In the DCF analysis a bin size of 50 days was used. Several other bin sizes from 30 to 100 days were tried, but 50 days seemed to give, on the average, the best results. The DCF analysis was complemented with visual inspection. The visual inspection was made by shifting light curves in respect to each other, trying different time lags. This analysis was done by using the whole light curves, and also by using only parts of the light curves. The visual inspection was performed both before the correlation analysis in order to find independent results, and after the analysis in order to confirm the results given by the DCF. The DCF was also calculated for the 37 and 22 GHz light curves. The correlation between the two radio frequencies was strong for all of the sources, the time lags were zero or small.

For most of the sources the model radio flare fit was successful. For sources that show only modest radio flux variations, reliable model flare fits are impossible.

At the epoch of each optical data point the phase and the brightness of concurrent model radio outbursts were calculated. The phase $\Phi$ gets values between [0,2], and the brightness of model radio outbursts, as well as the brightness of the composite model light curve, gets values from minimum flux to maximum flux that are calculated for each source individually. These ranges were divided into ten bins and the average of the optical flux level was calculated for each bin (see e.g. Fig. 4).

For each source we identified the radio phase at the time of the highest average optical flux level. In Table 2 the phase of the model radio flare is marked with "-'' if the phase $\Phi <0.7$ , "-'' when $0.7 \le \Phi <0.9$ , "0'' when $0.9 \le \Phi \le 1.1$ , "+'' when $1.1 <\Phi \le 1.3$ and "++'' when $\Phi >1.3$ . The "-'' signs therefore refer to developing stage of model radio outbursts and "+'' signs the decaying stage of the outbursts.

In comparing the average optical and radio flux levels we divided the sources into three categories based on the type of dependence found. In the first category, marked with "d/D'', are the sources in which the highest optical flux levels occur during the dimmest third of the radio flux level. In the second category ("m/M''), are the sources in which the highest optical flux levels occur during the middle third of the radio flux level. In the third category ("b/B'') are the sources in which the highest optical flux levels occur during the brightest third of the radio flux level. As an example see Fig. 8 and explanation in the text for the source.

In the following we discuss each source individually. In this paper we only show some of the figures, selected in order to show typical features for each step of the analysis. The figures for all the sources are available in electronic form.

4.1 S2 0109+224

This source has been intensively monitored since 1995 (Figs. 5-8). The DCF shows a strong correlation between the optical and the 37 GHz and the 22 GHz events with time lags of approximately 350 days and 400 days, respectively. The time lag between 37 GHz and 22 GHz is 0-50 days. Visual inspection confirms these results.

The model flare fit to radio light curves was successful. The brightest optical flux levels do not occur at the time of the brightest radio flux, but rather at 2/3 of the maximum radio flux levels (Fig. 8). This is expected, since the time lag between optical and radio is large.

4.2 0219+428 (3C 66A)

This source has been monitored optically frequently since 1993. Radio monitoring for this source started at the same time. In the optical bands this source is very active, showing several bright outbursts. After 1997 the optical activity settled down. The DCF shows a possible correlation between optical and both the radio bands with a time lag of c. 400 days. Visual inspection shows that this is also possible, although some possible simultaneous events are present as well (e.g. 1996). The time lag between the radio frequencies is 0-50 days.

The model flare fit is not very good, because the radio flux variations are small. Nevertheless, the fit shows the trends in variations fairly well. The brightest optical flux levels seem to coincide with fairly bright radio flux levels at the rising part of the radio flares (i.e., the shock is growing).

4.3 AO 0235+164

This source has been monitored intensively since 1986 at Metsähovi. The optical light curve reaches back to 1976. The source shows large variations in all the studied frequency bands. The DCF doesn't reveal strong correlations, but it shows a possible correlation with a small time lag between the optical and the 37 GHz light curves. Visual inspection shows the possible correlation as well. At least two radio outbursts coincide with simultaneous optical flares (1987 and 1998).

The model flare fits were successful, especially after 1986. The brightest optical flux levels occur, on the average, in the rising part of the radio flare. The optical flux levels tend to be high when the radio flux levels are high.

Balonek & Dent (1980) and B82 found correlations with no time lag. C95 agreed with these results. Webb & Malkan (2000) studied the outburst in 1997, and suggested that variations in the optical bands and in the radio frequencies at 14.5, 8.0 and 4.5 GHz were a result of a microlensing effect, while earlier outbursts were not. Raiteri et al. (2001) found optical variability correlated with radio with a time lag of 0-60 days.

4.4 PKS 0420-014

This source shows variability at all frequencies. Several bright optical outbursts have been observed since 1990. The radio light curve time coverage is good after 1987. The DCF doesn't reveal any strong correlation between optical and radio events with optical events leading radio events. A possible correlation can be seen with a time lag of 0-200 days, though. Visual inspection doesn't give any clear correlations, either. Some possible simultaneous events are present (e.g., the outbursts in 1985 and in Feb. 1991).

The model flare fits were successful after 1990. The brightest optical flux levels seem to occur, on the average, after the radio outburst peak at both the radio frequencies. The optical flux levels are high when radio flux levels are high, too.

Pomphrey et al. (1976) found a possible correlation with optical leading radio events by 0.2 years. Dent et al. (1979) found evidence for optical events leading radio events by 2.2 years. B82 found possible correlations between optical and radio with several time lags between 1-2 years. Also, one simultaneous event was recorded. The best correlation function result was with a 1.75 year time lag.

4.5 PKS 0422+004

The light curves of this source are fairly densely sampled. In the optical bands the source is very variable. At the radio frequencies the sampling is less frequent. The DCF didn't show any strong correlations. A weak correlation is seen with a time lag of c. 250 days. The highest DCF value is at 900-day time lag but that may not be a real feature. Visual inspection doesn't reveal clear correlations, although one possible simultaneous event is present (c. 1994.9).

The model flare fits were quite good, although the small variability in the radio light curves made fitting the model flares difficult. The brightest optical flux levels occur, on the average, in the rising part of the radio flares, and the radio flux levels are high, as well. This suggests that the possible time lag between optical and radio events is not large.

4.6 PKS 0735+178

The optical and radio light curves are extensive. The optical data used in this analysis reaches back to 1989 and the radio light curves start from 1980. The 22 GHz and 37 GHz events are strongly correlated without time lag. No mathematical correlation was found between optical and radio events for this source. Visual inspection shows a possible simultaneous event in 1991, and that both optical and radio activity decrease after 1992 (see Figs. 9-12).

The model flare fit was successful. The optical flux level is, on the average, highest when the radio flare has its peak at 37 GHz and reaching maximum at 22 GHz.

Pomphrey et al. (1976) found a possible correlation in one event, with radio events leading optical events by 0.88 years. B82 found a visual correlation between optical and radio variability in this source. Also, with post-1976.5 optical and radio data there is a clear correlation with a small or zero time lag. Hufnagel & Bregman (1992) found no significant correlations for this source between optical and radio. No clear correlation was found by T94 between optical and radio, but at least one optical flare coincided with a radio event (c. 1990). No correlation was found between optical and radio for this source by C95.

4.7 PKS 0736+017

There are too few optical data points to draw any conclusions about correlations between optical and radio in this source. A possible simultaneous event in optical and radio is present at the end of 1996.

The model flare fit is reasonably good but the optical data points are in three clumps which occur when the radio flux is bright at both radio frequencies. The analysis therefore gives a good correlation between the radio and optical flux levels, but the result may be misleading.

Neither Pomphrey et al. (1976), B82 nor T94 found any correlation between the optical and radio events for this source.

4.8 0754+100 (OI 090.4)

The model flare fit is good throughout the 1990's. The optical flux levels tend to be high, on the average, in the rising part of the radio flares, and the radio flux levels are still roughly half of the maximum value. This suggests a longer time lag and agrees with the DCF results.

4.9 0851+202 (OJ 287)

The model flare fits were succesful, although the problem with outbursts overlapping each other is seen here as well. Comparing the optical flux level with the radio model flare flux level and phase is hard, because it is impossible to estimate to which model radio flare a certain optical data point should be connected.

Pomphrey et al. (1976) found a good correlation between optical and radio (2.8 cm) for this object only. B82 found a signifiant correlation between optical and radio events, with no time lag before 1973, and with a one-month time lag since 1974. Valtaoja et al. (1987) studied the optical and 4-37 GHz radio light curves and found that the typical time lag between optical and radio is from 0.12 to 0.54 years. Hufnagel & Bregman (1992) found optical leading radio 8 GHz by 200-800 days. T94 found a fair correlation with a zero time lag, and with other time lags as well. C95 found optical and radio correlated by time lags of either 0-2 months or 9-15 months.

4.10 1156+295 (4C +29.45)

This source is quite intensively monitored in radio but the optical light curve contains long gaps. This source is very variable in both the optical and radio frequencies. The DCF shows peaks at time lags of 450 and 600 days, but these cannot be confirmed by visual inspection. Visual inspection reveals a possible correlation with a time lag of approximately 150 days (see Figs. 13-14).

The model flare fit is good after 1986. The optical points after 1990 only are taken into account for this analysis. The optical and radio flux levels are, on the average, high at same time. This cannot be confirmed by inspecting the optical flux levels against the phase of the radio flares.

T94 found a correlation between optical and radio data with a time lag between 500 and 600 days. C95 saw a major peak in DCF at a 600 days' time lag, but could not confirm the result with visual inspection.

4.11 1219+285 (ON 231)

This source was quiet in optical from the beginning of the 1970's to 1986. Since 1988 it started showing activity in optical. In radio the small activity seems to quiet down after 1993. The DCF doesn't show any correlations between optical and radio. Visual inspection reveals no correlations, either.

The model flare fit was successful after 1986. The optical flux level is brightest after the radio outburst peaks at 37 GHz. The result may be misleading, because the only bright optical outburst occurs just after the last radio outburst. The radio flux levels tend to be low when the optical flux levels are high. At 22 GHz the optical flux levels are high when radio flares are, on the average, at their last decaying stage.

4.12 1226+023 (3C 273)

The radio data sets are extensive and reach back to 1980. The optical light curve studied here is sparser and covers only about 2000 days from 1993. The correlation function has a peak at a time lag of 1000 days. Visual inspection shows that this is possible. More optical data points are needed in the future to confirm this.

The model flare fit was very successful, because both the radio data sets were extremely well sampled. On the average, the brightest optical flux level occurs after the model outburst peaks at 22 GHz, and at the end of the radio flare at 37 GHz. The model radio flare flux level is intermediate at both the radio frequencies when the optical flux level is, on the average, highest.

B82 found no correlation between optical and radio for this source, neither did T94 nor C95. There are other studies as well, like Valtaoja et al. (1991) and Robson et al. (1993), where clear correlations between some optical and radio events have been found.

4.13 1253-055 (3C 279)

This source has been monitored frequently in optical bands since the 1970's. Radio light curves reach back to 1984. The DCF doesn't show correlations between optical and radio. The high DCF value at a 950-day time lag is probably due to the length of the optical light curve. Visual inspection shows similar simultaneous behaviour between different frequency bands until the last optical outburst.

The model flare fits were succesful. The optical flux levels are high, on the average, at the end of the radio flare at 37 GHz and at the peak of the radio flare at 22 GHz. However, if the last optical outburst is excluded (because visual inspection indicates possible simultaneous behaviour until the last optical outburst), the brightest optical flux level coincides with the highest radio flux levels at both the radio frequencies at the peak of the radio flare.

B82 found no correlation between optical and radio. T94 found a strong correlation between optical and radio with no time lag.

4.14 1633+382 (4C +38.41)

The optical data set for this source is short. Radio events have been monitored since 1982. Visual inspection shows a simultaneous optical and radio event in 1995. The DCF doesn't reveal any correlations between optical and radio.

The model flare fit is fairly good. There is no clear correlation between the optical and radio flux levels. The optical flux level is, on the average, highest when the radio flux levels are low/intermediate at 37 GHz and high at 22 GHz. The phase of the radio flare at both the radio frequencies, when the optical flux level is high, indicates a long time lag, if there is a correlation.

4.15 1641+399 (3C 345)

The historical optical light curve of this source reaches back to 1970. The radio monitoring in Metsähovi started in 1980. There is a very active period in both optical and radio between 1990 and 1993. Some events are clearly simultaneous. The DCF doesn't give any single time lag, rather the DCF peak is quite broad. However, visually it is obvious that the optical events correlate well with the radio events.

The model flare fit was successful. There the simultaneity can easily be seen. The optical flux level is, on the average, highest when the radio flux level is high as well, and the radio flares have peaks at both frequencies.

B82 found no significant correlations between optical and radio. Hufnagel & Bregman (1992) found a possible correlation between the optical and the 14.5 GHz radio with a time lag of 100-400 days. T94 found simultaneous optical and radio events.

4.16 1749+096 (4C +09.57)

The optical light curve of this source is short, but optical outbursts seem to coincide with radio flickering at the same time, although the amplitude is different.

The flare fit is reasonably good after 1990. Because there are so few optical data points it is hard to say anything conclusive.

T94 found at least one simultaneous event between optical and 90 GHz. C95 found a correlation with optical events leading radio events by 2 months.

4.17 1807+698 (3C 371)

The optical light curve is, again, too short for effective analysis. The optical brightening at 1997 is simultaneous with a flux rise at 37 GHz.

Neither B82 nor T94 found correlations between optical and radio for this source.

4.18 2200+420 (BL Lac)

BL Lac is strongly variable at all the frequencies studied here (see Figs. 1-4). The optical light curve starts in 1971 and radio light curves in 1979. The variability is so fast and flares occur so often that DCF analysis is not very reliable. The DCF gives best correlation with a time lag of 700-750 days. There are also possible simultaneous events. The optical data set contains large gaps. This makes comparison between optical and radio difficult.

The model flare fit was successful. Comparing optical flux level with radio flux level is difficult, because the events in both are frequent and fast. It is hard to say to which radio outburst an optical data point associates with. The results, however, seem to agree with the DCF results. The radio outburst phase and flux level, when the optical flux level is highest, indicate a large time lag between optical and radio events. This agrees with the DCF results well.

Pomphrey et al. (1976) found no obvious overall correlation between optical and radio, but a possible time lag of 0.5 years with optical leading radio was seen. B82 found a correlation between optical and radio events in post-1980 data, but also claims that there is no significant overall correlation between these regimes. Hufnagel & Bregman (1992) found a weak correlation between optical B-band and radio (14.5 and 8 GHz) with a time lag of 1-2.5 years. T94 said that all optical events after 1980 were semi-simultaneous or simultaneous with radio outbursts. C95 found that the optical and radio data prior to 1977 are correlated with a time lag of two months, but after 1977 the correlations disappear.

4.19 2223-052 (3C 446)

The optical light curve of this source is long and radio light curves reach back to 1985. The optical activity coincides with activity at the radio frequencies. Simultaneous events are seen. The DCF gives a possible correlation between optical and radio with a time lag of about 200 or 300 days. Visual inspection doesn't rule these out, but cannot confirm them, either.

The model flare fit was successful. Comparison between the optical flux levels and the radio flux levels give different results for 37 and 22 GHz data. At 37 GHz the radio flux is, on the average, high or intermediate when the optical flux level is high in the rising part of the radio flare. At 22 GHz the highest optical flux levels occur at the lowest radio flux levels at the beginning of radio flares.

Pomphrey et al. (1976) found no significant correlation between optical and radio. B82 found a possible correlation with optical events leading radio events by ca. 500-600 days. Bregman et al. (1988) found optical-radio correlation with optical leading radio by 400-600 days. Hufnagel & Bregman (1992) found possible correlation between optical and radio with a time lag of 1-2 years with a mean variance correlation method. The DCF analysis showed no significant correlation. T94 found simultaneous or nearly simultaneous optical and radio events.

4.20 2251+158 (3C 454.3)

The data sets are very good in radio and fairly good in optical. In optical there are several gaps. Some events are clearly simultaneous, like the bright outburst in 1993. The DCF gives a strong correlation with no time lag between optical and radio (see Figs. 15-18).

The model flare fit is good. The comparison between optical flux levels and radio flux levels gives similar results with the DCF. The optical flux level is, on the average, high when the radio flux level is high. At 37 GHz the highest optical flux occurs at the peak of the radio flares. At 22 GHz there is no clear preferred phase of radio flares where the optical flux levels are high.

Pomphrey et al. (1976) found a possible correlation with optical leading radio events by 1.2 years. B82 found a possible correlation between optical and radio with time lags of 180, 285, and 310 days. T94 found simultaneous events. C95 didn't find correlations between radio and optical events.

4.21 Summary

Table 2: The results of radio-optical comparisons: $\tau$ is the time lag (in days) between optical and radio events. VIS means that at least one simultaneous optical and radio event was seen, and NC means not correlated. $\Phi$ is the phase, and S the flux level of radio flare, where on the average highest optical flux levels occur. Subindexes: "-'' means that on the average optical flux is strongest in the beginning of outburst ( $\Phi <0.7$ ), "-'' means just before peak ( $0.7 \le \Phi <0.9$ ), "0'' means the peak of model radio outburst ( $0.9 \le \Phi \le 1.1$ ) , "+'' means after peak ( $1.1 <\Phi \le 1.3$ ) and "++'' means the end of outburst ( $\Phi >1.3$ ). In columns S₂₂ and S₃₇ "d'' means dim, "m'' medium and "b'' bright, the corresponding values for the summed model light curve, columns S_37t and S_22t (t is for "total'') are denoted with capital letters. The references in "Results'' are the same as in Table 1.
Source $\tau$ (days) $\Phi _{37}$ $\Phi_{22}$ S₃₇ S₂₂ S_37t S_22t Results of other studies

0109+224 350-400 ++ 0 b m B B -

0219+428 ca. 400 VIS - - b b M B B82(NC)

0235+164 $\approx$ 0 VIS - - b b B D B82 and BD80 ( $\approx$ 0), C95 (0-2 months), W00 ( $\approx$ 0),R01 (0-2 months)

0420-014 0-200? VIS + + b b B B P76 (0.2 yr?), D79 (2.2 yr), B82 (1-2 yr, not constant)

0422+004 250? VIS? - 0 b b B B -

0735+178 VIS 0 - b b B B P76 (-0.88 yr?), B82 ( $\approx$ 0), HB92 (NC), T94 (VIS), C95 (NC)

0736+017 VIS? 0 - b m B B P76 (NC), B82 (NC), T94 (NC)

0754+100 200 or 500 - 0 d b D D -

0851+202 ca. 750? VIS ++ ++ b m M M P76 (0.875-0.60 and 0 yr), U79 (6 months), V87 (2-6 months), HB92 (400 days), T94 (0 d and others), C95 (0, 1-2 and 11 months)

1156+295 450 or 600? - ++ b b D B T94 (500-600 days), C95 (NC)

1219+285 NC 0 ++ m d M D B82 (NC), T94 (NC)

1226+023 1000? ++ 0 m m M B P76 (NC), B82 (NC), T94 (0-200 days?), C95 (NC)

1253-055 VIS 0 0 b b B B B82 (NC), T94 (0 days)

1633+382 VIS - - d b B M -

1641+399 0-300 VIS 0 - b b B B B82 (NC or 720 days), HB92 (100-400 days), T94 (0 days)

1749+096 VIS? 0 0 b b B B T94 (marginal?), C95 (2 months)

1807+698 VIS + + b b B B P76 (0.5 yr?), B82 (NC), T94 (NC)

2200+420 750-800 VIS - ++ d d D D P76 (NC), B82 (NC), Bregman et al. (1988) (400-600 days), HB92 (400-800 days), T94 (200-300 days)

2223-052 200? VIS - - m d B D P76 (NC), B82 (520-560 days), HB92 (NC), T94 ( $\le$ 100 days)

2251+158 0 VIS 0 - b m B B P76 (1.2 yr), B82 (180-310 days), T94 (0-100 days), C95 (NC)

**Table 2:** The results of radio-optical comparisons: $\tau$ is the time lag (in days) between optical and radio events. VIS means that at least one simultaneous optical and radio event was seen, and NC means not correlated. $\Phi$ is the phase, and S the flux level of radio flare, where on the average highest optical flux levels occur. Subindexes: "-'' means that on the average optical flux is strongest in the beginning of outburst ( $\Phi <0.7$ ), "-'' means just before peak ( $0.7 \le \Phi <0.9$ ), "0'' means the peak of model radio outburst ( $0.9 \le \Phi \le 1.1$ ) , "+'' means after peak ( $1.1 <\Phi \le 1.3$ ) and "++'' means the end of outburst ( $\Phi >1.3$ ). In columns S₂₂ and S₃₇ "d'' means dim, "m'' medium and "b'' bright, the corresponding values for the summed model light curve, columns S_37t and S_22t (t is for "total'') are denoted with capital letters. The references in "Results'' are the same as in Table 1.
Source	$\tau$ (days)	$\Phi _{37}$	$\Phi_{22}$	S₃₇	S₂₂	S_37t	S_22t	Results of other studies
0109+224	350-400	++	0	b	m	B	B	-
0219+428	ca. 400 VIS	-	-	b	b	M	B	B82(NC)
0235+164	$\approx$ 0 VIS	-	-	b	b	B	D	B82 and BD80 ( $\approx$ 0), C95 (0-2 months), W00 ( $\approx$ 0),R01 (0-2 months)
0420-014	0-200? VIS	+	+	b	b	B	B	P76 (0.2 yr?), D79 (2.2 yr), B82 (1-2 yr, not constant)
0422+004	250? VIS?	-	0	b	b	B	B	-
0735+178	VIS	0	-	b	b	B	B	P76 (-0.88 yr?), B82 ( $\approx$ 0), HB92 (NC), T94 (VIS), C95 (NC)
0736+017	VIS?	0	-	b	m	B	B	P76 (NC), B82 (NC), T94 (NC)
0754+100	200 or 500	-	0	d	b	D	D	-
0851+202	ca. 750? VIS	++	++	b	m	M	M	P76 (0.875-0.60 and 0 yr), U79 (6 months), V87 (2-6 months), HB92 (400 days), T94 (0 d and others), C95 (0, 1-2 and 11 months)
1156+295	450 or 600?	-	++	b	b	D	B	T94 (500-600 days), C95 (NC)
1219+285	NC	0	++	m	d	M	D	B82 (NC), T94 (NC)
1226+023	1000?	++	0	m	m	M	B	P76 (NC), B82 (NC), T94 (0-200 days?), C95 (NC)
1253-055	VIS	0	0	b	b	B	B	B82 (NC), T94 (0 days)
1633+382	VIS	-	-	d	b	B	M	-
1641+399	0-300 VIS	0	-	b	b	B	B	B82 (NC or 720 days), HB92 (100-400 days), T94 (0 days)
1749+096	VIS?	0	0	b	b	B	B	T94 (marginal?), C95 (2 months)
1807+698	VIS	+	+	b	b	B	B	P76 (0.5 yr?), B82 (NC), T94 (NC)
2200+420	750-800 VIS	-	++	d	d	D	D	P76 (NC), B82 (NC), Bregman et al. (1988) (400-600 days), HB92 (400-800 days), T94 (200-300 days)
2223-052	200? VIS	-	-	m	d	B	D	P76 (NC), B82 (520-560 days), HB92 (NC), T94 ( $\le$ 100 days)
2251+158	0 VIS	0	-	b	m	B	B	P76 (1.2 yr), B82 (180-310 days), T94 (0-100 days), C95 (NC)

The DCF gave a clear correlation for seven sources and a possible correlation for six more sources. For 12 sources, at least one clearly simultaneous outburst was seen in visual inspection. Of these, seven did not show significant correlations in the DCF analysis.

The new method we have introduced, a comparison between optical flux levels and radio model flares, can reveal statistical correlations between the two regimes in cases where the data is not sufficient for correlation analysis or visual inspection. A source with simultaneous radio and optical variations should in our analysis have its average optical flux strongest at the peak of the model radio outbursts (denoted by "0'' in Table 2). Furthermore, there should be a positive correlation between average optical and radio flux levels (denoted by "b/B'' in Table 2). A source for which optical variations precede the radio variations should have its average optical flux strongest during the rising part of the model radio outburst (denoted by "-'' or "-'' in case of a longer time delay). The highest average optical flux levels should correspond to medium (or dim) average radio flux levels (denoted by "m/M'' or "d/D'' in case of a longer time delay).

An example of "a nearly perfect case'' is 1253-055 (3C 279), which lacks only a significant DCF correlation.

For 11 sources the optical flux levels, on the average, are highest at the peak of the model radio outbursts (at least for one of the radio frequencies).

For 16 sources the highest optical flux level occurs during the highest radio flare flux levels at least for one of the radio frequencies. Sixteen sources also show a similar correlation to the modelled total radio flux.