**5. Time correlation analysis**

As a starting point in time cross correlation analysis we examine correlation coefficients single numbers that can be used as simple measure of cross correlation intensity between two variables. There are several coefficients named after their inventors such as Spearman, Pearson and Kendal [24] and they all indicate certain statistical properties that can relate two data series. **Table 5** summarizes standard cross coefficient between relative temperature and CO2 for individual cycles as well as for the entire Vostok data set. The intensity of the cross correlation is quite high, on average more than 0.8 for the entire set. If we split the cycles into up and down sub cycles we obtain **Table 6** which indicates cross correlation coefficients for up and down cycle parts. Overall these coefficients indicate bigger spread between up and down sub cycles, and are very sensitive to where the break between up and down parts is chosen.

Time correlation analysis produces a variety of useful information about periodicity and correlation strength among data samples of a given quantity. In particular autocorrelations produce the measure of self correlation of a data series, and the cross correlations indicate how two different data sets are correlated. We used standard programs such as R and Excel to generate those. One property of crosscorrelations Rxy is very useful in analysis of relative temperature vs. CO2 content and that is the estimate of the time delay between the two. In general one needs to locate the maximum value point for Rxy and locate the corresponding argument, time lag in our case, between relative temperature and CO2 content. **Figure 3** (with numerical values around zero lag) illustrates short term calculations for cross correlation between the two variables for the total C1234 cycle. We can read the value of the delay τdelay between temperature and CO2 as approximately equal to two lag units. Since the calculation is done for the entire data set, from **Table 4** we can read an average sampling time for C1234 as 1149 years, hence we can make a rough upper limit

*Kalman Filter Harmonic Bank for Vostok Ice Core Data Analysis and Climate Predictions*

*DOI: http://dx.doi.org/10.5772/intechopen.94263*

τdelay <2 times 1, 149 years 2, 300 years

*Total long term temperature and CO2 autocorrelations (left to right, above) and cross correlation (bellow)*

The calculation is approximate, primarily because the non-uniform distribution of the Vostok data. Some data is separated by only hundred years rather than thousands. This points to a need to make the data more uniform by inserting additional data. We address how to harmonize these data in Section 5. The total delay appears to be of order of 2000 or more years and not 100–200 years. That may be an important finding which can influence our thinking about the role of CO2 increase caused by human actions. **Figure 4** indicates entire data set auto and cross correlation. It is clear that the data exhibits some periodicity. To determine average time delays between relative temperature and CO2 for individual cycles we examine **Figures 5**–**8** which also have numerical values around zero lag. The first two diagrams in **Figure 5** are autocorrelations and they also indicate certain periodicity within the each cycle but obviously not as well as the entire data set. The third diagram shows cross correlation between two variables. The time delay can be read from cross correlation and for C1 it is less than one lag period but maybe more than zero lag, due to the non uniformity of data. We can estimate it as less than half of one period lag. From **Table 3** for both

approximation of the size of the delay for the entire data set to be:

**Figure 4.**

**45**

*indicating inherent data periodicity.*

In general, one of the coefficients (up or down) is considerably larger than the overall single cycle coefficient. This might indicate that the usefulness of the individual up and down cross correlation analysis may be limited of the current C1 cycle. In the context of machine learning methodology this points to putting less emphasis on the cycles from longer in the past compared to the ongoing C1 cycle. To complete this analysis, the down-up period (boldfaced) ratios in **Table 6** indicate that the descending period is on average 6 to 8.6 times longer than the ascending period during interglacials, given that the cooling period is that much longer than the warming part. See also **Figure 9** for C1 CO2.


**Table 5.**



#### **Table 6.**

*Cross correlation coefficients for individual sub cycles.*

#### *Kalman Filter Harmonic Bank for Vostok Ice Core Data Analysis and Climate Predictions DOI: http://dx.doi.org/10.5772/intechopen.94263*

Time correlation analysis produces a variety of useful information about periodicity and correlation strength among data samples of a given quantity. In particular autocorrelations produce the measure of self correlation of a data series, and the cross correlations indicate how two different data sets are correlated. We used standard programs such as R and Excel to generate those. One property of crosscorrelations Rxy is very useful in analysis of relative temperature vs. CO2 content and that is the estimate of the time delay between the two. In general one needs to locate the maximum value point for Rxy and locate the corresponding argument, time lag in our case, between relative temperature and CO2 content. **Figure 3** (with numerical values around zero lag) illustrates short term calculations for cross correlation between the two variables for the total C1234 cycle. We can read the value of the delay τdelay between temperature and CO2 as approximately equal to two lag units. Since the calculation is done for the entire data set, from **Table 4** we can read an average sampling time for C1234 as 1149 years, hence we can make a rough upper limit approximation of the size of the delay for the entire data set to be:
