.000434
22   0.071806  51.182092  0.000405
23   0.059462  52.370145  0.000446
24  -0.078581  54.452304  0.000371
25   0.014392  54.522396  0.000568
26  -0.004754  54.530072  0.000870
27  -0.081503  56.793746  0.000683
28   0.050981  57.682592  0.000798
29  -0.050678  58.564034  0.000929
30  -0.064736  60.007507  0.000919
31  -0.058183  61.177715  0.000978
32   0.067297  62.748932  0.000931
33   0.038756  63.271926  0.001174
34   0.011765  63.320293  0.001663
35   0.005979  63.332829  0.002343
36  -0.108496  67.476475  0.001142
37  -0.025842  67.712414  0.001523
38  -0.022721  67.895486  0.002035
39   0.000369  67.895534  0.002814
40  -0.002080  67.897080  0.003841
In [69]:
predict_sunspots1 = arima_mod1.predict('1990', '2012', dynamic=True)
print predict_sunspots1
1990-12-31    175.041558
1991-12-31    164.082608
1992-12-31    129.091771
1993-12-31     82.132355
1994-12-31     38.576035
1995-12-31     11.871561
1996-12-31      9.232555
1997-12-31     29.609714
1998-12-31     64.487621
1999-12-31    101.089254
2000-12-31    126.803664
2001-12-31    133.315806
2002-12-31    119.098931
2003-12-31     89.541196
2004-12-31     54.794996
2005-12-31     26.180332
2006-12-31     12.402623
2007-12-31     16.836970
2008-12-31     36.711921
2009-12-31     64.356302
2010-12-31     89.986320
2011-12-31    105.033134
2012-12-31    104.888644
Freq: A-DEC, dtype: float64
In [70]:
ax = dta.ix['1950':].plot(figsize=(12,8))
ax = predict_sunspots1.plot(ax=ax, style='r--', label='Dynamic Prediction');
ax.legend();
ax.axis((-20.0, 38.0, -4.0, 200.0));
ts-17
In [71]:
print "Metrics for ARIMA(3,0,2): ",mean_forecast_err(dta.SUNACTIVITY, predict_sunspots1),"MLE"
Metrics for ARIMA(3,0,2):  -13.3948141872 MLE
In [72]:
print "Metrics for ARIMA(3,0,2): ",mean_absolute_err(dta.SUNACTIVITY, predict_sunspots1),"MAE"
Metrics for ARIMA(3,0,2):  0.357429358606 MAE

Model Six: ARIMA(2,0,3)

Now let’s examine a sixth model.

In [73]:
arima_mod2 = sm.tsa.ARIMA(dta, (2,0,3)).fit()
In [74]:
print arima_mod2.params
const                49.718535
ar.L1.SUNACTIVITY     1.447478
ar.L2.SUNACTIVITY    -0.749029
ma.L1.SUNACTIVITY    -0.144607
ma.L2.SUNACTIVITY     0.067941
ma.L3.SUNACTIVITY     0.009734
dtype: float64
In [75]:
mean_forecast_err(dta.SUNACTIVITY, predict_sunspots30)
Out[75]:
5.63694046318347
In [76]:
print arima_mod2.aic, arima_mod2.bic, arima_mod2.hqic
2622.85169078 2648.98507972 2633.29986951
In [77]:
# show plots in the notebook
%matplotlib inline
fig = plt.figure(figsize=(12,8))
ax = fig.add_subplot(111)
ax = arima_mod2.resid.plot(ax=ax);
ts-18
In [78]:
resid2 = arima_mod2.resid
In [79]:
stats.normaltest(resid2)
Out[79]:
NormaltestResult(statistic=48.615275864226881, pvalue=2.7754089086771973e-11)
In [80]:
%matplotlib inline
fig = plt.figure(figsize=(12,8))
ax = fig.add_subplot(111)
fig = qqplot(resid2, line='q', ax=ax, fit=True)
ts-19
In [81]:
%matplotlib inline
fig = plt.figure(figsize=(12,8))
ax1 = fig.add_subplot(211)
fig = sm.graphics.tsa.plot_acf(resid2.values.squeeze(), lags=40, ax=ax1)
ax2 = fig.add_subplot(212)
fig = sm.graphics.tsa.plot_pacf(resid2, lags=40, ax=ax2)
ts-20
In [82]:
r,q,p = sm.tsa.acf(resid2.values.squeeze(), qstat=True)
data = np.c_[range(1,41), r[1:], q, p]
table = pd.DataFrame(data, columns=['lag', "AC", "Q", "Prob(>Q)"])
print table.set_index('lag')
           AC          Q      Prob(>Q)
lag                                   
1    0.003654   0.004166  9.485368e-01
2    0.016284   0.087173  9.573495e-01
3   -0.012200   0.133919  9.874772e-01
4    0.135484   5.917449  2.053986e-01
5    0.087475   8.336347  1.386472e-01
6    0.081117  10.423244  1.079229e-01
7    0.053309  11.327560  1.249555e-01
8   -0.029051  11.597006  1.701103e-01
9    0.177629  21.704026  9.865850e-03
10   0.210414  35.933812  8.640929e-05
11   0.198595  48.652409  1.091797e-06
12   0.113626  52.829935  4.416460e-07
13  -0.017531  52.929715  9.294815e-07
14   0.010195  52.963573  1.929801e-06
15  -0.028636  53.231609  3.528614e-06
16   0.076135  55.132751  3.381868e-06
17   0.034417  55.522597  5.646754e-06
18  -0.058278  56.644198  7.048564e-06
19   0.057447  57.737799  8.771678e-06
20  -0.011998  57.785664  1.558883e-05
21   0.074738  59.649517  1.442882e-05
22   0.127935  65.129954  3.773169e-06
23   0.092605  68.011458  2.456406e-06
24  -0.084264  70.405618  1.897762e-06
25   0.014395  70.475736  3.269389e-06
26  -0.025898  70.703497  5.237901e-06
27  -0.127168  76.214477  1.396141e-06
28   0.052248  77.148070  1.761348e-06
29  -0.037720  77.636389  2.555245e-06
30  -0.019569  77.768295  4.115004e-06
31  -0.019972  77.906185  6.513392e-06
32   0.106340  81.829283  3.036510e-06
33   0.042161  82.448210  4.093087e-06
34   0.010416  82.486126  6.560314e-06
35   0.015178  82.566924  1.022858e-05
36  -0.118302  87.493436  3.552541e-06
37  -0.036109  87.954084  4.926259e-06
38  -0.043925  88.638258  6.325244e-06
39  -0.017690  88.749642  9.574259e-06
40  -0.008921  88.778071  1.467656e-05
In [83]:
predict_sunspots2 = arima_mod2.predict('1990', '2012', dynamic=True)
print predict_sunspots2
1990-12-31    167.956436
1991-12-31    142.397104
1992-12-31     95.564926
1993-12-31     46.661219
1994-12-31     10.952848
1995-12-31     -4.103928
1996-12-31      0.848334
1997-12-31     19.294592
1998-12-31     42.285756
1999-12-31     61.748175
2000-12-31     72.698544
2001-12-31     73.971042
2002-12-31     67.610807
2003-12-31     57.451369
2004-12-31     47.509807
2005-12-31     40.729330
2006-12-31     38.361260
2007-12-31     40.012307
2008-12-31     44.175913
2009-12-31     48.965961
2010-12-31     52.780787
2011-12-31     54.714777
2012-12-31     54.656770
Freq: A-DEC, dtype: float64
In [84]:
print "Metrics for ARIMA(2,0,3): ",mean_forecast_err(dta.SUNACTIVITY, predict_sunspots2),"MLE"
Metrics for ARIMA(2,0,3):  5.27758730154 MLE
In [85]:
print "Metrics for ARIMA(2,0,3): ",mean_absolute_err(dta.SUNACTIVITY, predict_sunspots2),"MAE"
Metrics for ARIMA(2,0,3):  0.325297875678 MAE
In [86]:
ax = dta.ix['1950':].plot(figsize=(12,8))
ax = predict_sunspots2.plot(ax=ax, style='r--', label='Dynamic Prediction');
ax.legend();
ax.axis((-20.0, 38.0, -4.0, 200.0));
ts-21

Now let’s compare all six ARMA models using the metrics we have discussed.

In [87]:
print "Model 1: ARMA(2,0)     AIC", arma_mod20.aic, " BIC", arma_mod20.bic, " HQIC", arma_mod20.hqic
print "Model 2: ARMA(3,0)     AIC", arma_mod30.aic, " BIC", arma_mod30.bic, " HQIC", arma_mod30.hqic
print "Model 3: ARMA(2,1)     AIC", arma_mod40.aic, "BIC", arma_mod40.bic, " HQIC", arma_mod40.hqic
print "Model 4: ARMA(2,3)     AIC", arma_mod50.aic, " BIC", arma_mod50.bic, " HQIC", arma_mod50.hqic
print "Model 5: ARIMA(3,0,2)  AIC", arima_mod1.aic, " BIC", arima_mod1.bic, " HQIC", arima_mod1.hqic
print "Model 6: ARIMA(2,0,3)  AIC", arima_mod2.aic, " BIC", arima_mod2.bic, " HQIC", arima_mod2.hqic
Model 1: ARMA(2,0)     AIC 2622.63633806  BIC 2637.56970317  HQIC 2628.60672591
Model 2: ARMA(3,0)     AIC 2619.4036287  BIC 2638.07033508  HQIC 2626.8666135
Model 3: ARMA(2,1)     AIC 2620.27719156 BIC 2638.94389794  HQIC 2627.74017636
Model 4: ARMA(2,3)     AIC 2622.85169078  BIC 2648.98507972  HQIC 2633.29986951
Model 5: ARIMA(3,0,2)  AIC 2581.57229897  BIC 2607.7056879  HQIC 2592.0204777
Model 6: ARIMA(2,0,3)  AIC 2622.85169078  BIC 2648.98507972  HQIC 2633.29986951
In [88]:
print "Metrics for Model 1 ARMA(2,0):    ",mean_forecast_err(dta.SUNACTIVITY, predict_sunspots20)," MFE  ", mean_absolute_err(dta.SUNACTIVITY, predict_sunspots20),"MAE"
print "Metrics for Model 2 ARMA(3,0):    ",mean_forecast_err(dta.SUNACTIVITY, predict_sunspots30)," MFE  ", mean_absolute_err(dta.SUNACTIVITY, predict_sunspots30),"MAE"
print "Metrics for Model 3 ARMA(2,1):    ",mean_forecast_err(dta.SUNACTIVITY, predict_sunspots40)," MFE  ", mean_absolute_err(dta.SUNACTIVITY, predict_sunspots40),"MAE"
print "Metrics for Model 4 ARMA(2,3):    ",mean_forecast_err(dta.SUNACTIVITY, predict_sunspots50)," MFE  ", mean_absolute_err(dta.SUNACTIVITY, predict_sunspots50),"MAE"
print "Metrics for Model 5 ARIMA(3,0,2): ",mean_forecast_err(dta.SUNACTIVITY, predict_sunspots1)," MFE  ", mean_absolute_err(dta.SUNACTIVITY, predict_sunspots1),"MAE"
print "Metrics for Model 6 ARIMA(2,0,3): ",mean_forecast_err(dta.SUNACTIVITY, predict_sunspots2)," MFE  ", mean_absolute_err(dta.SUNACTIVITY, predict_sunspots2),"MAE"
Metrics for Model 1 ARMA(2,0):     4.73038833847  MFE   0.134688778061 MAE
Metrics for Model 2 ARMA(3,0):     5.63694046318  MFE   -0.141136496478 MAE
Metrics for Model 3 ARMA(2,1):     5.65195019619  MFE   4.82202232657 MAE
Metrics for Model 4 ARMA(2,3):     5.27758730154  MFE   0.325297875678 MAE
Metrics for Model 5 ARIMA(3,0,2):  -13.3948141872  MFE   0.357429358606 MAE
Metrics for Model 6 ARIMA(2,0,3):  5.27758730154  MFE   0.325297875678 MAE

Analysis

Clearly Model 5 minimizes (aic, bic, hqic). However, Model 1 has the smallest MFE and MAE. In terms of consistency, Model 2 minimizes AIC, BIC, HQIC, MFE and MAE. Also, ARIMA(3,0,2) has the lowest AIC, BIC and HQIC values, though it slightly overfits. Notice that ARIMA(2,0,3) and ARMA(2,3) are the same. Why?

Conclusion

Based on the analysis, I would be conservative and choose Model 1, ARMA(2,0), above the other models.

More Experimentation

Now lets build a few model models for comparison. Note we will renumber the models as well.

In [89]:
arma_mod10 = sm.tsa.ARMA(dta, (1,0)).fit()
arma_mod20 = sm.tsa.ARMA(dta, (2,0)).fit()
arma_mod30 = sm.tsa.ARMA(dta, (3,0)).fit()
arma_mod40 = sm.tsa.ARMA(dta, (2,1)).fit()
arma_mod50 = sm.tsa.ARMA(dta, (2,3)).fit()
arima_mod1 = sm.tsa.ARIMA(dta, (3,0,2)).fit()
arima_mod2 = sm.tsa.ARIMA(dta, (2,0,2)).fit()
arima_mod3 = sm.tsa.ARIMA(dta, (1,0,0)).fit()
arima_mod4 = sm.tsa.ARIMA(dta, (0,1,0)).fit()
arima_mod5 = sm.tsa.ARIMA(dta, (0,0,1)).fit()
arima_mod6 = sm.tsa.ARIMA(dta, (1,1,0)).fit()
arima_mod7 = sm.tsa.ARIMA(dta, (0,1,1)).fit()
arima_mod8 = sm.tsa.ARIMA(dta, (1,1,1)).fit()
arima_mod9 = sm.tsa.ARIMA(dta, (3,0,3)).fit()
arima_mod10= sm.tsa.ARIMA(dta, (1,0,6)).fit()
arima_mod11= sm.tsa.ARIMA(dta, (1,0,3)).fit()

We caclualue AIC, BIC, and HQIC.

In [90]:
print "Model 01: ARMA(1,0)     AIC", arma_mod20.aic, " BIC", arma_mod20.bic, " HQIC", arma_mod20.hqic
print "Model 02: ARMA(2,0)     AIC", arma_mod20.aic, " BIC", arma_mod20.bic, " HQIC", arma_mod20.hqic
print "Model 03: ARMA(3,0)     AIC", arma_mod30.aic, " BIC", arma_mod30.bic, " HQIC", arma_mod30.hqic
print "Model 04: ARMA(2,1)     AIC", arma_mod40.aic, "BIC", arma_mod40.bic, " HQIC", arma_mod40.hqic
print "Model 05: ARMA(2,3)     AIC", arma_mod50.aic, " BIC", arma_mod50.bic, " HQIC", arma_mod50.hqic
print "Model 06: ARIMA(3,0,2)  AIC", arima_mod1.aic, " BIC", arima_mod1.bic, " HQIC", arima_mod1.hqic
print "Model 07: ARIMA(2,0,2)  AIC", arima_mod2.aic, " BIC", arima_mod2.bic, " HQIC", arima_mod2.hqic
print "Model 08: ARIMA(1,0,0)  AIC", arima_mod3.aic, " BIC", arima_mod3.bic, " HQIC", arima_mod3.hqic
print "Model 09: ARIMA(0,1,0)  AIC", arima_mod4.aic, " BIC", arima_mod4.bic, " HQIC", arima_mod4.hqic
print "Model 10: ARIMA(0,0,1)  AIC", arima_mod5.aic, " BIC", arima_mod5.bic, " HQIC", arima_mod5.hqic
print "Model 11: ARIMA(1,1,0)  AIC", arima_mod6.aic, " BIC", arima_mod6.bic, " HQIC", arima_mod6.hqic
print "Model 12: ARIMA(0,1,1)  AIC", arima_mod7.aic, " BIC", arima_mod7.bic, " HQIC", arima_mod7.hqic
print "Model 13: ARIMA(1,1,1)  AIC", arima_mod8.aic, " BIC", arima_mod8.bic, " HQIC", arima_mod8.hqic
print "Model 14: ARIMA(3,0,3)  AIC", arima_mod9.aic, " BIC", arima_mod9.bic, " HQIC", arima_mod9.hqic
print "Model 15: ARIMA(1,0,6)  AIC", arima_mod10.aic, " BIC", arima_mod10.bic, " HQIC", arima_mod10.hqic
print "Model 16: ARIMA(1,0,3)  AIC", arima_mod11.aic, " BIC", arima_mod11.bic, " HQIC", arima_mod11.hqic
Model 01: ARMA(1,0)     AIC 2622.63633806  BIC 2637.56970317  HQIC 2628.60672591
Model 02: ARMA(2,0)     AIC 2622.63633806  BIC 2637.56970317  HQIC 2628.60672591
Model 03: ARMA(3,0)     AIC 2619.4036287  BIC 2638.07033508  HQIC 2626.8666135
Model 04: ARMA(2,1)     AIC 2620.27719156 BIC 2638.94389794  HQIC 2627.74017636
Model 05: ARMA(2,3)     AIC 2622.85169078  BIC 2648.98507972  HQIC 2633.29986951
Model 06: ARIMA(3,0,2)  AIC 2581.57229897  BIC 2607.7056879  HQIC 2592.0204777
Model 07: ARIMA(2,0,2)  AIC 2620.87269582  BIC 2643.27274348  HQIC 2629.82827759
Model 08: ARIMA(1,0,0)  AIC 2819.16915243  BIC 2830.36917626  HQIC 2823.64694331
Model 09: ARIMA(0,1,0)  AIC 2835.1157734  BIC 2842.57597297  HQIC 2838.09870518
Model 10: ARIMA(0,0,1)  AIC 2886.90066859  BIC 2898.10069242  HQIC 2891.37845948
Model 11: ARIMA(1,1,0)  AIC 2730.68977742  BIC 2741.88007677  HQIC 2735.16417509
Model 12: ARIMA(0,1,1)  AIC 2743.27827663  BIC 2754.46857598  HQIC 2747.7526743
Model 13: ARIMA(1,1,1)  AIC 2724.51170879  BIC 2739.43210792  HQIC 2730.47757235
Model 14: ARIMA(3,0,3)  AIC 2575.69564491  BIC 2605.56237513  HQIC 2587.6364206
Model 15: ARIMA(1,0,6)  AIC 2644.84706483  BIC 2678.44713632  HQIC 2658.28043748
Model 16: ARIMA(1,0,3)  AIC 2655.64444292  BIC 2678.04449059  HQIC 2664.60002469

Now we calculate predictions for each model.

In [91]:
predict_sunspots10 = arma_mod10.predict('1990', '2012', dynamic=True)
predict_sunspots20 = arma_mod20.predict('1990', '2012', dynamic=True)
predict_sunspots30 = arma_mod30.predict('1990', '2012', dynamic=True)
predict_sunspots40 = arma_mod40.predict('1990', '2012', dynamic=True)
predict_sunspots50 = arma_mod50.predict('1990', '2012', dynamic=True)
predict_sunspots1 = arima_mod1.predict('1990', '2012', dynamic=True)
predict_sunspots2 = arima_mod2.predict('1990', '2012', dynamic=True)
predict_sunspots3 = arima_mod3.predict('1990', '2012', dynamic=True)
predict_sunspots4 = arima_mod4.predict('1990', '2012', dynamic=True)
predict_sunspots5 = arima_mod5.predict('1990', '2012', dynamic=True)
predict_sunspots6 = arima_mod6.predict('1990', '2012', dynamic=True)
predict_sunspots7 = arima_mod7.predict('1990', '2012', dynamic=True)
predict_sunspots8 = arima_mod8.predict('1990', '2012', dynamic=True)
predict_sunspots9 = arima_mod9.predict('1990', '2012', dynamic=True)
predict_sunspots10 = arima_mod10.predict('1990', '2012', dynamic=True)
predict_sunspots11 = arima_mod11.predict('1990', '2012', dynamic=True)

Now we calculate MFE and MAE for each model.

In [92]:
print "Metrics for Model 01 ARMA(1,0):    ",mean_forecast_err(dta.SUNACTIVITY, predict_sunspots10)," MFE  ", mean_absolute_err(dta.SUNACTIVITY, predict_sunspots10),"MAE"
print "Metrics for Model 02 ARMA(2,0):    ",mean_forecast_err(dta.SUNACTIVITY, predict_sunspots20)," MFE  ", mean_absolute_err(dta.SUNACTIVITY, predict_sunspots20),"MAE"
print "Metrics for Model 03 ARMA(3,0):    ",mean_forecast_err(dta.SUNACTIVITY, predict_sunspots30)," MFE  ", mean_absolute_err(dta.SUNACTIVITY, predict_sunspots30),"MAE"
print "Metrics for Model 04 ARMA(2,1):    ",mean_forecast_err(dta.SUNACTIVITY, predict_sunspots40)," MFE  ", mean_absolute_err(dta.SUNACTIVITY, predict_sunspots40),"MAE"
print "Metrics for Model 05 ARMA(2,3):    ",mean_forecast_err(dta.SUNACTIVITY, predict_sunspots50)," MFE  ", mean_absolute_err(dta.SUNACTIVITY, predict_sunspots50),"MAE"
print "Metrics for Model 06 ARIMA(3,0,2): ",mean_forecast_err(dta.SUNACTIVITY, predict_sunspots1)," MFE  ", mean_absolute_err(dta.SUNACTIVITY, predict_sunspots1),"MAE"
print "Metrics for Model 07 ARIMA(2,0,2): ",mean_forecast_err(dta.SUNACTIVITY, predict_sunspots2)," MFE  ", mean_absolute_err(dta.SUNACTIVITY, predict_sunspots2),"MAE"
print "Metrics for Model 08 ARIMA(1,0,0): ",mean_forecast_err(dta.SUNACTIVITY, predict_sunspots3)," MFE  ", mean_absolute_err(dta.SUNACTIVITY, predict_sunspots3),"MAE"
print "Metrics for Model 09 ARIMA(0,1,0): ",mean_forecast_err(dta.SUNACTIVITY, predict_sunspots4)," MFE  ", mean_absolute_err(dta.SUNACTIVITY, predict_sunspots4),"MAE"
print "Metrics for Model 10 ARIMA(0,0,1): ",mean_forecast_err(dta.SUNACTIVITY, predict_sunspots5)," MFE  ", mean_absolute_err(dta.SUNACTIVITY, predict_sunspots5),"MAE"
print "Metrics for Model 11 ARIMA(1,1,0): ",mean_forecast_err(dta.SUNACTIVITY, predict_sunspots6)," MFE  ", mean_absolute_err(dta.SUNACTIVITY, predict_sunspots6),"MAE"
print "Metrics for Model 12 ARIMA(0,1,1): ",mean_forecast_err(dta.SUNACTIVITY, predict_sunspots7)," MFE  ", mean_absolute_err(dta.SUNACTIVITY, predict_sunspots7),"MAE"
print "Metrics for Model 13 ARIMA(1,1,1): ",mean_forecast_err(dta.SUNACTIVITY, predict_sunspots8)," MFE  ", mean_absolute_err(dta.SUNACTIVITY, predict_sunspots8),"MAE"
print "Metrics for Model 14 ARIMA(3,0,3): ",mean_forecast_err(dta.SUNACTIVITY, predict_sunspots9)," MFE  ", mean_absolute_err(dta.SUNACTIVITY, predict_sunspots9),"MAE"
print "Metrics for Model 15 ARIMA(1,0,6): ",mean_forecast_err(dta.SUNACTIVITY, predict_sunspots10)," MFE  ", mean_absolute_err(dta.SUNACTIVITY, predict_sunspots10),"MAE"
print "Metrics for Model 16 ARIMA(1,0,3): ",mean_forecast_err(dta.SUNACTIVITY, predict_sunspots11)," MFE  ", mean_absolute_err(dta.SUNACTIVITY, predict_sunspots11),"MAE"
Metrics for Model 01 ARMA(1,0):     2.46349253974  MFE   0.0491634383748 MAE
Metrics for Model 02 ARMA(2,0):     4.73038833847  MFE   0.134688778061 MAE
Metrics for Model 03 ARMA(3,0):     5.63694046318  MFE   -0.141136496478 MAE
Metrics for Model 04 ARMA(2,1):     5.65195019619  MFE   4.82202232657 MAE
Metrics for Model 05 ARMA(2,3):     5.27758730154  MFE   0.325297875678 MAE
Metrics for Model 06 ARIMA(3,0,2):  -13.3948141872  MFE   0.357429358606 MAE
Metrics for Model 07 ARIMA(2,0,2):  5.26199149696  MFE   0.215826413782 MAE
Metrics for Model 08 ARIMA(1,0,0):  -13.3102092512  MFE   0.207465788606 MAE
Metrics for Model 09 ARIMA(0,1,0):  61.396291866  MFE   -9004.78947368 MAE
Metrics for Model 10 ARIMA(0,0,1):  9.30138026093  MFE   0.183541896273 MAE
Metrics for Model 11 ARIMA(1,1,0):  57.8502628003  MFE   11612.9783655 MAE
Metrics for Model 12 ARIMA(0,1,1):  60.7306048362  MFE   -17940.4779952 MAE
Metrics for Model 13 ARIMA(1,1,1):  59.019277972  MFE   10926.8374441 MAE
Metrics for Model 14 ARIMA(3,0,3):  -10.2668182636  MFE   0.281033130874 MAE
Metrics for Model 15 ARIMA(1,0,6):  2.46349253974  MFE   0.0491634383748 MAE
Metrics for Model 16 ARIMA(1,0,3):  -3.61421431343  MFE   0.0652529614671 MAE

Conclusion

We had assumed from our preliminary data analysis that the model would have at leat 2 autoregression terms. So, we will first examine and compare the models ARMA(2,0), ARMA(2,1), ARMA(2,3) and ARIMA(2,0,3)

Model 14 ARIMA(2,0,3) has relatively good values for AIC, BIC, HQIC, and MAE, but not MFE and underfits (MFE = 5.2779 > 4).

Model 02 ARMA(2,0) has the next best MAE and a relatively good BIC, but slightly underfits (MFE = 4.7303 > 4).

Model 01 ARMA(2,1) and ARMA(2,3) are not as good as ARMA(2,0).

However, look at ARMA(1,0). It has the same AIX, BIC, and HQIC, but it minimizes MFE and MAE.

We choose Model 01 ARMA(1,0) as the best model based on these results. However, I would take the best performing model and plot their prediction and compare them.

 Now try to build ARNA(3,2) on your own and compare it with the best model.

Jeffrey StricklandAuthored by:
Jeffrey Strickland, Ph.D.

Jeffrey Strickland, Ph.D., is the Author of “Predictive Analytics Using R” and a Senior Analytics Scientist with Clarity Solution Group. He has performed predictive modeling, simulation and analysis for the Department of Defense, NASA, the Missile Defense Agency, and the Financial and Insurance Industries for over 20 years. Jeff is a Certified Modeling and Simulation professional (CMSP) and an Associate Systems Engineering Professional. He has published nearly 200 blogs on LinkedIn, is also a frequently invited guest speaker and the author of 20 books including:

  • Discrete Event simulation using ExtendSim
  • Crime Analysis and Mapping
  • Missile Flight Simulation, Second Edition
  • Mathematical modeling of Warfare and Combat Phenomenon
  • Predictive Modeling and Analytics
  • Using Math to Defeat the Enemy
  • Verification and Validation for Modeling and Simulation
  • Simulation Conceptual Modeling

Connect with Jeffrey Strickland
Contact Jeffrey Strickland

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