## 4.1 Time series analysis in SPSS (17 and higher)

### 4.1.1 Nonlinear Growth curves in SPSS

- Open the file Growthregression.sav, it contains two variables:
`Time`

and`Y(t)`

.

This is data from an iteration of the logistic growth differential equation you are familiar with by now, but let’s pretend it’s data from one subject measured on 100 occasions.

- Plot Y(t) against Time Recognize the shape?
- To get the growth parameter we’ll try to fit the solution of the logistic flow with SPSS nonlinear regression
- Select nonlinear… from the
`Analysis`

>>`Regression`

menu. - Here we can build the solution equation. We need three parameters: a.
**Yzero**, the initial condition. b.*K*, the carrying capacity. c.*r*, the growth rate. - Fill these in where it says
`parameters`

give all parameters a starting value of \(0.01\)

- Select nonlinear… from the
- Take a good look at the analytic solution of the (stilized) logistic flow:

\[ Y(t) = \frac{K * Y_0}{Y_0 + \left(K-Y_{0}\right) * e^{(-K*r*t)} } \]

Tr to build this equation, the function fo \(e\) is called `EXP`

in `SPSS`

(`Function Group`

>> `Arithmetic`

) Group terms by using parentheses as shown in the equation.

- If you think you have built the model correctly, click on
`Save`

choose`predicted values`

. Then paste your syntax and run it!- Check the estimated parameter values.
- Check \(R^2\)!!!

Plot a line graph of both the original data and the predicted values. (Smile)

- A polynomial fishing expedition:
Create time-varying covariates of \(Y(t)\):

`COMPUTE T1=Yt * Time. COMPUTE T2=Yt * (Time ** 2). COMPUTE T3=Yt * (Time ** 3). COMPUTE T4=Yt * (Time ** 4). EXECUTE.`

- Use these variables as predictors of \(Y(t)\) in a regular linear regression analysis. This is called a
*polynomial regression*: Fitting combinations of curves of different shapes on the data. Before you run the analysis: Click

`Save`

Choose`Predicted Values: Unstandardized`

Look at \(R^2\). This is also almost 1. Which model is better? Think about this: Based o the results o the linear regression what can yo tell about the

*growth rate*, the*carrying capacity*or the*initial condition*?Create a line graph of \(Y(t)\), plot the predicted values of the nonlinear regression and the unstandardized predicted values of the linear polynomial regression against

`time`

in one figure.Now you can see that the shape is approximated by the polynomials, but it is not quite the same. Is this really a model of a growth process as we could encounter it in nature?

### 4.1.2 Correlation functions and AR-MA models

Download the file

`series.sav`

from blackboard. It contains three time series`TS_1`

,`TS_2`

and`TS_3`

. As a first step look at the mean and the standard deviation (`Analyze`

>>`Descriptives`

). Suppose these were time series from three subjects in an experiment, what would you conclude based on the means and SD’s?Let’s visualize these data. Go to

`Forecasting`

>>`Time Series`

>>`Sequence Charts`

. Check the box One chart per variable and move all the variables to Variables. Are they really the same?- Let’s look at the
`ACF`

and`PCF`

- Go to
`Analyze`

>>`Forecasting`

>>`Autocorrelations`

. - Enter all the variables and make sure both
*Autocorrelations*(ACF) and*Partial autocorrelations*(PACF) boxes are checked. Click`Options`

, and change the`Maximum Number of Lags`

to`30`

. - Use the table to characterize the time series:

- Go to

SHAPE | INDICATED MODEL |
---|---|

Exponential, decaying to zero | Autoregressive model. Use the partial autocorrelation plot to identify the order of the autoregressive model |

Alternating positive and negative, decaying to zero | Autoregressive model. Use the partial autocorrelation plot to help identify the order. |

One or more spikes, rest are essentially zero | Moving average model, order identified by where plot becomes zero. |

Decay, starting after a few lags | Mixed autoregressive and moving average model. |

All zero or close to zero | Data is essentially random. |

High values at fixed intervals | Include seasonal autoregressive term. |

No decay to zero | Series is not stationary. |

- You should have identified just one time series with autocorrelations:
`TS_2`

. Try to fit an`ARIMA(p,0,q)`

model on this time series.- Go to
`Analyze`

>>`Forecasting`

>>`Create Model`

, and at`Method`

(Expert modeler) choose`ARIMA`

. - Look back at the
`PACF`

to identify which order (`p`

) you need (last lag value at which the correlation is still significant). This lag value should go in the Autocorrelation p box. - Start with a Moving Average
`q`

of one. The time series variable`TS_2`

is the`Dependent`

. - You can check the statistical significance of the parameters in the output under
`Statistics`

, by checking the box`Parameter Estimates`

. - This value for
`p`

is probably too high, because not all AR parameters are significant. - Run ARIMA again and decrease the number of AR parameters by leaving out the non-significant ones.

- Go to
By default

`SPSS`

saves the predicted values and 95% confidence limits (check the data file). We can now check how well the prediction is: Go to`Graphs`

>>`Legacy Dialogs`

>>`Line.`

Select`Multiple`

and`Summaries of Separate Variables`

. Now enter`TS_2`

,`Fit_X`

,`LCL_X`

and`UCL_X`

in`Lines Represent`

.`X`

should be the number of the last (best) model you fitted, probably 2. Enter`TIME`

as the`Category Axis`

.- In the simulation part of this course we have learned a very simple way to explore the dynamics of a system: The return plot. The time series is plotted against itself shifted by 1 step in time.
Create return plots (use a Scatterplot) for the three time series. Tip: You can easily create a

`t+1`

version of the time series by using the LAG function in a`COMPUTE`

statement. For instance:`COMPUTE TS_1_lag1 = LAG(TS_1)`

Are your conclusions about the time series the same as in 3. after interpreting these return plots?