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Trends of brand perceptions

An earlier postdescribes how you can use correspondence analysis to analyze trends . I have repeated one of the scatterplots from this earlier post below. It shows the change in people’s perceptions of technology brands from 2012 to 2017.

The plot has. Thismeans that the distances between row points and the distances between column points are meaningful, but not necessarily the distance between row and column points. Click here for a further explanation about interpretation and normalization .

Separating core and supplementary points

An alternative way to analyze the same data is to consider 2017 as the “ground truth” and plot 2012 as supplementary points. This means that the 2017 data determine the dimensions and axes of the map. You can add 2012 data after as supplementary points. In technical correspondence analysis terminology, the 2012 rows have zero .

We can see below that while the output shows the same themes as the first chart, it is different in the detail.andare now closer together.We can now deduce that on the basis of 2017 data,andhave more similar meanings.

Yet another perspective is to consider 2012 the ground truth and to then plot the 2017 points as supplementary. This produces the results below whereandare further apart than in the original chart. Evidently, the association between innovation and ease of use is a more recent phenomenon.

All three charts are equally valid views of the data. They differ in their emphasis. For instance, the second chart would be most relevant for a study on the state of the technology market in 2017. In this case, the 2012 data is added for context but does not influence the positioning of the 2017 points.

Note that the first chart from the previous post is an “average” (in a strictly non-technical, hand-waving sense!) of the 2012 and 2017 charts.

Focusing on a subset of data

The second example below is the correspondence analysis resulting from a table of 14 car models. Let’s say we wanted to study the 4 German brands. They form a line across the top from on the left, through then . The chart has normalizationThis means that is it valid to compare distances between row points. It is also valid to measure the association between rows and columns by their scalar products.

We might be tempted to say that the was , the and are and the is. Before doing so, note that thetotal explained variance is only 53%. This means there is information hidden in the dimensions that are not plotted.

Let’s repeat theanalysis, this time treating all the non-German cars as supplementary. Now we see that the is very near the center of the plot. This means that it is not strongly associated with any of the characteristics. We can conclude that amongst all 14 cars the is considered a luxury car, but amongst the German cars, it is not. Note also that the total explained variance below is now almost 97%. This means that we can be more confident about our conclusions.

There is also a close relationship between andEvidently, the German cars discriminate relatively little between those characteristics.

Finally, we can check the result above by removing the supplementary points. This produces the chart below, which is the same except we can no longer see how the German cars relate to the non-Germans.


You can add data to a “core” correspondence analysis as supplementary points. The advantage of supplementary points over just analyzing all the data together is that supplementarypoints do not influence the placement of core data points. As the name implies, they are added after the core data has determined the map. Supplementary data points are an excellent way to provide additional context to an analysis that is driven entirely by another part of the data set.

TRY IT OUT Allthe analysisin this post was conducted in Displayr. Review the worked example from this post or run your own analysis by clicking through to this correspondence analysis example (just sign into Displayr first). The supplementary points are specified in the panel, seen on the right after clicking on any map.

The flipDimensionReduction package ( available on GitHub ) was used, which itself uses the ca package for correspondence analysis.

About JakeHoare

After escaping from physics to a career in banking, then escaping from banking, I decided to go back to BASIC and study computing. This led me to rediscover artificial intelligence and data science. I now get to indulge myself at Displayr working in the Data Science team, sometimes on machine learning.

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Worked Example

Wild Sight

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When you hear that a beach has been voted as one of the 10 most beautiful in the world it’s fair to say that expectations are raised a bit, and Wineglass Bay doesn’t disappoint – the white quartz beach curving in a perfect arc beneath granite peaks is one of the iconic images of Tasmania, but this is only one aspect of one of Australia’s most picturesque national parks.

Built on one of the southern outcrops of a huge mass of granite that stretches across Bass Strait all the way to Wilsons Promontory, Freycinet is a beguiling mixture of rugged peaks, rocky headlands and isolated beaches that anywhere else in the world would be covered by tourists, but here you can often have all to yourself. The park has a long history of bushwalking and conservation – in 1916 it was declared as Tasmania’s first national park, along with Mount Field to the southwest – and an excellent network of walking tracks can take you on treks that last anything from 10 minutes to four days.

As a bonus, the location of the peninsula on the east coast of Tasmania shelters it from much of the bad weather that typifies the island’s other walking destinations; it is quite possible to drive from heavy rain or even snow at Cradle Mountain to find yourself arriving here in perfect sunshine.

Wineglass Bay Lookout walk Distance: 3km Time: 1 – 1½ hours Start/Finish: Wineglass Bay Car Park

Look through any book on Tasmania and you will undoubtedly see a picture of Wineglass Bay taken from the lookout. Of course with fame comes popularity and you won’t have this track to yourself but brave the crowds and you’ll be rewarded, as places like this become icons for a reason. The track starts at the car park and almost immediately begins the 250-metre climb to the saddle between Mount Amos and Mount Mayson where the lookout is located (much to the surprise of many visitors, who arrive at the “Wineglass Bay Car Park” expecting the bay to be on their doorstep).

The parks service has done a lot of work grading the track so that it is not as steep as it once was but it is still strenuous; if you have doubts about your health we recommend travelling to Cape Tourville where the much easier walk still offers a glimpse of the bay. As you climb higher views open up below of the water and the town of Coles Bay – a formal lookout has been built at the best location. When you finally reach the saddle, which is studded by huge eroded granite boulders, a side track leads to the lookout with its iconic view. Return to the car park via the same track.

Hazards Beach – Wineglass Bay Circuit Distance: 11km Time: 4 – 5 hours Start/Finish: Wineglass Bay Car Park

This is the classic Freycinet day walk, combining two destinations that are worthy of icon status and adding in mountains, forest and a sense of wilderness that is hard to find less than a day’s trek from civilization. From the car park follow the signposted track to Hazards Beach through dry eucalypt forest below the rocky slopes of Mount Mayson, the easternmost of the Hazards. Close up the size of the peak is impressive, as are the rocky outcrops and cliffs that decorate the slopes above – it must be quite a scramble to get to the top. The frequent gaps in the vegetation provide good views to the north and west across Great Oyster Bay and the town of Coles Bay beyond.

After a couple of hours the track rounds the southwestern slopes of the mountain and drops to sea level, emerging abruptly onto the beach, which stretches away to the south where Mount Freycinet looms over the bay. Walk along the beach, keeping an eye out for the numerous shell middens left behind by the Aboriginal people who lived here for tens of thousands of years before Europeans arrived, and after 1km take the track left into the forest. The change in environment is quite startling; the soils of the isthmus are able to support more substantial forest than the granite of the Hazards, and the many tall trees create a feeling of seclusion for the half-hour walk between the bays.

Wineglass Bay, when you do arrive, is well worth the wait – a perfect arc of white sand which slopes steeply down to (depending on the day) calm turquoise waters or raging surf. There are usually quite a lot of people here, so if you have time it’s worth strolling to the other end of the beach or at least down a little way until the crowds thin out. To complete the circuit take the track from the northern end of the beach which climbs steeply uphill to Wineglass Bay lookout, then follow the lookout track back to the car park.

Cape Tourville Distance: 500m Time: 20 minutes Start/Finish: Cape Tourville Car Park

Cape Tourville has commanding views of the east coast of Freycinet, which made it the ideal spot to build a lighthouse and, more recently, a walking track. The latter is suitable for wheelchairs.

Starting at the car park take the track to the right, which immediately becomes a boardwalk over the coastal scrub. After only 100 metres or so you arrive at the first view, a grand vista that takes in much of the southern extent of the peninsula. To your right are the precipitous granite slopes of Mount Parsons, in the middle distance the peaks of Mount Graham and Freycinet – the highest in the park – and peeking in between is a sliver of sand that is the southern end of Wineglass Bay.

As you follow the boardwalk around the headland the vast expanse of the Tasman Sea becomes apparent. This is a good spot for seeing whales as they cruise past the cape on their annual migration. A short detour to the lighthouse – built in 1971 and never manned – requires negotiating a couple of small steps. After reaching its eastern-most point the boardwalk is replaced my a more conventional walking track which pauses for a view over The Nuggets, a group of small rocky islets that are home to hundreds of seabirds, and then winds its way back to the car park while providing grand views of the rugged coastline to the north and the Friendly Beaches beyond.

Sleepy Bay and Little Gravelly Beach Distance: 1.5km Time: 30 minutes Start/Finish: Sleepy Bay Car Park

Nestled between Cape Tourville and the Hazards, Sleepy Bay provides a close-up look at the rugged coastline shaped by the wild weather that can batter Tasmania’s east coast. A set of gently graded steps leads from the car park to a lookout over a rocky inlet, featuring a parks information panel on the life under the waves. The colours here on a sunny day are quite striking, with boulders of Freycinet’s distinctive granite, tinged pink by iron, fringed below the water by rich beds of kelp and above the waves by reddish lichen that thrives in the salt spray. The water is a mottled mixture of deep greens from the kelp under the water and turquoise where this gives away to sand, fading into a deep blue further out to sea. The track continues to the right, contouring above the shoreline through groves of she-oaks, before descending through a lush gully to the aptly named Little Gravelly Beach.

Unlike other beaches in the area this is made from chips of granite broken off from the surrounding cliffs and rounded by the surf into pebbles a few millimetres across. From a distance these fade into an overall pinkish-grey colour but hold a few in your palm and the individual mineral grains in each pebble are clearly visible, interlocking together in the order that they crystallized out of molten rock hundreds of millions of years ago. The cove is a deeply indented inlet under the rocky slopes of Mount Parsons, which combined with the rocky shoreline and absence of any human presence (the track into the bay is not visible from the shore) give it a wild atmosphere that is quite something. Return by the same track to the car park.

Mount Amos Distance: 4km Time: 3 hours Start/Finish: Wineglass Bay Car Park

The view from the Wineglass Bay lookout is spectacular enough for most visitors, but it is really just the poor cousin to the vista from the top of Mt Amos, the highest peak in the Hazards. The peak provides a birds-eye view of Wineglass Bay and its crescent beach and across the Isthmus to Hazards Beach and the southern peaks of the Freycinet Peninsula. The view the other way – north over Honeymoon Bay – isn’t quite as famous but still more than justifies the difficult scramble to the top. Follow the track to the Wineglass Bay lookout for a short distance before leaving the main path on a rough track pointing to the summit, 450 metres above. Signs warn the route is rough and very slippery and dangerous in wet conditions – take these warnings seriously. Sturdy hiking shoes are highly recommended.

A gentle walk through open forest soon gives way to a climb on a series of switchbacks. The track is badly eroded in some areas then disappears altogether; from here walking is over giant slabs of bare rock. Watch for painted arrows – it’s a good idea to note the location of each one as you go because there can be some distance between them and the route is easy to lose. At one point the arrows point straight up a steep slope where hand-holds come in very handy. Even a small amount of water trickling over the slope can make this section extremely slippery – take care.

Nearing the top the track levels out and passes through a gully before emerging on the wide open summit. If you have any breath left from the climb, the views will take it away. The views are especially brilliant in late afternoon light but you should leave plenty of time to get back to the car park before dark. Retrace your steps down the mountain – take extra care going down the steep rock and picking up the track again on the other side.

Bluestone Bay Distance: 5km Time: 2 hours Start/Finish: Cape Tourville Road

The walking track to Bluestone Bay is not one you will tell your grandchildren about – it’s a dusty, pot-holed four-wheel drive track. But the destination is infinitely more memorable: a bay lined with pink and blue boulders smoothed by the ocean, backed by spectacular cliffs. The rough vehicle track leaves the Cape Tourville Road on the left (north) about 1km from the lighthouse car park. Follow this as it winds downhill, sticking to the main track. A rock-climbing campsite is passed at White Water Wall before the track drops suddenly to Bluestone Bay 2.5km from Cape Tourville Road. The bay, at its most stunning early in the morning when sunlight from the east lights up the cliffs, is worth exploring before following the same track back to the bitumen.

Mount Graham Distance: 16km (approx) Time: 8 – 10 hours Start/Finish: Wineglass Bay Car Park

At 570 metres Mount Graham is the second-highest peak on the Freycinet Peninsula, providing panoramic views over virtually the entire national park. It’s distance from Coles Bay means that most walkers who clamber to the top do so with multi-day packs on their back, but if you give yourself an early start it’s possible to climb it and return in a day.

From Wineglass Bay Car Park take the track to the Wineglass Bay Lookout (see above). From here Mount Graham is clearly visible as one of two peaks whose mass dominates the southern portion of the Freycinet Peninsula (the other being the slightly higher Mount Freycinet). It is also obvious that it is still a long way away, so make sure you don’t delay too long admiring the views before continuing along the track as it descends to Wineglas Bay. The beach makes a pleasant change from the steep rocky slopes you have been negotiating for the past hour and a half, although the sand is quite soft and slopes quite steeply towards the sea making this section a bit tougher than you might expect. Still, the views of the pure white sand and turquise water contrasting with the pink and grey granite of the peaks at either end make it well worthwhile. As you approach the southern end of the beach take the track inland, passing through a walkers’ campsite which can make a good rest spot or indeed a campsite if you want to turn this long day walk into a short overnighter with the Mount Graham summit as a side trip.

After passing though the campsite the track begins to climb remorselessly, contouring up a creek valley with occasional glimpses of the sea and beach below. After an hour and a half or so it levels off, passing briefly through a buttongrass plain – a familiar sight for anyone who has been walking in central or southwestern Tasmana but certainly not something you would expect to see on the drier and warmer east coast.

The peak of Mount Graham is perched on the ridge, and another bout of climbing putsyou at the top and in a position to see those outstanding views (you will have earnt them). To the north Wineglass Bay curves gently behind the Tasman Sea, backed by the peaks of the Hazards, while to the south the Peninsula stretches away to Schouten Island. In the distance you can make out the rugged shape of Maria Island. After taking a bunch of photos and congratulting yourself on a view few people get to see, return the way you came.


Coles Bay, the town just outside the national park boundaries, is a good place to base yourself for a walking holiday in Freycinet. You can also hire a kayak or just soak in the view of the Hazards while you eat dinner or drink coffee at one of the strategically placed cafes or restauants.

The Friendly Beaches are another great site to check out while you are in the area. Stretching along the eastern coastline of the Peninsula to the north of Coles Bay, they are reached via a turnoff from the main road 19km from the town and 11km from the highway. The unsealed road winds through the bush before plunging steeply to the coastline and a rough campsite with several access points to the pure white quartz beaches beyond. You can spend a couple of hours strolling along the beach and enjoying the solitude.

Last but not least, while you are in the national park pop into the visitor information centre. Not only is it a great place to source information and souvenirs, you can also say hi to the staff from Bruce since he used to work here.


The drier climate of the east coast is reflected in the vegetation, with none of the rainforest that is so distinctive in other Tasmanian wilderness areas. Instead the forests are made up of eucalypts, with a healthy smattering of Oyster Bay pines and she-oaks which are adapted to the dry coastal conditions. It is not uncommon to see Bennett’s wallabies and Pademelons grazing amongst the coastal scrub or even hopping along the beach (make sure you resist the temptation to feed them, no matter how cute they look).

The most spectacular of the park’s residents, though, are found in the ocean – humpback and southern right whales both pass along this coast in autumn and spring on their annual migrations, while pilot whales, dolphins and seals can be seen around the coast at any time of year.


From Hobart take the Tasman Highway east to Sorell, turning left and following the highway as it winds through the Prosser River valley then north along the coast (keep an eye out for the views of rugged Maria Island from Orford and Triabunna). Two hours from Hobart turn right at the sign-posted turnoff and drive south for another 30 minutes to Coles Bay, the picturesque town nestled at the entrance to the park.From Launceston drive south along the Midland Highway in the direction of Hobart, turning left at Campbell Town and driving past Lake Leake to the intersection with the Tasman Highway. Turn left and follow the directions above.


Sunny and warm (at least in comparison to the rest of Tasmania) is the rule in Freycinet; average daily temperatures range from 13 degrees in July to 22 in January, with only a dozen rainy days a month even in winter. You’ll still need a raincoat, though – it is Tassie, after all.


Coles Bay has a full range of accommodation, from caravan parks to hotels and resorts, while Freycinet Lodge to the south of town also runs a range of guided walks and activities. Camping is also available in the national park, although due to the high visitor numbers a ballot system operates during the summer and Easter holidays. For more information contact the national parks visitor centre on (03) 6256 7000.


Many of the walks traverse rocky coastlines and cliffs, so take care. Caution should also be exercised when swimming, as the beaches are often home to rips or other dangerous swimming conditions. Contact the parks visitor centre for more information and for the latest track conditions.


The standard of walking tracks varies a lot depending on the visitor numbers, from boardwalks and heavily graded pathways at Cape Tourville and (most of) the climb to the Wineglass Bay lookout, to rough trails and faded arrows on rock slabs on the more strenuous and challenging walks. If in doubt ask the rangers for a recommendation based on your fitness and experience.


Tasmanian Parks and Wildlife Service: Freycinet

Wineglass Bay Tourism

Photography guide

G h d i s s i p u : uDissip ( local to MOM_VI_HDISSIP.F )
G h d i s s i p v : vDissip ( local to MOM_VI_HDISSIP.F )

Currently, this is exactly the same code as the flux form equations.

(2.140) G v d i s s u = 1 Δ r f h w δ k τ 13
(2.141) G v d i s s v = 1 Δ r f h s δ k τ 23
τ 13 = A v 1 Δ r c δ k u τ 23 = A v 1 Δ r c δ k v
τ 13 , τ 23 : ( local to MOM_VECINV.F )

The basic discretization used for the tracer equations is the second order piece-wise constant finite volume form of the forced advection-diffusion equations. There are many alternatives to second order method for advection and alternative parameterizations for the sub-grid scale processes. The Gent-McWilliams eddy parameterization, KPP mixing scheme and PV flux parameterization are all dealt with in separate sections. The basic discretization of the advection-diffusion part of the tracer equations and the various advection schemes will be described here.

Note that in addition to avoiding the somewhat clumsy "Tree" syntax, the postponed evaluation of annotations will also speed up your code, since type hints are not executed. Forward references are already supported by .

By far, the most common use of annotations is type hinting. Still, you have full access to the annotations at runtime and can use them as you see fit. If you are handling annotations directly, you need to deal with the possible forward references explicitly.

Let us create some admittedly silly examples that show when annotations are evaluated. First we do it old-style, so annotations are evaluated at import time. Let contain the following code:

Note that the annotation of name is print() . This is only to see exactly when the annotation is evaluated. Import the new module:

As you can see, the annotation was evaluated at import time. Note that name ends up annotated with None because that is the return value of print() .

Add the __future__ import to enable postponed evaluation of annotations:

Importing this updated code will not evaluate the annotation:

Note that Now! is never printed and the annotation is kept as a string literal in the __annotations__ dictionary. In order to evaluate the annotation, use typing.get_type_hints() or eval() :

Observe that the __annotations__ dictionary is never updated, so you need to evaluate the annotation every time you use it.

In Python 3.7, the module gains some new functions as described in PEP 564 . In particular, the following six functions are added:

clock_gettime_ns() : clock_settime_ns() : monotonic_ns() : perf_counter_ns() process_time_ns() : time_ns()

In a sense, there is no new functionality added. Each function is similar to an already existing function without the _ns suffix. The difference being that the new functions return a number of nanoseconds as an int instead of a number of seconds as a float .

For most applications, the difference between these new nanosecond functions and their old counterpart will not be appreciable. However, the new functions are easier to reason about because they rely on int instead of float . Floating point numbers are by nature inaccurate :

This is not an issue with Python but rather a consequence of computers needing to represent infinite decimal numbers using a finite number of bits.

A Python float follows the IEEE 754 standard and uses 53 significant bits. The result is that any time greater than about 104 days (2⁵³ or approximately Report Arvey Black Exotic Shoes CY86M
) cannot be expressed as a float with nanosecond precision. In contrast, a Python int is unlimited , so an integer number of nanoseconds will always have nanosecond precision independent of the time value.

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